Subgroups — C₂×C₅×#3496 #11985484

Order 1280.

12378 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₅×#3496	C₂	1	32	yes	maximal, direct product factor
3	C₂×C₅×#547	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#546	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×D₄²	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#537	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#534	C₂	1	4	yes	maximal, complemented
8	C₂×C₅×#532	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×#121	C₂	1	4	yes	maximal, complemented
10	C₂²×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×H(5,𝔽₂)	—	2	32	no	—
12	C₂×C₅×H(5,𝔽₂)	C₂²	1	32	yes	complemented
13	C₂²×C₅×#41	C₂²	1	4	yes	complemented
14	C₂²×C₅×#41	C₂²	1	8	yes	complemented
15	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₅×D₄	C₂²	1	8	yes	complemented
17	C₂³×C₅×D₄	C₂²	1	8	yes	complemented
18	C₂³×C₅×D₄	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₅	C₂²	1	2	yes	complemented
20	C₅×#547	C₂²	1	64	yes	complemented
21	C₅×#546	C₂²	1	128	yes	complemented
22	C₅×D₄²	C₂²	1	64	yes	complemented
23	C₅×#537	C₂²	1	32	yes	complemented
24	C₅×#534	C₂²	1	64	yes	complemented
25	C₅×#532	C₂²	1	32	yes	complemented
26	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
27	C₂×C₅×#127	C₂²	1	2	yes	complemented
28	C₂×C₅×#124	C₂²	1	4	yes	complemented
29	C₂×C₅×#123	C₂²	1	16	yes	complemented
30	C₂×C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
31	C₂×C₅×#121	C₂²	1	4	yes	complemented
32	C₂×C₅×#121	C₂²	1	8	yes	complemented
33	C₂×C₅×#121	C₂²	1	8	yes	complemented
34	C₂×C₅×#121	C₂²	1	8	yes	complemented
35	C₂×C₅×#121	C₂²	1	4	yes	complemented
36	C₂×C₅×#121	C₂²	1	8	yes	complemented
37	C₂×C₅×#121	C₂²	1	64	yes	complemented
38	C₂×C₅×#120	C₂²	1	16	yes	complemented
39	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
40	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
41	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
42	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
43	C₂×C₄×C₅×D₄	C₂²	1	16	yes	complemented
44	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
45	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
46	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
47	C₂²×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
48	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
49	C₂×#3496	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁵×C₅	C₂³	1	8	yes	—
51	C₂⁵×C₅	C₂³	1	2	yes	—
52	C₅×H(5,𝔽₂)	—	2	256	no	—
53	C₂×C₅×#41	—	2	64	no	—
54	C₂×C₅×#41	—	2	32	no	—
55	C₂×C₅×#41	C₂³	1	32	yes	complemented
56	C₂×C₅×#41	C₂³	1	64	yes	complemented
57	C₂²×C₅×Q₈	C₂³	1	4	yes	—
58	C₂²×C₅×D₄	—	2	64	no	—
59	C₂²×C₅×D₄	—	2	64	no	—
60	C₂²×C₅×D₄	—	4	16	no	—
61	C₂²×C₅×D₄	D₄	1	16	yes	complemented
62	C₂²×C₅×D₄	C₂³	1	4	yes	—
63	C₂²×C₅×D₄	C₂³	1	16	yes	—
64	C₂²×C₅×D₄	C₂³	1	8	yes	—
65	C₂²×C₅×D₄	C₂³	1	4	yes	—
66	C₂²×C₅×D₄	C₂³	1	4	yes	—
67	C₂²×C₅×D₄	C₂³	1	8	yes	—
68	C₂²×C₅×D₄	C₂³	1	4	yes	—
69	C₂²×C₅×D₄	C₂³	1	8	yes	—
70	C₂²×C₅×D₄	C₂³	1	64	yes	complemented
71	C₂²×C₅×D₄	C₂³	1	64	yes	complemented
72	C₂²×C₅×D₄	C₂³	1	32	yes	complemented
73	C₂³×C₄×C₅	—	4	8	no	—
74	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅	C₂³	1	8	yes	—
76	C₂³×C₄×C₅	C₂³	1	4	yes	—
77	C₂³×C₄×C₅	C₂³	1	8	yes	—
78	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅	C₂³	1	4	yes	—
80	C₂³×C₄×C₅	C₂³	1	2	yes	—
81	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₅	C₂³	1	16	yes	complemented
83	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
84	C₅×#127	C₂³	1	16	yes	complemented
85	C₅×#124	C₂³	1	32	yes	complemented
86	C₅×#123	C₂³	1	128	yes	complemented
87	C₅×C₂²⋊Q₈	C₂³	1	64	yes	complemented
88	C₅×#121	C₂³	1	32	yes	complemented
89	C₅×#121	C₂³	1	64	yes	complemented
90	C₅×#121	C₂³	1	64	yes	complemented
91	C₅×#121	C₂³	1	64	yes	complemented
92	C₅×#121	C₂³	1	32	yes	complemented
93	C₅×#121	C₂³	1	64	yes	complemented
94	C₅×#120	C₂³	1	128	yes	complemented
95	C₄×C₅×D₄	C₂³	1	32	yes	complemented
96	C₄×C₅×D₄	C₂³	1	32	yes	complemented
97	C₄×C₅×D₄	C₂³	1	32	yes	complemented
98	C₄×C₅×D₄	C₂³	1	32	yes	complemented
99	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
101	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
102	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	complemented
104	C₂×C₅×C₄⋊C₄	C₂³	1	32	yes	complemented
105	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
107	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
108	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	complemented
110	C₂×C₅×C₂²⋊C₄	C₂³	1	64	yes	complemented
111	C₂×C₄²×C₅	C₂³	1	2	yes	—
112	C₂×C₄²×C₅	C₂³	1	2	yes	—
113	C₂×C₄²×C₅	C₂³	1	8	yes	complemented
114	C₂²×H(5,𝔽₂)	C₂×C₅	1	2	yes	complemented
115	#3496	C₂×C₅	1	32	yes	direct product factor
116	C₂×#547	C₂×C₅	1	4	yes	complemented
117	C₂×#546	C₂×C₅	1	8	yes	complemented
118	C₂×D₄²	C₂×C₅	1	4	yes	complemented
119	C₂×#537	C₂×C₅	1	2	yes	complemented
120	C₂×#534	C₂×C₅	1	4	yes	complemented
121	C₂×#532	C₂×C₅	1	2	yes	complemented
122	C₂²×#121	C₂×C₅	1	4	yes	complemented
123	C₂²×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
124	C₂⁴×C₅	—	2	32	no	—
125	C₂⁴×C₅	—	4	4	no	—
126	C₂⁴×C₅	—	4	4	no	—
127	C₂⁴×C₅	—	4	16	no	—
128	C₂⁴×C₅	—	4	16	no	—
129	C₂⁴×C₅	C₂×D₄	1	16	yes	complemented
130	C₂⁴×C₅	C₂⁴	1	8	yes	—
131	C₂⁴×C₅	C₂⁴	1	4	yes	—
132	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
133	C₂⁴×C₅	C₂⁴	1	32	yes	—
134	C₂⁴×C₅	C₂⁴	1	8	yes	—
135	C₅×#41	—	2	256	no	—
136	C₅×#41	—	2	128	no	—
137	C₂×C₅×Q₈	—	2	16	no	—
138	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
139	C₂×C₅×D₄	—	2	64	no	—
140	C₂×C₅×D₄	—	2	32	no	—
141	C₂×C₅×D₄	—	2	256	no	—
142	C₂×C₅×D₄	—	2	256	no	—
143	C₂×C₅×D₄	—	2	16	no	—
144	C₂×C₅×D₄	—	2	16	no	—
145	C₂×C₅×D₄	—	4	64	no	—
146	C₂×C₅×D₄	—	4	16	no	—
147	C₂×C₅×D₄	—	4	8	no	—
148	C₂×C₅×D₄	C₂×D₄	1	32	yes	complemented
149	C₂×C₅×D₄	C₂×D₄	1	64	yes	complemented
150	C₂×C₅×D₄	C₂⁴	1	16	yes	—
151	C₂×C₅×D₄	C₂⁴	1	64	yes	—
152	C₂×C₅×D₄	C₂⁴	1	32	yes	—
153	C₂×C₅×D₄	C₂⁴	1	16	yes	—
154	C₂×C₅×D₄	C₂⁴	1	16	yes	—
155	C₂×C₅×D₄	C₂⁴	1	32	yes	—
156	C₂×C₅×D₄	C₂⁴	1	16	yes	—
157	C₂²×C₄×C₅	—	2	32	no	—
158	C₂²×C₄×C₅	—	2	32	no	—
159	C₂²×C₄×C₅	—	2	4	no	—
160	C₂²×C₄×C₅	—	2	16	no	—
161	C₂²×C₄×C₅	—	4	32	no	—
162	C₂²×C₄×C₅	—	4	8	no	—
163	C₂²×C₄×C₅	—	4	4	no	—
164	C₂²×C₄×C₅	—	4	2	no	—
165	C₂²×C₄×C₅	C₂×D₄	1	8	yes	complemented
166	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅	C₂⁴	1	32	yes	—
168	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
169	C₂²×C₄×C₅	C₂⁴	1	32	yes	—
170	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
176	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
177	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
180	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
181	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
182	C₅×C₄⋊C₄	C₂⁴	1	32	yes	—
183	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
184	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
185	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
186	C₅×C₂²⋊C₄	C₂⁴	1	64	yes	—
187	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
188	C₄²×C₅	C₂⁴	1	8	yes	—
189	C₄²×C₅	C₂⁴	1	8	yes	—
190	C₂×H(5,𝔽₂)	—	2	32	no	—
191	C₂×H(5,𝔽₂)	C₂²×C₅	1	32	yes	complemented
192	C₂²×#41	C₂²×C₅	1	4	yes	complemented
193	C₂²×#41	C₂²×C₅	1	8	yes	complemented
194	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
195	C₂³×D₄	C₂²×C₅	1	8	yes	complemented
196	C₂³×D₄	C₂²×C₅	1	8	yes	complemented
197	C₂³×D₄	C₂²×C₅	1	4	yes	complemented
198	C₂⁴×C₄	C₂²×C₅	1	2	yes	complemented
199	#547	C₂²×C₅	1	64	yes	complemented
200	#546	C₂²×C₅	1	128	yes	complemented
201	D₄²	C₂²×C₅	1	64	yes	complemented
202	#537	C₂²×C₅	1	32	yes	complemented
203	#534	C₂²×C₅	1	64	yes	complemented
204	#532	C₂²×C₅	1	32	yes	complemented
205	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
206	C₂×#127	C₂²×C₅	1	2	yes	complemented
207	C₂×#124	C₂²×C₅	1	4	yes	complemented
208	C₂×#123	C₂²×C₅	1	16	yes	complemented
209	C₂×C₂²⋊Q₈	C₂²×C₅	1	8	yes	complemented
210	C₂×#121	C₂²×C₅	1	4	yes	complemented
211	C₂×#121	C₂²×C₅	1	8	yes	complemented
212	C₂×#121	C₂²×C₅	1	8	yes	complemented
213	C₂×#121	C₂²×C₅	1	8	yes	complemented
214	C₂×#121	C₂²×C₅	1	4	yes	complemented
215	C₂×#121	C₂²×C₅	1	8	yes	complemented
216	C₂×#121	C₂²×C₅	1	64	yes	complemented
217	C₂×#120	C₂²×C₅	1	16	yes	complemented
218	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
219	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
220	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
221	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
222	C₂×C₄×D₄	C₂²×C₅	1	16	yes	complemented
223	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
224	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
225	C₂²×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
226	C₂²×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
227	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
228	C₂³×C₅	—	2	2	no	—
229	C₂³×C₅	—	2	8	no	—
230	C₂³×C₅	—	2	2	no	—
231	C₂³×C₅	—	2	8	no	—
232	C₂³×C₅	—	2	16	no	—
233	C₂³×C₅	—	2	16	no	—
234	C₂³×C₅	—	2	16	no	—
235	C₂³×C₅	—	2	64	no	—
236	C₂³×C₅	—	4	8	no	—
237	C₂³×C₅	—	4	32	no	—
238	C₂³×C₅	—	4	8	no	—
239	C₂³×C₅	—	4	32	no	—
240	C₂³×C₅	—	4	32	no	—
241	C₂³×C₅	—	8	64	no	—
242	C₂³×C₅	C₂²×D₄	1	16	yes	—
243	C₂³×C₅	C₂²×D₄	1	4	yes	—
244	C₂³×C₅	C₂²×D₄	1	32	yes	complemented
245	C₂³×C₅	C₂⁵	1	16	yes	—
246	C₂³×C₅	C₂⁵	1	8	yes	—
247	C₂³×C₅	C₂⁵	1	2	yes	—
248	C₂³×C₅	C₂⁵	1	1	yes	characteristic
249	C₅×Q₈	—	2	32	no	—
250	C₅×D₄	—	2	128	no	—
251	C₅×D₄	—	2	64	no	—
252	C₅×D₄	—	2	32	no	—
253	C₅×D₄	—	2	32	no	—
254	C₅×D₄	—	4	32	no	—
255	C₅×D₄	—	4	16	no	—
256	C₅×D₄	C₂²×D₄	1	64	yes	complemented
257	C₂×C₄×C₅	—	2	16	no	—
258	C₂×C₄×C₅	—	2	64	no	—
259	C₂×C₄×C₅	—	2	2	no	—
260	C₂×C₄×C₅	—	2	8	no	—
261	C₂×C₄×C₅	—	2	4	no	—
262	C₂×C₄×C₅	—	2	64	no	—
263	C₂×C₄×C₅	—	2	8	no	—
264	C₂×C₄×C₅	—	2	32	no	—
265	C₂×C₄×C₅	—	4	16	no	—
266	C₂×C₄×C₅	—	4	8	no	—
267	C₂×C₄×C₅	—	4	4	no	—
268	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
269	C₂×C₄×C₅	C₂²×D₄	1	16	yes	complemented
270	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	complemented
271	C₂×C₄×C₅	C₂⁵	1	4	yes	—
272	C₂×C₄×C₅	C₂⁵	1	2	yes	—
273	C₂×C₄×C₅	C₂⁵	1	8	yes	—
274	C₂×C₄×C₅	C₂⁵	1	16	yes	—
275	C₂×C₄×C₅	C₂⁵	1	4	yes	—
276	C₂×C₄×C₅	C₂⁵	1	2	yes	—
277	C₂⁵	C₂³×C₅	1	8	yes	—
278	C₂⁵	C₂³×C₅	1	2	yes	—
279	H(5,𝔽₂)	—	2	256	no	—
280	C₂×#41	—	2	64	no	—
281	C₂×#41	—	2	32	no	—
282	C₂×#41	C₂³×C₅	1	32	yes	complemented
283	C₂×#41	C₂³×C₅	1	64	yes	complemented
284	C₂²×Q₈	C₂³×C₅	1	4	yes	—
285	C₂²×D₄	—	2	64	no	—
286	C₂²×D₄	—	2	64	no	—
287	C₂²×D₄	—	4	16	no	—
288	C₂²×D₄	C₅×D₄	1	16	yes	complemented
289	C₂²×D₄	C₂³×C₅	1	4	yes	—
290	C₂²×D₄	C₂³×C₅	1	16	yes	—
291	C₂²×D₄	C₂³×C₅	1	8	yes	—
292	C₂²×D₄	C₂³×C₅	1	4	yes	—
293	C₂²×D₄	C₂³×C₅	1	4	yes	—
294	C₂²×D₄	C₂³×C₅	1	8	yes	—
295	C₂²×D₄	C₂³×C₅	1	4	yes	—
296	C₂²×D₄	C₂³×C₅	1	8	yes	—
297	C₂²×D₄	C₂³×C₅	1	64	yes	complemented
298	C₂²×D₄	C₂³×C₅	1	64	yes	complemented
299	C₂²×D₄	C₂³×C₅	1	32	yes	complemented
300	C₂³×C₄	—	4	8	no	—
301	C₂³×C₄	C₂³×C₅	1	8	yes	—
302	C₂³×C₄	C₂³×C₅	1	4	yes	—
303	C₂³×C₄	C₂³×C₅	1	8	yes	—
304	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
305	C₂³×C₄	C₂³×C₅	1	4	yes	—
306	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₅	1	2	yes	—
308	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₅	1	16	yes	complemented
310	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
311	#127	C₂³×C₅	1	16	yes	complemented
312	#124	C₂³×C₅	1	32	yes	complemented
313	#123	C₂³×C₅	1	128	yes	complemented
314	C₂²⋊Q₈	C₂³×C₅	1	64	yes	complemented
315	#121	C₂³×C₅	1	32	yes	complemented
316	#121	C₂³×C₅	1	64	yes	complemented
317	#121	C₂³×C₅	1	64	yes	complemented
318	#121	C₂³×C₅	1	64	yes	complemented
319	#121	C₂³×C₅	1	32	yes	complemented
320	#121	C₂³×C₅	1	64	yes	complemented
321	#120	C₂³×C₅	1	128	yes	complemented
322	C₄×D₄	C₂³×C₅	1	32	yes	complemented
323	C₄×D₄	C₂³×C₅	1	32	yes	complemented
324	C₄×D₄	C₂³×C₅	1	32	yes	complemented
325	C₄×D₄	C₂³×C₅	1	32	yes	complemented
326	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
327	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
328	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
329	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
330	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	complemented
331	C₂×C₄⋊C₄	C₂³×C₅	1	32	yes	complemented
332	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	complemented
337	C₂×C₂²⋊C₄	C₂³×C₅	1	64	yes	complemented
338	C₂×C₄²	C₂³×C₅	1	2	yes	—
339	C₂×C₄²	C₂³×C₅	1	2	yes	—
340	C₂×C₄²	C₂³×C₅	1	8	yes	complemented
341	C₂²×C₅	—	2	2	no	—
342	C₂²×C₅	—	2	8	no	—
343	C₂²×C₅	—	2	2	no	—
344	C₂²×C₅	—	2	8	no	—
345	C₂²×C₅	—	2	8	no	—
346	C₂²×C₅	—	2	32	no	—
347	C₂²×C₅	—	2	16	no	—
348	C₂²×C₅	—	2	16	no	—
349	C₂²×C₅	—	2	16	no	—
350	C₂²×C₅	—	4	32	no	—
351	C₂²×C₅	—	4	32	no	—
352	C₂²×C₅	—	8	64	no	—
353	C₂²×C₅	C₂³×D₄	1	16	yes	—
354	C₂²×C₅	C₂³×D₄	1	4	yes	—
355	C₂²×C₅	C₂³×D₄	1	2	yes	—
356	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
357	C₂²×C₅	#585	1	2	yes	—
358	C₂²×C₅	C₂⁶	1	1	yes	characteristic
359	C₄×C₅	—	2	16	no	—
360	C₄×C₅	—	2	2	no	—
361	C₄×C₅	—	2	8	no	—
362	C₄×C₅	—	2	4	no	—
363	C₄×C₅	C₂³×D₄	1	8	yes	—
364	C₄×C₅	C₂×H(5,𝔽₂)	1	4	yes	complemented
365	C₂⁴	—	2	32	no	—
366	C₂⁴	—	4	4	no	—
367	C₂⁴	—	4	4	no	—
368	C₂⁴	—	4	16	no	—
369	C₂⁴	—	4	16	no	—
370	C₂⁴	C₂×C₅×D₄	1	16	yes	complemented
371	C₂⁴	C₂⁴×C₅	1	8	yes	—
372	C₂⁴	C₂⁴×C₅	1	4	yes	—
373	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
374	C₂⁴	C₂⁴×C₅	1	32	yes	—
375	C₂⁴	C₂⁴×C₅	1	8	yes	—
376	#41	—	2	256	no	—
377	#41	—	2	128	no	—
378	C₂×Q₈	—	2	16	no	—
379	C₂×Q₈	C₂⁴×C₅	1	16	yes	—
380	C₂×D₄	—	2	64	no	—
381	C₂×D₄	—	2	32	no	—
382	C₂×D₄	—	2	256	no	—
383	C₂×D₄	—	2	256	no	—
384	C₂×D₄	—	2	16	no	—
385	C₂×D₄	—	2	16	no	—
386	C₂×D₄	—	4	64	no	—
387	C₂×D₄	—	4	16	no	—
388	C₂×D₄	—	4	8	no	—
389	C₂×D₄	C₂×C₅×D₄	1	32	yes	complemented
390	C₂×D₄	C₂×C₅×D₄	1	64	yes	complemented
391	C₂×D₄	C₂⁴×C₅	1	16	yes	—
392	C₂×D₄	C₂⁴×C₅	1	64	yes	—
393	C₂×D₄	C₂⁴×C₅	1	32	yes	—
394	C₂×D₄	C₂⁴×C₅	1	16	yes	—
395	C₂×D₄	C₂⁴×C₅	1	16	yes	—
396	C₂×D₄	C₂⁴×C₅	1	32	yes	—
397	C₂×D₄	C₂⁴×C₅	1	16	yes	—
398	C₂²×C₄	—	2	32	no	—
399	C₂²×C₄	—	2	32	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	16	no	—
402	C₂²×C₄	—	4	32	no	—
403	C₂²×C₄	—	4	8	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	—	4	2	no	—
406	C₂²×C₄	C₂×C₅×D₄	1	8	yes	complemented
407	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
412	C₂²×C₄	C₂⁴×C₅	1	32	yes	—
413	C₂²×C₄	C₂⁴×C₅	1	16	yes	—
414	C₂²×C₄	C₂⁴×C₅	1	32	yes	—
415	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
416	C₂²×C₄	C₂⁴×C₅	1	16	yes	—
417	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
418	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
420	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
421	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
422	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
423	C₄⋊C₄	C₂⁴×C₅	1	32	yes	—
424	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
425	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
426	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
427	C₂²⋊C₄	C₂⁴×C₅	1	64	yes	—
428	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
429	C₄²	C₂⁴×C₅	1	8	yes	—
430	C₄²	C₂⁴×C₅	1	8	yes	—
431	C₂×C₅	—	2	4	no	—
432	C₂×C₅	—	2	16	no	—
433	C₂×C₅	—	4	16	no	—
434	C₂×C₅	#3496	1	4	yes	direct product factor
435	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
436	C₂×C₅	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
437	C₂×C₅	C₂×#585	1	1	yes	characteristic
438	C₂³	—	2	2	no	—
439	C₂³	—	2	8	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	8	no	—
442	C₂³	—	2	16	no	—
443	C₂³	—	2	16	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	64	no	—
446	C₂³	—	4	8	no	—
447	C₂³	—	4	32	no	—
448	C₂³	—	4	8	no	—
449	C₂³	—	4	32	no	—
450	C₂³	—	4	32	no	—
451	C₂³	—	8	64	no	—
452	C₂³	C₂²×C₅×D₄	1	16	yes	—
453	C₂³	C₂²×C₅×D₄	1	4	yes	—
454	C₂³	C₂²×C₅×D₄	1	32	yes	complemented
455	C₂³	C₂⁵×C₅	1	16	yes	—
456	C₂³	C₂⁵×C₅	1	8	yes	—
457	C₂³	C₂⁵×C₅	1	2	yes	—
458	C₂³	C₂⁵×C₅	1	1	yes	characteristic
459	Q₈	—	2	32	no	—
460	D₄	—	2	128	no	—
461	D₄	—	2	64	no	—
462	D₄	—	2	32	no	—
463	D₄	—	2	32	no	—
464	D₄	—	4	32	no	—
465	D₄	—	4	16	no	—
466	D₄	C₂²×C₅×D₄	1	64	yes	complemented
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	2	64	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	64	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	32	no	—
475	C₂×C₄	—	4	16	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
479	C₂×C₄	C₂²×C₅×D₄	1	16	yes	complemented
480	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	complemented
481	C₂×C₄	C₂⁵×C₅	1	4	yes	—
482	C₂×C₄	C₂⁵×C₅	1	2	yes	—
483	C₂×C₄	C₂⁵×C₅	1	8	yes	—
484	C₂×C₄	C₂⁵×C₅	1	16	yes	—
485	C₂×C₄	C₂⁵×C₅	1	4	yes	—
486	C₂×C₄	C₂⁵×C₅	1	2	yes	—
487	C₅	C₂×#3496	1	1	yes	characteristic, direct product factor, Sylow
488	C₂²	—	2	2	no	—
489	C₂²	—	2	8	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	8	no	—
493	C₂²	—	2	32	no	—
494	C₂²	—	2	16	no	—
495	C₂²	—	2	16	no	—
496	C₂²	—	2	16	no	—
497	C₂²	—	4	32	no	—
498	C₂²	—	4	32	no	—
499	C₂²	—	8	64	no	—
500	C₂²	C₂³×C₅×D₄	1	16	yes	—
501	C₂²	C₂³×C₅×D₄	1	4	yes	—
502	C₂²	C₂³×C₅×D₄	1	2	yes	—
503	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
504	C₂²	C₅×#585	1	2	yes	—
505	C₂²	C₂⁶×C₅	1	1	yes	characteristic
506	C₄	—	2	16	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	8	no	—
509	C₄	—	2	4	no	—
510	C₄	C₂³×C₅×D₄	1	8	yes	—
511	C₄	C₂×C₅×H(5,𝔽₂)	1	4	yes	complemented
512	C₂	—	2	4	no	—
513	C₂	—	2	16	no	—
514	C₂	—	4	16	no	—
515	C₂	C₅×#3496	1	4	yes	direct product factor
516	C₂	C₂⁴×C₅×D₄	1	1	yes	characteristic
517	C₂	C₂²×C₅×H(5,𝔽₂)	1	1	yes	characteristic
518	C₂	C₂×C₅×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₅×H(5,𝔽₂)	C₂	semidirect
2. C₅×#3496	C₂	direct (N × H)
3. C₂×C₅×#547	C₂	semidirect
4. C₂×C₅×#546	C₂	semidirect
5. C₂×C₅×D₄²	C₂	semidirect
6. C₂×C₅×#537	C₂	semidirect
7. C₂×C₅×#534	C₂	semidirect
8. C₂×C₅×#532	C₂	semidirect
9. C₂²×C₅×#121	C₂	semidirect
10. C₂²×C₄×C₅×D₄	C₂	semidirect
12. C₂×C₅×H(5,𝔽₂)	C₂²	semidirect
13. C₂²×C₅×#41	C₂²	semidirect
14. C₂²×C₅×#41	C₂²	semidirect
16. C₂³×C₅×D₄	C₂²	semidirect
17. C₂³×C₅×D₄	C₂²	semidirect
18. C₂³×C₅×D₄	C₂²	semidirect
19. C₂⁴×C₄×C₅	C₂²	semidirect
20. C₅×#547	C₂²	semidirect
21. C₅×#546	C₂²	semidirect
22. C₅×D₄²	C₂²	semidirect
23. C₅×#537	C₂²	semidirect
24. C₅×#534	C₂²	semidirect
25. C₅×#532	C₂²	semidirect
26. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
27. C₂×C₅×#127	C₂²	semidirect
28. C₂×C₅×#124	C₂²	semidirect
29. C₂×C₅×#123	C₂²	semidirect
30. C₂×C₅×C₂²⋊Q₈	C₂²	semidirect
31. C₂×C₅×#121	C₂²	semidirect
32. C₂×C₅×#121	C₂²	semidirect
33. C₂×C₅×#121	C₂²	semidirect
34. C₂×C₅×#121	C₂²	semidirect
35. C₂×C₅×#121	C₂²	semidirect
36. C₂×C₅×#121	C₂²	semidirect
37. C₂×C₅×#121	C₂²	semidirect
38. C₂×C₅×#120	C₂²	semidirect
39. C₂×C₄×C₅×D₄	C₂²	semidirect
40. C₂×C₄×C₅×D₄	C₂²	semidirect
41. C₂×C₄×C₅×D₄	C₂²	semidirect
42. C₂×C₄×C₅×D₄	C₂²	semidirect
43. C₂×C₄×C₅×D₄	C₂²	semidirect
44. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
45. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
46. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
47. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
48. C₂²×C₄²×C₅	C₂²	semidirect
49. C₂×#3496	C₅	direct (N × H)
55. C₂×C₅×#41	C₂³	semidirect
56. C₂×C₅×#41	C₂³	semidirect
61. C₂²×C₅×D₄	D₄	semidirect
70. C₂²×C₅×D₄	C₂³	semidirect
71. C₂²×C₅×D₄	C₂³	semidirect
72. C₂²×C₅×D₄	C₂³	semidirect
82. C₂³×C₄×C₅	C₂³	semidirect
83. C₅×C₄⋊Q₈	C₂³	semidirect
84. C₅×#127	C₂³	semidirect
85. C₅×#124	C₂³	semidirect
86. C₅×#123	C₂³	semidirect
87. C₅×C₂²⋊Q₈	C₂³	semidirect
88. C₅×#121	C₂³	semidirect
89. C₅×#121	C₂³	semidirect
90. C₅×#121	C₂³	semidirect
91. C₅×#121	C₂³	semidirect
92. C₅×#121	C₂³	semidirect
93. C₅×#121	C₂³	semidirect
94. C₅×#120	C₂³	semidirect
95. C₄×C₅×D₄	C₂³	semidirect
96. C₄×C₅×D₄	C₂³	semidirect
97. C₄×C₅×D₄	C₂³	semidirect
98. C₄×C₅×D₄	C₂³	semidirect
103. C₂×C₅×C₄⋊C₄	C₂³	semidirect
104. C₂×C₅×C₄⋊C₄	C₂³	semidirect
109. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
110. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
113. C₂×C₄²×C₅	C₂³	semidirect
114. C₂²×H(5,𝔽₂)	C₂×C₅	semidirect
115. #3496	C₂×C₅	direct (N × H)
116. C₂×#547	C₂×C₅	semidirect
117. C₂×#546	C₂×C₅	semidirect
118. C₂×D₄²	C₂×C₅	semidirect
119. C₂×#537	C₂×C₅	semidirect
120. C₂×#534	C₂×C₅	semidirect
121. C₂×#532	C₂×C₅	semidirect
122. C₂²×#121	C₂×C₅	semidirect
123. C₂²×C₄×D₄	C₂×C₅	semidirect
129. C₂⁴×C₅	C₂×D₄	semidirect
148. C₂×C₅×D₄	C₂×D₄	semidirect
149. C₂×C₅×D₄	C₂×D₄	semidirect
165. C₂²×C₄×C₅	C₂×D₄	semidirect
191. C₂×H(5,𝔽₂)	C₂²×C₅	semidirect
192. C₂²×#41	C₂²×C₅	semidirect
193. C₂²×#41	C₂²×C₅	semidirect
195. C₂³×D₄	C₂²×C₅	semidirect
196. C₂³×D₄	C₂²×C₅	semidirect
197. C₂³×D₄	C₂²×C₅	semidirect
198. C₂⁴×C₄	C₂²×C₅	semidirect
199. #547	C₂²×C₅	semidirect
200. #546	C₂²×C₅	semidirect
201. D₄²	C₂²×C₅	semidirect
202. #537	C₂²×C₅	semidirect
203. #534	C₂²×C₅	semidirect
204. #532	C₂²×C₅	semidirect
205. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
206. C₂×#127	C₂²×C₅	semidirect
207. C₂×#124	C₂²×C₅	semidirect
208. C₂×#123	C₂²×C₅	semidirect
209. C₂×C₂²⋊Q₈	C₂²×C₅	semidirect
210. C₂×#121	C₂²×C₅	semidirect
211. C₂×#121	C₂²×C₅	semidirect
212. C₂×#121	C₂²×C₅	semidirect
213. C₂×#121	C₂²×C₅	semidirect
214. C₂×#121	C₂²×C₅	semidirect
215. C₂×#121	C₂²×C₅	semidirect
216. C₂×#121	C₂²×C₅	semidirect
217. C₂×#120	C₂²×C₅	semidirect
218. C₂×C₄×D₄	C₂²×C₅	semidirect
219. C₂×C₄×D₄	C₂²×C₅	semidirect
220. C₂×C₄×D₄	C₂²×C₅	semidirect
221. C₂×C₄×D₄	C₂²×C₅	semidirect
222. C₂×C₄×D₄	C₂²×C₅	semidirect
223. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
224. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
225. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
226. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
227. C₂²×C₄²	C₂²×C₅	semidirect
244. C₂³×C₅	C₂²×D₄	semidirect
256. C₅×D₄	C₂²×D₄	semidirect
269. C₂×C₄×C₅	C₂²×D₄	semidirect
270. C₂×C₄×C₅	H(5,𝔽₂)	semidirect
282. C₂×#41	C₂³×C₅	semidirect
283. C₂×#41	C₂³×C₅	semidirect
288. C₂²×D₄	C₅×D₄	semidirect
297. C₂²×D₄	C₂³×C₅	semidirect
298. C₂²×D₄	C₂³×C₅	semidirect
299. C₂²×D₄	C₂³×C₅	semidirect
309. C₂³×C₄	C₂³×C₅	semidirect
310. C₄⋊Q₈	C₂³×C₅	semidirect
311. #127	C₂³×C₅	semidirect
312. #124	C₂³×C₅	semidirect
313. #123	C₂³×C₅	semidirect
314. C₂²⋊Q₈	C₂³×C₅	semidirect
315. #121	C₂³×C₅	semidirect
316. #121	C₂³×C₅	semidirect
317. #121	C₂³×C₅	semidirect
318. #121	C₂³×C₅	semidirect
319. #121	C₂³×C₅	semidirect
320. #121	C₂³×C₅	semidirect
321. #120	C₂³×C₅	semidirect
322. C₄×D₄	C₂³×C₅	semidirect
323. C₄×D₄	C₂³×C₅	semidirect
324. C₄×D₄	C₂³×C₅	semidirect
325. C₄×D₄	C₂³×C₅	semidirect
330. C₂×C₄⋊C₄	C₂³×C₅	semidirect
331. C₂×C₄⋊C₄	C₂³×C₅	semidirect
336. C₂×C₂²⋊C₄	C₂³×C₅	semidirect
337. C₂×C₂²⋊C₄	C₂³×C₅	semidirect
340. C₂×C₄²	C₂³×C₅	semidirect
364. C₄×C₅	C₂×H(5,𝔽₂)	semidirect
370. C₂⁴	C₂×C₅×D₄	semidirect
389. C₂×D₄	C₂×C₅×D₄	semidirect
390. C₂×D₄	C₂×C₅×D₄	semidirect
406. C₂²×C₄	C₂×C₅×D₄	semidirect
434. C₂×C₅	#3496	direct (N × H)
454. C₂³	C₂²×C₅×D₄	semidirect
466. D₄	C₂²×C₅×D₄	semidirect
479. C₂×C₄	C₂²×C₅×D₄	semidirect
480. C₂×C₄	C₅×H(5,𝔽₂)	semidirect
487. C₅	C₂×#3496	direct (N × H)
511. C₄	C₂×C₅×H(5,𝔽₂)	semidirect
515. C₂	C₅×#3496	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×H(5,𝔽₂)	2
C₅×#3496	2
C₂×C₅×#547	2
C₂×C₅×#546	2
C₂×C₅×D₄²	2
C₂×C₅×#537	2
C₂×C₅×#534	2
C₂×C₅×#532	2
C₂²×C₅×#121	2
C₂²×C₄×C₅×D₄	2
C₂×#3496	5

Subgroups — C2×C5×#3496 #11985484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3496 #11985484