Subgroups — C₂×C₅×#3299 #11983837

Order 1280.

2158 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×D₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#3299	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#484	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#467	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#451	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₅×#103	C₂	1	2	yes	maximal, complemented
13	C₂³×C₅×Q₈	C₂²	1	2	yes	—
14	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
16	C₂³×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#554	—	2	4	no	—
18	C₅×#554	C₂²	1	8	yes	—
19	C₅×D₄×Q₈	—	2	4	no	—
20	C₅×D₄×Q₈	C₂²	1	8	yes	—
21	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₅×#125	—	2	1	no	—
25	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
27	C₂×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₅×D₄	—	2	1	no	—
32	C₂×C₄×C₅×D₄	—	2	1	no	—
33	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
34	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
35	C₅×#487	C₂²	1	8	yes	complemented
36	C₅×#484	C₂²	1	16	yes	complemented
37	C₅×#477	C₂²	1	8	yes	complemented
38	C₅×#467	C₂²	1	16	yes	complemented
39	C₅×#462	C₂²	1	8	yes	complemented
40	C₅×#451	C₂²	1	16	yes	complemented
41	C₄×C₅×Q₁₆	C₂²	1	8	yes	complemented
42	C₅×C₈×D₄	C₂²	1	8	yes	—
43	C₂×C₅×#107	C₂²	1	2	yes	—
44	C₂×C₅×#107	C₂²	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₅×#103	C₂²	1	2	yes	—
47	C₂×C₅×#103	C₂²	1	2	yes	—
48	C₂×C₅×#103	C₂²	1	2	yes	—
49	C₂×C₅×#103	C₂²	1	1	yes	characteristic
50	C₂×C₅×#103	C₂²	1	2	yes	—
51	C₂×C₅×#103	C₂²	1	1	yes	characteristic
52	C₂×C₅×#103	C₂²	1	16	yes	complemented
53	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
55	C₂×#3299	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₅×Q₈	—	2	2	no	—
57	C₂²×C₅×Q₈	—	2	4	no	—
58	C₂²×C₅×Q₈	—	2	1	no	—
59	C₂²×C₅×Q₈	—	2	8	no	—
60	C₂²×C₅×Q₈	—	4	4	no	—
61	C₂²×C₅×Q₈	C₂³	1	2	yes	—
62	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂²×C₅×Q₈	C₂³	1	8	yes	—
64	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
65	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₅	—	2	2	no	—
67	C₂³×C₄×C₅	—	2	2	no	—
68	C₂³×C₄×C₅	C₂³	1	2	yes	—
69	C₂×C₅×Q₁₆	—	2	4	no	—
70	C₂×C₅×Q₁₆	—	4	4	no	—
71	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
72	C₂×C₅×Q₁₆	C₂³	1	8	yes	—
73	C₂²×C₅×C₈	—	4	4	no	—
74	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₅×C₈	C₂³	1	2	yes	—
77	C₂²×C₅×C₈	C₂³	1	8	yes	—
78	C₅×C₄⋊Q₈	—	2	8	no	—
79	C₅×C₄⋊Q₈	—	2	2	no	—
80	C₅×C₄⋊Q₈	—	2	2	no	—
81	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
83	C₅×#125	—	2	8	no	—
84	C₅×C₂²⋊Q₈	—	2	4	no	—
85	C₅×C₂²⋊Q₈	—	2	4	no	—
86	C₅×C₂²⋊Q₈	—	2	16	no	—
87	C₅×C₂²⋊Q₈	—	2	16	no	—
88	C₅×C₂²⋊Q₈	C₂³	1	8	yes	—
89	C₅×C₂²⋊Q₈	C₂³	1	8	yes	—
90	C₄×C₅×Q₈	—	2	2	no	—
91	C₄×C₅×Q₈	—	2	2	no	—
92	C₄×C₅×Q₈	C₂³	1	4	yes	—
93	C₄×C₅×Q₈	C₂³	1	4	yes	—
94	C₄×C₅×D₄	—	2	8	no	—
95	C₄×C₅×D₄	—	2	8	no	—
96	C₄×C₅×D₄	D₄	1	4	yes	—
97	C₄×C₅×D₄	C₂³	1	4	yes	—
98	C₂×C₅×C₄⋊C₄	—	2	2	no	—
99	C₂×C₅×C₄⋊C₄	—	2	4	no	—
100	C₂×C₅×C₄⋊C₄	—	2	1	no	—
101	C₂×C₅×C₄⋊C₄	—	2	1	no	—
102	C₂×C₅×C₄⋊C₄	—	2	1	no	—
103	C₂×C₅×C₄⋊C₄	—	2	1	no	—
104	C₂×C₅×C₄⋊C₄	—	2	1	no	—
105	C₂×C₅×C₄⋊C₄	—	2	2	no	—
106	C₂×C₅×C₄⋊C₄	—	2	2	no	—
107	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
108	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
109	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
111	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
112	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
114	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
115	C₂×C₄²×C₅	—	2	1	no	—
116	C₂×C₄²×C₅	—	2	1	no	—
117	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
118	C₅×#107	C₂³	1	4	yes	—
119	C₅×#107	C₂³	1	8	yes	—
120	C₅×C₄⋊C₈	C₂³	1	4	yes	—
121	C₅×#103	C₂³	1	4	yes	—
122	C₅×#103	C₂³	1	8	yes	—
123	C₅×#103	C₂³	1	8	yes	—
124	C₅×#103	C₂³	1	8	yes	—
125	C₅×#103	C₂³	1	4	yes	—
126	C₅×#103	C₂³	1	8	yes	—
127	C₅×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₄×C₅×C₈	C₂³	1	4	yes	—
129	C₂×#554	C₂×C₅	1	1	yes	characteristic
130	C₂×D₄×Q₈	C₂×C₅	1	1	yes	characteristic
131	#3299	C₂×C₅	1	16	yes	direct product factor
132	C₂×#487	C₂×C₅	1	1	yes	characteristic, complemented
133	C₂×#484	C₂×C₅	1	2	yes	complemented
134	C₂×#477	C₂×C₅	1	1	yes	characteristic, complemented
135	C₂×#467	C₂×C₅	1	2	yes	complemented
136	C₂×#462	C₂×C₅	1	1	yes	characteristic, complemented
137	C₂×#451	C₂×C₅	1	2	yes	complemented
138	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
139	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
140	C₂²×#103	C₂×C₅	1	2	yes	complemented
141	C₂⁴×C₅	C₂×D₄	1	2	yes	—
142	C₂×C₅×Q₈	—	2	1	no	—
143	C₂×C₅×Q₈	—	2	1	no	—
144	C₂×C₅×Q₈	—	2	8	no	—
145	C₂×C₅×Q₈	—	2	2	no	—
146	C₂×C₅×Q₈	—	2	4	no	—
147	C₂×C₅×Q₈	—	2	2	no	—
148	C₂×C₅×Q₈	—	2	16	no	—
149	C₂×C₅×Q₈	—	2	2	no	—
150	C₂×C₅×Q₈	—	2	4	no	—
151	C₂×C₅×Q₈	—	2	2	no	—
152	C₂×C₅×Q₈	—	2	4	no	—
153	C₂×C₅×Q₈	—	4	8	no	—
154	C₂×C₅×Q₈	—	4	8	no	—
155	C₂×C₅×Q₈	—	4	16	no	—
156	C₂×C₅×Q₈	—	8	2	no	—
157	C₂×C₅×Q₈	#41	1	4	yes	—
158	C₂×C₅×Q₈	C₂⁴	1	2	yes	—
159	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
160	C₂×C₅×Q₈	C₂⁴	1	2	yes	—
161	C₂×C₅×D₄	Q₁₆	1	8	yes	complemented
162	C₂×C₅×D₄	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₅	—	2	1	no	—
164	C₂²×C₄×C₅	—	2	4	no	—
165	C₂²×C₄×C₅	—	2	4	no	—
166	C₂²×C₄×C₅	—	2	8	no	—
167	C₂²×C₄×C₅	—	2	1	no	—
168	C₂²×C₄×C₅	—	2	1	no	—
169	C₂²×C₄×C₅	—	2	1	no	—
170	C₂²×C₄×C₅	—	2	2	no	—
171	C₂²×C₄×C₅	—	2	2	no	—
172	C₂²×C₄×C₅	—	2	8	no	—
173	C₂²×C₄×C₅	—	4	8	no	—
174	C₂²×C₄×C₅	—	8	4	no	—
175	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
181	C₅×Q₁₆	—	2	8	no	—
182	C₅×Q₁₆	—	4	8	no	—
183	C₂×C₅×C₈	—	2	2	no	—
184	C₂×C₅×C₈	—	2	2	no	—
185	C₂×C₅×C₈	—	2	4	no	—
186	C₂×C₅×C₈	—	4	8	no	—
187	C₂×C₅×C₈	C₂⁴	1	2	yes	—
188	C₂×C₅×C₈	C₂⁴	1	4	yes	—
189	C₂×C₅×C₈	C₂⁴	1	2	yes	—
190	C₅×C₄⋊C₄	—	2	1	no	—
191	C₅×C₄⋊C₄	—	2	8	no	—
192	C₅×C₄⋊C₄	—	2	2	no	—
193	C₅×C₄⋊C₄	—	2	4	no	—
194	C₅×C₄⋊C₄	—	2	4	no	—
195	C₅×C₄⋊C₄	—	2	4	no	—
196	C₅×C₄⋊C₄	—	2	4	no	—
197	C₅×C₄⋊C₄	—	2	4	no	—
198	C₅×C₄⋊C₄	—	2	8	no	—
199	C₅×C₄⋊C₄	—	2	8	no	—
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
202	C₅×C₄⋊C₄	C₂⁴	1	2	yes	—
203	C₅×C₄⋊C₄	C₂⁴	1	4	yes	—
204	C₅×C₄⋊C₄	C₂⁴	1	2	yes	—
205	C₅×C₂²⋊C₄	—	2	8	no	—
206	C₅×C₂²⋊C₄	—	2	8	no	—
207	C₅×C₂²⋊C₄	C₂×D₄	1	8	yes	—
208	C₄²×C₅	—	2	4	no	—
209	C₄²×C₅	—	2	4	no	—
210	C₄²×C₅	C₂×D₄	1	2	yes	—
211	C₄²×C₅	C₂⁴	1	2	yes	—
212	C₂³×Q₈	C₂²×C₅	1	2	yes	—
213	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
214	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
215	C₂³×C₈	C₂²×C₅	1	2	yes	—
216	#554	—	2	4	no	—
217	#554	C₂²×C₅	1	8	yes	—
218	D₄×Q₈	—	2	4	no	—
219	D₄×Q₈	C₂²×C₅	1	8	yes	—
220	C₂×C₄⋊Q₈	—	2	1	no	—
221	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
223	C₂×#125	—	2	1	no	—
224	C₂×C₂²⋊Q₈	—	2	2	no	—
225	C₂×C₂²⋊Q₈	—	2	2	no	—
226	C₂×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
227	C₂×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
228	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
229	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
230	C₂×C₄×D₄	—	2	1	no	—
231	C₂×C₄×D₄	—	2	1	no	—
232	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
233	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
234	#487	C₂²×C₅	1	8	yes	complemented
235	#484	C₂²×C₅	1	16	yes	complemented
236	#477	C₂²×C₅	1	8	yes	complemented
237	#467	C₂²×C₅	1	16	yes	complemented
238	#462	C₂²×C₅	1	8	yes	complemented
239	#451	C₂²×C₅	1	16	yes	complemented
240	C₄×Q₁₆	C₂²×C₅	1	8	yes	complemented
241	C₈×D₄	C₂²×C₅	1	8	yes	—
242	C₂×#107	C₂²×C₅	1	2	yes	—
243	C₂×#107	C₂²×C₅	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
245	C₂×#103	C₂²×C₅	1	2	yes	—
246	C₂×#103	C₂²×C₅	1	2	yes	—
247	C₂×#103	C₂²×C₅	1	2	yes	—
248	C₂×#103	C₂²×C₅	1	2	yes	—
249	C₂×#103	C₂²×C₅	1	1	yes	characteristic
250	C₂×#103	C₂²×C₅	1	1	yes	characteristic
251	C₂×#103	C₂²×C₅	1	16	yes	complemented
252	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
253	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
254	C₂³×C₅	—	2	4	no	—
255	C₂³×C₅	—	2	4	no	—
256	C₂³×C₅	C₂×Q₁₆	1	8	yes	—
257	C₂³×C₅	C₂²×D₄	1	4	yes	—
258	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
259	C₅×Q₈	—	2	4	no	—
260	C₅×Q₈	—	2	2	no	—
261	C₅×Q₈	—	2	2	no	—
262	C₅×Q₈	—	2	4	no	—
263	C₅×Q₈	—	2	2	no	—
264	C₅×Q₈	—	2	4	no	—
265	C₅×Q₈	—	4	16	no	—
266	C₅×Q₈	—	4	2	no	—
267	C₅×Q₈	—	4	4	no	—
268	C₅×Q₈	—	8	4	no	—
269	C₅×Q₈	C₂×#41	1	4	yes	—
270	C₅×D₄	C₂×Q₁₆	1	16	yes	complemented
271	C₂×C₄×C₅	—	2	4	no	—
272	C₂×C₄×C₅	—	2	4	no	—
273	C₂×C₄×C₅	—	2	2	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	2	8	no	—
276	C₂×C₄×C₅	—	2	4	no	—
277	C₂×C₄×C₅	—	2	4	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	4	no	—
282	C₂×C₄×C₅	—	2	4	no	—
283	C₂×C₄×C₅	—	4	16	no	—
284	C₂×C₄×C₅	—	4	4	no	—
285	C₂×C₄×C₅	—	4	2	no	—
286	C₂×C₄×C₅	—	4	2	no	—
287	C₂×C₄×C₅	—	4	2	no	—
288	C₂×C₄×C₅	—	4	4	no	—
289	C₂×C₄×C₅	—	8	8	no	—
290	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
291	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₅	C₂×#41	1	2	yes	—
296	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
297	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
298	C₅×C₈	—	2	2	no	—
299	C₅×C₈	—	2	2	no	—
300	C₅×C₈	—	2	4	no	—
301	C₂²×Q₈	—	2	2	no	—
302	C₂²×Q₈	—	2	4	no	—
303	C₂²×Q₈	—	2	1	no	—
304	C₂²×Q₈	—	2	8	no	—
305	C₂²×Q₈	—	4	4	no	—
306	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₅	1	2	yes	—
308	C₂²×Q₈	C₂³×C₅	1	8	yes	—
309	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
310	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
311	C₂³×C₄	—	2	2	no	—
312	C₂³×C₄	—	2	2	no	—
313	C₂³×C₄	C₂³×C₅	1	2	yes	—
314	C₂×Q₁₆	—	2	4	no	—
315	C₂×Q₁₆	—	4	4	no	—
316	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
317	C₂×Q₁₆	C₂³×C₅	1	8	yes	—
318	C₂²×C₈	—	4	4	no	—
319	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₅	1	2	yes	—
321	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₅	1	8	yes	—
323	C₄⋊Q₈	—	2	8	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
327	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
328	#125	—	2	8	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	4	no	—
331	C₂²⋊Q₈	—	2	16	no	—
332	C₂²⋊Q₈	—	2	16	no	—
333	C₂²⋊Q₈	C₂³×C₅	1	8	yes	—
334	C₂²⋊Q₈	C₂³×C₅	1	8	yes	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	C₂³×C₅	1	4	yes	—
338	C₄×Q₈	C₂³×C₅	1	4	yes	—
339	C₄×D₄	—	2	8	no	—
340	C₄×D₄	—	2	8	no	—
341	C₄×D₄	C₅×D₄	1	4	yes	—
342	C₄×D₄	C₂³×C₅	1	4	yes	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	—	2	4	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
356	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
363	#107	C₂³×C₅	1	4	yes	—
364	#107	C₂³×C₅	1	8	yes	—
365	C₄⋊C₈	C₂³×C₅	1	4	yes	—
366	#103	C₂³×C₅	1	4	yes	—
367	#103	C₂³×C₅	1	8	yes	—
368	#103	C₂³×C₅	1	8	yes	—
369	#103	C₂³×C₅	1	8	yes	—
370	#103	C₂³×C₅	1	4	yes	—
371	#103	C₂³×C₅	1	8	yes	—
372	C₂²⋊C₈	C₂³×C₅	1	8	yes	—
373	C₄×C₈	C₂³×C₅	1	4	yes	—
374	C₂²×C₅	—	2	4	no	—
375	C₂²×C₅	—	2	16	no	—
376	C₂²×C₅	—	2	4	no	—
377	C₂²×C₅	#546	1	2	yes	—
378	C₂²×C₅	C₂²×Q₁₆	1	8	yes	—
379	C₂²×C₅	C₂²×Q₁₆	1	2	yes	—
380	C₂²×C₅	#577	1	2	yes	—
381	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
382	C₄×C₅	—	2	4	no	—
383	C₄×C₅	—	2	4	no	—
384	C₄×C₅	—	2	2	no	—
385	C₄×C₅	—	4	4	no	—
386	C₄×C₅	—	4	2	no	—
387	C₄×C₅	—	4	2	no	—
388	C₄×C₅	—	4	2	no	—
389	C₄×C₅	—	4	4	no	—
390	C₄×C₅	#546	1	2	yes	—
391	C₄×C₅	C₂²×Q₁₆	1	4	yes	—
392	C₄×C₅	C₂²×#41	1	1	yes	characteristic
393	C₄×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
394	C₂⁴	C₂×C₅×D₄	1	2	yes	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	16	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	4	8	no	—
407	C₂×Q₈	—	4	8	no	—
408	C₂×Q₈	—	4	16	no	—
409	C₂×Q₈	—	8	2	no	—
410	C₂×Q₈	C₅×#41	1	4	yes	—
411	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
412	C₂×Q₈	C₂⁴×C₅	1	2	yes	—
413	C₂×Q₈	C₂⁴×C₅	1	2	yes	—
414	C₂×D₄	C₅×Q₁₆	1	8	yes	complemented
415	C₂×D₄	C₂×C₅×D₄	1	4	yes	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	4	8	no	—
427	C₂²×C₄	—	8	4	no	—
428	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
429	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
430	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
431	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
434	Q₁₆	—	2	8	no	—
435	Q₁₆	—	4	8	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	4	no	—
439	C₂×C₈	—	4	8	no	—
440	C₂×C₈	C₂⁴×C₅	1	2	yes	—
441	C₂×C₈	C₂⁴×C₅	1	2	yes	—
442	C₂×C₈	C₂⁴×C₅	1	4	yes	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₅	1	2	yes	—
456	C₄⋊C₄	C₂⁴×C₅	1	4	yes	—
457	C₄⋊C₄	C₂⁴×C₅	1	2	yes	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	C₂×C₅×D₄	1	8	yes	—
461	C₄²	—	2	4	no	—
462	C₄²	—	2	4	no	—
463	C₄²	C₂×C₅×D₄	1	2	yes	—
464	C₄²	C₂⁴×C₅	1	2	yes	—
465	C₂×C₅	—	2	8	no	—
466	C₂×C₅	#3299	1	4	yes	direct product factor
467	C₂×C₅	C₂×#546	1	1	yes	characteristic
468	C₂×C₅	C₂³×Q₁₆	1	1	yes	characteristic
469	C₂×C₅	C₂×#577	1	1	yes	characteristic
470	C₂³	—	2	4	no	—
471	C₂³	—	2	4	no	—
472	C₂³	C₂×C₅×Q₁₆	1	8	yes	—
473	C₂³	C₂²×C₅×D₄	1	4	yes	—
474	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
475	Q₈	—	2	4	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	2	2	no	—
480	Q₈	—	2	4	no	—
481	Q₈	—	4	16	no	—
482	Q₈	—	4	4	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	8	4	no	—
485	Q₈	C₂×C₅×#41	1	4	yes	—
486	D₄	C₂×C₅×Q₁₆	1	16	yes	complemented
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	8	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	16	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	8	8	no	—
506	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
507	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
508	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
509	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
511	C₂×C₄	C₂×C₅×#41	1	2	yes	—
512	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
513	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
514	C₈	—	2	2	no	—
515	C₈	—	2	2	no	—
516	C₈	—	2	4	no	—
517	C₅	C₂×#3299	1	1	yes	characteristic, direct product factor, Sylow
518	C₂²	—	2	16	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	4	no	—
521	C₂²	C₅×#546	1	2	yes	—
522	C₂²	C₂²×C₅×Q₁₆	1	8	yes	—
523	C₂²	C₂²×C₅×Q₁₆	1	2	yes	—
524	C₂²	C₅×#577	1	2	yes	—
525	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
526	C₄	—	2	4	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	4	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	2	no	—
534	C₄	C₅×#546	1	2	yes	—
535	C₄	C₂²×C₅×Q₁₆	1	4	yes	—
536	C₄	C₂²×C₅×#41	1	1	yes	characteristic
537	C₄	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
538	C₂	—	2	8	no	—
539	C₂	C₅×#3299	1	4	yes	direct product factor
540	C₂	C₂×C₅×#546	1	1	yes	characteristic
541	C₂	C₂³×C₅×Q₁₆	1	1	yes	characteristic
542	C₂	C₂×C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#3299	C₂	direct (N × H)
4. C₂×C₅×#487	C₂	semidirect
5. C₂×C₅×#484	C₂	semidirect
6. C₂×C₅×#477	C₂	semidirect
7. C₂×C₅×#467	C₂	semidirect
8. C₂×C₅×#462	C₂	semidirect
9. C₂×C₅×#451	C₂	semidirect
10. C₂×C₄×C₅×Q₁₆	C₂	semidirect
12. C₂²×C₅×#103	C₂	semidirect
35. C₅×#487	C₂²	semidirect
36. C₅×#484	C₂²	semidirect
37. C₅×#477	C₂²	semidirect
38. C₅×#467	C₂²	semidirect
39. C₅×#462	C₂²	semidirect
40. C₅×#451	C₂²	semidirect
41. C₄×C₅×Q₁₆	C₂²	semidirect
52. C₂×C₅×#103	C₂²	semidirect
55. C₂×#3299	C₅	direct (N × H)
131. #3299	C₂×C₅	direct (N × H)
132. C₂×#487	C₂×C₅	semidirect
133. C₂×#484	C₂×C₅	semidirect
134. C₂×#477	C₂×C₅	semidirect
135. C₂×#467	C₂×C₅	semidirect
136. C₂×#462	C₂×C₅	semidirect
137. C₂×#451	C₂×C₅	semidirect
138. C₂×C₄×Q₁₆	C₂×C₅	semidirect
140. C₂²×#103	C₂×C₅	semidirect
161. C₂×C₅×D₄	Q₁₆	semidirect
234. #487	C₂²×C₅	semidirect
235. #484	C₂²×C₅	semidirect
236. #477	C₂²×C₅	semidirect
237. #467	C₂²×C₅	semidirect
238. #462	C₂²×C₅	semidirect
239. #451	C₂²×C₅	semidirect
240. C₄×Q₁₆	C₂²×C₅	semidirect
251. C₂×#103	C₂²×C₅	semidirect
270. C₅×D₄	C₂×Q₁₆	semidirect
414. C₂×D₄	C₅×Q₁₆	semidirect
466. C₂×C₅	#3299	direct (N × H)
486. D₄	C₂×C₅×Q₁₆	semidirect
517. C₅	C₂×#3299	direct (N × H)
539. C₂	C₅×#3299	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#554	2
C₂×C₅×D₄×Q₈	2
C₅×#3299	2
C₂×C₅×#487	2
C₂×C₅×#484	2
C₂×C₅×#477	2
C₂×C₅×#467	2
C₂×C₅×#462	2
C₂×C₅×#451	2
C₂×C₄×C₅×Q₁₆	2
C₂×C₅×C₈×D₄	2
C₂²×C₅×#103	2
C₂×#3299	5

Subgroups — C2×C5×#3299 #11983837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3299 #11983837