Subgroups — C₅×#34258 #11956978

Order 1280.

5682 conjugacy classes of subgroups.

The subgroup lattice has 458 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	1	yes	maximal, characteristic
2	C₂×C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2341	C₂	1	4	yes	maximal, complemented
5	C₅×D₄×C₂²⋊C₄	C₂	1	12	yes	maximal, complemented
6	C₅×#2311	C₂	1	3	yes	maximal, complemented
7	C₅×#2309	C₂	1	3	yes	maximal, complemented
8	C₅×#2297	C₂	1	3	yes	maximal, complemented
9	C₂×C₅×#390	C₂	1	3	yes	maximal, complemented
10	C₂×C₅×H(5,𝔽₂)	—	2	8	no	—
11	C₂×C₅×H(5,𝔽₂)	—	2	8	no	—
12	C₂×C₅×H(5,𝔽₂)	C₄	1	16	yes	complemented
13	C₂²×C₅×#41	C₂²	1	3	yes	—
14	C₂²×C₅×#41	C₂²	1	2	yes	—
15	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
16	C₂³×C₅×D₄	C₂²	1	3	yes	—
17	C₂³×C₅×D₄	C₂²	1	6	yes	—
18	C₅×#520	—	2	8	no	—
19	C₄×C₅×#41	—	2	8	no	—
20	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
21	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
22	C₂×C₄×C₅×D₄	C₂²	1	6	yes	complemented
23	C₂×C₄×C₅×D₄	C₂²	1	6	yes	complemented
24	C₂×C₅×#117	C₂²	1	3	yes	complemented
25	C₂×C₅×#117	C₂²	1	3	yes	complemented
26	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	complemented
27	C₂²×C₅×C₂²⋊C₄	C₂²	1	12	yes	complemented
28	C₂²×C₄²×C₅	C₂²	1	3	yes	complemented
29	C₅×#391	C₂²	1	1	yes	characteristic, complemented
30	C₅×#391	C₂²	1	3	yes	complemented
31	C₅×#390	C₂²	1	3	yes	complemented
32	C₅×#390	C₂²	1	6	yes	complemented
33	C₅×#390	C₂²	1	12	yes	complemented
34	C₅×#390	C₂²	1	3	yes	complemented
35	C₅×#390	C₂²	1	12	yes	complemented
36	C₅×#386	C₂²	1	12	yes	complemented
37	C₅×#386	C₂²	1	12	yes	complemented
38	C₅×#380	C₂²	1	12	yes	complemented
39	C₅×#379	C₂²	1	12	yes	complemented
40	C₄×C₅×C₂²⋊C₄	C₂²	1	12	yes	complemented
41	#34258	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁵×C₅	C₂³	1	6	yes	—
43	C₅×H(5,𝔽₂)	—	2	128	no	—
44	C₂×C₅×#41	—	2	8	no	—
45	C₂×C₅×#41	—	2	8	no	—
46	C₂×C₅×#41	—	2	12	no	—
47	C₂×C₅×#41	—	2	12	no	—
48	C₂×C₅×#41	C₂×C₄	1	8	yes	—
49	C₂×C₅×#41	C₂×C₄	1	16	yes	complemented
50	C₂×C₅×#41	C₂×C₄	1	24	yes	complemented
51	C₂×C₅×#41	D₄	1	8	yes	—
52	C₂×C₅×#41	D₄	1	8	yes	—
53	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
54	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
55	C₂²×C₅×D₄	—	2	24	no	—
56	C₂²×C₅×D₄	—	2	24	no	—
57	C₂²×C₅×D₄	—	2	12	no	—
58	C₂²×C₅×D₄	—	2	12	no	—
59	C₂²×C₅×D₄	C₂×C₄	1	48	yes	complemented
60	C₂²×C₅×D₄	C₂×C₄	1	24	yes	complemented
61	C₂²×C₅×D₄	C₂³	1	6	yes	—
62	C₂²×C₅×D₄	C₂³	1	3	yes	—
63	C₂²×C₅×D₄	C₂³	1	6	yes	—
64	C₂²×C₅×D₄	C₂³	1	3	yes	—
65	C₂³×C₄×C₅	C₂³	1	6	yes	—
66	C₂³×C₄×C₅	C₂³	1	6	yes	—
67	C₂³×C₄×C₅	C₂³	1	6	yes	—
68	C₂³×C₄×C₅	C₂³	1	3	yes	—
69	C₄×C₅×Q₈	—	2	8	no	—
70	C₄×C₅×Q₈	—	2	8	no	—
71	C₄×C₅×D₄	—	2	24	no	—
72	C₄×C₅×D₄	—	2	24	no	—
73	C₅×#117	—	2	12	no	—
74	C₅×#117	—	2	12	no	—
75	C₂×C₅×C₄⋊C₄	—	2	12	no	—
76	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
77	C₂×C₅×C₄⋊C₄	C₂³	1	6	yes	—
78	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
79	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₂×C₅×C₂²⋊C₄	—	4	24	no	—
81	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
82	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
83	C₂×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
84	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
85	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	—
86	C₂×C₄²×C₅	—	2	12	no	—
87	C₂×C₄²×C₅	C₂³	1	6	yes	—
88	C₂×C₄²×C₅	C₂³	1	6	yes	—
89	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅	C₂³	1	3	yes	—
91	C₅×#95	C₂³	1	4	yes	—
92	C₅×#95	C₂³	1	12	yes	—
93	C₂²×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic
94	C₂×#520	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
96	#2341	C₂×C₅	1	4	yes	complemented
97	D₄×C₂²⋊C₄	C₂×C₅	1	12	yes	complemented
98	#2311	C₂×C₅	1	3	yes	complemented
99	#2309	C₂×C₅	1	3	yes	complemented
100	#2297	C₂×C₅	1	3	yes	complemented
101	C₂×#390	C₂×C₅	1	3	yes	complemented
102	C₂⁴×C₅	—	2	12	no	—
103	C₂⁴×C₅	—	2	12	no	—
104	C₂⁴×C₅	—	4	12	no	—
105	C₂⁴×C₅	—	4	12	no	—
106	C₂⁴×C₅	C₂²×C₄	1	24	yes	—
107	C₂⁴×C₅	C₂⁴	1	6	yes	—
108	C₂⁴×C₅	C₂⁴	1	3	yes	—
109	C₅×#41	—	2	64	no	—
110	C₅×#41	—	2	96	no	—
111	C₅×#41	C₂²⋊C₄	1	64	yes	complemented
112	C₂×C₅×Q₈	—	2	2	no	—
113	C₂×C₅×Q₈	—	2	2	no	—
114	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
115	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
116	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
117	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
118	C₂×C₅×D₄	—	2	12	no	—
119	C₂×C₅×D₄	—	2	12	no	—
120	C₂×C₅×D₄	—	2	192	no	—
121	C₂×C₅×D₄	—	2	96	no	—
122	C₂×C₅×D₄	—	2	6	no	—
123	C₂×C₅×D₄	—	2	12	no	—
124	C₂×C₅×D₄	—	2	12	no	—
125	C₂×C₅×D₄	—	2	6	no	—
126	C₂×C₅×D₄	C₂²×C₄	1	24	yes	—
127	C₂×C₅×D₄	C₂²×C₄	1	24	yes	—
128	C₂×C₅×D₄	C₂²×C₄	1	12	yes	—
129	C₂×C₅×D₄	C₂²×C₄	1	12	yes	—
130	C₂×C₅×D₄	C₂×D₄	1	12	yes	—
131	C₂×C₅×D₄	C₂×D₄	1	12	yes	—
132	C₂²×C₄×C₅	—	2	12	no	—
133	C₂²×C₄×C₅	—	2	12	no	—
134	C₂²×C₄×C₅	—	2	12	no	—
135	C₂²×C₄×C₅	—	2	12	no	—
136	C₂²×C₄×C₅	—	2	12	no	—
137	C₂²×C₄×C₅	—	4	6	no	—
138	C₂²×C₄×C₅	—	4	12	no	—
139	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
140	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
141	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
142	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
143	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
145	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
146	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
150	C₅×C₄⋊C₄	—	2	6	no	—
151	C₅×C₄⋊C₄	—	2	12	no	—
152	C₅×C₄⋊C₄	—	2	2	no	—
153	C₅×C₄⋊C₄	—	2	12	no	—
154	C₅×C₂²⋊C₄	—	2	12	no	—
155	C₅×C₂²⋊C₄	—	2	12	no	—
156	C₅×C₂²⋊C₄	—	2	24	no	—
157	C₅×C₂²⋊C₄	—	4	24	no	—
158	C₅×C₂²⋊C₄	—	4	12	no	—
159	C₄²×C₅	—	2	2	no	—
160	C₄²×C₅	—	2	6	no	—
161	C₄²×C₅	—	2	12	no	—
162	C₄²×C₅	—	2	12	no	—
163	C₂×H(5,𝔽₂)	—	2	8	no	—
164	C₂×H(5,𝔽₂)	—	2	8	no	—
165	C₂×H(5,𝔽₂)	C₄×C₅	1	16	yes	complemented
166	C₂²×#41	C₂²×C₅	1	3	yes	—
167	C₂²×#41	C₂²×C₅	1	2	yes	—
168	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
169	C₂³×D₄	C₂²×C₅	1	3	yes	—
170	C₂³×D₄	C₂²×C₅	1	6	yes	—
171	#520	—	2	8	no	—
172	C₄×#41	—	2	8	no	—
173	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
174	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
175	C₂×C₄×D₄	C₂²×C₅	1	6	yes	complemented
176	C₂×C₄×D₄	C₂²×C₅	1	6	yes	complemented
177	C₂×#117	C₂²×C₅	1	3	yes	complemented
178	C₂×#117	C₂²×C₅	1	3	yes	complemented
179	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	complemented
180	C₂²×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
181	C₂²×C₄²	C₂²×C₅	1	3	yes	complemented
182	#391	C₂²×C₅	1	3	yes	complemented
183	#391	C₂²×C₅	1	1	yes	characteristic, complemented
184	#390	C₂²×C₅	1	3	yes	complemented
185	#390	C₂²×C₅	1	6	yes	complemented
186	#390	C₂²×C₅	1	12	yes	complemented
187	#390	C₂²×C₅	1	12	yes	complemented
188	#390	C₂²×C₅	1	3	yes	complemented
189	#386	C₂²×C₅	1	12	yes	complemented
190	#386	C₂²×C₅	1	12	yes	complemented
191	#380	C₂²×C₅	1	12	yes	complemented
192	#379	C₂²×C₅	1	12	yes	complemented
193	C₄×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
194	C₂³×C₅	—	2	6	no	—
195	C₂³×C₅	—	2	6	no	—
196	C₂³×C₅	—	2	6	no	—
197	C₂³×C₅	—	2	6	no	—
198	C₂³×C₅	—	2	12	no	—
199	C₂³×C₅	—	2	12	no	—
200	C₂³×C₅	—	2	48	no	—
201	C₂³×C₅	—	4	48	no	—
202	C₂³×C₅	—	8	24	no	—
203	C₂³×C₅	—	8	12	no	—
204	C₂³×C₅	—	8	12	no	—
205	C₂³×C₅	C₂³×C₄	1	12	yes	—
206	C₂³×C₅	C₂³×C₄	1	6	yes	—
207	C₂³×C₅	C₂²×D₄	1	6	yes	—
208	C₂³×C₅	C₂²×D₄	1	6	yes	—
209	C₂³×C₅	C₂⁵	1	1	yes	characteristic
210	C₅×Q₈	—	2	8	no	—
211	C₅×Q₈	C₂×C₂²⋊C₄	1	16	yes	complemented
212	C₅×D₄	—	2	48	no	—
213	C₅×D₄	—	2	48	no	—
214	C₅×D₄	—	2	24	no	—
215	C₅×D₄	C₂×C₂²⋊C₄	1	48	yes	complemented
216	C₂×C₄×C₅	—	2	2	no	—
217	C₂×C₄×C₅	—	2	2	no	—
218	C₂×C₄×C₅	—	2	2	no	—
219	C₂×C₄×C₅	—	2	2	no	—
220	C₂×C₄×C₅	—	2	48	no	—
221	C₂×C₄×C₅	—	2	2	no	—
222	C₂×C₄×C₅	—	2	6	no	—
223	C₂×C₄×C₅	—	2	6	no	—
224	C₂×C₄×C₅	—	2	12	no	—
225	C₂×C₄×C₅	—	2	6	no	—
226	C₂×C₄×C₅	—	2	4	no	—
227	C₂×C₄×C₅	—	2	6	no	—
228	C₂×C₄×C₅	—	4	24	no	—
229	C₂×C₄×C₅	—	8	12	no	—
230	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	48	yes	complemented
231	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
232	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
233	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
234	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
236	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
237	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	complemented
239	C₂⁵	C₂³×C₅	1	6	yes	—
240	H(5,𝔽₂)	—	2	128	no	—
241	C₂×#41	—	2	8	no	—
242	C₂×#41	—	2	8	no	—
243	C₂×#41	—	2	12	no	—
244	C₂×#41	—	2	12	no	—
245	C₂×#41	C₂×C₄×C₅	1	8	yes	—
246	C₂×#41	C₂×C₄×C₅	1	16	yes	complemented
247	C₂×#41	C₂×C₄×C₅	1	24	yes	complemented
248	C₂×#41	C₅×D₄	1	8	yes	—
249	C₂×#41	C₅×D₄	1	8	yes	—
250	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
251	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
252	C₂²×D₄	—	2	24	no	—
253	C₂²×D₄	—	2	24	no	—
254	C₂²×D₄	—	2	12	no	—
255	C₂²×D₄	—	2	12	no	—
256	C₂²×D₄	C₂×C₄×C₅	1	48	yes	complemented
257	C₂²×D₄	C₂×C₄×C₅	1	24	yes	complemented
258	C₂²×D₄	C₂³×C₅	1	6	yes	—
259	C₂²×D₄	C₂³×C₅	1	3	yes	—
260	C₂²×D₄	C₂³×C₅	1	6	yes	—
261	C₂²×D₄	C₂³×C₅	1	3	yes	—
262	C₂³×C₄	C₂³×C₅	1	6	yes	—
263	C₂³×C₄	C₂³×C₅	1	6	yes	—
264	C₂³×C₄	C₂³×C₅	1	6	yes	—
265	C₂³×C₄	C₂³×C₅	1	3	yes	—
266	C₄×Q₈	—	2	8	no	—
267	C₄×Q₈	—	2	8	no	—
268	C₄×D₄	—	2	24	no	—
269	C₄×D₄	—	2	24	no	—
270	#117	—	2	12	no	—
271	#117	—	2	12	no	—
272	C₂×C₄⋊C₄	—	2	12	no	—
273	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
274	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
275	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₅	1	6	yes	—
277	C₂×C₂²⋊C₄	—	4	24	no	—
278	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
279	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
280	C₂×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₅	1	24	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₅	1	12	yes	—
283	C₂×C₄²	—	2	12	no	—
284	C₂×C₄²	C₂³×C₅	1	3	yes	—
285	C₂×C₄²	C₂³×C₅	1	6	yes	—
286	C₂×C₄²	C₂³×C₅	1	6	yes	—
287	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
288	#95	C₂³×C₅	1	4	yes	—
289	#95	C₂³×C₅	1	12	yes	—
290	C₂²×C₅	—	2	12	no	—
291	C₂²×C₅	—	2	12	no	—
292	C₂²×C₅	—	2	6	no	—
293	C₂²×C₅	—	2	6	no	—
294	C₂²×C₅	—	2	12	no	—
295	C₂²×C₅	—	2	24	no	—
296	C₂²×C₅	—	4	12	no	—
297	C₂²×C₅	—	4	6	no	—
298	C₂²×C₅	—	4	6	no	—
299	C₂²×C₅	—	8	48	no	—
300	C₂²×C₅	C₂²×C₂²⋊C₄	1	12	yes	—
301	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
302	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
303	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
304	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
305	C₂²×C₅	#585	1	2	yes	—
306	C₄×C₅	—	2	4	no	—
307	C₄×C₅	—	2	4	no	—
308	C₄×C₅	—	4	6	no	—
309	C₄×C₅	—	4	6	no	—
310	C₄×C₅	—	4	2	no	—
311	C₄×C₅	C₂²×C₂²⋊C₄	1	12	yes	—
312	C₄×C₅	C₂²×C₂²⋊C₄	1	4	yes	—
313	C₂⁴	—	2	12	no	—
314	C₂⁴	—	2	12	no	—
315	C₂⁴	—	4	12	no	—
316	C₂⁴	—	4	12	no	—
317	C₂⁴	C₂²×C₄×C₅	1	24	yes	—
318	C₂⁴	C₂⁴×C₅	1	6	yes	—
319	C₂⁴	C₂⁴×C₅	1	3	yes	—
320	#41	—	2	64	no	—
321	#41	—	2	96	no	—
322	#41	C₅×C₂²⋊C₄	1	64	yes	complemented
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
326	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
327	C₂×Q₈	C₂×C₅×D₄	1	4	yes	—
328	C₂×Q₈	C₂×C₅×D₄	1	4	yes	—
329	C₂×D₄	—	2	12	no	—
330	C₂×D₄	—	2	12	no	—
331	C₂×D₄	—	2	192	no	—
332	C₂×D₄	—	2	96	no	—
333	C₂×D₄	—	2	6	no	—
334	C₂×D₄	—	2	12	no	—
335	C₂×D₄	—	2	12	no	—
336	C₂×D₄	—	2	6	no	—
337	C₂×D₄	C₂²×C₄×C₅	1	24	yes	—
338	C₂×D₄	C₂²×C₄×C₅	1	24	yes	—
339	C₂×D₄	C₂²×C₄×C₅	1	12	yes	—
340	C₂×D₄	C₂²×C₄×C₅	1	12	yes	—
341	C₂×D₄	C₂×C₅×D₄	1	12	yes	—
342	C₂×D₄	C₂×C₅×D₄	1	12	yes	—
343	C₂²×C₄	—	2	12	no	—
344	C₂²×C₄	—	2	12	no	—
345	C₂²×C₄	—	2	12	no	—
346	C₂²×C₄	—	2	12	no	—
347	C₂²×C₄	—	2	12	no	—
348	C₂²×C₄	—	4	6	no	—
349	C₂²×C₄	—	4	12	no	—
350	C₂²×C₄	C₂²×C₄×C₅	1	24	yes	—
351	C₂²×C₄	C₂²×C₄×C₅	1	12	yes	—
352	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
353	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
354	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
356	C₂²×C₄	C₂⁴×C₅	1	6	yes	—
357	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
359	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
360	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
361	C₄⋊C₄	—	2	6	no	—
362	C₄⋊C₄	—	2	12	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	12	no	—
365	C₂²⋊C₄	—	2	12	no	—
366	C₂²⋊C₄	—	2	12	no	—
367	C₂²⋊C₄	—	2	24	no	—
368	C₂²⋊C₄	—	4	24	no	—
369	C₂²⋊C₄	—	4	12	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	6	no	—
372	C₄²	—	2	12	no	—
373	C₄²	—	2	12	no	—
374	C₂×C₅	—	2	12	no	—
375	C₂×C₅	—	4	12	no	—
376	C₂×C₅	C₂³×C₂²⋊C₄	1	1	yes	characteristic
377	C₂×C₅	C₄×H(5,𝔽₂)	1	2	yes	—
378	C₂×C₅	#3488	1	2	yes	—
379	C₂×C₅	#3498	1	1	yes	characteristic
380	C₂×C₅	#3500	1	1	yes	characteristic
381	C₂³	—	2	6	no	—
382	C₂³	—	2	6	no	—
383	C₂³	—	2	6	no	—
384	C₂³	—	2	6	no	—
385	C₂³	—	2	12	no	—
386	C₂³	—	2	12	no	—
387	C₂³	—	2	48	no	—
388	C₂³	—	4	48	no	—
389	C₂³	—	8	24	no	—
390	C₂³	—	8	12	no	—
391	C₂³	—	8	12	no	—
392	C₂³	C₂³×C₄×C₅	1	12	yes	—
393	C₂³	C₂³×C₄×C₅	1	6	yes	—
394	C₂³	C₂²×C₅×D₄	1	6	yes	—
395	C₂³	C₂²×C₅×D₄	1	6	yes	—
396	C₂³	C₂⁵×C₅	1	1	yes	characteristic
397	Q₈	—	2	8	no	—
398	Q₈	C₂×C₅×C₂²⋊C₄	1	16	yes	complemented
399	D₄	—	2	48	no	—
400	D₄	—	2	48	no	—
401	D₄	—	2	24	no	—
402	D₄	C₂×C₅×C₂²⋊C₄	1	48	yes	complemented
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	48	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	6	no	—
410	C₂×C₄	—	2	6	no	—
411	C₂×C₄	—	2	12	no	—
412	C₂×C₄	—	2	6	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	6	no	—
415	C₂×C₄	—	4	24	no	—
416	C₂×C₄	—	8	12	no	—
417	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	48	yes	complemented
418	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
419	C₂×C₄	C₂³×C₄×C₅	1	6	yes	—
420	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
421	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
422	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
423	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
424	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
425	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	complemented
426	C₅	#34258	1	1	yes	characteristic, direct product factor, Sylow
427	C₂²	—	2	12	no	—
428	C₂²	—	2	12	no	—
429	C₂²	—	2	6	no	—
430	C₂²	—	2	6	no	—
431	C₂²	—	2	12	no	—
432	C₂²	—	2	24	no	—
433	C₂²	—	4	12	no	—
434	C₂²	—	4	6	no	—
435	C₂²	—	4	6	no	—
436	C₂²	—	8	48	no	—
437	C₂²	C₂²×C₅×C₂²⋊C₄	1	12	yes	—
438	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
439	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
440	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
441	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
442	C₂²	C₅×#585	1	2	yes	—
443	C₄	—	2	4	no	—
444	C₄	—	2	4	no	—
445	C₄	—	4	6	no	—
446	C₄	—	4	6	no	—
447	C₄	—	4	2	no	—
448	C₄	C₂²×C₅×C₂²⋊C₄	1	12	yes	—
449	C₄	C₂²×C₅×C₂²⋊C₄	1	4	yes	—
450	C₂	—	2	12	no	—
451	C₂	—	4	12	no	—
452	C₂	C₂³×C₅×C₂²⋊C₄	1	1	yes	characteristic
453	C₂	C₄×C₅×H(5,𝔽₂)	1	2	yes	—
454	C₂	C₅×#3488	1	2	yes	—
455	C₂	C₅×#3498	1	1	yes	characteristic
456	C₂	C₅×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂²×C₅×H(5,𝔽₂)	C₂
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
2. C₂×C₅×#520	C₂	semidirect
3. C₂×C₄×C₅×#41	C₂	semidirect
4. C₅×#2341	C₂	semidirect
5. C₅×D₄×C₂²⋊C₄	C₂	semidirect
6. C₅×#2311	C₂	semidirect
7. C₅×#2309	C₂	semidirect
8. C₅×#2297	C₂	semidirect
9. C₂×C₅×#390	C₂	semidirect
12. C₂×C₅×H(5,𝔽₂)	C₄	semidirect
20. C₂×C₄×C₅×Q₈	C₂²	semidirect
21. C₂×C₄×C₅×Q₈	C₂²	semidirect
22. C₂×C₄×C₅×D₄	C₂²	semidirect
23. C₂×C₄×C₅×D₄	C₂²	semidirect
24. C₂×C₅×#117	C₂²	semidirect
25. C₂×C₅×#117	C₂²	semidirect
26. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
27. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
28. C₂²×C₄²×C₅	C₂²	semidirect
29. C₅×#391	C₂²	semidirect
30. C₅×#391	C₂²	semidirect
31. C₅×#390	C₂²	semidirect
32. C₅×#390	C₂²	semidirect
33. C₅×#390	C₂²	semidirect
34. C₅×#390	C₂²	semidirect
35. C₅×#390	C₂²	semidirect
36. C₅×#386	C₂²	semidirect
37. C₅×#386	C₂²	semidirect
38. C₅×#380	C₂²	semidirect
39. C₅×#379	C₂²	semidirect
40. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
41. #34258	C₅	direct (N × H)
49. C₂×C₅×#41	C₂×C₄	semidirect
50. C₂×C₅×#41	C₂×C₄	semidirect
59. C₂²×C₅×D₄	C₂×C₄	semidirect
60. C₂²×C₅×D₄	C₂×C₄	semidirect
94. C₂×#520	C₂×C₅	semidirect
95. C₂×C₄×#41	C₂×C₅	semidirect
96. #2341	C₂×C₅	semidirect
97. D₄×C₂²⋊C₄	C₂×C₅	semidirect
98. #2311	C₂×C₅	semidirect
99. #2309	C₂×C₅	semidirect
100. #2297	C₂×C₅	semidirect
101. C₂×#390	C₂×C₅	semidirect
111. C₅×#41	C₂²⋊C₄	semidirect
165. C₂×H(5,𝔽₂)	C₄×C₅	semidirect
173. C₂×C₄×Q₈	C₂²×C₅	semidirect
174. C₂×C₄×Q₈	C₂²×C₅	semidirect
175. C₂×C₄×D₄	C₂²×C₅	semidirect
176. C₂×C₄×D₄	C₂²×C₅	semidirect
177. C₂×#117	C₂²×C₅	semidirect
178. C₂×#117	C₂²×C₅	semidirect
179. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
180. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
181. C₂²×C₄²	C₂²×C₅	semidirect
182. #391	C₂²×C₅	semidirect
183. #391	C₂²×C₅	semidirect
184. #390	C₂²×C₅	semidirect
185. #390	C₂²×C₅	semidirect
186. #390	C₂²×C₅	semidirect
187. #390	C₂²×C₅	semidirect
188. #390	C₂²×C₅	semidirect
189. #386	C₂²×C₅	semidirect
190. #386	C₂²×C₅	semidirect
191. #380	C₂²×C₅	semidirect
192. #379	C₂²×C₅	semidirect
193. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
211. C₅×Q₈	C₂×C₂²⋊C₄	semidirect
215. C₅×D₄	C₂×C₂²⋊C₄	semidirect
230. C₂×C₄×C₅	C₂×C₂²⋊C₄	semidirect
238. C₂×C₄×C₅	H(5,𝔽₂)	semidirect
246. C₂×#41	C₂×C₄×C₅	semidirect
247. C₂×#41	C₂×C₄×C₅	semidirect
256. C₂²×D₄	C₂×C₄×C₅	semidirect
257. C₂²×D₄	C₂×C₄×C₅	semidirect
322. #41	C₅×C₂²⋊C₄	semidirect
398. Q₈	C₂×C₅×C₂²⋊C₄	semidirect
402. D₄	C₂×C₅×C₂²⋊C₄	semidirect
417. C₂×C₄	C₂×C₅×C₂²⋊C₄	semidirect
425. C₂×C₄	C₅×H(5,𝔽₂)	semidirect
426. C₅	#34258	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
C₂×C₅×#520	2
C₂×C₄×C₅×#41	2
C₅×#2341	2
C₅×D₄×C₂²⋊C₄	2
C₅×#2311	2
C₅×#2309	2
C₅×#2297	2
C₂×C₅×#390	2
#34258	5

Subgroups — C5×#34258 #11956978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34258 #11956978