Subgroups — #13107165

Order 1600.

912 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717266	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717264	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
5	C₂×#87806	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717207	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717201	C₂	1	1	yes	maximal, characteristic
13	#11717201	C₂	1	1	yes	maximal, characteristic
14	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
19	C₂²×#1039	C₂²	1	1	yes	characteristic
20	C₂²×#1039	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
24	C₂×#6045	C₂²	1	1	yes	characteristic
25	C₂×#6045	C₂²	1	1	yes	characteristic
26	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂²⋊#1039	C₂²	1	1	yes	characteristic
32	#87808	C₂²	1	1	yes	characteristic
33	#87808	C₂²	1	1	yes	characteristic
34	#87808	C₂²	1	1	yes	characteristic
35	C₄⋊#1039	C₂²	1	1	yes	characteristic
36	C₄⋊#1039	C₂²	1	1	yes	characteristic
37	#87806	C₂²	1	1	yes	characteristic
38	#87806	C₂²	1	1	yes	characteristic
39	#87806	C₂²	1	1	yes	characteristic
40	#87806	C₂²	1	1	yes	characteristic
41	#87806	C₂²	1	1	yes	characteristic
42	#87806	C₂²	1	1	yes	characteristic
43	#87806	C₂²	1	1	yes	characteristic
44	#87806	C₂²	1	1	yes	characteristic
45	#87806	C₂²	1	1	yes	characteristic
46	#87806	C₂²	1	1	yes	characteristic
47	#87806	C₂²	1	1	yes	characteristic
48	C₄×#1039	C₂²	1	1	yes	characteristic
49	C₄×#1039	C₂²	1	1	yes	characteristic
50	C₄×#1039	C₂²	1	1	yes	characteristic
51	#65970	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
54	C₅²×Q₈	—	4	1	no	—
55	C₂×C₄×C₅²	—	4	1	no	—
56	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
62	C₂²⋊(C₅⋊D₅)	Q₈	1	4	yes	complemented
63	C₂×#1039	—	2	2	no	—
64	C₂×#1039	—	4	1	no	—
65	C₂×#1039	C₂³	1	1	yes	characteristic
66	C₂×#1039	C₂³	1	1	yes	characteristic
67	C₂×#1039	C₂³	1	1	yes	characteristic
68	C₂×#1039	C₂³	1	1	yes	characteristic
69	C₂×#1039	C₂³	1	1	yes	characteristic
70	C₂×#1039	C₂³	1	1	yes	characteristic
71	C₂×#1039	C₂³	1	1	yes	characteristic
72	C₄×C₅⋊D₅	—	2	2	no	—
73	C₄×C₅⋊D₅	—	4	1	no	—
74	C₄×C₅⋊D₅	—	4	1	no	—
75	#6045	—	2	2	no	—
76	#6045	—	4	1	no	—
77	C₅×C₂²⋊Q₈	D₅	1	6	yes	complemented
78	#4092	—	5	6	no	—
79	#4090	—	5	6	no	—
80	C₄×C₂²⋊D₅	—	5	6	no	—
81	C₂×#683	—	5	6	no	—
82	#4042	—	5	6	no	—
83	#4042	—	5	6	no	—
84	D₅×C₄⋊C₄	—	5	6	no	—
85	#4035	—	5	6	no	—
86	#4035	—	5	6	no	—
87	#4033	—	5	6	no	—
88	#4027	—	5	6	no	—
89	#4027	—	5	6	no	—
90	#4024	—	5	6	no	—
91	#4024	—	5	6	no	—
92	C₂²×C₅²	—	2	1	no	—
93	C₂²×C₅²	—	2	1	no	—
94	C₂²×C₅²	C₂×Q₈	1	2	yes	—
95	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
96	C₂×C₅⋊D₅	—	2	1	no	—
97	C₂×C₅⋊D₅	—	2	1	no	—
98	C₂×C₅⋊D₅	C₂×Q₈	1	2	yes	—
99	C₄×C₅²	—	2	1	no	—
100	C₄×C₅²	—	2	1	no	—
101	C₄×C₅²	—	2	1	no	—
102	C₄×C₅²	—	2	1	no	—
103	C₄×C₅²	—	2	1	no	—
104	C₄×C₅²	—	2	1	no	—
105	#1039	—	2	1	no	—
106	#1039	—	2	1	no	—
107	#1039	—	2	1	no	—
108	#1039	—	2	1	no	—
109	#1039	—	2	1	no	—
110	#1039	C₂×Q₈	1	2	yes	—
111	#1039	#41	1	2	yes	—
112	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
113	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
114	C₂×C₂²⋊D₅	—	5	6	no	—
115	C₂²×Dic₅	—	5	6	no	—
116	C₂²×Dic₅	—	5	6	no	—
117	C₂×C₄×D₅	—	5	6	no	—
118	C₂×C₄×D₅	—	5	6	no	—
119	C₂×C₄×D₅	—	5	6	no	—
120	C₂×Dic₁₀	—	5	6	no	—
121	C₂×Dic₁₀	—	5	6	no	—
122	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
123	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
124	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
125	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
126	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
127	C₂²⋊Dic₅	—	5	6	no	—
128	#685	—	5	6	no	—
129	#685	—	5	6	no	—
130	#685	—	5	6	no	—
131	C₄⋊Dic₅	—	5	6	no	—
132	C₄⋊Dic₅	—	5	6	no	—
133	#683	—	5	6	no	—
134	#683	—	5	6	no	—
135	#683	—	5	6	no	—
136	#683	—	5	6	no	—
137	#683	—	5	6	no	—
138	#683	—	5	6	no	—
139	#683	—	5	6	no	—
140	#683	—	5	6	no	—
141	#683	—	5	6	no	—
142	#683	—	5	6	no	—
143	#683	—	5	6	no	—
144	C₄×Dic₅	—	5	6	no	—
145	C₄×Dic₅	—	5	6	no	—
146	C₄×Dic₅	—	5	6	no	—
147	#554	—	25	1	no	Sylow
148	C₂×C₅²	—	2	2	no	—
149	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₅²	C₂×#41	1	1	yes	characteristic
151	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
152	C₅⋊D₅	—	2	2	no	—
153	C₂³×C₅	C₂²×D₅	1	6	yes	—
154	C₂²×D₅	—	5	6	no	—
155	C₅×Q₈	—	4	6	no	—
156	C₂×C₄×C₅	—	4	6	no	—
157	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
158	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
159	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
160	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
161	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
162	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
163	C₂²⋊D₅	—	5	24	no	—
164	C₂×Dic₅	—	5	6	no	—
165	C₂×Dic₅	—	5	6	no	—
166	C₂×Dic₅	—	5	6	no	—
167	C₂×Dic₅	—	5	6	no	—
168	C₂×Dic₅	—	5	6	no	—
169	C₂×Dic₅	—	5	6	no	—
170	C₂×Dic₅	—	5	6	no	—
171	C₂×Dic₅	—	10	12	no	—
172	C₂×Dic₅	—	20	6	no	—
173	C₄×D₅	—	10	12	no	—
174	C₄×D₅	—	20	6	no	—
175	C₄×D₅	—	20	6	no	—
176	Dic₁₀	—	10	12	no	—
177	Dic₁₀	—	20	6	no	—
178	C₄⋊Q₈	—	25	1	no	—
179	#125	—	25	1	no	—
180	#125	—	25	1	no	—
181	C₂²⋊Q₈	—	25	1	no	—
182	C₂²⋊Q₈	—	25	1	no	—
183	C₂²⋊Q₈	—	25	1	no	—
184	C₂²⋊Q₈	—	25	1	no	—
185	C₂²⋊Q₈	—	25	1	no	—
186	C₂²⋊Q₈	C₅⋊D₅	1	1	yes	characteristic, complemented
187	C₄×Q₈	—	25	1	no	—
188	C₄×D₄	—	25	1	no	—
189	C₄×D₄	—	25	1	no	—
190	C₄×D₄	—	25	1	no	—
191	C₂×C₄⋊C₄	—	25	1	no	—
192	C₂×C₄⋊C₄	—	25	1	no	—
193	C₅²	#554	1	1	yes	characteristic, complemented, Sylow
194	C₂²×C₅	—	2	6	no	—
195	C₂²×C₅	—	2	6	no	—
196	C₂²×C₅	Q₈×D₅	1	12	yes	—
197	C₂²×C₅	C₂³×D₅	1	6	yes	—
198	C₂×D₅	—	5	12	no	—
199	C₂×D₅	—	10	6	no	—
200	C₂×D₅	—	10	6	no	—
201	C₄×C₅	—	2	6	no	—
202	C₄×C₅	—	2	6	no	—
203	C₄×C₅	—	2	6	no	—
204	C₄×C₅	—	2	6	no	—
205	C₄×C₅	—	2	6	no	—
206	C₄×C₅	—	2	6	no	—
207	Dic₅	—	5	12	no	—
208	Dic₅	—	5	12	no	—
209	Dic₅	—	10	6	no	—
210	Dic₅	—	10	6	no	—
211	Dic₅	—	10	6	no	—
212	Dic₅	—	10	6	no	—
213	Dic₅	—	10	6	no	—
214	C₂×Q₈	—	25	1	no	—
215	C₂×Q₈	—	25	1	no	—
216	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
217	C₂×D₄	—	25	1	no	—
218	C₂²×C₄	—	25	1	no	—
219	C₂²×C₄	—	25	1	no	—
220	C₂²×C₄	—	25	1	no	—
221	C₂²×C₄	—	25	1	no	—
222	C₂²×C₄	—	25	1	no	—
223	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
224	C₄⋊C₄	—	25	1	no	—
225	C₄⋊C₄	—	25	1	no	—
226	C₄⋊C₄	—	25	1	no	—
227	C₄⋊C₄	—	25	1	no	—
228	C₄⋊C₄	—	25	1	no	—
229	C₄⋊C₄	—	25	1	no	—
230	C₄⋊C₄	—	25	1	no	—
231	C₄⋊C₄	—	25	1	no	—
232	C₄⋊C₄	—	25	1	no	—
233	C₄⋊C₄	—	25	1	no	—
234	C₄⋊C₄	—	25	1	no	—
235	C₄⋊C₄	—	25	1	no	—
236	C₄⋊C₄	—	25	1	no	—
237	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
240	C₂²⋊C₄	—	25	1	no	—
241	C₂²⋊C₄	—	25	1	no	—
242	C₂²⋊C₄	—	25	1	no	—
243	C₂²⋊C₄	—	25	1	no	—
244	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
245	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
246	C₄²	—	25	1	no	—
247	C₄²	—	25	1	no	—
248	C₄²	—	25	1	no	—
249	C₂×C₅	—	2	12	no	—
250	C₂×C₅	#4144	1	6	yes	—
251	C₂×C₅	C₂×Q₈×D₅	1	6	yes	—
252	C₂×C₅	D₅×#41	1	6	yes	—
253	D₅	—	10	12	no	—
254	C₂³	—	25	1	no	—
255	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
256	Q₈	—	4	1	no	—
257	Q₈	—	50	2	no	—
258	Q₈	—	100	1	no	—
259	D₄	—	25	4	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	25	1	no	—
262	C₂×C₄	—	25	1	no	—
263	C₂×C₄	—	25	1	no	—
264	C₂×C₄	—	25	1	no	—
265	C₂×C₄	—	25	1	no	—
266	C₂×C₄	—	25	1	no	—
267	C₂×C₄	—	25	1	no	—
268	C₂×C₄	—	50	2	no	—
269	C₂×C₄	—	50	2	no	—
270	C₂×C₄	—	100	1	no	—
271	C₂×C₄	—	100	1	no	—
272	C₂×C₄	—	100	1	no	—
273	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
279	C₅	#65970	1	6	yes	complemented
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	25	2	no	—
283	C₂²	—	50	1	no	—
284	C₂²	—	50	1	no	—
285	C₂²	Q₈×C₅⋊D₅	1	2	yes	—
286	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	25	2	no	—
294	C₄	—	25	2	no	—
295	C₄	—	50	1	no	—
296	C₄	—	50	1	no	—
297	C₄	—	50	1	no	—
298	C₄	—	50	1	no	—
299	C₄	—	50	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	50	2	no	—
302	C₂	#11717517	1	1	yes	characteristic
303	C₂	C₂×Q₈×C₅⋊D₅	1	1	yes	characteristic
304	C₂	#41×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊(C₅⋊D₅)	C₂²
hypercenter	C₂²⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#554
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₅²×C₂²⋊Q₈	C₂	semidirect
2. #11717266	C₂	semidirect
3. #11717264	C₂	semidirect
5. C₂×#87806	C₂	semidirect
6. #11717216	C₂	semidirect
7. #11717216	C₂	semidirect
8. C₄⋊C₄×C₅⋊D₅	C₂	semidirect
9. #11717209	C₂	semidirect
10. #11717209	C₂	semidirect
11. #11717207	C₂	semidirect
14. #11717198	C₂	semidirect
15. #11717198	C₂	semidirect
62. C₂²⋊(C₅⋊D₅)	Q₈	semidirect
77. C₅×C₂²⋊Q₈	D₅	semidirect
186. C₂²⋊Q₈	C₅⋊D₅	semidirect
193. C₅²	#554	semidirect
279. C₅	#65970	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₂²⋊Q₈	2
#11717266	2
#11717264	2
C₄×C₂²⋊(C₅⋊D₅)	2
C₂×#87806	2
#11717216	2
C₄⋊C₄×C₅⋊D₅	2
#11717209	2
#11717207	2
#11717201	2
#11717198	2
#65970	5