Subgroups — #13098963

Order 1600.

327 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716508	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716508	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716491	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716460	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716456	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716455	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716451	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716451	C₂	1	1	yes	maximal, characteristic, complemented
12	#11716446	C₂	1	1	yes	maximal, characteristic
13	#11716445	C₂	1	1	yes	maximal, characteristic, complemented
14	#11716445	C₂	1	1	yes	maximal, characteristic, complemented
15	#11716441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×C₂₅	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₅	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₂₅	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×C₂₅	C₂²	1	1	yes	characteristic
30	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
31	C₂²⋊Dic₂₅	C₂²	1	1	yes	characteristic
32	#87720	C₂²	1	1	yes	characteristic
33	#87720	C₂²	1	1	yes	characteristic
34	#87720	C₂²	1	1	yes	characteristic
35	#87720	C₂²	1	1	yes	characteristic
36	#87720	C₂²	1	1	yes	characteristic
37	#87720	C₂²	1	1	yes	characteristic
38	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
39	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
40	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
41	#87718	C₂²	1	1	yes	characteristic
42	#87718	C₂²	1	1	yes	characteristic
43	#87718	C₂²	1	1	yes	characteristic
44	#87718	C₂²	1	1	yes	characteristic
45	#87718	C₂²	1	1	yes	characteristic
46	#87718	C₂²	1	1	yes	characteristic
47	#87718	C₂²	1	1	yes	characteristic
48	#87718	C₂²	1	1	yes	characteristic
49	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
50	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
51	#65972	—	5	1	no	maximal
52	C₂³×C₂₅	C₂³	1	1	yes	characteristic
53	C₂²×D₂₅	C₂³	1	1	yes	characteristic
54	C₂²×D₂₅	C₂³	1	1	yes	characteristic
55	Q₈×C₂₅	—	4	1	no	—
56	C₂×C₄×C₂₅	—	4	1	no	—
57	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
63	C₂²⋊D₂₅	—	2	2	no	—
64	C₂²⋊D₂₅	—	4	1	no	—
65	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
68	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
69	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
70	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
71	D₁₀₀	—	2	2	no	—
72	C₄×D₂₅	—	2	2	no	—
73	C₄×D₂₅	—	4	1	no	—
74	C₄×D₂₅	—	4	1	no	—
75	C₄×D₂₅	—	4	1	no	—
76	C₅×C₂²⋊Q₈	D₅	1	1	yes	characteristic
77	#4092	—	5	1	no	—
78	#4092	—	5	1	no	—
79	#4075	—	5	1	no	—
80	C₄×C₂²⋊D₅	—	5	1	no	—
81	#4044	—	5	1	no	—
82	#4040	—	5	1	no	—
83	#4039	—	5	1	no	—
84	D₅×C₄⋊C₄	—	5	1	no	—
85	#4035	—	5	1	no	—
86	#4035	—	5	1	no	—
87	#4030	—	5	1	no	—
88	#4029	—	5	1	no	—
89	#4029	—	5	1	no	—
90	#4025	—	5	1	no	—
91	C₂²×C₂₅	—	2	1	no	—
92	C₂²×C₂₅	—	2	1	no	—
93	C₂²×C₂₅	—	2	1	no	—
94	C₂²×C₂₅	C₂⁴	1	1	yes	characteristic
95	C₂×D₂₅	—	2	1	no	—
96	C₂×D₂₅	—	2	1	no	—
97	C₂×D₂₅	—	2	1	no	—
98	C₂×D₂₅	—	2	1	no	—
99	C₂×D₂₅	—	2	1	no	—
100	C₂×D₂₅	#41	1	2	yes	—
101	C₄×C₂₅	—	2	1	no	—
102	C₄×C₂₅	—	2	1	no	—
103	C₄×C₂₅	—	2	1	no	—
104	C₄×C₂₅	—	2	1	no	—
105	C₄×C₂₅	—	2	1	no	—
106	C₄×C₂₅	—	2	1	no	—
107	Dic₂₅	—	2	1	no	—
108	Dic₂₅	—	2	1	no	—
109	Dic₂₅	—	2	1	no	—
110	Dic₂₅	—	2	1	no	—
111	Dic₂₅	—	2	1	no	—
112	Dic₂₅	—	2	1	no	—
113	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
115	C₂×C₂²⋊D₅	—	5	1	no	—
116	C₂×C₂²⋊D₅	—	5	1	no	—
117	C₂×D₂₀	—	5	1	no	—
118	C₂×C₄×D₅	—	5	1	no	—
119	C₂×C₄×D₅	—	5	1	no	—
120	C₂×C₄×D₅	—	5	1	no	—
121	C₂×C₄×D₅	—	5	1	no	—
122	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
123	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
124	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
125	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
126	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
127	C₂²⋊Dic₅	—	5	1	no	—
128	C₂²⋊Dic₅	—	5	1	no	—
129	#685	—	5	1	no	—
130	#685	—	5	1	no	—
131	#685	—	5	1	no	—
132	#685	—	5	1	no	—
133	#685	—	5	1	no	—
134	#685	—	5	1	no	—
135	C₄⋊Dic₅	—	5	1	no	—
136	C₄⋊Dic₅	—	5	1	no	—
137	C₄⋊Dic₅	—	5	1	no	—
138	#683	—	5	1	no	—
139	#683	—	5	1	no	—
140	#683	—	5	1	no	—
141	#683	—	5	1	no	—
142	#683	—	5	1	no	—
143	#683	—	5	1	no	—
144	#683	—	5	1	no	—
145	#683	—	5	1	no	—
146	C₄×Dic₅	—	5	1	no	—
147	C₄×Dic₅	—	5	1	no	—
148	#539	—	25	1	no	Sylow
149	C₂×C₂₅	—	4	1	no	—
150	C₂×C₂₅	C₂×#41	1	1	yes	characteristic
151	C₂×C₂₅	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₂₅	2₋¹⁺⁴	1	1	yes	characteristic
153	D₂₅	—	2	2	no	—
154	D₂₅	—	4	1	no	—
155	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
156	C₂²×D₅	—	5	1	no	—
157	C₂²×D₅	—	5	1	no	—
158	C₅×Q₈	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
166	C₂²⋊D₅	—	10	2	no	—
167	C₂²⋊D₅	—	20	1	no	—
168	C₂×Dic₅	—	5	1	no	—
169	C₂×Dic₅	—	5	1	no	—
170	C₂×Dic₅	—	5	1	no	—
171	C₂×Dic₅	—	5	1	no	—
172	C₂×Dic₅	—	5	1	no	—
173	C₂×Dic₅	—	5	1	no	—
174	D₂₀	—	10	2	no	—
175	C₄×D₅	—	10	2	no	—
176	C₄×D₅	—	20	1	no	—
177	C₄×D₅	—	20	1	no	—
178	C₄×D₅	—	20	1	no	—
179	#126	—	25	1	no	—
180	#126	—	25	1	no	—
181	#125	—	25	1	no	—
182	#125	—	25	1	no	—
183	#123	—	25	1	no	—
184	#123	—	25	1	no	—
185	#123	—	25	1	no	—
186	#123	—	25	1	no	—
187	C₂²⋊Q₈	—	25	1	no	—
188	C₂²⋊Q₈	—	25	1	no	—
189	C₂²⋊Q₈	D₂₅	1	1	yes	characteristic, complemented
190	#121	—	25	1	no	—
191	C₄×D₄	—	25	1	no	—
192	C₄×D₄	—	25	1	no	—
193	C₂×C₄⋊C₄	—	25	1	no	—
194	C₂₅	#539	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
199	C₂×D₅	—	5	2	no	—
200	C₂×D₅	—	10	1	no	—
201	C₂×D₅	—	10	1	no	—
202	C₂×D₅	—	10	1	no	—
203	C₂×D₅	—	10	1	no	—
204	C₂×D₅	—	10	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	1	no	—
211	Dic₅	—	10	1	no	—
212	Dic₅	—	10	1	no	—
213	Dic₅	—	10	1	no	—
214	Dic₅	—	10	1	no	—
215	Dic₅	—	10	1	no	—
216	Dic₅	—	10	1	no	—
217	C₂×Q₈	C₂×D₂₅	1	1	yes	characteristic
218	C₂×D₄	—	25	1	no	—
219	C₂×D₄	—	25	1	no	—
220	C₂×D₄	—	25	1	no	—
221	C₂²×C₄	—	25	1	no	—
222	C₂²×C₄	—	25	1	no	—
223	C₂²×C₄	—	25	1	no	—
224	C₂²×C₄	—	25	1	no	—
225	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
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×D₂₅	1	1	yes	characteristic
238	C₄⋊C₄	C₂×D₂₅	1	1	yes	characteristic
239	C₄⋊C₄	C₂×D₂₅	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	—	25	1	no	—
245	C₂²⋊C₄	—	25	1	no	—
246	C₂²⋊C₄	—	25	1	no	—
247	C₂²⋊C₄	—	25	1	no	—
248	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
249	C₂²⋊C₄	C₂×D₂₅	1	1	yes	characteristic
250	C₄²	—	25	1	no	—
251	C₄²	—	25	1	no	—
252	C₂×C₅	—	4	1	no	—
253	C₂×C₅	#4144	1	1	yes	characteristic
254	C₂×C₅	#4147	1	1	yes	characteristic
255	C₂×C₅	D₅×#41	1	1	yes	characteristic
256	D₅	—	10	2	no	—
257	D₅	—	20	1	no	—
258	C₂³	—	25	1	no	—
259	C₂³	—	25	1	no	—
260	C₂³	C₂²×D₂₅	1	1	yes	characteristic
261	Q₈	—	4	1	no	—
262	D₄	—	50	2	no	—
263	D₄	—	50	2	no	—
264	D₄	—	100	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	25	1	no	—
267	C₂×C₄	—	25	1	no	—
268	C₂×C₄	—	25	1	no	—
269	C₂×C₄	—	25	1	no	—
270	C₂×C₄	—	25	1	no	—
271	C₂×C₄	—	25	1	no	—
272	C₂×C₄	—	50	2	no	—
273	C₂×C₄	—	100	1	no	—
274	C₂×C₄	—	100	1	no	—
275	C₂×C₄	—	100	1	no	—
276	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
282	C₅	#65972	1	1	yes	characteristic
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	25	2	no	—
287	C₂²	—	50	1	no	—
288	C₂²	—	50	1	no	—
289	C₂²	—	50	1	no	—
290	C₂²	—	50	1	no	—
291	C₂²	—	50	1	no	—
292	C₂²	C₂³×D₂₅	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	50	1	no	—
300	C₄	—	50	1	no	—
301	C₄	—	50	1	no	—
302	C₄	—	50	1	no	—
303	C₄	—	50	1	no	—
304	C₄	—	50	1	no	—
305	C₂	—	4	1	no	—
306	C₂	—	50	2	no	—
307	C₂	—	100	1	no	—
308	C₂	#11716563	1	1	yes	characteristic
309	C₂	#11716566	1	1	yes	characteristic
310	C₂	#41×D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂⁴
Frattini, socle	C₂²×C₅	C₂³×D₅
Fitting	C₂₅×C₂²⋊Q₈	C₂
prime radical	#11716446	C₂
hypercenter	C₂²⋊Q₈	D₂₅
hypocenter	C₂₅	#539
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₅×C₂²⋊Q₈	C₂	semidirect
2. #11716508	C₂	semidirect
3. #11716508	C₂	semidirect
4. #11716491	C₂	semidirect
5. C₄×C₂²⋊D₂₅	C₂	semidirect
6. #11716460	C₂	semidirect
7. #11716456	C₂	semidirect
8. #11716455	C₂	semidirect
9. C₄⋊C₄×D₂₅	C₂	semidirect
10. #11716451	C₂	semidirect
11. #11716451	C₂	semidirect
13. #11716445	C₂	semidirect
14. #11716445	C₂	semidirect
15. #11716441	C₂	semidirect
189. C₂²⋊Q₈	D₂₅	semidirect
194. C₂₅	#539	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×C₂²⋊Q₈	2
#11716508	2
#11716491	2
C₄×C₂²⋊D₂₅	2
#11716460	2
#11716456	2
#11716455	2
C₄⋊C₄×D₂₅	2
#11716451	2
#11716446	2
#11716445	2
#11716441	2
#65972	5