Subgroups — #38756104

Order 25216.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161703	C₂	1	4	yes	maximal, complemented
2	#27161692	C₂	1	2	yes	maximal, complemented
3	#27161685	C₂	1	2	yes	maximal, complemented
4	#27161678	C₂	1	4	yes	maximal, complemented
5	#27161669	C₂	1	4	yes	maximal, complemented
6	#27161629	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161616	C₂	1	8	yes	maximal, complemented
8	#27161591	C₂	1	2	yes	maximal, complemented
9	#27161547	C₂	1	2	yes	maximal, complemented
10	C₂×#20263283	C₂	1	1	yes	maximal, characteristic, complemented
11	#533×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
14	#20263326	C₂²	1	1	yes	characteristic, complemented
15	#20263323	C₂²	1	1	yes	characteristic, complemented
16	#20263321	C₂²	1	8	yes	—
17	#20263321	C₂²	1	8	yes	complemented
18	#20263317	C₂²	1	8	yes	complemented
19	#20263315	C₂²	1	8	yes	—
20	#20263314	C₂²	1	2	yes	complemented
21	#20263311	C₂²	1	8	yes	—
22	#20263311	C₂²	1	8	yes	complemented
23	#20263310	C₂²	1	4	yes	—
24	#20263306	C₂²	1	4	yes	—
25	#20263302	C₂²	1	8	yes	complemented
26	#20263301	C₂²	1	8	yes	—
27	#20263301	C₂²	1	4	yes	—
28	#20263300	C₂²	1	4	yes	—
29	#20263296	C₂²	1	4	yes	—
30	#20263291	C₂²	1	4	yes	complemented
31	C₄×D₇₈₈	C₂²	1	8	yes	—
32	C₄×D₇₈₈	C₂²	1	4	yes	complemented
33	#20263288	C₂²	1	4	yes	complemented
34	#20263286	C₂²	1	8	yes	—
35	#20263283	C₂²	1	4	yes	—
36	#20263283	C₂²	1	2	yes	—
37	#20263283	C₂²	1	2	yes	—
38	#20263283	C₂²	1	4	yes	complemented
39	C₂×#15691337	C₂²	1	2	yes	—
40	C₂×#15691335	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
42	C₄²×D₁₉₇	C₂²	1	4	yes	complemented
43	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	complemented
44	#125×C₁₉₇	C₂²	1	2	yes	complemented
45	C₂²⋊Q₈×C₁₉₇	C₂²	1	4	yes	—
46	C₄×Q₈×C₁₉₇	C₂²	1	4	yes	complemented
47	#117×C₁₉₇	C₂²	1	2	yes	—
48	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
49	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
50	#15691358	—	2	8	no	—
51	C₂×Dic₃₉₄	C₂³	1	2	yes	—
52	C₂×D₇₈₈	C₂³	1	2	yes	—
53	C₂×D₇₈₈	C₂³	1	8	yes	—
54	C₂×D₇₈₈	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₉₇	C₂³	1	4	yes	—
56	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
58	C₂×C₄×D₁₉₇	—	2	4	no	—
59	C₂×C₄×D₁₉₇	C₂³	1	4	yes	—
60	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
61	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
62	C₂×C₄×D₁₉₇	C₂³	1	8	yes	—
63	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
64	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
65	#15691337	C₂³	1	4	yes	—
66	#15691337	C₂³	1	4	yes	—
67	#15691337	C₂³	1	8	yes	—
68	#15691335	C₂³	1	4	yes	—
69	#15691335	C₂³	1	4	yes	—
70	#15691335	C₂³	1	8	yes	—
71	#15691335	C₂³	1	8	yes	—
72	#15691335	C₂³	1	8	yes	—
73	#15691335	C₂³	1	8	yes	—
74	C₄×Dic₁₉₇	C₂³	1	2	yes	—
75	C₄×Dic₁₉₇	C₂³	1	8	yes	—
76	C₄×Dic₁₉₇	C₂³	1	2	yes	—
77	C₂×Q₈×C₁₉₇	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
81	C₄⋊C₄×C₁₉₇	C₂³	1	8	yes	—
82	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₁₉₇	C₂³	1	4	yes	—
84	C₄²×C₁₉₇	C₂³	1	2	yes	—
85	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₉₇	C₂³	1	4	yes	—
88	C₂²×D₁₉₇	—	2	2	no	—
89	C₂²×D₁₉₇	—	4	2	no	—
90	C₂²×D₁₉₇	C₂⁴	1	2	yes	—
91	C₂²×D₁₉₇	C₂⁴	1	4	yes	—
92	Dic₃₉₄	—	2	4	no	—
93	D₇₈₈	—	2	4	no	—
94	D₇₈₈	—	4	4	no	—
95	D₇₈₈	—	4	8	no	—
96	C₂²⋊D₁₉₇	—	2	8	no	—
97	C₂×Dic₁₉₇	—	2	2	no	—
98	C₂×Dic₁₉₇	—	4	2	no	—
99	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
100	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
101	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
102	C₄×D₁₉₇	—	2	4	no	—
103	C₄×D₁₉₇	—	2	4	no	—
104	C₄×D₁₉₇	—	2	8	no	—
105	C₄×D₁₉₇	—	4	8	no	—
106	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
107	Q₈×C₁₉₇	—	4	4	no	—
108	C₂×C₄×C₁₉₇	—	4	2	no	—
109	C₂×C₄×C₁₉₇	#41	1	4	yes	—
110	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₉₇	C₂⁴	1	8	yes	—
113	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
115	C₂²×C₁₉₇	—	2	2	no	—
116	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
117	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₉₇	—	2	2	no	—
119	C₂×D₁₉₇	—	2	4	no	—
120	C₂×D₁₉₇	—	2	8	no	—
121	C₂×D₁₉₇	—	2	4	no	—
122	C₂×D₁₉₇	—	4	4	no	—
123	C₄×C₁₉₇	—	2	2	no	—
124	C₄×C₁₉₇	—	2	2	no	—
125	C₄×C₁₉₇	—	2	8	no	—
126	C₄×C₁₉₇	C₂×#41	1	2	yes	—
127	C₄×C₁₉₇	C₂×#41	1	2	yes	—
128	Dic₁₉₇	—	2	2	no	—
129	Dic₁₉₇	—	2	4	no	—
130	Dic₁₉₇	—	2	4	no	—
131	C₂×C₁₉₇	—	2	2	no	—
132	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
133	C₂×C₁₉₇	#585	1	2	yes	—
134	D₁₉₇	—	4	4	no	—
135	D₁₉₇	—	4	2	no	—
136	C₁₉₇	#3455	1	1	yes	characteristic, complemented, Sylow
137	#3455	—	197	1	no	maximal, Sylow
138	#542	—	197	4	no	—
139	#541	—	197	2	no	—
140	#540	—	197	2	no	—
141	#539	—	197	4	no	—
142	#538	—	197	4	no	—
143	#533	D₁₉₇	1	1	yes	characteristic, complemented
144	#532	—	197	1	no	—
145	#529	—	197	8	no	—
146	#525	—	197	2	no	—
147	C₄×#41	—	197	2	no	—
148	C₂×#117	—	197	1	no	—
149	C₂×#41	—	197	2	no	—
150	C₂³×C₄	—	197	1	no	—
151	C₄⋊Q₈	—	197	1	no	—
152	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	complemented
153	#127	—	197	1	no	—
154	#126	—	197	8	no	—
155	#126	—	197	8	no	—
156	#125	—	197	8	no	—
157	#125	C₂×D₁₉₇	1	2	yes	complemented
158	#124	—	197	8	no	—
159	#124	—	197	2	no	—
160	#123	—	197	8	no	—
161	#123	—	197	8	no	—
162	#123	—	197	4	no	—
163	C₂²⋊Q₈	—	197	8	no	—
164	C₂²⋊Q₈	—	197	4	no	—
165	C₂²⋊Q₈	C₂×D₁₉₇	1	4	yes	—
166	#121	—	197	4	no	—
167	#121	—	197	4	no	—
168	#121	—	197	8	no	—
169	#120	—	197	4	no	—
170	C₄×Q₈	—	197	4	no	—
171	C₄×Q₈	C₂×D₁₉₇	1	4	yes	complemented
172	C₄×D₄	—	197	4	no	—
173	C₄×D₄	—	197	8	no	—
174	C₄×D₄	—	197	8	no	—
175	C₄×D₄	—	197	4	no	—
176	#117	—	197	4	no	—
177	#117	—	197	2	no	—
178	#117	—	197	2	no	—
179	#117	—	197	4	no	—
180	#117	C₂×D₁₉₇	1	2	yes	—
181	C₂×C₄⋊C₄	—	197	2	no	—
182	C₂×C₂²⋊C₄	—	197	2	no	—
183	C₂×C₄²	—	197	1	no	—
184	C₂×C₄²	—	197	4	no	—
185	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
186	C₂⁴	—	197	1	no	—
187	#41	—	394	8	no	—
188	C₂×Q₈	—	197	2	no	—
189	C₂×Q₈	C₂²×D₁₉₇	1	4	yes	—
190	C₂×D₄	—	197	4	no	—
191	C₂×D₄	—	197	8	no	—
192	C₂×D₄	—	197	4	no	—
193	C₂×D₄	—	197	2	no	—
194	C₂²×C₄	—	394	4	no	—
195	C₂²×C₄	—	197	2	no	—
196	C₂²×C₄	—	197	8	no	—
197	C₂²×C₄	—	197	1	no	—
198	C₂²×C₄	—	197	2	no	—
199	C₂²×C₄	—	197	4	no	—
200	C₂²×C₄	—	197	2	no	—
201	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
203	C₄⋊C₄	—	197	4	no	—
204	C₄⋊C₄	—	197	4	no	—
205	C₄⋊C₄	—	197	8	no	—
206	C₄⋊C₄	—	197	4	no	—
207	C₄⋊C₄	—	197	8	no	—
208	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
209	C₄⋊C₄	C₂²×D₁₉₇	1	8	yes	—
210	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
211	C₂²⋊C₄	—	197	4	no	—
212	C₂²⋊C₄	—	197	4	no	—
213	C₂²⋊C₄	—	197	8	no	—
214	C₂²⋊C₄	—	197	8	no	—
215	C₂²⋊C₄	—	197	8	no	—
216	C₂²⋊C₄	—	197	8	no	—
217	C₂²⋊C₄	C₂²×D₁₉₇	1	4	yes	—
218	C₄²	—	197	2	no	—
219	C₄²	—	197	8	no	—
220	C₄²	—	197	2	no	—
221	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
222	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉₇	1	4	yes	—
224	C₄²	C₂²×D₁₉₇	1	2	yes	—
225	C₂³	—	788	2	no	—
226	C₂³	—	394	2	no	—
227	C₂³	—	197	2	no	—
228	C₂³	—	197	4	no	—
229	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	394	4	no	—
232	D₄	—	788	4	no	—
233	D₄	—	788	8	no	—
234	D₄	—	394	8	no	—
235	D₄	—	394	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	788	8	no	—
238	C₂×C₄	—	788	2	no	—
239	C₂×C₄	—	394	2	no	—
240	C₂×C₄	—	394	4	no	—
241	C₂×C₄	—	394	4	no	—
242	C₂×C₄	—	394	8	no	—
243	C₂×C₄	—	197	4	no	—
244	C₂×C₄	—	197	4	no	—
245	C₂×C₄	—	197	2	no	—
246	C₂×C₄	#15691359	1	4	yes	—
247	C₂×C₄	C₂³×D₁₉₇	1	8	yes	—
248	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
250	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	788	4	no	—
254	C₂²	—	394	2	no	—
255	C₂²	—	394	4	no	—
256	C₂²	—	394	8	no	—
257	C₂²	—	394	4	no	—
258	C₂²	C₂×#15691359	1	2	yes	—
259	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	394	2	no	—
264	C₄	—	394	4	no	—
265	C₄	—	394	4	no	—
266	C₄	C₂×#15691359	1	2	yes	—
267	C₄	C₂×#15691359	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	788	2	no	—
270	C₂	—	788	4	no	—
271	C₂	C₂²×#15691359	1	1	yes	characteristic
272	C₂	#27161915	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	#533×C₁₉₇	C₂
hypercenter	#533	D₁₉₇
hypocenter	C₁₉₇	#3455
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #27161703	C₂	semidirect
2. #27161692	C₂	semidirect
3. #27161685	C₂	semidirect
4. #27161678	C₂	semidirect
5. #27161669	C₂	semidirect
6. #27161629	C₂	semidirect
7. #27161616	C₂	semidirect
8. #27161591	C₂	semidirect
9. #27161547	C₂	semidirect
10. C₂×#20263283	C₂	semidirect
11. #533×C₁₉₇	C₂	semidirect
14. #20263326	C₂²	semidirect
15. #20263323	C₂²	semidirect
17. #20263321	C₂²	semidirect
18. #20263317	C₂²	semidirect
20. #20263314	C₂²	semidirect
22. #20263311	C₂²	semidirect
25. #20263302	C₂²	semidirect
30. #20263291	C₂²	semidirect
32. C₄×D₇₈₈	C₂²	semidirect
33. #20263288	C₂²	semidirect
38. #20263283	C₂²	semidirect
42. C₄²×D₁₉₇	C₂²	semidirect
43. C₄⋊Q₈×C₁₉₇	C₂²	semidirect
44. #125×C₁₉₇	C₂²	semidirect
46. C₄×Q₈×C₁₉₇	C₂²	semidirect
136. C₁₉₇	#3455	semidirect
143. #533	D₁₉₇	semidirect
152. C₄⋊Q₈	C₂×D₁₉₇	semidirect
157. #125	C₂×D₁₉₇	semidirect
171. C₄×Q₈	C₂×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161703	2
#27161692	2
#27161685	2
#27161678	2
#27161669	2
#27161629	2
#27161616	2
#27161591	2
#27161547	2
C₂×#20263283	2
#533×C₁₉₇	2
#3455	197