Subgroups — #38756007

Order 25216.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161636	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161634	C₂	1	2	yes	maximal, complemented
3	#27161627	C₂	1	2	yes	maximal, complemented
4	#27161611	C₂	1	4	yes	maximal, complemented
5	#27161610	C₂	1	4	yes	maximal, complemented
6	#27161608	C₂	1	4	yes	maximal, complemented
7	#27161589	C₂	1	4	yes	maximal, complemented
8	#27161586	C₂	1	2	yes	maximal, complemented
9	#41×Dic₁₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×#15691357	C₂	1	4	yes	maximal, complemented
11	C₂×C₄²×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#532×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#15691357	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
16	#20263326	C₂²	1	2	yes	complemented
17	#20263319	C₂²	1	8	yes	complemented
18	#20263318	C₂²	1	2	yes	complemented
19	#20263317	C₂²	1	4	yes	complemented
20	#20263313	C₂²	1	4	yes	complemented
21	#20263308	C₂²	1	8	yes	—
22	#20263307	C₂²	1	4	yes	—
23	#20263306	C₂²	1	4	yes	—
24	#20263305	C₂²	1	4	yes	—
25	#20263302	C₂²	1	4	yes	—
26	#20263297	C₂²	1	4	yes	—
27	#20263297	C₂²	1	4	yes	—
28	#20263294	C₂²	1	4	yes	—
29	Q₈×Dic₁₉₇	C₂²	1	4	yes	complemented
30	C₄×Dic₃₉₄	C₂²	1	4	yes	complemented
31	#20263291	C₂²	1	4	yes	complemented
32	D₄×Dic₁₉₇	C₂²	1	8	yes	—
33	D₄×Dic₁₉₇	C₂²	1	4	yes	complemented
34	D₄×Dic₁₉₇	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊D₁₉₇	C₂²	1	8	yes	—
36	#20263286	C₂²	1	8	yes	—
37	#20263285	C₂²	1	2	yes	—
38	#20263285	C₂²	1	4	yes	complemented
39	#20263284	C₂²	1	4	yes	—
40	#20263282	C₂²	1	8	yes	complemented
41	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
42	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×Dic₁₉₇	C₂²	1	4	yes	complemented
44	C₄²×D₁₉₇	C₂²	1	2	yes	—
45	C₄²×D₁₉₇	C₂²	1	2	yes	—
46	C₄²×D₁₉₇	C₂²	1	2	yes	—
47	C₄²×D₁₉₇	C₂²	1	2	yes	—
48	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
49	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
50	#127×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
51	#124×C₁₉₇	C₂²	1	2	yes	complemented
52	#121×C₁₉₇	C₂²	1	4	yes	—
53	C₄×D₄×C₁₉₇	C₂²	1	4	yes	complemented
54	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
55	C₂³×D₁₉₇	C₂³	1	1	yes	characteristic
56	#15691357	—	2	16	no	—
57	C₂×Dic₃₉₄	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₉₇	C₂³	1	8	yes	—
59	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
60	C₂²×Dic₁₉₇	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₉₇	C₂³	1	8	yes	—
62	C₂×C₄×D₁₉₇	—	2	8	no	—
63	C₂×C₄×D₁₉₇	—	2	4	no	—
64	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
65	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
66	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
67	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
68	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
69	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
70	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
71	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
72	C₄⋊Dic₁₉₇	C₂³	1	8	yes	—
73	#15691337	C₂³	1	8	yes	—
74	#15691337	C₂³	1	8	yes	—
75	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₉₇	C₂³	1	4	yes	—
77	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
78	#15691335	C₂³	1	8	yes	—
79	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
80	C₄×Dic₁₉₇	C₂³	1	2	yes	—
81	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
82	C₄×Dic₁₉₇	C₂³	1	2	yes	—
83	C₄×Dic₁₉₇	C₂³	1	2	yes	—
84	C₄×Dic₁₉₇	C₂³	1	2	yes	—
85	C₄×Dic₁₉₇	C₂³	1	2	yes	—
86	C₄×Dic₁₉₇	C₂³	1	8	yes	—
87	C₄×Dic₁₉₇	C₂³	1	4	yes	—
88	#41×C₁₉₇	—	2	8	no	—
89	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
90	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
91	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
92	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
97	C₂²⋊C₄×C₁₉₇	C₂³	1	8	yes	—
98	C₄²×C₁₉₇	C₂³	1	2	yes	—
99	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
100	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
101	C₂²×D₁₉₇	—	2	2	no	—
102	C₂²×D₁₉₇	—	2	4	no	—
103	C₂²×D₁₉₇	C₂⁴	1	2	yes	—
104	Dic₃₉₄	—	2	8	no	—
105	C₂²⋊D₁₉₇	—	2	16	no	—
106	C₂×Dic₁₉₇	—	2	2	no	—
107	C₂×Dic₁₉₇	—	2	4	no	—
108	C₂×Dic₁₉₇	—	2	16	no	—
109	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
110	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
111	C₂×Dic₁₉₇	C₂⁴	1	8	yes	—
112	C₂×Dic₁₉₇	C₂⁴	1	2	yes	—
113	C₄×D₁₉₇	—	2	4	no	—
114	C₄×D₁₉₇	—	2	4	no	—
115	C₄×D₁₉₇	—	2	4	no	—
116	C₄×D₁₉₇	—	2	4	no	—
117	C₄×D₁₉₇	—	2	4	no	—
118	C₄×D₁₉₇	#41	1	8	yes	complemented
119	C₂³×C₁₉₇	C₂⁴	1	4	yes	—
120	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
121	Q₈×C₁₉₇	—	2	4	no	—
122	D₄×C₁₉₇	—	2	8	no	—
123	D₄×C₁₉₇	—	2	4	no	—
124	D₄×C₁₉₇	—	2	8	no	—
125	C₂×C₄×C₁₉₇	—	2	8	no	—
126	C₂×C₄×C₁₉₇	—	2	4	no	—
127	C₂×C₄×C₁₉₇	#41	1	4	yes	complemented
128	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
129	C₂×C₄×C₁₉₇	C₂⁴	1	4	yes	—
130	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
131	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
133	C₂²×C₁₉₇	—	2	2	no	—
134	C₂²×C₁₉₇	—	2	4	no	—
135	C₂²×C₁₉₇	—	2	8	no	—
136	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
137	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
138	C₂×D₁₉₇	—	2	8	no	—
139	C₂×D₁₉₇	—	2	2	no	—
140	C₂×D₁₉₇	—	2	2	no	—
141	C₂×D₁₉₇	C₂×#41	1	4	yes	—
142	C₄×C₁₉₇	—	2	4	no	—
143	C₄×C₁₉₇	—	2	2	no	—
144	C₄×C₁₉₇	C₂×#41	1	4	yes	—
145	C₄×C₁₉₇	C₂×#41	1	2	yes	—
146	C₄×C₁₉₇	C₂×#41	1	2	yes	—
147	Dic₁₉₇	—	2	2	no	—
148	Dic₁₉₇	—	2	2	no	—
149	Dic₁₉₇	—	2	8	no	—
150	Dic₁₉₇	C₂×#41	1	4	yes	—
151	C₂×C₁₉₇	—	2	2	no	—
152	C₂×C₁₉₇	—	4	4	no	—
153	C₂×C₁₉₇	C₂²×#41	1	2	yes	—
154	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
155	D₁₉₇	—	2	4	no	—
156	C₁₉₇	#3444	1	1	yes	characteristic, complemented, Sylow
157	#3444	—	197	1	no	maximal, Sylow
158	#533	—	197	2	no	—
159	#533	—	197	1	no	—
160	#532	—	197	2	no	—
161	#532	D₁₉₇	1	1	yes	characteristic, complemented
162	#529	—	197	4	no	—
163	#529	—	197	4	no	—
164	#529	—	197	4	no	—
165	#525	—	197	2	no	—
166	#525	—	197	4	no	—
167	C₄×#41	—	197	2	no	—
168	C₄×#41	—	197	4	no	—
169	C₂²×C₄²	—	197	1	no	—
170	C₂×#41	—	197	4	no	—
171	C₂×#41	C₂×D₁₉₇	1	2	yes	—
172	C₂³×C₄	—	197	2	no	—
173	C₂³×C₄	—	197	1	no	—
174	C₄⋊Q₈	—	197	2	no	—
175	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
176	#127	C₂×D₁₉₇	1	1	yes	characteristic, complemented
177	#126	—	197	8	no	—
178	#125	—	197	4	no	—
179	#125	—	197	2	no	—
180	#124	—	197	4	no	—
181	#124	C₂×D₁₉₇	1	2	yes	complemented
182	#123	—	197	4	no	—
183	#123	—	197	8	no	—
184	C₂²⋊Q₈	—	197	4	no	—
185	C₂²⋊Q₈	—	197	4	no	—
186	C₂²⋊Q₈	—	197	4	no	—
187	#121	—	197	4	no	—
188	#121	—	197	4	no	—
189	#121	C₂×D₁₉₇	1	4	yes	—
190	#120	—	197	4	no	—
191	C₄×Q₈	—	197	4	no	—
192	C₄×Q₈	—	197	4	no	—
193	C₄×Q₈	—	197	4	no	—
194	C₄×D₄	—	197	8	no	—
195	C₄×D₄	—	197	4	no	—
196	C₄×D₄	—	197	4	no	—
197	C₄×D₄	—	197	8	no	—
198	C₄×D₄	—	197	8	no	—
199	C₄×D₄	C₂×D₁₉₇	1	4	yes	complemented
200	#117	—	197	2	no	—
201	#117	—	197	4	no	—
202	#117	—	197	4	no	—
203	#117	—	197	8	no	—
204	C₂×C₄²	—	197	4	no	—
205	C₂×C₄²	—	197	1	no	—
206	C₂×C₄²	—	197	2	no	—
207	C₂×C₄²	—	197	2	no	—
208	C₂×C₄²	—	197	2	no	—
209	C₂×C₄²	—	197	2	no	—
210	C₂×C₄²	—	197	2	no	—
211	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
212	C₂⁴	—	197	1	no	—
213	#41	—	2	8	no	—
214	#41	—	394	16	no	—
215	C₂×Q₈	—	197	4	no	—
216	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
217	C₂×D₄	—	197	8	no	—
218	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
219	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
220	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
221	C₂²×C₄	—	394	8	no	—
222	C₂²×C₄	—	394	4	no	—
223	C₂²×C₄	—	197	2	no	—
224	C₂²×C₄	—	197	1	no	—
225	C₂²×C₄	—	197	8	no	—
226	C₂²×C₄	—	197	2	no	—
227	C₂²×C₄	—	197	2	no	—
228	C₂²×C₄	—	197	2	no	—
229	C₂²×C₄	—	197	2	no	—
230	C₂²×C₄	—	197	2	no	—
231	C₂²×C₄	—	197	2	no	—
232	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
233	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
234	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
235	C₄⋊C₄	—	197	8	no	—
236	C₄⋊C₄	—	197	8	no	—
237	C₄⋊C₄	—	197	4	no	—
238	C₄⋊C₄	—	197	4	no	—
239	C₄⋊C₄	—	197	8	no	—
240	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
241	C₂²⋊C₄	—	197	4	no	—
242	C₂²⋊C₄	—	197	8	no	—
243	C₂²⋊C₄	—	197	8	no	—
244	C₂²⋊C₄	—	197	8	no	—
245	C₂²⋊C₄	C₂²×D₁₉₇	1	8	yes	—
246	C₄²	—	197	2	no	—
247	C₄²	—	197	2	no	—
248	C₄²	—	197	2	no	—
249	C₄²	—	197	2	no	—
250	C₄²	—	197	2	no	—
251	C₄²	—	197	8	no	—
252	C₄²	—	197	1	no	—
253	C₄²	—	197	1	no	—
254	C₄²	—	197	4	no	—
255	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
256	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
257	C₄²	C₂²×D₁₉₇	1	2	yes	—
258	C₂³	—	394	4	no	—
259	C₂³	—	394	2	no	—
260	C₂³	—	197	2	no	—
261	C₂³	C₂³×D₁₉₇	1	4	yes	—
262	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	394	8	no	—
265	D₄	—	2	8	no	—
266	D₄	—	2	8	no	—
267	D₄	—	2	4	no	—
268	D₄	—	394	16	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	394	4	no	—
272	C₂×C₄	—	394	4	no	—
273	C₂×C₄	—	394	2	no	—
274	C₂×C₄	—	394	4	no	—
275	C₂×C₄	—	394	16	no	—
276	C₂×C₄	—	394	4	no	—
277	C₂×C₄	—	394	4	no	—
278	C₂×C₄	—	394	4	no	—
279	C₂×C₄	—	197	2	no	—
280	C₂×C₄	—	197	2	no	—
281	C₂×C₄	—	197	8	no	—
282	C₂×C₄	—	197	2	no	—
283	C₂×C₄	—	197	8	no	—
284	C₂×C₄	#15691357	1	4	yes	complemented
285	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
286	C₂×C₄	C₂³×D₁₉₇	1	4	yes	—
287	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
288	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	394	8	no	—
294	C₂²	—	394	2	no	—
295	C₂²	—	394	2	no	—
296	C₂²	—	197	4	no	—
297	C₂²	C₂×#15691357	1	2	yes	—
298	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	394	2	no	—
302	C₄	—	394	8	no	—
303	C₄	—	394	2	no	—
304	C₄	—	197	4	no	—
305	C₄	#41×D₁₉₇	1	4	yes	—
306	C₄	C₂×#15691357	1	2	yes	—
307	C₄	C₂×#15691357	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	4	4	no	—
310	C₂	—	394	4	no	—
311	C₂	C₂×#41×D₁₉₇	1	2	yes	—
312	C₂	C₂²×#15691357	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	#532×C₁₉₇	C₂
hypercenter	#532	D₁₉₇
hypocenter	C₁₉₇	#3444
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #27161636	C₂	semidirect
2. #27161634	C₂	semidirect
3. #27161627	C₂	semidirect
4. #27161611	C₂	semidirect
5. #27161610	C₂	semidirect
6. #27161608	C₂	semidirect
7. #27161589	C₂	semidirect
8. #27161586	C₂	semidirect
9. #41×Dic₁₉₇	C₂	semidirect
10. C₄×#15691357	C₂	semidirect
11. C₂×C₄²×D₁₉₇	C₂	semidirect
12. #532×C₁₉₇	C₂	semidirect
16. #20263326	C₂²	semidirect
17. #20263319	C₂²	semidirect
18. #20263318	C₂²	semidirect
19. #20263317	C₂²	semidirect
20. #20263313	C₂²	semidirect
29. Q₈×Dic₁₉₇	C₂²	semidirect
30. C₄×Dic₃₉₄	C₂²	semidirect
31. #20263291	C₂²	semidirect
33. D₄×Dic₁₉₇	C₂²	semidirect
34. D₄×Dic₁₉₇	C₂²	semidirect
38. #20263285	C₂²	semidirect
40. #20263282	C₂²	semidirect
43. C₂×C₄×Dic₁₉₇	C₂²	semidirect
49. C₄⋊Q₈×C₁₉₇	C₂²	semidirect
50. #127×C₁₉₇	C₂²	semidirect
51. #124×C₁₉₇	C₂²	semidirect
53. C₄×D₄×C₁₉₇	C₂²	semidirect
118. C₄×D₁₉₇	#41	semidirect
127. C₂×C₄×C₁₉₇	#41	semidirect
156. C₁₉₇	#3444	semidirect
161. #532	D₁₉₇	semidirect
175. C₄⋊Q₈	C₂×D₁₉₇	semidirect
176. #127	C₂×D₁₉₇	semidirect
181. #124	C₂×D₁₉₇	semidirect
199. C₄×D₄	C₂×D₁₉₇	semidirect
284. C₂×C₄	#15691357	semidirect

Maximal subgroups

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

subgroup	index
#27161636	2
#27161634	2
#27161627	2
#27161611	2
#27161610	2
#27161608	2
#27161589	2
#27161586	2
#41×Dic₁₉₇	2
C₄×#15691357	2
C₂×C₄²×D₁₉₇	2
#532×C₁₉₇	2
#3444	197