Subgroups — #466×D₁₉₇ #38752276

Order 25216.

616 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	C₂×D₈×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×D₁₉₇	C₂	1	2	yes	maximal, complemented
4	#27161316	C₂	1	1	yes	maximal, characteristic, complemented
5	#27161315	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161314	C₂	1	2	yes	maximal, complemented
7	#27161313	C₂	1	2	yes	maximal, complemented
8	#27161312	C₂	1	1	yes	maximal, characteristic, complemented
9	#107×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₉₇	C₂	1	2	yes	maximal, complemented
11	#466×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₉₇	—	2	2	no	—
13	C₂×D₄×D₁₉₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#15691343	C₂²	1	2	yes	complemented
17	D₈×D₁₉₇	—	2	4	no	—
18	C₂²×#13096802	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₈×D₁₉₇	—	2	4	no	—
20	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic, complemented
24	#20263301	C₂²	1	2	yes	complemented
25	#20263300	C₂²	1	2	yes	complemented
26	#20263299	—	2	2	no	—
27	#20263298	—	2	2	no	—
28	#20263297	C₂²	1	2	yes	complemented
29	C₄⋊C₄×D₁₉₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×D₁₉₇	—	2	2	no	—
31	#20263260	C₂²	1	1	yes	characteristic, complemented
32	#20263259	C₂²	1	2	yes	complemented
33	#20263248	C₂²	1	2	yes	complemented
34	#20263247	C₂²	1	2	yes	complemented
35	#20263246	C₂²	1	2	yes	complemented
36	C₂×D₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
38	#121×C₁₉₇	C₂²	1	2	yes	complemented
39	#107×C₁₉₇	C₂²	1	1	yes	characteristic, complemented
40	#102×C₁₉₇	C₂²	1	2	yes	complemented
41	C₂³×D₁₉₇	—	2	2	no	—
42	C₂³×D₁₉₇	D₄	1	1	yes	characteristic
43	C₂×D₇₈₈	—	2	2	no	—
44	C₂×D₇₈₈	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
46	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
47	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
48	D₄×D₁₉₇	—	2	4	no	—
49	D₄×D₁₉₇	—	2	4	no	—
50	D₄×D₁₉₇	—	4	8	no	—
51	C₂²×Dic₁₉₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₉₇	—	2	2	no	—
53	C₂×C₄×D₁₉₇	—	2	2	no	—
54	C₂×C₄×D₁₉₇	—	2	2	no	—
55	C₂×C₄×D₁₉₇	D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₉₇	D₄	1	1	yes	characteristic
57	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
59	D₁₅₇₆	—	2	2	no	—
60	#15691343	—	2	4	no	—
61	C₂×#13096802	—	2	2	no	—
62	C₂×#13096802	C₂³	1	1	yes	characteristic
63	C₂×#13096802	C₂³	1	1	yes	characteristic
64	C₈×D₁₉₇	—	2	2	no	—
65	C₈×D₁₉₇	—	2	2	no	—
66	C₈×D₁₉₇	—	2	2	no	—
67	C₈×D₁₉₇	D₄	1	4	yes	complemented
68	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
69	#15691337	—	2	2	no	—
70	C₂²⋊Dic₁₉₇	—	2	2	no	—
71	#15691335	—	2	2	no	—
72	#15691335	—	2	2	no	—
73	C₂×D₄×C₁₉₇	—	2	2	no	—
74	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
76	D₈×C₁₉₇	—	2	2	no	—
77	C₂×C₈×C₁₉₇	—	2	2	no	—
78	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
82	C₂²×D₁₉₇	—	2	2	no	—
83	C₂²×D₁₉₇	—	2	2	no	—
84	C₂²×D₁₉₇	—	2	2	no	—
85	C₂²×D₁₉₇	—	4	4	no	—
86	C₂²×D₁₉₇	—	4	4	no	—
87	C₂²×D₁₉₇	—	4	4	no	—
88	C₂²×D₁₉₇	D₈	1	4	yes	complemented
89	C₂²×D₁₉₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₉₇	C₂×D₄	1	1	yes	characteristic
91	D₇₈₈	—	2	2	no	—
92	D₇₈₈	—	2	2	no	—
93	D₇₈₈	—	4	4	no	—
94	C₂²⋊D₁₉₇	—	4	4	no	—
95	C₂²⋊D₁₉₇	—	4	4	no	—
96	C₂²⋊D₁₉₇	—	4	4	no	—
97	C₂×Dic₁₉₇	—	2	2	no	—
98	C₂×Dic₁₉₇	—	2	2	no	—
99	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₁₉₇	C₂×D₄	1	1	yes	characteristic
101	C₄×D₁₉₇	—	2	2	no	—
102	C₄×D₁₉₇	—	2	1	no	—
103	C₄×D₁₉₇	—	2	1	no	—
104	C₄×D₁₉₇	—	2	2	no	—
105	C₄×D₁₉₇	—	4	4	no	—
106	C₄×D₁₉₇	C₂×D₄	1	2	yes	—
107	C₄×D₁₉₇	#41	1	2	yes	—
108	#13096802	—	2	1	no	—
109	#13096802	C₂×D₄	1	2	yes	complemented
110	C₂³×C₁₉₇	—	2	2	no	—
111	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
112	D₄×C₁₉₇	—	2	2	no	—
113	D₄×C₁₉₇	—	2	2	no	—
114	D₄×C₁₉₇	—	4	4	no	—
115	C₂×C₄×C₁₉₇	—	2	1	no	—
116	C₂×C₄×C₁₉₇	—	2	1	no	—
117	C₂×C₄×C₁₉₇	—	2	2	no	—
118	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
120	C₈×C₁₉₇	—	2	1	no	—
121	C₈×C₁₉₇	C₂×D₄	1	2	yes	complemented
122	C₂²×C₁₉₇	—	2	1	no	—
123	C₂²×C₁₉₇	—	2	1	no	—
124	C₂²×C₁₉₇	—	4	2	no	—
125	C₂²×C₁₉₇	—	4	2	no	—
126	C₂²×C₁₉₇	—	4	2	no	—
127	C₂²×C₁₉₇	C₂×D₈	1	2	yes	complemented
128	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×D₁₉₇	—	2	1	no	—
130	C₂×D₁₉₇	—	2	1	no	—
131	C₂×D₁₉₇	—	2	8	no	—
132	C₂×D₁₉₇	—	4	2	no	—
133	C₂×D₁₉₇	—	4	2	no	—
134	C₂×D₁₉₇	—	4	2	no	—
135	C₂×D₁₉₇	—	8	8	no	—
136	C₂×D₁₉₇	#121	1	2	yes	—
137	C₂×D₁₉₇	C₂×D₈	1	2	yes	—
138	C₂×D₁₉₇	C₂×D₈	1	2	yes	complemented
139	C₂×D₁₉₇	#135	1	2	yes	—
140	C₄×C₁₉₇	—	2	1	no	—
141	C₄×C₁₉₇	—	4	2	no	—
142	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
144	Dic₁₉₇	—	2	1	no	—
145	Dic₁₉₇	—	4	2	no	—
146	Dic₁₉₇	#121	1	2	yes	—
147	C₂×C₁₉₇	—	2	2	no	—
148	C₂×C₁₉₇	—	8	2	no	—
149	C₂×C₁₉₇	C₂×#121	1	1	yes	characteristic
150	C₂×C₁₉₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
152	D₁₉₇	—	2	2	no	—
153	D₁₉₇	—	8	2	no	—
154	D₁₉₇	#466	1	4	yes	direct product factor
155	C₁₉₇	C₂×#466	1	1	yes	characteristic, complemented, Sylow
156	C₂×#466	—	197	1	no	maximal, Sylow
157	C₂²×D₈	—	197	1	no	—
158	C₂³×C₈	—	197	1	no	—
159	C₂×#121	—	197	2	no	—
160	#466	—	197	1	no	—
161	#466	—	197	1	no	—
162	#466	—	197	2	no	—
163	#466	—	197	1	no	—
164	#466	—	197	2	no	—
165	#466	D₁₉₇	1	1	yes	characteristic, direct product factor
166	C₂×#107	—	197	1	no	—
167	C₂×#102	—	197	2	no	—
168	C₂²×D₄	—	394	2	no	—
169	C₂²×D₄	—	197	2	no	—
170	C₂³×C₄	—	197	1	no	—
171	C₂×D₈	—	394	4	no	—
172	C₂×D₈	—	197	2	no	—
173	C₂×D₈	—	197	1	no	—
174	C₂×D₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
175	C₂²×C₈	—	394	4	no	—
176	C₂²×C₈	—	197	1	no	—
177	C₂²×C₈	—	197	1	no	—
178	C₂²×C₈	—	197	1	no	—
179	C₂²×C₈	—	197	1	no	—
180	C₂²×C₈	—	197	1	no	—
181	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic, complemented
182	#121	—	394	2	no	—
183	#121	—	394	2	no	—
184	#121	—	197	2	no	—
185	#121	—	197	2	no	—
186	#121	—	197	2	no	—
187	#121	C₂×D₁₉₇	1	2	yes	complemented
188	C₂×C₄⋊C₄	—	197	2	no	—
189	C₂×C₂²⋊C₄	—	394	2	no	—
190	#107	—	197	1	no	—
191	#107	—	197	2	no	—
192	#107	C₂×D₁₉₇	1	1	yes	characteristic, complemented
193	#102	—	197	2	no	—
194	#102	—	197	2	no	—
195	#102	—	197	2	no	—
196	#102	C₂×D₁₉₇	1	2	yes	complemented
197	C₂⁴	—	394	2	no	—
198	C₂⁴	—	197	1	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	788	8	no	—
201	C₂×D₄	—	394	2	no	—
202	C₂×D₄	—	394	2	no	—
203	C₂×D₄	—	394	2	no	—
204	C₂×D₄	—	394	4	no	—
205	C₂×D₄	—	394	4	no	—
206	C₂×D₄	—	394	2	no	—
207	C₂×D₄	—	197	2	no	—
208	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
209	C₂²×C₄	—	394	2	no	—
210	C₂²×C₄	—	394	2	no	—
211	C₂²×C₄	—	394	2	no	—
212	C₂²×C₄	—	197	1	no	—
213	C₂²×C₄	—	197	1	no	—
214	C₂²×C₄	—	197	1	no	—
215	C₂²×C₄	—	197	1	no	—
216	C₂²×C₄	—	197	1	no	—
217	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
218	D₈	—	2	2	no	—
219	D₈	—	394	4	no	—
220	D₈	—	394	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	394	2	no	—
223	C₂×C₈	—	394	2	no	—
224	C₂×C₈	—	394	2	no	—
225	C₂×C₈	—	394	2	no	—
226	C₂×C₈	—	197	1	no	—
227	C₂×C₈	—	197	1	no	—
228	C₂×C₈	—	197	4	no	—
229	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
231	C₄⋊C₄	—	394	2	no	—
232	C₄⋊C₄	—	197	2	no	—
233	C₄⋊C₄	C₂²×D₁₉₇	1	2	yes	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	394	2	no	—
236	C₂²⋊C₄	—	394	2	no	—
237	C₂²⋊C₄	—	394	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	788	4	no	—
240	C₂³	—	788	4	no	—
241	C₂³	—	788	4	no	—
242	C₂³	—	394	2	no	—
243	C₂³	—	394	2	no	—
244	C₂³	—	394	2	no	—
245	C₂³	—	197	4	no	—
246	C₂³	—	197	1	no	—
247	C₂³	—	197	1	no	—
248	C₂³	D₄×D₁₉₇	1	1	yes	characteristic
249	D₄	—	2	2	no	—
250	D₄	—	2	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	788	4	no	—
253	D₄	—	788	4	no	—
254	D₄	—	788	4	no	—
255	D₄	—	788	4	no	—
256	D₄	—	394	2	no	—
257	D₄	—	394	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	788	4	no	—
262	C₂×C₄	—	394	2	no	—
263	C₂×C₄	—	394	2	no	—
264	C₂×C₄	—	394	2	no	—
265	C₂×C₄	—	394	2	no	—
266	C₂×C₄	—	394	1	no	—
267	C₂×C₄	—	394	1	no	—
268	C₂×C₄	—	197	1	no	—
269	C₂×C₄	—	197	1	no	—
270	C₂×C₄	—	197	2	no	—
271	C₂×C₄	—	197	2	no	—
272	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	394	1	no	—
276	C₈	—	197	2	no	—
277	C₈	D₄×D₁₉₇	1	2	yes	complemented
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	788	2	no	—
284	C₂²	—	788	2	no	—
285	C₂²	—	788	2	no	—
286	C₂²	—	1576	8	no	—
287	C₂²	—	394	8	no	—
288	C₂²	—	394	1	no	—
289	C₂²	—	394	1	no	—
290	C₂²	—	197	2	no	—
291	C₂²	—	197	2	no	—
292	C₂²	—	197	2	no	—
293	C₂²	—	197	2	no	—
294	C₂²	D₈×D₁₉₇	1	2	yes	complemented
295	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	788	2	no	—
299	C₄	—	394	1	no	—
300	C₄	—	197	2	no	—
301	C₄	C₂×D₄×D₁₉₇	1	1	yes	characteristic
302	C₄	#41×D₁₉₇	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	1576	2	no	—
306	C₂	—	394	2	no	—
307	C₂	—	197	4	no	—
308	C₂	#121×D₁₉₇	1	1	yes	characteristic
309	C₂	C₂×D₈×D₁₉₇	1	1	yes	characteristic
310	C₂	#135×D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#466×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	#466	D₁₉₇
hypocenter	C₁₉₇	C₂×#466
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×D₁₉₇	C₂	semidirect
2. C₂²×C₈×D₁₉₇	C₂	semidirect
3. #121×D₁₉₇	C₂	semidirect
4. #27161316	C₂	semidirect
5. #27161315	C₂	semidirect
6. #27161314	C₂	semidirect
7. #27161313	C₂	semidirect
8. #27161312	C₂	semidirect
9. #107×D₁₉₇	C₂	semidirect
10. #102×D₁₉₇	C₂	semidirect
11. #466×C₁₉₇	C₂	semidirect
15. C₂×D₁₅₇₆	C₂²	semidirect
16. C₂×#15691343	C₂²	semidirect
18. C₂²×#13096802	C₂²	semidirect
22. C₂×C₈×D₁₉₇	C₂²	semidirect
23. C₂×C₈×D₁₉₇	C₂²	semidirect
24. #20263301	C₂²	semidirect
25. #20263300	C₂²	semidirect
28. #20263297	C₂²	semidirect
31. #20263260	C₂²	semidirect
32. #20263259	C₂²	semidirect
33. #20263248	C₂²	semidirect
34. #20263247	C₂²	semidirect
35. #20263246	C₂²	semidirect
36. C₂×D₈×C₁₉₇	C₂²	semidirect
37. C₂²×C₈×C₁₉₇	C₂²	semidirect
38. #121×C₁₉₇	C₂²	semidirect
39. #107×C₁₉₇	C₂²	semidirect
40. #102×C₁₉₇	C₂²	semidirect
67. C₈×D₁₉₇	D₄	semidirect
88. C₂²×D₁₉₇	D₈	semidirect
109. #13096802	C₂×D₄	semidirect
121. C₈×C₁₉₇	C₂×D₄	semidirect
127. C₂²×C₁₉₇	C₂×D₈	semidirect
138. C₂×D₁₉₇	C₂×D₈	semidirect
154. D₁₉₇	#466	direct (N × H)
155. C₁₉₇	C₂×#466	semidirect
165. #466	D₁₉₇	direct (N × H)
174. C₂×D₈	C₂×D₁₉₇	semidirect
181. C₂²×C₈	C₂×D₁₉₇	semidirect
187. #121	C₂×D₁₉₇	semidirect
192. #107	C₂×D₁₉₇	semidirect
196. #102	C₂×D₁₉₇	semidirect
277. C₈	D₄×D₁₉₇	semidirect
294. C₂²	D₈×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₁₉₇	2
C₂²×C₈×D₁₉₇	2
#121×D₁₉₇	2
#27161316	2
#27161315	2
#27161314	2
#27161313	2
#27161312	2
#107×D₁₉₇	2
#102×D₁₉₇	2
#466×C₁₉₇	2
C₂×#466	197

Subgroups — #466×D197 #38752276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #466×D₁₉₇ #38752276