Subgroups — #38751434

Order 25216.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161586	C₂	1	1	yes	maximal, characteristic
3	#27161155	C₂	1	2	yes	maximal, complemented
4	#27161150	C₂	1	2	yes	maximal, complemented
5	#27161148	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₁₉₇	C₂	1	2	yes	maximal, complemented
7	#27161105	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#20263238	C₂	1	1	yes	maximal, characteristic, complemented
10	#27161037	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#131×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×#15691342	C₂²	1	1	yes	characteristic
14	C₂×#15691341	C₂²	1	2	yes	—
15	M₁₆×D₁₉₇	—	2	4	no	—
16	C₂²×#13096802	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₉₇	C₂²	1	2	yes	—
18	#20263307	C₄	1	2	yes	—
19	#20263302	C₄	1	1	yes	characteristic
20	#20263302	C₄	1	1	yes	characteristic
21	#20263297	C₄	1	1	yes	characteristic
22	#20263297	C₄	1	1	yes	characteristic
23	#20263294	C₄	1	2	yes	—
24	C₄×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
26	#20263285	C₂²	1	1	yes	characteristic
27	#20263255	C₂²	1	2	yes	complemented
28	#20263255	C₂²	1	2	yes	complemented
29	C₂²⋊#13096802	C₂²	1	2	yes	complemented
30	C₂²⋊#13096802	C₂²	1	2	yes	complemented
31	#20263238	C₂²	1	1	yes	characteristic
32	#20263238	C₂²	1	1	yes	characteristic
33	#20263238	C₂²	1	1	yes	characteristic
34	#20263238	C₂²	1	1	yes	characteristic
35	#20263236	C₂²	1	2	yes	complemented
36	C₈×Dic₁₉₇	C₂²	1	2	yes	complemented
37	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
38	#131×C₁₉₇	—	2	4	no	—
39	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	complemented
41	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	complemented
43	C₂³×D₁₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₁₉₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₉₇	—	2	2	no	—
48	C₂×C₄×D₁₉₇	—	2	2	no	—
49	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
53	#15691342	—	2	2	no	—
54	#15691341	—	2	4	no	—
55	C₂×#13096802	—	2	2	no	—
56	C₂×#13096802	C₂³	1	1	yes	characteristic
57	C₂×#13096802	C₂³	1	1	yes	characteristic
58	C₂×#13096802	C₂³	1	2	yes	—
59	C₈×D₁₉₇	—	2	4	no	—
60	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
62	#15691337	C₂×C₄	1	2	yes	—
63	#15691337	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
67	#15691335	C₂×C₄	1	2	yes	—
68	#15691335	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₉₇	C₂³	1	2	yes	—
70	#41×C₁₉₇	—	2	2	no	—
71	#41×C₁₉₇	—	2	2	no	—
72	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
73	C₂×D₄×C₁₉₇	C₂×C₄	1	2	yes	—
74	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
77	M₁₆×C₁₉₇	—	2	4	no	—
78	M₁₆×C₁₉₇	D₄	1	4	yes	complemented
79	C₂×C₈×C₁₉₇	—	2	2	no	—
80	C₂×C₈×C₁₉₇	—	2	4	no	—
81	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
84	C₂²×D₁₉₇	—	2	2	no	—
85	C₂²×D₁₉₇	—	2	2	no	—
86	C₂²×D₁₉₇	—	4	1	no	—
87	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊D₁₉₇	—	2	4	no	—
90	C₂²⋊D₁₉₇	—	4	2	no	—
91	C₂×Dic₁₉₇	—	2	2	no	—
92	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₉₇	—	2	2	no	—
97	C₄×D₁₉₇	—	2	2	no	—
98	C₄×D₁₉₇	—	2	1	no	—
99	C₄×D₁₉₇	—	2	1	no	—
100	C₄×D₁₉₇	—	2	1	no	—
101	C₄×D₁₉₇	—	2	1	no	—
102	#13096802	—	2	2	no	—
103	#13096802	—	2	1	no	—
104	#13096802	—	2	1	no	—
105	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
106	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₁₉₇	—	2	2	no	—
108	D₄×C₁₉₇	—	2	4	no	—
109	D₄×C₁₉₇	—	2	2	no	—
110	C₂×C₄×C₁₉₇	—	2	2	no	—
111	C₂×C₄×C₁₉₇	—	2	2	no	—
112	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₉₇	#41	1	2	yes	—
116	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
117	C₈×C₁₉₇	—	2	1	no	—
118	C₈×C₁₉₇	—	2	1	no	—
119	C₈×C₁₉₇	C₂×D₄	1	4	yes	—
120	C₂²×C₁₉₇	—	2	1	no	—
121	C₂²×C₁₉₇	—	2	1	no	—
122	C₂²×C₁₉₇	—	2	2	no	—
123	C₂²×C₁₉₇	—	2	2	no	—
124	C₂²×C₁₉₇	—	2	2	no	—
125	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
126	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₁₉₇	—	2	1	no	—
128	C₂×D₁₉₇	—	2	1	no	—
129	C₂×D₁₉₇	—	2	1	no	—
130	C₂×D₁₉₇	—	2	1	no	—
131	C₂×D₁₉₇	—	2	1	no	—
132	C₂×D₁₉₇	—	4	4	no	—
133	C₂×D₁₉₇	#131	1	2	yes	complemented
134	C₄×C₁₉₇	—	2	2	no	—
135	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
136	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
138	Dic₁₉₇	—	2	1	no	—
139	Dic₁₉₇	—	2	1	no	—
140	Dic₁₉₇	—	2	2	no	—
141	Dic₁₉₇	—	2	2	no	—
142	C₂×C₁₉₇	—	2	2	no	—
143	C₂×C₁₉₇	—	4	2	no	—
144	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
146	C₂×C₁₉₇	#568	1	1	yes	characteristic
147	D₁₉₇	—	2	2	no	—
148	D₁₉₇	—	4	1	no	—
149	C₁₉₇	#2933	1	1	yes	characteristic, complemented, Sylow
150	#2933	—	197	1	no	maximal, Sylow
151	C₂×#131	D₁₉₇	1	1	yes	characteristic, complemented
152	C₂²×M₁₆	—	197	1	no	—
153	#525	—	197	1	no	—
154	#436	—	197	2	no	—
155	#435	—	197	2	no	—
156	#435	—	197	2	no	—
157	C₈×D₄	—	197	2	no	—
158	#424	—	197	1	no	—
159	#408	—	197	1	no	—
160	C₂×C₂²⋊C₈	—	197	1	no	—
161	#405	—	197	1	no	—
162	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
163	C₂³×C₄	—	197	1	no	—
164	#131	—	2	4	no	—
165	C₂×M₁₆	—	394	4	no	—
166	C₂×M₁₆	—	197	1	no	—
167	C₂×M₁₆	—	197	2	no	—
168	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	complemented
170	C₂²×C₈	—	197	1	no	—
171	C₂²×C₈	—	197	2	no	—
172	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
173	C₂²×C₈	C₂×D₁₉₇	1	2	yes	complemented
174	#123	—	197	2	no	—
175	C₂²⋊Q₈	—	197	1	no	—
176	C₂²⋊Q₈	—	197	1	no	—
177	#121	—	197	1	no	—
178	#121	—	197	1	no	—
179	#120	—	197	2	no	—
180	C₄×D₄	—	197	2	no	—
181	C₄×D₄	—	197	2	no	—
182	#117	—	197	1	no	—
183	C₄⋊C₈	—	197	2	no	—
184	C₄⋊C₈	—	197	2	no	—
185	C₂²⋊C₈	—	197	2	no	—
186	C₂²⋊C₈	—	197	2	no	—
187	C₂²⋊C₈	—	197	1	no	—
188	C₂²⋊C₈	—	197	1	no	—
189	C₂²⋊C₈	—	197	1	no	—
190	C₂²⋊C₈	—	197	1	no	—
191	#97	—	197	2	no	—
192	C₄×C₈	—	197	2	no	—
193	C₂⁴	—	197	1	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
197	C₂×D₄	—	197	2	no	—
198	C₂×D₄	—	197	2	no	—
199	C₂×D₄	C₄×D₁₉₇	1	2	yes	—
200	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
201	C₂²×C₄	—	394	2	no	—
202	C₂²×C₄	—	394	2	no	—
203	C₂²×C₄	—	197	1	no	—
204	C₂²×C₄	—	197	1	no	—
205	C₂²×C₄	—	197	1	no	—
206	C₂²×C₄	—	197	1	no	—
207	C₂²×C₄	—	197	1	no	—
208	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	394	2	no	—
212	M₁₆	—	394	4	no	—
213	M₁₆	C₂²⋊D₁₉₇	1	4	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	394	2	no	—
217	C₂×C₈	—	394	4	no	—
218	C₂×C₈	—	197	1	no	—
219	C₂×C₈	—	197	1	no	—
220	C₂×C₈	—	197	2	no	—
221	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
224	C₄⋊C₄	—	197	2	no	—
225	C₄⋊C₄	—	197	2	no	—
226	C₄⋊C₄	—	197	1	no	—
227	C₄⋊C₄	—	197	1	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	—	197	2	no	—
233	C₄²	—	197	2	no	—
234	C₂³	—	788	1	no	—
235	C₂³	—	394	2	no	—
236	C₂³	—	394	2	no	—
237	C₂³	—	197	1	no	—
238	C₂³	—	197	1	no	—
239	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
240	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	788	2	no	—
245	D₄	—	394	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	394	2	no	—
249	C₂×C₄	—	394	2	no	—
250	C₂×C₄	—	394	2	no	—
251	C₂×C₄	—	394	1	no	—
252	C₂×C₄	—	394	1	no	—
253	C₂×C₄	—	394	1	no	—
254	C₂×C₄	—	394	1	no	—
255	C₂×C₄	—	197	2	no	—
256	C₂×C₄	—	197	2	no	—
257	C₂×C₄	—	197	1	no	—
258	C₂×C₄	—	197	1	no	—
259	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
262	C₂×C₄	#15691358	1	2	yes	—
263	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	394	2	no	—
267	C₈	—	394	1	no	—
268	C₈	—	394	1	no	—
269	C₈	C₂×C₂²⋊D₁₉₇	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	788	4	no	—
276	C₂²	—	394	1	no	—
277	C₂²	—	394	1	no	—
278	C₂²	—	394	1	no	—
279	C₂²	—	394	1	no	—
280	C₂²	—	394	1	no	—
281	C₂²	—	197	2	no	—
282	C₂²	C₄×C₂²⋊D₁₉₇	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	394	1	no	—
286	C₄	—	394	1	no	—
287	C₄	—	394	2	no	—
288	C₄	—	394	2	no	—
289	C₄	C₄×C₂²⋊D₁₉₇	1	2	yes	—
290	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
291	C₄	C₂×#15691358	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	4	2	no	—
294	C₂	—	788	1	no	—
295	C₂	—	394	2	no	—
296	C₂	C₂×C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
297	C₂	#131×D₁₉₇	1	1	yes	characteristic
298	C₂	#27161844	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	C₂×#131×C₁₉₇	C₂
prime radical	#27161586	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂×#131	D₁₉₇
hypocenter	C₁₉₇	#2933
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₉₇	C₂	semidirect
3. #27161155	C₂	semidirect
4. #27161150	C₂	semidirect
5. #27161148	C₂	semidirect
6. C₈×C₂²⋊D₁₉₇	C₂	semidirect
7. #27161105	C₂	semidirect
8. #27161046	C₂	semidirect
9. C₂×#20263238	C₂	semidirect
10. #27161037	C₂	semidirect
11. C₂×#131×C₁₉₇	C₂	semidirect
27. #20263255	C₂²	semidirect
28. #20263255	C₂²	semidirect
29. C₂²⋊#13096802	C₂²	semidirect
30. C₂²⋊#13096802	C₂²	semidirect
35. #20263236	C₂²	semidirect
36. C₈×Dic₁₉₇	C₂²	semidirect
40. C₂×M₁₆×C₁₉₇	C₂²	semidirect
42. C₂²×C₈×C₁₉₇	C₂²	semidirect
78. M₁₆×C₁₉₇	D₄	semidirect
133. C₂×D₁₉₇	#131	semidirect
149. C₁₉₇	#2933	semidirect
151. C₂×#131	D₁₉₇	semidirect
169. C₂×M₁₆	C₂×D₁₉₇	semidirect
173. C₂²×C₈	C₂×D₁₉₇	semidirect
213. M₁₆	C₂²⋊D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₉₇	2
#27161586	2
#27161155	2
#27161150	2
#27161148	2
C₈×C₂²⋊D₁₉₇	2
#27161105	2
#27161046	2
C₂×#20263238	2
#27161037	2
C₂×#131×C₁₉₇	2
#2933	197