Subgroups — #38751611

Order 25216.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161877	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20263333	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161198	C₂	1	4	yes	maximal, complemented
5	#27161193	C₂	1	4	yes	maximal, complemented
6	#27161127	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161097	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161035	C₂	1	1	yes	maximal, characteristic, complemented
9	#421×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15691358	C₂²	1	1	yes	characteristic
11	#20263357	C₄	1	2	yes	complemented
12	#20263356	C₄	1	2	yes	complemented
13	#20263352	C₄	1	2	yes	complemented
14	#20263351	C₄	1	2	yes	complemented
15	#20263346	C₂²	1	4	yes	—
16	#20263334	—	2	1	no	—
17	#20263334	—	2	1	no	—
18	#20263334	C₂²	1	2	yes	complemented
19	#20263334	C₂²	1	2	yes	complemented
20	#20263333	—	2	1	no	—
21	#20263333	—	2	1	no	—
22	#20263333	C₂²	1	2	yes	complemented
23	#20263333	C₂²	1	2	yes	complemented
24	C₂×#15691342	—	2	1	no	—
25	C₂×#15691342	C₂²	1	1	yes	characteristic
26	C₂×#15691342	C₂²	1	1	yes	characteristic
27	C₂×#15691341	—	2	1	no	—
28	C₂²×#13096802	—	2	1	no	—
29	C₂²×#13096802	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₉₇	—	2	1	no	—
31	#20263261	C₂²	1	4	yes	complemented
32	#20263254	C₂²	1	4	yes	complemented
33	#20263237	C₂²	1	2	yes	complemented
34	C₄×#13096802	C₂²	1	2	yes	complemented
35	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
37	#117×C₁₉₇	C₂²	1	1	yes	characteristic
38	#104×C₁₉₇	C₂²	1	4	yes	complemented
39	#15691358	—	2	1	no	—
40	#15691358	—	2	1	no	—
41	#15691358	C₂×C₄	1	2	yes	—
42	#15691358	C₂³	1	2	yes	—
43	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
44	C₂×D₇₈₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
46	C₂×C₄×D₁₉₇	—	2	1	no	—
47	#15691348	C₂×C₄	1	4	yes	—
48	#15691346	C₂×C₄	1	4	yes	—
49	#15691345	C₂×C₄	1	4	yes	—
50	#15691343	C₂×C₄	1	4	yes	—
51	#15691342	—	2	2	no	—
52	#15691342	—	2	2	no	—
53	#15691342	—	2	1	no	—
54	#15691342	C₂×C₄	1	2	yes	—
55	#15691342	C₂×C₄	1	2	yes	—
56	#15691342	C₂³	1	2	yes	—
57	#15691341	—	2	2	no	—
58	#15691341	—	2	2	no	—
59	C₂×#13096802	—	2	2	no	—
60	C₂×#13096802	—	2	2	no	—
61	C₂×#13096802	—	2	1	no	—
62	C₂×#13096802	D₄	1	1	yes	characteristic
63	C₂×#13096802	D₄	1	1	yes	characteristic
64	C₂×#13096802	D₄	1	1	yes	characteristic
65	C₂×#13096802	D₄	1	1	yes	characteristic
66	C₂×#13096802	C₂³	1	2	yes	—
67	C₂×#13096802	C₂³	1	1	yes	characteristic
68	C₂×#13096802	C₂³	1	1	yes	characteristic
69	C₈×D₁₉₇	—	2	2	no	—
70	C₈×D₁₉₇	—	2	2	no	—
71	#41×C₁₉₇	—	2	1	no	—
72	#41×C₁₉₇	—	2	1	no	—
73	#41×C₁₉₇	C₂×C₄	1	2	yes	—
74	#41×C₁₉₇	C₂³	1	2	yes	—
75	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
76	C₂×D₄×C₁₉₇	—	2	1	no	—
77	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉₇	—	2	1	no	—
79	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₉₇	—	2	1	no	—
81	M₁₆×C₁₉₇	C₂³	1	2	yes	—
82	C₂×C₈×C₁₉₇	—	2	1	no	—
83	C₄⋊C₄×C₁₉₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₉₇	—	2	1	no	—
85	C₄²×C₁₉₇	C₂³	1	2	yes	—
86	C₂²×D₁₉₇	—	2	1	no	—
87	Dic₃₉₄	—	2	1	no	—
88	Dic₃₉₄	C₂²×C₄	1	2	yes	—
89	D₇₈₈	—	2	1	no	—
90	D₇₈₈	C₂²×C₄	1	2	yes	—
91	C₂²⋊D₁₉₇	—	2	2	no	—
92	C₂²⋊D₁₉₇	—	2	2	no	—
93	C₂×Dic₁₉₇	—	2	1	no	—
94	C₄×D₁₉₇	—	2	2	no	—
95	C₄×D₁₉₇	—	2	2	no	—
96	#13096802	—	2	2	no	—
97	#13096802	C₂²×C₄	1	4	yes	—
98	#13096802	C₂×D₄	1	1	yes	characteristic
99	#13096802	C₂×D₄	1	1	yes	characteristic
100	#13096802	C₂×D₄	1	1	yes	characteristic
101	#13096802	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₁₉₇	—	2	1	no	—
103	C₂³×C₁₉₇	#41	1	1	yes	characteristic
104	Q₈×C₁₉₇	—	2	1	no	—
105	Q₈×C₁₉₇	C₂²×C₄	1	2	yes	—
106	D₄×C₁₉₇	—	2	2	no	—
107	D₄×C₁₉₇	—	2	2	no	—
108	D₄×C₁₉₇	—	2	1	no	—
109	D₄×C₁₉₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₉₇	—	2	1	no	—
111	C₂×C₄×C₁₉₇	—	2	2	no	—
112	C₂×C₄×C₁₉₇	—	2	2	no	—
113	C₂×C₄×C₁₉₇	—	2	2	no	—
114	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
119	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
120	C₈×C₁₉₇	—	2	2	no	—
121	C₂²×C₁₉₇	—	2	2	no	—
122	C₂²×C₁₉₇	—	4	1	no	—
123	C₂²×C₁₉₇	—	4	2	no	—
124	C₂²×C₁₉₇	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₉₇	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₉₇	—	2	2	no	—
128	C₂×D₁₉₇	—	4	2	no	—
129	C₄×C₁₉₇	—	2	2	no	—
130	C₄×C₁₉₇	—	4	2	no	—
131	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
133	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
135	Dic₁₉₇	—	2	2	no	—
136	C₂×C₁₉₇	—	2	1	no	—
137	C₂×C₁₉₇	—	2	1	no	—
138	C₂×C₁₉₇	—	2	1	no	—
139	C₂×C₁₉₇	—	4	2	no	—
140	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
141	D₁₉₇	—	4	2	no	—
142	C₁₉₇	#2955	1	1	yes	characteristic, complemented, Sylow
143	#2955	—	197	1	no	maximal, Sylow
144	#575	—	197	1	no	—
145	C₂×#131	—	197	1	no	—
146	C₂×#131	—	197	1	no	—
147	#444	—	197	4	no	—
148	#443	—	197	4	no	—
149	#430	—	197	1	no	—
150	#421	—	197	1	no	—
151	#421	D₁₉₇	1	1	yes	characteristic, complemented
152	#405	—	197	1	no	—
153	C₂×#41	—	197	1	no	—
154	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
155	#137	—	197	2	no	—
156	#137	—	197	2	no	—
157	AGL(1,ℤ/8)	—	197	2	no	—
158	AGL(1,ℤ/8)	—	197	2	no	—
159	#135	—	197	4	no	—
160	#131	—	394	1	no	—
161	#131	—	394	1	no	—
162	#131	—	394	1	no	—
163	#131	—	394	1	no	—
164	#131	—	197	2	no	—
165	#131	—	197	2	no	—
166	#131	—	197	2	no	—
167	#131	—	197	2	no	—
168	C₂×M₁₆	—	394	1	no	—
169	C₂×M₁₆	—	394	1	no	—
170	C₂×M₁₆	—	197	1	no	—
171	C₂×M₁₆	—	197	1	no	—
172	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
173	C₂²×C₈	—	394	1	no	—
174	C₂²×C₈	—	394	1	no	—
175	C₂²×C₈	—	197	1	no	—
176	#117	C₂×D₁₉₇	1	1	yes	characteristic
177	#108	—	197	4	no	—
178	#104	—	197	4	no	—
179	#104	C₂×D₁₉₇	1	4	yes	complemented
180	#97	—	197	2	no	—
181	C₄×C₈	—	197	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	394	1	no	—
185	#41	—	394	1	no	—
186	#41	—	197	2	no	—
187	#41	—	197	2	no	—
188	#41	C₄×D₁₉₇	1	2	yes	—
189	#41	C₂²×D₁₉₇	1	2	yes	—
190	C₂×Q₈	—	197	1	no	—
191	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	394	1	no	—
194	C₂×D₄	—	197	1	no	—
195	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	394	1	no	—
198	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
199	Q₁₆	—	197	4	no	—
200	SD₁₆	—	197	4	no	—
201	SD₁₆	—	197	4	no	—
202	D₈	—	197	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	394	2	no	—
205	M₁₆	—	394	2	no	—
206	M₁₆	—	394	1	no	—
207	M₁₆	—	394	2	no	—
208	M₁₆	—	394	2	no	—
209	M₁₆	—	197	2	no	—
210	M₁₆	—	197	2	no	—
211	M₁₆	—	197	2	no	—
212	M₁₆	C₂²×D₁₉₇	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	394	2	no	—
215	C₂×C₈	—	394	2	no	—
216	C₂×C₈	—	394	1	no	—
217	C₂×C₈	—	394	2	no	—
218	C₂×C₈	—	394	2	no	—
219	C₂×C₈	—	197	1	no	—
220	C₂×C₈	—	197	1	no	—
221	C₂×C₈	—	197	1	no	—
222	C₂×C₈	—	197	1	no	—
223	C₂×C₈	—	197	1	no	—
224	C₂×C₈	—	197	2	no	—
225	C₂×C₈	—	197	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	C₂²×D₁₉₇	1	2	yes	—
229	C₂³	—	2	1	no	—
230	C₂³	—	394	1	no	—
231	C₂³	#15691357	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	394	1	no	—
234	Q₈	—	197	2	no	—
235	Q₈	C₂×C₄×D₁₉₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	394	2	no	—
240	D₄	—	394	2	no	—
241	D₄	—	394	1	no	—
242	D₄	—	197	2	no	—
243	D₄	C₂×C₄×D₁₉₇	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	394	1	no	—
249	C₂×C₄	—	394	2	no	—
250	C₂×C₄	—	394	2	no	—
251	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
253	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
254	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
255	C₂×C₄	#15691357	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	394	2	no	—
259	C₈	—	197	4	no	—
260	C₈	—	197	1	no	—
261	C₈	—	197	1	no	—
262	C₈	—	197	1	no	—
263	C₈	—	197	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	788	2	no	—
268	C₂²	—	394	2	no	—
269	C₂²	#20263286	1	1	yes	characteristic
270	C₂²	#20263286	1	1	yes	characteristic
271	C₂²	C₂×#15691357	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	394	2	no	—
275	C₄	#20263286	1	1	yes	characteristic
276	C₄	#20263286	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
278	C₄	C₂×D₄×D₁₉₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	788	2	no	—
284	C₂	C₂×#20263286	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#421×C₁₉₇	C₂
socle	C₂×C₁₉₇	C₂×C₄×D₄
prime radical	#27161877	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#421	D₁₉₇
hypocenter	C₁₉₇	#2955
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. C₂×#20263334	C₂	semidirect
3. C₂×#20263333	C₂	semidirect
4. #27161198	C₂	semidirect
5. #27161193	C₂	semidirect
6. #27161127	C₂	semidirect
7. #27161097	C₂	semidirect
8. #27161035	C₂	semidirect
9. #421×C₁₉₇	C₂	semidirect
11. #20263357	C₄	semidirect
12. #20263356	C₄	semidirect
13. #20263352	C₄	semidirect
14. #20263351	C₄	semidirect
18. #20263334	C₂²	semidirect
19. #20263334	C₂²	semidirect
22. #20263333	C₂²	semidirect
23. #20263333	C₂²	semidirect
31. #20263261	C₂²	semidirect
32. #20263254	C₂²	semidirect
33. #20263237	C₂²	semidirect
34. C₄×#13096802	C₂²	semidirect
38. #104×C₁₉₇	C₂²	semidirect
142. C₁₉₇	#2955	semidirect
151. #421	D₁₉₇	semidirect
179. #104	C₂×D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161877	2
C₂×#20263334	2
C₂×#20263333	2
#27161198	2
#27161193	2
#27161127	2
#27161097	2
#27161035	2
#421×C₁₉₇	2
#2955	197