Subgroups — #38752502

Order 25216.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₄×#15691358	C₂	1	1	yes	maximal, characteristic
3	#27161387	C₂	1	2	yes	maximal, complemented
4	#27161380	C₂	1	2	yes	maximal, complemented
5	#27161375	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161370	C₂	1	1	yes	maximal, characteristic, complemented
7	#27161365	C₂	1	1	yes	maximal, characteristic, complemented
8	#27161360	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161117	C₂	1	2	yes	maximal, complemented
10	#27161076	C₂	1	2	yes	maximal
11	C₂×C₄⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691358	C₂²	1	1	yes	characteristic
13	#20263326	C₂²	1	1	yes	characteristic
14	#20263326	C₂²	1	1	yes	characteristic
15	#20263318	C₂²	1	2	yes	—
16	#20263305	—	2	1	no	—
17	#20263305	—	2	1	no	—
18	C₄×Dic₃₉₄	—	2	1	no	—
19	C₄×Dic₃₉₄	—	2	1	no	—
20	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₉₄	C₂²	1	1	yes	characteristic
22	C₄×D₇₈₈	C₂²	1	1	yes	characteristic
23	C₄×D₇₈₈	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₉₇	—	2	1	no	—
25	C₄×C₂²⋊D₁₉₇	—	2	1	no	—
26	#20263285	C₂²	1	1	yes	characteristic
27	#20263285	C₂²	1	2	yes	—
28	#20263283	—	2	1	no	—
29	C₄²×D₁₉₇	—	2	1	no	—
30	#20263260	C₂²	1	2	yes	—
31	#20263260	C₂²	1	2	yes	—
32	#20263258	C₂²	1	2	yes	—
33	#20263258	C₂²	1	2	yes	—
34	#20263251	C₂²	1	2	yes	—
35	#20263251	C₂²	1	2	yes	—
36	#20263248	C₂²	1	2	yes	—
37	#20263248	C₂²	1	2	yes	—
38	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
39	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
43	#15691358	—	2	2	no	—
44	#15691358	Q₈	1	4	yes	complemented
45	C₂×Dic₃₉₄	—	2	1	no	—
46	C₂×Dic₃₉₄	—	2	1	no	—
47	C₂×Dic₃₉₄	C₂³	1	1	yes	characteristic
48	C₂×D₇₈₈	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
50	C₂×C₄×D₁₉₇	—	2	1	no	—
51	C₂×C₄×D₁₉₇	—	2	1	no	—
52	C₂×C₄×D₁₉₇	—	2	1	no	—
53	C₄⋊Dic₁₉₇	—	2	1	no	—
54	C₄⋊Dic₁₉₇	—	2	1	no	—
55	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
56	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
57	C₄⋊Dic₁₉₇	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₉₇	C₂³	1	1	yes	characteristic
59	#15691337	—	2	1	no	—
60	#15691337	—	2	1	no	—
61	#15691337	—	2	2	no	—
62	#15691337	—	2	2	no	—
63	C₂²⋊Dic₁₉₇	—	2	1	no	—
64	C₂²⋊Dic₁₉₇	—	2	2	no	—
65	#15691335	—	2	1	no	—
66	#15691335	—	2	1	no	—
67	C₄×Dic₁₉₇	—	2	1	no	—
68	C₄×Dic₁₉₇	—	2	1	no	—
69	C₄×Dic₁₉₇	—	2	1	no	—
70	C₄×Dic₁₉₇	—	2	2	no	—
71	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉₇	—	2	4	no	—
75	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
76	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
77	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
81	C₂²×D₁₉₇	—	2	1	no	—
82	Dic₃₉₄	—	2	1	no	—
83	Dic₃₉₄	—	4	2	no	—
84	Dic₃₉₄	—	4	2	no	—
85	Dic₃₉₄	C₂×Q₈	1	2	yes	—
86	D₇₈₈	—	2	1	no	—
87	D₇₈₈	C₂×Q₈	1	2	yes	—
88	C₂²⋊D₁₉₇	—	2	4	no	—
89	C₂×Dic₁₉₇	—	2	1	no	—
90	C₂×Dic₁₉₇	—	2	1	no	—
91	C₂×Dic₁₉₇	—	2	1	no	—
92	C₂×Dic₁₉₇	—	2	2	no	—
93	C₂×Dic₁₉₇	—	2	2	no	—
94	C₄×D₁₉₇	—	2	4	no	—
95	C₄×D₁₉₇	—	4	2	no	—
96	C₄×D₁₉₇	—	4	2	no	—
97	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉₇	—	2	2	no	—
99	C₂×C₄×C₁₉₇	—	4	1	no	—
100	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₉₇	#41	1	2	yes	—
107	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₉₇	—	2	2	no	—
109	C₈×C₁₉₇	—	2	2	no	—
110	C₂²×C₁₉₇	—	2	1	no	—
111	C₂²×C₁₉₇	—	2	1	no	—
112	C₂²×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₉₇	—	2	2	no	—
115	C₂×D₁₉₇	—	4	1	no	—
116	C₂×D₁₉₇	—	4	1	no	—
117	C₄×C₁₉₇	—	2	1	no	—
118	C₄×C₁₉₇	—	2	1	no	—
119	C₄×C₁₉₇	—	2	1	no	—
120	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
121	C₄×C₁₉₇	#135	1	2	yes	—
122	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
123	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
124	Dic₁₉₇	—	2	2	no	—
125	Dic₁₉₇	—	4	1	no	—
126	Dic₁₉₇	—	4	2	no	—
127	Dic₁₉₇	—	4	1	no	—
128	Dic₁₉₇	—	4	2	no	—
129	C₂×C₁₉₇	—	2	2	no	—
130	C₂×C₁₉₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
132	C₂×C₁₉₇	#575	1	1	yes	characteristic
133	D₁₉₇	—	4	2	no	—
134	C₁₉₇	#3076	1	1	yes	characteristic, complemented, Sylow
135	#3076	—	197	1	no	maximal, Sylow
136	#533	—	197	1	no	—
137	C₄×#41	—	197	1	no	—
138	#479	—	197	2	no	—
139	#478	—	197	2	no	—
140	#477	—	197	1	no	—
141	#476	—	197	1	no	—
142	#475	—	197	1	no	—
143	#474	—	197	1	no	—
144	#427	—	197	2	no	—
145	C₂×C₄⋊C₈	D₁₉₇	1	1	yes	characteristic, complemented
146	#416	—	197	2	no	—
147	C₂×#41	—	197	1	no	—
148	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
149	C₄⋊Q₈	—	197	1	no	—
150	C₄⋊Q₈	—	197	1	no	—
151	#125	—	197	2	no	—
152	C₂²⋊Q₈	—	394	1	no	—
153	C₂²⋊Q₈	—	394	1	no	—
154	C₄×Q₈	—	394	1	no	—
155	C₄×Q₈	—	394	1	no	—
156	C₄×Q₈	—	197	1	no	—
157	C₄×Q₈	—	197	1	no	—
158	C₄×D₄	—	394	1	no	—
159	C₄×D₄	—	394	1	no	—
160	C₄×D₄	—	197	1	no	—
161	C₄×D₄	—	197	1	no	—
162	#117	—	394	1	no	—
163	#117	—	197	1	no	—
164	#117	—	197	2	no	—
165	C₂×C₄²	—	394	1	no	—
166	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
167	#107	—	197	2	no	—
168	#107	—	197	2	no	—
169	#106	—	197	2	no	—
170	#106	—	197	2	no	—
171	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
172	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
173	C₄⋊C₈	C₂×D₁₉₇	1	2	yes	—
174	#103	—	197	2	no	—
175	#103	—	197	2	no	—
176	#102	—	197	2	no	—
177	#102	—	197	2	no	—
178	#41	—	394	2	no	—
179	#41	—	197	4	no	—
180	C₂×Q₈	—	394	1	no	—
181	C₂×Q₈	—	394	1	no	—
182	C₂×Q₈	—	197	1	no	—
183	C₂×D₄	—	394	1	no	—
184	C₂×D₄	—	197	1	no	—
185	C₂²×C₄	—	394	1	no	—
186	C₂²×C₄	—	394	1	no	—
187	C₂²×C₄	—	394	1	no	—
188	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
189	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
193	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
194	C₄⋊C₄	—	394	1	no	—
195	C₄⋊C₄	—	394	1	no	—
196	C₄⋊C₄	—	394	2	no	—
197	C₄⋊C₄	—	394	2	no	—
198	C₄⋊C₄	—	394	1	no	—
199	C₄⋊C₄	—	394	1	no	—
200	C₄⋊C₄	—	197	2	no	—
201	C₄⋊C₄	—	197	2	no	—
202	C₄⋊C₄	—	197	1	no	—
203	C₄⋊C₄	—	197	1	no	—
204	C₂²⋊C₄	—	394	1	no	—
205	C₂²⋊C₄	—	394	2	no	—
206	C₂²⋊C₄	—	394	1	no	—
207	C₂²⋊C₄	—	394	1	no	—
208	C₄²	—	394	1	no	—
209	C₄²	—	394	1	no	—
210	C₄²	—	394	1	no	—
211	C₄²	—	394	2	no	—
212	C₄²	D₇₈₈	1	1	yes	characteristic
213	C₄²	D₇₈₈	1	1	yes	characteristic
214	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
215	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
216	C₂³	—	394	1	no	—
217	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
218	Q₈	—	788	2	no	—
219	Q₈	—	788	2	no	—
220	Q₈	—	394	1	no	—
221	Q₈	—	197	2	no	—
222	D₄	—	394	4	no	—
223	D₄	—	394	1	no	—
224	D₄	—	197	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	788	2	no	—
228	C₂×C₄	—	788	2	no	—
229	C₂×C₄	—	394	1	no	—
230	C₂×C₄	—	394	1	no	—
231	C₂×C₄	—	394	2	no	—
232	C₂×C₄	—	394	2	no	—
233	C₂×C₄	—	394	1	no	—
234	C₂×C₄	—	394	4	no	—
235	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
236	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
237	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
238	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
239	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
241	C₂×C₄	#15691357	1	2	yes	—
242	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	788	1	no	—
248	C₂²	—	788	1	no	—
249	C₂²	—	394	2	no	—
250	C₂²	#20263306	1	2	yes	—
251	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	788	1	no	—
256	C₄	—	788	2	no	—
257	C₄	—	788	1	no	—
258	C₄	—	788	2	no	—
259	C₄	—	394	2	no	—
260	C₄	#20263306	1	2	yes	—
261	C₄	#20263348	1	2	yes	—
262	C₄	C₂×Q₈×D₁₉₇	1	1	yes	characteristic
263	C₄	C₂×#15691357	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	788	2	no	—
266	C₂	C₂×#20263306	1	1	yes	characteristic
267	C₂	C₂×#20263348	1	1	yes	characteristic
268	C₂	#27161879	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₈₈
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×C₄⋊C₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂×#15691358	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₉₇
hypocenter	C₁₉₇	#3076
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #27161636	C₂	semidirect
3. #27161387	C₂	semidirect
4. #27161380	C₂	semidirect
5. #27161375	C₂	semidirect
6. #27161370	C₂	semidirect
7. #27161365	C₂	semidirect
8. #27161360	C₂	semidirect
9. #27161117	C₂	semidirect
11. C₂×C₄⋊C₈×C₁₉₇	C₂	semidirect
44. #15691358	Q₈	semidirect
134. C₁₉₇	#3076	semidirect
145. C₂×C₄⋊C₈	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161636	2
C₄×#15691358	2
#27161387	2
#27161380	2
#27161375	2
#27161370	2
#27161365	2
#27161360	2
#27161117	2
#27161076	2
C₂×C₄⋊C₈×C₁₉₇	2
#3076	197