Subgroups — #38753282

Order 25216.

380 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161613	C₂	1	2	yes	maximal, complemented
2	#27161481	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161463	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161434	C₂	1	2	yes	maximal, complemented
5	#27161320	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161315	C₂	1	1	yes	maximal, characteristic
7	#27161308	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₇₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15691347	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₅₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₇₈₈	C₂²	1	1	yes	characteristic
13	C₂×#15691347	C₂²	1	2	yes	—
14	C₂×D₁₅₇₆	C₂²	1	1	yes	characteristic
15	#20263320	—	2	2	no	—
16	#20263318	C₂²	1	2	yes	—
17	#20263314	C₂²	1	2	yes	—
18	#20263309	—	2	2	no	—
19	#20263305	C₂²	1	2	yes	—
20	#20263300	C₂²	1	2	yes	—
21	C₄×Dic₃₉₄	C₂²	1	2	yes	—
22	C₄×D₇₈₈	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₁₉₇	—	2	2	no	—
24	#20263283	—	2	2	no	—
25	#20263260	C₂²	1	1	yes	characteristic
26	#20263260	C₂²	1	1	yes	characteristic
27	#20263258	C₂²	1	2	yes	—
28	#20263251	C₂²	1	2	yes	—
29	#20263251	C₂²	1	2	yes	—
30	#20263248	C₂²	1	2	yes	—
31	#20263248	C₂²	1	2	yes	—
32	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₉₄	C₂³	1	2	yes	—
40	C₂×D₇₈₈	C₂³	1	2	yes	—
41	C₂×C₂²⋊D₁₉₇	—	2	2	no	—
42	C₂×C₄×D₁₉₇	—	2	2	no	—
43	Dic₇₈₈	—	2	2	no	—
44	#15691347	—	2	4	no	—
45	D₁₅₇₆	—	2	2	no	—
46	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
47	C₄⋊Dic₁₉₇	C₂³	1	2	yes	—
48	#15691337	—	2	2	no	—
49	#15691337	—	2	2	no	—
50	#15691337	—	2	2	no	—
51	#15691337	—	2	2	no	—
52	C₂²⋊Dic₁₉₇	—	2	2	no	—
53	#15691335	—	2	2	no	—
54	#15691335	—	2	2	no	—
55	#15691335	—	2	2	no	—
56	#15691335	—	2	2	no	—
57	C₄×Dic₁₉₇	—	2	2	no	—
58	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₉₇	—	2	2	no	—
62	C₂×C₈×C₁₉₇	—	2	2	no	—
63	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
67	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
68	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
69	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
71	C₂²×D₁₉₇	—	2	2	no	—
72	Dic₃₉₄	—	2	2	no	—
73	Dic₃₉₄	—	2	2	no	—
74	D₇₈₈	—	2	2	no	—
75	D₇₈₈	—	2	2	no	—
76	C₂²⋊D₁₉₇	—	4	4	no	—
77	C₂×Dic₁₉₇	—	2	2	no	—
78	C₂×Dic₁₉₇	—	2	2	no	—
79	C₂×Dic₁₉₇	—	2	2	no	—
80	C₄×D₁₉₇	—	4	4	no	—
81	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉₇	—	2	2	no	—
83	C₂×C₄×C₁₉₇	—	2	2	no	—
84	C₂×C₄×C₁₉₇	—	2	1	no	—
85	C₂×C₄×C₁₉₇	—	2	1	no	—
86	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₉₇	—	2	1	no	—
93	C₈×C₁₉₇	—	2	1	no	—
94	C₈×C₁₉₇	#41	1	2	yes	—
95	C₈×C₁₉₇	#41	1	2	yes	—
96	C₂²×C₁₉₇	—	2	1	no	—
97	C₂²×C₁₉₇	—	2	1	no	—
98	C₂²×C₁₉₇	#135	1	2	yes	—
99	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₁₉₇	—	4	2	no	—
101	C₂×D₁₉₇	—	4	2	no	—
102	C₂×D₁₉₇	—	4	2	no	—
103	C₄×C₁₉₇	—	2	1	no	—
104	C₄×C₁₉₇	—	2	1	no	—
105	C₄×C₁₉₇	—	2	1	no	—
106	C₄×C₁₉₇	—	2	1	no	—
107	C₄×C₁₉₇	#135	1	2	yes	—
108	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
109	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
110	Dic₁₉₇	—	4	2	no	—
111	Dic₁₉₇	—	4	2	no	—
112	Dic₁₉₇	—	4	2	no	—
113	C₂×C₁₉₇	—	2	2	no	—
114	C₂×C₁₉₇	#532	1	1	yes	characteristic
115	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
116	C₂×C₁₉₇	C₂×#135	1	1	yes	characteristic
117	D₁₉₇	—	8	2	no	—
118	C₁₉₇	#3168	1	1	yes	characteristic, complemented, Sylow
119	#3168	—	197	1	no	maximal, Sylow
120	#529	—	197	2	no	—
121	#499	—	197	1	no	—
122	#495	—	197	1	no	—
123	#488	—	197	2	no	—
124	#467	—	197	1	no	—
125	#466	—	197	1	no	—
126	#465	—	197	2	no	—
127	C₄×Q₁₆	—	197	1	no	—
128	C₄×SD₁₆	—	197	2	no	—
129	C₄×D₈	—	197	1	no	—
130	C₂×C₄×C₈	D₁₉₇	1	1	yes	characteristic, complemented
131	C₂×Q₁₆	—	197	1	no	—
132	C₂×SD₁₆	—	197	2	no	—
133	C₂×D₈	—	197	1	no	—
134	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
136	#126	—	394	2	no	—
137	#125	—	197	2	no	—
138	#124	—	197	2	no	—
139	#123	—	394	2	no	—
140	C₂²⋊Q₈	—	197	2	no	—
141	#121	—	197	2	no	—
142	C₄×Q₈	—	197	2	no	—
143	C₄×D₄	—	394	2	no	—
144	C₄×D₄	—	197	2	no	—
145	#117	—	394	2	no	—
146	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
147	#107	—	197	1	no	—
148	#107	—	197	1	no	—
149	#106	—	197	2	no	—
150	#103	—	197	2	no	—
151	#103	—	197	2	no	—
152	#102	—	197	2	no	—
153	#102	—	197	2	no	—
154	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
158	C₂×Q₈	—	197	2	no	—
159	C₂×D₄	—	394	2	no	—
160	C₂×D₄	—	197	2	no	—
161	C₂²×C₄	—	394	2	no	—
162	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
163	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
165	Q₁₆	—	394	2	no	—
166	SD₁₆	—	394	4	no	—
167	D₈	—	394	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
174	C₄⋊C₄	—	394	2	no	—
175	C₄⋊C₄	—	394	2	no	—
176	C₄⋊C₄	—	394	2	no	—
177	C₄⋊C₄	—	394	2	no	—
178	C₄⋊C₄	—	197	2	no	—
179	C₄⋊C₄	—	197	2	no	—
180	C₂²⋊C₄	—	394	2	no	—
181	C₂²⋊C₄	—	394	2	no	—
182	C₂²⋊C₄	—	394	2	no	—
183	C₂²⋊C₄	—	394	2	no	—
184	C₂²⋊C₄	—	394	2	no	—
185	C₄²	—	394	2	no	—
186	C₄²	D₇₈₈	1	1	yes	characteristic
187	C₄²	D₇₈₈	1	1	yes	characteristic
188	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
189	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
190	C₂³	—	394	2	no	—
191	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
192	Q₈	—	394	2	no	—
193	Q₈	—	394	2	no	—
194	D₄	—	788	4	no	—
195	D₄	—	394	2	no	—
196	D₄	—	394	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	788	4	no	—
202	C₂×C₄	—	394	2	no	—
203	C₂×C₄	—	394	2	no	—
204	C₂×C₄	—	394	2	no	—
205	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
206	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
207	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#15691358	1	2	yes	—
214	C₈	#15691358	1	2	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	788	2	no	—
218	C₂²	—	788	2	no	—
219	C₂²	—	788	2	no	—
220	C₂²	#20263348	1	2	yes	—
221	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	788	2	no	—
227	C₄	—	788	2	no	—
228	C₄	—	788	2	no	—
229	C₄	#20263348	1	2	yes	—
230	C₄	C₂×#15691358	1	1	yes	characteristic
231	C₄	C₂×#15691358	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	1576	2	no	—
234	C₂	#27161629	1	1	yes	characteristic
235	C₂	C₂×#20263348	1	1	yes	characteristic
236	C₂	C₂×#20263348	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	#27161315	C₂
hypercenter	C₂×C₄×C₈	D₁₉₇
hypocenter	C₁₉₇	#3168
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. #27161613	C₂	semidirect
2. #27161481	C₂	semidirect
3. #27161463	C₂	semidirect
4. #27161434	C₂	semidirect
5. #27161320	C₂	semidirect
7. #27161308	C₂	semidirect
8. C₄×Dic₇₈₈	C₂	semidirect
9. C₄×#15691347	C₂	semidirect
10. C₄×D₁₅₇₆	C₂	semidirect
11. C₂×C₄×C₈×C₁₉₇	C₂	semidirect
118. C₁₉₇	#3168	semidirect
130. 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
#27161613	2
#27161481	2
#27161463	2
#27161434	2
#27161320	2
#27161315	2
#27161308	2
C₄×Dic₇₈₈	2
C₄×#15691347	2
C₄×D₁₅₇₆	2
C₂×C₄×C₈×C₁₉₇	2
#3168	197