Subgroups — #38745612

Order 25216.

776 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263288	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄×D₁₉₇	C₂	1	2	yes	maximal, complemented
4	#27160975	C₂	1	2	yes	maximal, complemented
5	#27160950	C₂	1	4	yes	maximal, complemented
6	#27160936	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#384×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₉₇	C₂²	1	4	yes	—
11	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
12	C₄×D₇₈₈	C₄	1	8	yes	complemented
13	#20263288	—	2	8	no	—
14	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×#15691337	C₂²	1	4	yes	—
18	C₄⋊C₄×D₁₉₇	—	2	8	no	—
19	C₂×#15691335	C₂²	1	4	yes	—
20	C₂×#15691335	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
22	#20263233	C₂²	1	2	yes	—
23	#20263233	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
28	C₂³×D₁₉₇	C₂³	1	2	yes	—
29	C₂×D₇₈₈	C₂×C₄	1	4	yes	—
30	C₂×D₇₈₈	D₄	1	4	yes	—
31	C₂×D₇₈₈	Q₈	1	4	yes	—
32	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
33	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
34	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
35	C₂×C₄×D₁₉₇	—	2	8	no	—
36	C₂×C₄×D₁₉₇	—	2	8	no	—
37	C₂×C₄×D₁₉₇	—	2	4	no	—
38	C₂×C₄×D₁₉₇	—	2	4	no	—
39	C₂×C₄×D₁₉₇	—	4	4	no	—
40	C₂×C₄×D₁₉₇	C₂×C₄	1	8	yes	complemented
41	C₄⋊Dic₁₉₇	—	2	2	no	—
42	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	complemented
43	C₄⋊Dic₁₉₇	D₄	1	4	yes	complemented
44	#15691337	—	2	8	no	—
45	#15691335	—	2	8	no	—
46	#15691335	C₂×C₄	1	8	yes	complemented
47	C₄×Dic₁₉₇	—	2	4	no	—
48	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₉₇	—	2	4	no	—
53	C₄⋊C₄×C₁₉₇	—	2	4	no	—
54	C₄⋊C₄×C₁₉₇	—	4	2	no	—
55	C₄²×C₁₉₇	C₂×C₄	1	4	yes	complemented
56	C₂²×D₁₉₇	—	2	2	no	—
57	C₂²×D₁₉₇	—	2	2	no	—
58	C₂²×D₁₉₇	—	2	2	no	—
59	C₂²×D₁₉₇	—	2	2	no	—
60	C₂²×D₁₉₇	C₂²×C₄	1	4	yes	—
61	C₂²×D₁₉₇	C₂×D₄	1	4	yes	—
62	C₂²×D₁₉₇	C₂×Q₈	1	4	yes	—
63	D₇₈₈	C₄⋊C₄	1	16	yes	complemented
64	C₂×Dic₁₉₇	—	2	2	no	—
65	C₂×Dic₁₉₇	—	2	2	no	—
66	C₂×Dic₁₉₇	—	2	2	no	—
67	C₂×Dic₁₉₇	—	2	2	no	—
68	C₂×Dic₁₉₇	—	2	2	no	—
69	C₂×Dic₁₉₇	—	2	2	no	—
70	C₂×Dic₁₉₇	—	2	2	no	—
71	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₉₇	C₂×D₄	1	4	yes	—
73	C₄×D₁₉₇	—	2	16	no	—
74	C₄×D₁₉₇	—	4	16	no	—
75	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₄×C₁₉₇	—	2	1	no	—
77	C₂×C₄×C₁₉₇	—	2	1	no	—
78	C₂×C₄×C₁₉₇	—	2	4	no	—
79	C₂×C₄×C₁₉₇	—	2	4	no	—
80	C₂×C₄×C₁₉₇	—	2	4	no	—
81	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₁₉₇	#41	1	2	yes	—
88	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
89	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
93	C₂²×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
95	C₂×D₁₉₇	—	2	4	no	—
96	C₂×D₁₉₇	—	2	4	no	—
97	C₂×D₁₉₇	—	2	4	no	—
98	C₂×D₁₉₇	—	2	4	no	—
99	C₂×D₁₉₇	—	2	4	no	—
100	C₂×D₁₉₇	—	2	4	no	—
101	C₂×D₁₉₇	C₂×C₄⋊C₄	1	8	yes	complemented
102	C₄×C₁₉₇	—	2	2	no	—
103	C₄×C₁₉₇	—	4	4	no	—
104	C₄×C₁₉₇	C₂×C₄⋊C₄	1	4	yes	complemented
105	C₄×C₁₉₇	C₄×D₄	1	4	yes	complemented
106	Dic₁₉₇	—	2	4	no	—
107	Dic₁₉₇	—	2	4	no	—
108	Dic₁₉₇	—	4	2	no	—
109	C₂×C₁₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₁₉₇	#520	1	1	yes	characteristic
112	C₂×C₁₉₇	D₄²	1	1	yes	characteristic
113	C₂×C₁₉₇	#547	1	1	yes	characteristic
114	C₂×C₁₉₇	D₄×Q₈	1	1	yes	characteristic
115	C₂×C₁₉₇	#554	1	1	yes	characteristic
116	D₁₉₇	—	2	8	no	—
117	C₁₉₇	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
118	D₄×C₄⋊C₄	—	197	1	no	maximal, Sylow
119	C₂×C₄×D₄	—	197	2	no	—
120	C₂×C₄×D₄	—	197	1	no	—
121	C₂²×C₄⋊C₄	—	197	2	no	—
122	#389	—	197	2	no	—
123	#385	—	197	4	no	—
124	#384	D₁₉₇	1	1	yes	characteristic, complemented
125	#380	—	197	2	no	—
126	C₄×C₄⋊C₄	—	197	1	no	—
127	C₂²×D₄	—	197	1	no	—
128	C₂³×C₄	—	197	4	no	—
129	C₂³×C₄	—	197	2	no	—
130	C₄×D₄	—	394	8	no	—
131	C₄×D₄	—	197	8	no	—
132	C₂×C₄⋊C₄	—	394	8	no	—
133	C₂×C₄⋊C₄	—	197	4	no	—
134	C₂×C₄⋊C₄	—	197	1	no	—
135	C₂×C₄⋊C₄	—	197	1	no	—
136	C₂×C₄⋊C₄	—	197	1	no	—
137	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
140	C₂×C₂²⋊C₄	—	197	4	no	—
141	C₂×C₂²⋊C₄	—	197	2	no	—
142	C₂×C₄²	—	197	2	no	—
143	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
144	#95	—	197	2	no	—
145	#95	—	197	4	no	—
146	C₂⁴	—	197	2	no	—
147	C₂×D₄	—	197	4	no	—
148	C₂×D₄	—	197	4	no	—
149	C₂×D₄	—	197	4	no	—
150	C₂²×C₄	—	788	4	no	—
151	C₂²×C₄	—	394	8	no	—
152	C₂²×C₄	—	394	8	no	—
153	C₂²×C₄	—	394	4	no	—
154	C₂²×C₄	—	394	4	no	—
155	C₂²×C₄	—	197	2	no	—
156	C₂²×C₄	—	197	2	no	—
157	C₂²×C₄	—	197	2	no	—
158	C₂²×C₄	—	197	8	no	—
159	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
160	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	394	8	no	—
167	C₄⋊C₄	—	394	2	no	—
168	C₄⋊C₄	—	197	4	no	—
169	C₄⋊C₄	—	197	4	no	—
170	C₂²⋊C₄	—	394	8	no	—
171	C₂²⋊C₄	—	197	8	no	—
172	C₄²	—	394	4	no	—
173	C₄²	C₄×D₁₉₇	1	4	yes	complemented
174	C₂³	—	394	2	no	—
175	C₂³	—	394	2	no	—
176	C₂³	—	394	2	no	—
177	C₂³	—	394	2	no	—
178	C₂³	—	197	4	no	—
179	C₂³	—	197	4	no	—
180	C₂³	—	197	4	no	—
181	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
182	D₄	—	197	16	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	788	16	no	—
189	C₂×C₄	—	394	2	no	—
190	C₂×C₄	—	394	2	no	—
191	C₂×C₄	—	394	2	no	—
192	C₂×C₄	—	394	2	no	—
193	C₂×C₄	—	394	2	no	—
194	C₂×C₄	—	394	2	no	—
195	C₂×C₄	—	394	2	no	—
196	C₂×C₄	—	394	16	no	—
197	C₂×C₄	—	197	4	no	—
198	C₂×C₄	—	197	4	no	—
199	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
201	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
202	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
203	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
204	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
205	C₂×C₄	#15691359	1	2	yes	—
206	C₂²	—	394	4	no	—
207	C₂²	—	394	4	no	—
208	C₂²	—	394	4	no	—
209	C₂²	—	394	4	no	—
210	C₂²	—	394	4	no	—
211	C₂²	—	394	4	no	—
212	C₂²	—	197	8	no	—
213	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
214	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
215	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
216	C₂²	C₂×Q₈×D₁₉₇	1	1	yes	characteristic
217	C₂²	C₂×#15691359	1	1	yes	characteristic
218	C₂²	#20263371	1	1	yes	characteristic
219	C₂²	#20263374	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	4	4	no	—
222	C₄	—	788	2	no	—
223	C₄	—	394	4	no	—
224	C₄	—	394	4	no	—
225	C₄	C₄⋊C₄×D₁₉₇	1	4	yes	complemented
226	C₄	#20263288	1	4	yes	complemented
227	C₂	—	394	8	no	—
228	C₂	C₂×C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
229	C₂	C₂×#20263288	1	1	yes	characteristic
230	C₂	#27161560	1	1	yes	characteristic
231	C₂	#27161714	1	1	yes	characteristic
232	C₂	#27161732	1	1	yes	characteristic
233	C₂	#27161744	1	1	yes	characteristic
234	C₂	#27161790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Fitting	#384×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂²×D₇₈₈	C₂²
hypercenter	#384	D₁₉₇
hypocenter	C₁₉₇	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₄×D₁₉₇	C₂	semidirect
4. #27160975	C₂	semidirect
5. #27160950	C₂	semidirect
6. #27160936	C₂	semidirect
7. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
8. #384×C₁₉₇	C₂	semidirect
12. C₄×D₇₈₈	C₄	semidirect
40. C₂×C₄×D₁₉₇	C₂×C₄	semidirect
42. C₄⋊Dic₁₉₇	C₂×C₄	semidirect
43. C₄⋊Dic₁₉₇	D₄	semidirect
46. #15691335	C₂×C₄	semidirect
55. C₄²×C₁₉₇	C₂×C₄	semidirect
63. D₇₈₈	C₄⋊C₄	semidirect
101. C₂×D₁₉₇	C₂×C₄⋊C₄	semidirect
104. C₄×C₁₉₇	C₂×C₄⋊C₄	semidirect
105. C₄×C₁₉₇	C₄×D₄	semidirect
117. C₁₉₇	D₄×C₄⋊C₄	semidirect
124. #384	D₁₉₇	semidirect
173. C₄²	C₄×D₁₉₇	semidirect
225. C₄	C₄⋊C₄×D₁₉₇	semidirect
226. C₄	#20263288	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₈₈	2
C₂×#20263288	2
C₂×C₄⋊C₄×D₁₉₇	2
#27160975	2
#27160950	2
#27160936	2
C₄×C₄⋊Dic₁₉₇	2
#384×C₁₉₇	2
D₄×C₄⋊C₄	197