Subgroups — #27161220

Order 12608.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15691347	C₂	1	2	yes	maximal, complemented
3	#20263300	C₂	1	1	yes	maximal, characteristic
4	#20263251	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₄	—	2	4	no	—
7	C₂×Dic₃₉₄	—	2	1	no	—
8	C₂×Dic₃₉₄	C₂²	1	2	yes	—
9	C₂×D₇₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
11	C₂²×Dic₁₉₇	—	2	1	no	—
12	#15691347	—	2	4	no	—
13	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
14	#15691335	—	2	1	no	—
15	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₉₇	C₂²	1	2	yes	—
17	C₂²×D₁₉₇	—	2	1	no	—
18	Dic₃₉₄	—	2	4	no	—
19	Dic₃₉₄	—	2	2	no	—
20	Dic₃₉₄	D₄	1	4	yes	complemented
21	D₇₈₈	—	2	2	no	—
22	C₂²⋊D₁₉₇	—	4	2	no	—
23	C₂×Dic₁₉₇	—	2	4	no	—
24	C₂×Dic₁₉₇	—	2	2	no	—
25	C₂×Dic₁₉₇	—	2	1	no	—
26	C₂³×C₁₉₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₇	—	2	2	no	—
28	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
30	C₈×C₁₉₇	—	2	2	no	—
31	C₂²×C₁₉₇	—	2	1	no	—
32	C₂²×C₁₉₇	—	2	1	no	—
33	C₂²×C₁₉₇	SD₁₆	1	2	yes	complemented
34	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₉₇	—	4	1	no	—
36	C₂×D₁₉₇	—	4	1	no	—
37	C₂×D₁₉₇	—	4	1	no	—
38	C₄×C₁₉₇	—	2	1	no	—
39	C₄×C₁₉₇	C₂×D₄	1	2	yes	—
40	Dic₁₉₇	—	2	4	no	—
41	Dic₁₉₇	—	4	1	no	—
42	C₂×C₁₉₇	—	2	2	no	—
43	C₂×C₁₉₇	#120	1	1	yes	characteristic
44	C₂×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
45	C₂×C₁₉₇	#137	1	1	yes	characteristic
46	D₁₉₇	—	8	1	no	—
47	C₁₉₇	#448	1	1	yes	characteristic, complemented, Sylow
48	#448	—	197	1	no	maximal, Sylow
49	C₂²×Q₈	—	197	1	no	—
50	C₂×SD₁₆	—	197	2	no	—
51	#121	—	197	1	no	—
52	#103	—	197	2	no	—
53	C₂²⋊C₈	D₁₉₇	1	1	yes	characteristic, complemented
54	C₂×Q₈	—	394	4	no	—
55	C₂×Q₈	—	394	1	no	—
56	C₂×Q₈	—	197	2	no	—
57	C₂×D₄	—	394	1	no	—
58	C₂×D₄	—	197	1	no	—
59	C₂²×C₄	—	394	1	no	—
60	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
61	SD₁₆	—	394	4	no	—
62	C₂×C₈	C₂×D₁₉₇	1	2	yes	—
63	C₄⋊C₄	—	197	1	no	—
64	C₂²⋊C₄	—	394	1	no	—
65	C₂³	—	394	1	no	—
66	C₂³	D₇₈₈	1	1	yes	characteristic
67	Q₈	—	394	4	no	—
68	Q₈	—	394	2	no	—
69	Q₈	—	197	4	no	—
70	D₄	—	788	2	no	—
71	D₄	—	394	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	394	1	no	—
74	C₂×C₄	—	394	4	no	—
75	C₂×C₄	—	394	2	no	—
76	C₂×C₄	D₇₈₈	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	788	1	no	—
82	C₂²	—	788	1	no	—
83	C₂²	—	788	1	no	—
84	C₂²	#15691347	1	2	yes	complemented
85	C₂²	C₂×D₇₈₈	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	788	1	no	—
88	C₄	—	394	4	no	—
89	C₄	D₄×D₁₉₇	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	1576	1	no	—
92	C₂	#20263296	1	1	yes	characteristic
93	C₂	C₂×#15691347	1	1	yes	characteristic
94	C₂	#20263359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₈₈
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₂²⋊C₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₄
prime radical	#20263300	C₂
hypercenter	C₂²⋊C₈	D₁₉₇
hypocenter	C₁₉₇	#448
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₃₉₄	C₂	semidirect
2. C₂×#15691347	C₂	semidirect
4. #20263251	C₂	semidirect
5. C₂²⋊C₈×C₁₉₇	C₂	semidirect
20. Dic₃₉₄	D₄	semidirect
33. C₂²×C₁₉₇	SD₁₆	semidirect
47. C₁₉₇	#448	semidirect
53. C₂²⋊C₈	D₁₉₇	semidirect
84. C₂²	#15691347	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₉₄	2
C₂×#15691347	2
#20263300	2
#20263251	2
C₂²⋊C₈×C₁₉₇	2
#448	197