Subgroups — #38740744

Order 25216.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161444	C₂	1	1	yes	maximal, characteristic
2	#27160857	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263323	C₄	1	1	yes	characteristic
5	#20263318	C₄	1	1	yes	characteristic
6	#20263266	—	2	1	no	—
7	#20263248	—	2	1	no	—
8	#20263248	—	2	1	no	—
9	M₃₂×C₁₉₇	—	2	1	no	—
10	#97×C₁₉₇	C₂²	1	1	yes	characteristic
11	C₂×D₇₈₈	—	2	1	no	—
12	C₂×D₇₈₈	—	4	1	no	—
13	C₄⋊Dic₁₉₇	—	2	1	no	—
14	#15691337	—	4	1	no	—
15	#15691333	—	2	2	no	—
16	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
18	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₉₇	—	2	2	no	—
20	C₂²×D₁₉₇	—	4	1	no	—
21	D₇₈₈	—	4	1	no	—
22	D₇₈₈	—	4	1	no	—
23	D₇₈₈	—	8	2	no	—
24	C₂×Dic₁₉₇	—	4	1	no	—
25	C₂×C₄×C₁₉₇	—	2	1	no	—
26	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₉₇	—	2	1	no	—
28	C₈×C₁₉₇	—	2	1	no	—
29	C₂²×C₁₉₇	#100	1	1	yes	characteristic
30	C₂×D₁₉₇	—	8	2	no	—
31	C₂×D₁₉₇	—	8	1	no	—
32	C₄×C₁₉₇	—	4	1	no	—
33	C₄×C₁₉₇	#104	1	1	yes	characteristic
34	C₄×C₁₉₇	#104	1	1	yes	characteristic
35	Dic₁₉₇	—	8	1	no	—
36	C₂×C₁₉₇	—	2	1	no	—
37	C₂×C₁₉₇	#329	1	1	yes	characteristic
38	D₁₉₇	—	16	1	no	—
39	C₁₉₇	#1355	1	1	yes	characteristic, complemented, Sylow
40	#1355	—	197	1	no	maximal, Sylow
41	#490	—	197	1	no	—
42	#347	—	197	1	no	—
43	#347	D₁₉₇	1	1	yes	characteristic, complemented
44	#127	—	197	1	no	—
45	#125	—	197	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	394	1	no	—
48	#102	—	394	1	no	—
49	#102	—	394	1	no	—
50	#97	C₂×D₁₉₇	1	1	yes	characteristic
51	C₂×D₄	—	788	1	no	—
52	C₂×D₄	—	394	1	no	—
53	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
54	C₂×C₈	D₇₈₈	1	1	yes	characteristic
55	C₄⋊C₄	—	788	1	no	—
56	C₄⋊C₄	—	394	1	no	—
57	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	394	2	no	—
60	C₂³	—	788	1	no	—
61	D₄	—	788	1	no	—
62	D₄	—	788	1	no	—
63	D₄	—	1576	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	788	1	no	—
66	C₂×C₄	#15691335	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	1576	2	no	—
70	C₂²	—	1576	1	no	—
71	C₂²	#20263242	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	1576	1	no	—
74	C₄	#20263252	1	1	yes	characteristic
75	C₄	#20263254	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	3152	1	no	—
78	C₂	#27160818	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27160818
commutator	C₄²×C₁₉₇	C₂×C₄
Frattini	#97	C₂×D₁₉₇
Fitting	#347×C₁₉₇	C₂
socle	C₂×C₁₉₇	#329
prime radical	#20263323	C₄
cosocle	#97×C₁₉₇	C₂²
hypercenter	#347	D₁₉₇
hypocenter	C₁₉₇	#1355
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #27160857	C₂	semidirect
3. #347×C₁₉₇	C₂	semidirect
39. C₁₉₇	#1355	semidirect
43. #347	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161444	2
#27160857	2
#347×C₁₉₇	2
#1355	197