Subgroups — #38744505

Order 25216.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263265	C₂	1	1	yes	maximal, characteristic
2	#108×D₁₉₇	C₂	1	2	yes	maximal
3	#27160909	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160908	C₂	1	2	yes	maximal, complemented
5	#368×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×D₁₉₇	—	2	2	no	—
7	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
8	#20263265	C₄	1	4	yes	—
9	C₂×#15691333	C₂²	1	1	yes	characteristic
10	#20263262	C₂²	1	2	yes	—
11	#20263261	—	2	2	no	—
12	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
13	#108×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₉₇	D₄	1	1	yes	characteristic
15	#15691342	—	2	2	no	—
16	#15691341	—	4	2	no	—
17	C₂×#13096802	D₄	1	1	yes	characteristic
18	C₈×D₁₉₇	—	4	2	no	—
19	C₈×D₁₉₇	C₂×C₄	1	2	yes	—
20	C₈×D₁₉₇	Q₈	1	2	yes	—
21	#15691333	C₂×C₄	1	2	yes	—
22	M₁₆×C₁₉₇	—	2	2	no	—
23	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₉₇	C₂×C₄	1	2	yes	—
25	C₂²×D₁₉₇	Q₁₆	1	1	yes	characteristic
26	C₂×Dic₁₉₇	Q₁₆	1	1	yes	characteristic
27	C₄×D₁₉₇	C₄⋊C₄	1	2	yes	—
28	C₄×D₁₉₇	D₈	1	2	yes	—
29	#13096802	—	4	2	no	—
30	#13096802	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₉₇	—	4	2	no	—
33	C₈×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×D₁₉₇	—	4	1	no	—
37	C₂×D₁₉₇	#107	1	2	yes	—
38	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
40	Dic₁₉₇	#107	1	2	yes	—
41	C₂×C₁₉₇	—	2	1	no	—
42	C₂×C₁₉₇	C₂×#107	1	1	yes	characteristic
43	D₁₉₇	—	2	2	no	—
44	C₁₉₇	#2161	1	1	yes	characteristic, complemented, Sylow
45	#2161	—	197	1	no	maximal, Sylow
46	C₂×M₃₂	—	197	1	no	—
47	C₂×#108	—	197	2	no	—
48	#368	—	197	1	no	—
49	#368	—	197	2	no	—
50	#368	D₁₉₇	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	394	2	no	—
52	C₂²×C₈	—	197	1	no	—
53	M₃₂	—	197	4	no	—
54	C₂×C₁₆	—	197	1	no	—
55	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
56	#108	—	394	2	no	—
57	#108	—	197	2	no	—
58	#108	C₂×D₁₉₇	1	2	yes	—
59	C₂²×C₄	—	197	1	no	—
60	M₁₆	—	2	2	no	—
61	M₁₆	—	788	2	no	—
62	M₁₆	—	394	2	no	—
63	C₂×C₈	—	788	2	no	—
64	C₂×C₈	—	197	1	no	—
65	C₂×C₈	—	197	2	no	—
66	C₂×C₈	—	197	2	no	—
67	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
68	C₁₆	—	197	2	no	—
69	C₁₆	C₄×D₁₉₇	1	2	yes	—
70	C₂³	—	197	1	no	—
71	C₂×C₄	—	197	1	no	—
72	C₂×C₄	—	197	2	no	—
73	C₂×C₄	—	197	2	no	—
74	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
75	C₈	—	4	2	no	—
76	C₈	—	788	2	no	—
77	C₈	—	197	2	no	—
78	C₈	C₂×C₄×D₁₉₇	1	1	yes	characteristic
79	C₈	Q₈×D₁₉₇	1	1	yes	characteristic
80	C₂²	—	788	1	no	—
81	C₂²	—	197	2	no	—
82	C₂²	Q₁₆×D₁₉₇	1	1	yes	characteristic
83	C₄	—	197	2	no	—
84	C₄	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
85	C₄	D₈×D₁₉₇	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	394	2	no	—
88	C₂	#107×D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₉₇
commutator	C₈×C₁₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#368×C₁₉₇	C₂
socle	C₂×C₁₉₇	C₂×#107
prime radical	C₂²×D₁₉₇	Q₁₆
cosocle	C₂×C₈×C₁₉₇	C₂³
hypercenter	#368	D₁₉₇
hypocenter	C₁₉₇	#2161
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
3. #27160909	C₂	semidirect
4. #27160908	C₂	semidirect
5. #368×C₁₉₇	C₂	semidirect
44. C₁₉₇	#2161	semidirect
50. #368	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263265	2
#108×D₁₉₇	2
#27160909	2
#27160908	2
#368×C₁₉₇	2
#2161	197