Subgroups — #38740684

Order 25216.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161197	C₂	1	1	yes	maximal, characteristic
2	C₈×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263348	C₄	1	1	yes	characteristic
5	#20263332	—	2	1	no	—
6	#20263262	C₄	1	1	yes	characteristic
7	#20263254	—	2	1	no	—
8	C₄×#13096802	—	2	1	no	—
9	M₃₂×C₁₉₇	—	2	1	no	—
10	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
11	#15691358	—	2	1	no	—
12	Dic₇₈₈	C₈	1	1	yes	characteristic, complemented
13	#15691347	—	2	1	no	—
14	D₁₅₇₆	C₈	1	1	yes	characteristic, complemented
15	#15691342	—	2	1	no	—
16	#15691341	—	4	1	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₈×D₁₉₇	—	4	1	no	—
19	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
21	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₉₇	—	4	1	no	—
23	Dic₃₉₄	—	2	1	no	—
24	D₇₈₈	—	2	1	no	—
25	C₂²⋊D₁₉₇	—	4	1	no	—
26	C₄×D₁₉₇	—	4	1	no	—
27	#13096802	—	2	4	no	—
28	#13096802	—	4	1	no	—
29	C₂×C₄×C₁₉₇	—	2	1	no	—
30	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₉₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₉₇	M₁₆	1	1	yes	characteristic
33	C₈×C₁₉₇	D₈	1	1	yes	characteristic
34	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₉₇	#104	1	1	yes	characteristic
36	C₂×D₁₉₇	—	4	1	no	—
37	C₄×C₁₉₇	—	2	2	no	—
38	C₄×C₁₉₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₉₇	#102	1	1	yes	characteristic
40	Dic₁₉₇	—	4	1	no	—
41	C₂×C₁₉₇	—	2	1	no	—
42	C₂×C₁₉₇	D₄⋊C₈	1	1	yes	characteristic
43	D₁₉₇	—	8	1	no	—
44	C₁₉₇	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	197	1	no	maximal, Sylow
46	#443	—	197	1	no	—
47	#364	D₁₉₇	1	1	yes	characteristic, complemented
48	C₈²	—	197	1	no	—
49	#135	—	197	1	no	—
50	#131	—	394	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	197	1	no	—
53	#104	—	394	1	no	—
54	C₄×C₈	—	394	1	no	—
55	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
56	#41	—	394	1	no	—
57	Q₁₆	—	197	1	no	—
58	SD₁₆	—	394	1	no	—
59	D₈	—	197	1	no	—
60	M₁₆	—	788	1	no	—
61	M₁₆	—	394	1	no	—
62	C₂×C₈	—	788	1	no	—
63	C₂×C₈	—	394	2	no	—
64	C₂×C₈	C₄×D₁₉₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
66	C₄²	D₇₈₈	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	394	1	no	—
69	D₄	—	788	1	no	—
70	D₄	—	394	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	788	1	no	—
73	C₂×C₄	#15691335	1	1	yes	characteristic
74	C₈	—	788	1	no	—
75	C₈	—	394	4	no	—
76	C₈	C₈×D₁₉₇	1	1	yes	characteristic
77	C₈	#15691341	1	1	yes	characteristic
78	C₈	#15691343	1	1	yes	characteristic
79	C₈	#15691345	1	1	yes	characteristic
80	C₂²	—	788	1	no	—
81	C₂²	#20263252	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	788	1	no	—
84	C₄	#20263238	1	1	yes	characteristic
85	C₄	#20263246	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1576	1	no	—
88	C₂	#27160806	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15691343
commutator	C₈×C₁₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	#364×C₁₉₇	C₂
socle	C₂×C₁₉₇	D₄⋊C₈
prime radical	#20263348	C₄
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	#364	D₁₉₇
hypocenter	C₁₉₇	#1335
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×#13096802	C₂	semidirect
3. #364×C₁₉₇	C₂	semidirect
12. Dic₇₈₈	C₈	semidirect
14. D₁₅₇₆	C₈	semidirect
44. C₁₉₇	#1335	semidirect
47. #364	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161197	2
C₈×#13096802	2
#364×C₁₉₇	2
#1335	197