Subgroups — #38740539

Order 25216.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13096802	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13096802	C₄	1	4	yes	—
4	C₄×#13096802	—	2	2	no	—
5	C₄×#13096802	—	2	2	no	—
6	C₂²×C₈×C₁₉₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₉₇	—	2	2	no	—
9	C₂×#13096802	—	2	4	no	—
10	C₂×#13096802	—	2	4	no	—
11	C₂×#13096802	C₈	1	8	yes	complemented
12	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₉₇	—	2	2	no	—
15	C₂×C₈×C₁₉₇	—	2	2	no	—
16	C₂×C₈×C₁₉₇	C₈	1	4	yes	complemented
17	C₄²×C₁₉₇	D₄	1	2	yes	—
18	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₉₇	Q₈	1	1	yes	characteristic
20	#13096802	—	2	8	no	—
21	C₂³×C₁₉₇	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₇	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₁₉₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₉₇	M₁₆	1	4	yes	—
29	C₂×C₄×C₁₉₇	M₁₆	1	2	yes	—
30	C₈×C₁₉₇	—	2	4	no	—
31	C₂²×C₁₉₇	#95	1	1	yes	characteristic
32	C₂²×C₁₉₇	C₄×C₈	1	2	yes	—
33	C₂²×C₁₉₇	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₁₉₇	#97	1	2	yes	—
35	C₂²×C₁₉₇	#97	1	1	yes	characteristic
36	C₄×C₁₉₇	C₂²⋊C₈	1	4	yes	—
37	C₄×C₁₉₇	C₂²⋊C₈	1	2	yes	—
38	C₄×C₁₉₇	C₄⋊C₈	1	4	yes	—
39	C₄×C₁₉₇	C₄⋊C₈	1	2	yes	—
40	C₂×C₁₉₇	C₈²	1	1	yes	characteristic
41	C₂×C₁₉₇	#322	1	1	yes	characteristic
42	C₂×C₁₉₇	#322	1	2	yes	—
43	C₂×C₁₉₇	#336	1	2	yes	—
44	C₂×C₁₉₇	#336	1	1	yes	characteristic
45	C₁₉₇	#1273	1	1	yes	characteristic, complemented, Sylow
46	#1273	—	197	1	no	maximal, Sylow
47	C₂×C₄×C₈	—	197	2	no	—
48	C₂×C₄×C₈	D₁₉₇	1	1	yes	characteristic
49	C₂²×C₈	—	197	4	no	—
50	C₂²×C₈	Dic₁₉₇	1	2	yes	—
51	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	394	2	no	—
54	C₄×C₈	—	394	2	no	—
55	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
56	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	394	4	no	—
60	C₂×C₈	—	394	4	no	—
61	C₂×C₈	—	197	8	no	—
62	C₂×C₈	#13096802	1	4	yes	complemented
63	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
64	C₄²	D₇₈₈	1	1	yes	characteristic
65	C₄²	Dic₃₉₄	1	1	yes	characteristic
66	C₂³	C₄×Dic₁₉₇	1	1	yes	characteristic
67	C₂×C₄	#15691335	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
69	C₂×C₄	#15691337	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
71	C₂×C₄	C₈×D₁₉₇	1	4	yes	—
72	C₂×C₄	C₂×#13096802	1	2	yes	—
73	C₂×C₄	#15691341	1	4	yes	—
74	C₂×C₄	#15691342	1	2	yes	—
75	C₈	—	2	4	no	—
76	C₈	—	394	8	no	—
77	C₂²	#20263233	1	1	yes	characteristic
78	C₂²	C₈×Dic₁₉₇	1	2	yes	—
79	C₂²	C₄×#13096802	1	1	yes	characteristic
80	C₂²	#20263236	1	2	yes	—
81	C₂²	#20263237	1	1	yes	characteristic
82	C₄	#20263238	1	4	yes	—
83	C₄	C₂²⋊#13096802	1	2	yes	—
84	C₄	#20263255	1	4	yes	—
85	C₄	C₄⋊#13096802	1	2	yes	—
86	C₂	C₈×#13096802	1	1	yes	characteristic
87	C₂	#27160800	1	1	yes	characteristic
88	C₂	#27160801	1	2	yes	—
89	C₂	#27160833	1	2	yes	—
90	C₂	#27160834	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×D₁₉₇
commutator	C₂×C₁₉₇	C₈²
Fitting	C₂×C₄×C₈×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₄²
cosocle	C₂×C₄²×C₁₉₇	C₂²
hypercenter	C₂×C₄×C₈	D₁₉₇
hypocenter	C₁₉₇	#1273
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
11. C₂×#13096802	C₈	semidirect
16. C₂×C₈×C₁₉₇	C₈	semidirect
45. C₁₉₇	#1273	semidirect
62. C₂×C₈	#13096802	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13096802	2
C₂×C₄×C₈×C₁₉₇	2
#1273	197