Subgroups — C₂×#27160855 #38744302

Order 25216.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691333	C₂	1	2	yes	maximal
2	#27160855	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691333	C₄	1	8	yes	complemented
5	C₂×#15691333	C₂²	1	4	yes	—
6	C₂²×C₈×C₁₉₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₉₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₉₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₉₇	C₂²	1	2	yes	—
11	#15691333	C₂×C₄	1	8	yes	complemented
12	C₂²×C₄×C₁₉₇	C₈	1	2	yes	—
13	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₉₇	C₈	1	4	yes	—
15	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
19	C₄²×C₁₉₇	C₈	1	2	yes	—
20	C₄²×C₁₉₇	C₂×C₄	1	2	yes	—
21	C₂³×C₁₉₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₁₉₇	—	2	4	no	—
23	C₂×C₄×C₁₉₇	C₄²	1	2	yes	—
24	C₂×C₄×C₁₉₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₁₉₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₉₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₉₇	C₄²	1	2	yes	—
29	C₈×C₁₉₇	C₂×C₈	1	4	yes	—
30	C₈×C₁₉₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₁₉₇	C₄×C₈	1	2	yes	—
32	C₂²×C₁₉₇	M₃₂	1	4	yes	—
33	C₂²×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₁₉₇	—	2	4	no	—
35	C₄×C₁₉₇	C₄×C₈	1	2	yes	—
36	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₁₉₇	#346	1	4	yes	complemented
39	C₂×C₁₉₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₉₇	C₂×M₃₂	1	2	yes	—
41	C₁₉₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
42	C₂×#346	—	197	1	no	maximal, Sylow
43	C₂²×C₁₆	—	197	2	no	—
44	C₂×C₄×C₈	D₁₉₇	1	1	yes	characteristic
45	#346	—	197	4	no	—
46	C₂²×C₈	Dic₁₉₇	1	1	yes	characteristic
47	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
48	C₂×C₄²	Dic₁₉₇	1	1	yes	characteristic
49	C₂×C₁₆	—	197	8	no	—
50	C₂×C₁₆	—	197	4	no	—
51	C₄×C₈	Dic₁₉₇	1	2	yes	—
52	C₄×C₈	C₂×D₁₉₇	1	2	yes	—
53	C₂²×C₄	#13096802	1	2	yes	—
54	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
55	C₂×C₈	#13096802	1	4	yes	—
56	C₂×C₈	C₄×D₁₉₇	1	4	yes	—
57	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
58	C₂×C₈	C₂×Dic₁₉₇	1	2	yes	—
59	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
60	C₄²	#13096802	1	2	yes	—
61	C₄²	C₂×Dic₁₉₇	1	2	yes	—
62	C₁₆	—	197	8	no	—
63	C₂³	C₂×#13096802	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄×Dic₁₉₇	1	2	yes	—
66	C₂×C₄	C₂×#13096802	1	4	yes	—
67	C₂×C₄	C₂×#13096802	1	1	yes	characteristic
68	C₂×C₄	C₂×#13096802	1	2	yes	—
69	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
70	C₈	C₄×Dic₁₉₇	1	2	yes	—
71	C₈	C₂×#13096802	1	4	yes	—
72	C₈	C₂×C₄×D₁₉₇	1	2	yes	—
73	C₂²	C₄×#13096802	1	2	yes	—
74	C₂²	#20263266	1	4	yes	—
75	C₂²	C₂²×#13096802	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×#13096802	1	2	yes	—
78	C₄	C₂×C₄×Dic₁₉₇	1	1	yes	characteristic
79	C₄	C₂²×#13096802	1	1	yes	characteristic
80	C₂	#27160855	1	4	yes	direct product factor
81	C₂	C₂×C₄×#13096802	1	1	yes	characteristic
82	C₂	C₂×#20263266	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₉₇
commutator	C₂×C₁₉₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	C₂×C₄×C₈×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×C₈
cosocle	C₂×C₈×C₁₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#346
LCS	G ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #27160855	C₂	direct (N × H)
4. C₂×#15691333	C₄	semidirect
11. #15691333	C₂×C₄	semidirect
38. C₂×C₁₉₇	#346	semidirect
41. C₁₉₇	C₂×#346	semidirect
80. C₂	#27160855	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15691333	2
#27160855	2
C₂×C₄×C₈×C₁₉₇	2
C₂×#346	197

Subgroups — C2×#27160855 #38744302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27160855 #38744302