Subgroups — #27163958

Order 12736.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₉₉⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₁₉₉⋊#95	C₂	1	2	yes	maximal
4	C₁₉₉⋊#95	C₂	1	2	yes	maximal
5	#95×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
8	C₂²×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₉	—	2	2	no	—
10	Dic₁₉₉⋊C₄	—	2	2	no	—
11	C₂²×C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₉	—	2	1	no	—
14	C₂×Dic₁₉₉	—	2	1	no	—
15	C₂×Dic₁₉₉	—	2	1	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂×Dic₁₉₉	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	4	no	—
19	C₂×Dic₁₉₉	Q₈	1	2	yes	—
20	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	—	2	2	no	—
22	C₂×C₄×C₁₉₉	—	2	2	no	—
23	C₂×C₄×C₁₉₉	—	2	2	no	—
24	C₂×C₄×C₁₉₉	—	2	2	no	—
25	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
26	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₁₉₉	#41	1	1	yes	characteristic
29	C₂²×C₁₉₉	#41	1	2	yes	—
30	C₂²×C₁₉₉	#41	1	2	yes	—
31	C₄×C₁₉₉	—	2	2	no	—
32	C₄×C₁₉₉	—	4	2	no	—
33	Dic₁₉₉	—	2	2	no	—
34	Dic₁₉₉	—	4	1	no	—
35	Dic₁₉₉	—	4	2	no	—
36	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₉₉	#123	1	1	yes	characteristic
38	C₂×C₁₉₉	#124	1	1	yes	characteristic
39	C₂×C₁₉₉	#125	1	2	yes	—
40	C₂×C₁₉₉	#126	1	2	yes	—
41	C₁₉₉	#400	1	1	yes	characteristic, complemented, Sylow
42	#400	—	199	1	no	maximal, Sylow
43	C₂×C₄⋊C₄	—	199	1	no	—
44	C₂×C₄⋊C₄	—	199	1	no	—
45	#95	—	199	2	no	—
46	#95	—	199	2	no	—
47	#95	D₁₉₉	1	1	yes	characteristic
48	C₂²×C₄	—	199	1	no	—
49	C₂²×C₄	—	199	2	no	—
50	C₂²×C₄	—	199	1	no	—
51	C₂²×C₄	C₂×D₁₉₉	1	2	yes	—
52	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
53	C₄⋊C₄	—	398	2	no	—
54	C₄⋊C₄	—	398	2	no	—
55	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	398	1	no	—
61	C₂×C₄	—	398	2	no	—
62	C₂×C₄	—	398	2	no	—
63	C₂×C₄	—	398	4	no	—
64	C₂×C₄	—	398	1	no	—
65	C₂×C₄	—	398	1	no	—
66	C₂×C₄	—	199	2	no	—
67	C₂×C₄	D₇₉₆	1	2	yes	—
68	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
69	C₂²	Q₈×D₁₉₉	1	1	yes	characteristic
70	C₂²	#15691885	1	2	yes	—
71	C₂²	#15691885	1	1	yes	characteristic
72	C₂²	#15691886	1	2	yes	—
73	C₄	—	2	2	no	—
74	C₄	—	4	2	no	—
75	C₄	—	796	1	no	—
76	C₄	—	796	2	no	—
77	C₄	—	398	2	no	—
78	C₂	#20263935	1	1	yes	characteristic
79	C₂	#20263939	1	1	yes	characteristic
80	C₂	#20263943	1	1	yes	characteristic
81	C₂	#20263945	1	2	yes	—
82	C₂	#20263948	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₉
commutator, socle, prime radical, cosocle	C₂³×C₁₉₉	C₂³
Fitting	#95×C₁₉₉	C₂
hypercenter	#95	D₁₉₉
hypocenter	C₁₉₉	#400
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
41. C₁₉₉	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₉₉	2
C₂×Dic₁₉₉⋊C₄	2
C₁₉₉⋊#95	2
#95×C₁₉₉	2
#400	199