Subgroups — #27163924

Order 12736.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉₉⋊#95	C₂	1	4	yes	maximal
4	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₉₈	—	2	2	no	—
6	C₂×Dic₃₉₈	—	2	2	no	—
7	C₂×Dic₃₉₈	C₄	1	4	yes	complemented
8	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
10	C₄⋊Dic₁₉₉	—	2	2	no	—
11	C₂²×C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₉	—	2	2	no	—
14	Dic₃₉₈	—	2	8	no	—
15	C₂×Dic₁₉₉	—	2	2	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂×Dic₁₉₉	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	4	no	—
19	C₂×Dic₁₉₉	C₂×C₄	1	4	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	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
25	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
26	C₂×C₄×C₁₉₉	Q₈	1	4	yes	complemented
27	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₁₉₉	C₂×Q₈	1	2	yes	—
31	C₂²×C₁₉₉	#41	1	2	yes	—
32	C₄×C₁₉₉	—	2	4	no	—
33	C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	—
34	Dic₁₉₉	—	2	4	no	—
35	Dic₁₉₉	—	4	2	no	—
36	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₁₉₉	C₄×Q₈	1	2	yes	—
38	C₂×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₁₉₉	#124	1	1	yes	characteristic
40	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
41	C₁₉₉	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	199	1	no	maximal, Sylow
43	C₂²×Q₈	—	199	1	no	—
44	C₂×C₄⋊C₄	—	199	1	no	—
45	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
46	#95	—	199	4	no	—
47	C₂×Q₈	—	398	2	no	—
48	C₂×Q₈	—	398	2	no	—
49	C₂×Q₈	—	199	4	no	—
50	C₂²×C₄	—	199	2	no	—
51	C₂²×C₄	—	199	2	no	—
52	C₂²×C₄	C₂×D₁₉₉	1	2	yes	—
53	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
54	C₄⋊C₄	—	398	2	no	—
55	C₄²	—	2	2	no	—
56	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
57	Q₈	—	398	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	398	2	no	—
61	C₂×C₄	—	398	4	no	—
62	C₂×C₄	—	398	2	no	—
63	C₂×C₄	—	398	2	no	—
64	C₂×C₄	—	199	4	no	—
65	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
66	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
67	C₂×C₄	D₇₉₆	1	2	yes	—
68	C₂×C₄	Dic₃₉₈	1	4	yes	complemented
69	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
72	C₂²	C₂×Dic₃₉₈	1	2	yes	—
73	C₂²	#15691886	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	796	2	no	—
76	C₄	—	398	4	no	—
77	C₄	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
78	C₂	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
79	C₂	C₄×Dic₃₉₈	1	2	yes	—
80	C₂	#20263934	1	2	yes	—
81	C₂	#20263943	1	1	yes	characteristic
82	C₂	#20263955	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₉
commutator	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₂×C₄²×C₁₉₉	C₂
socle, prime radical, cosocle	C₂³×C₁₉₉	C₂³
hypercenter	C₂×C₄²	D₁₉₉
hypocenter	C₁₉₉	#391
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₂×Dic₃₉₈	C₄	semidirect
26. C₂×C₄×C₁₉₉	Q₈	semidirect
41. C₁₉₉	#391	semidirect
68. C₂×C₄	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₉₈	2
C₂×C₄⋊Dic₁₉₉	2
C₁₉₉⋊#95	2
C₂×C₄²×C₁₉₉	2
#391	199