Subgroups — #27163986

Order 12736.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691886	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₉₉)⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#15691886	—	2	2	no	—
6	#15691886	—	2	2	no	—
7	C₂×Dic₃₉₈	C₄	1	1	yes	characteristic
8	C₂×D₇₉₆	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₉₉	C₄	1	2	yes	—
10	C₂×C₄×D₁₉₉	C₂²	1	2	yes	—
11	C₂×C₁₉₉⋊C₈	—	2	2	no	—
12	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₉₉	—	2	2	no	—
15	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
16	C₂²×D₁₉₉	C₂×C₄	1	2	yes	—
17	Dic₃₉₈	—	2	2	no	—
18	D₇₉₆	—	2	2	no	—
19	C₂²⋊D₁₉₉	—	2	4	no	—
20	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
21	C₄×D₁₉₉	—	2	2	no	—
22	C₄×D₁₉₉	—	2	2	no	—
23	C₁₉₉⋊C₈	—	2	2	no	—
24	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
27	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
28	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₈×C₁₉₉	—	2	2	no	—
30	C₂²×C₁₉₉	—	2	2	no	—
31	C₂²×C₁₉₉	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₁₉₉	—	2	2	no	—
34	C₂×D₁₉₉	—	2	4	no	—
35	C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
36	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	Dic₁₉₉	—	2	2	no	—
39	C₂×C₁₉₉	—	2	2	no	—
40	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₁₉₉	#131	1	2	yes	—
42	D₁₉₉	—	4	2	no	—
43	C₁₉₉	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	199	1	no	maximal, Sylow
45	C₂×#41	—	199	1	no	—
46	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic, complemented
47	C₂²×C₈	—	199	1	no	—
48	C₂²⋊C₈	—	199	4	no	—
49	#41	—	398	2	no	—
50	#41	—	398	2	no	—
51	C₂×Q₈	—	199	1	no	—
52	C₂×D₄	—	199	2	no	—
53	C₂×D₄	—	199	1	no	—
54	C₂²×C₄	—	199	2	no	—
55	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
56	M₁₆	—	2	2	no	—
57	C₂×C₈	—	398	2	no	—
58	C₂×C₈	—	199	2	no	—
59	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
60	C₂³	—	199	2	no	—
61	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
62	Q₈	—	398	2	no	—
63	D₄	—	398	4	no	—
64	D₄	—	398	2	no	—
65	C₂×C₄	—	398	2	no	—
66	C₂×C₄	—	398	2	no	—
67	C₂×C₄	—	199	2	no	—
68	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
70	C₂×C₄	D₇₉₆	1	2	yes	—
71	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	398	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	398	4	no	—
76	C₂²	—	398	2	no	—
77	C₂²	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
78	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
79	C₄	—	398	2	no	—
80	C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
81	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
82	C₄	C₂×D₇₉₆	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	796	2	no	—
85	C₂	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
86	C₂	#20263963	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator, socle	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₂×M₁₆×C₁₉₉	C₂
prime radical	C₂×#15691886	C₂
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#408
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₉₉⋊C₈	C₂	semidirect
3. (C₂×D₁₉₉)⋊C₈	C₂	semidirect
4. C₂×M₁₆×C₁₉₉	C₂	semidirect
43. C₁₉₉	#408	semidirect
46. C₂×M₁₆	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691886	2
C₂²×C₁₉₉⋊C₈	2
(C₂×D₁₉₉)⋊C₈	2
C₂×M₁₆×C₁₉₉	2
#408	199