Subgroups — #27164156

Order 12736.

158 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, complemented
2	C₂×#15691874	C₂	1	2	yes	maximal, complemented
3	#20263878	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#121×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	—	2	4	no	—
7	C₂×Dic₃₉₈	—	2	1	no	—
8	C₂×Dic₃₉₈	C₂²	1	2	yes	—
9	C₂²×Dic₁₉₉	—	2	1	no	—
10	#15691874	—	2	4	no	—
11	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
12	C₂×D₄×C₁₉₉	—	2	1	no	—
13	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
17	Dic₃₉₈	—	2	4	no	—
18	Dic₃₉₈	—	2	2	no	—
19	Dic₃₉₈	D₄	1	4	yes	complemented
20	C₂×Dic₁₉₉	—	2	4	no	—
21	C₂×Dic₁₉₉	—	2	2	no	—
22	C₁₉₉⋊C₈	—	2	2	no	—
23	C₂³×C₁₉₉	—	2	1	no	—
24	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
25	D₄×C₁₉₉	—	2	2	no	—
26	D₄×C₁₉₉	—	4	2	no	—
27	C₂×C₄×C₁₉₉	—	2	2	no	—
28	C₂×C₄×C₁₉₉	—	2	1	no	—
29	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
31	C₂²×C₁₉₉	—	2	1	no	—
32	C₂²×C₁₉₉	—	2	1	no	—
33	C₂²×C₁₉₉	—	4	1	no	—
34	C₂²×C₁₉₉	—	4	1	no	—
35	C₂²×C₁₉₉	—	4	1	no	—
36	C₂²×C₁₉₉	SD₁₆	1	2	yes	complemented
37	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₉₉	—	2	1	no	—
39	C₄×C₁₉₉	—	4	1	no	—
40	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
41	Dic₁₉₉	—	2	4	no	—
42	C₂×C₁₉₉	—	2	2	no	—
43	C₂×C₁₉₉	—	8	1	no	—
44	C₂×C₁₉₉	#120	1	1	yes	characteristic
45	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₁₉₉	#137	1	1	yes	characteristic
47	C₁₉₉	#448	1	1	yes	characteristic, complemented, Sylow
48	#448	—	199	1	no	maximal, Sylow
49	C₂²×Q₈	—	199	1	no	—
50	C₂×SD₁₆	—	199	2	no	—
51	#121	D₁₉₉	1	1	yes	characteristic
52	#103	—	199	2	no	—
53	C₂²⋊C₈	—	199	1	no	—
54	C₂×Q₈	—	398	4	no	—
55	C₂×Q₈	—	398	1	no	—
56	C₂×Q₈	—	199	2	no	—
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
59	C₂²×C₄	—	398	1	no	—
60	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
61	SD₁₆	—	398	4	no	—
62	C₂×C₈	—	199	2	no	—
63	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂³	—	2	1	no	—
66	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
67	Q₈	—	398	4	no	—
68	Q₈	—	398	2	no	—
69	Q₈	—	199	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	4	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	398	4	no	—
75	C₂×C₄	—	398	2	no	—
76	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
78	C₈	—	398	2	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	#15691874	1	2	yes	complemented
85	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	398	4	no	—
89	C₄	D₄×D₁₉₉	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	—	8	1	no	—
92	C₂	#20263923	1	1	yes	characteristic
93	C₂	C₂×#15691874	1	1	yes	characteristic
94	C₂	#20263985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting, prime radical	#121×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
hypercenter	#121	D₁₉₉
hypocenter	C₁₉₉	#448
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₃₉₈	C₂	semidirect
2. C₂×#15691874	C₂	semidirect
3. #20263878	C₂	semidirect
4. C₂²⋊(C₁₉₉⋊C₈)	C₂	semidirect
19. Dic₃₉₈	D₄	semidirect
36. C₂²×C₁₉₉	SD₁₆	semidirect
47. C₁₉₉	#448	semidirect
84. C₂²	#15691874	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×#15691874	2
#20263878	2
C₂²⋊(C₁₉₉⋊C₈)	2
#121×C₁₉₉	2
#448	199