Subgroups — #27164122

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691876	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263920	C₂	1	1	yes	maximal, characteristic
4	#20263886	C₂	1	1	yes	maximal, characteristic
5	#20263880	C₂	1	1	yes	maximal, characteristic
6	#20263865	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
9	#15691876	C₄	1	4	yes	complemented
10	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	—	2	1	no	—
12	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
13	Dic₁₉₉⋊C₄	—	2	1	no	—
14	C₄×Dic₁₉₉	—	2	1	no	—
15	C₄×Dic₁₉₉	—	2	1	no	—
16	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
20	Dic₃₉₈	—	2	1	no	—
21	Dic₃₉₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₉₉	—	2	1	no	—
23	C₂×Dic₁₉₉	—	2	1	no	—
24	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
26	C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
27	Q₈×C₁₉₉	—	2	1	no	—
28	Q₈×C₁₉₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₁₉₉	—	2	1	no	—
30	C₂×C₄×C₁₉₉	—	2	1	no	—
31	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₈×C₁₉₉	—	2	1	no	—
33	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₉₉	—	2	2	no	—
35	C₄×C₁₉₉	—	4	1	no	—
36	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₁₉₉	#41	1	1	yes	characteristic
38	Dic₁₉₉	—	2	2	no	—
39	Dic₁₉₉	—	2	1	no	—
40	Dic₁₉₉	—	2	1	no	—
41	Dic₁₉₉	—	4	1	no	—
42	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
43	C₂×C₁₉₉	#137	1	1	yes	characteristic
44	C₂×C₁₉₉	#137	1	1	yes	characteristic
45	C₁₉₉	#441	1	1	yes	characteristic, complemented, Sylow
46	#441	—	199	1	no	maximal, Sylow
47	C₂×Q₁₆	—	199	1	no	—
48	C₄×Q₈	—	199	1	no	—
49	C₄×Q₈	—	199	1	no	—
50	#106	—	199	1	no	—
51	#103	—	199	1	no	—
52	#103	D₁₉₉	1	1	yes	characteristic
53	#97	—	199	1	no	—
54	C₂×Q₈	—	199	1	no	—
55	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
56	Q₁₆	—	199	4	no	—
57	C₂×C₈	—	199	1	no	—
58	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
59	C₄⋊C₄	—	398	1	no	—
60	C₄⋊C₄	—	398	1	no	—
61	C₄⋊C₄	—	199	1	no	—
62	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
63	C₄²	—	398	1	no	—
64	C₄²	—	398	1	no	—
65	C₄²	—	199	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	398	1	no	—
68	Q₈	—	199	2	no	—
69	Q₈	C₄×D₁₉₉	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	398	1	no	—
73	C₂×C₄	—	398	1	no	—
74	C₂×C₄	—	199	1	no	—
75	C₂×C₄	—	199	1	no	—
76	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	199	2	no	—
79	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	796	1	no	—
83	C₄	—	398	1	no	—
84	C₄	—	398	1	no	—
85	C₄	—	398	2	no	—
86	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
87	C₄	#15691885	1	1	yes	characteristic
88	C₂	(C₂²⋊D₁₉₉)⋊C₄	1	1	yes	characteristic
89	C₂	#20263984	1	1	yes	characteristic
90	C₂	#20263987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#103×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#103	D₁₉₉
hypocenter	C₁₉₉	#441
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #15691876	C₄	semidirect
45. C₁₉₉	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691876	2
Q₈×Dic₁₉₉	2
#20263920	2
#20263886	2
#20263880	2
#20263865	2
#103×C₁₉₉	2
#441	199