Subgroups — #27164218

Order 12736.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691873	C₂	1	2	yes	maximal, complemented
2	#20263952	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263875	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	—	2	1	no	—
7	C₂×D₇₉₆	—	2	1	no	—
8	C₂×D₇₉₆	C₂²	1	2	yes	—
9	#15691873	—	2	4	no	—
10	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄²×C₁₉₉	—	2	1	no	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₉	—	2	2	no	—
17	D₇₉₆	—	2	2	no	—
18	D₇₉₆	—	2	2	no	—
19	D₇₉₆	—	4	2	no	—
20	D₇₉₆	—	4	2	no	—
21	C₁₉₉⋊C₈	—	2	2	no	—
22	Q₈×C₁₉₉	—	2	1	no	—
23	Q₈×C₁₉₉	D₄	1	2	yes	complemented
24	C₂×C₄×C₁₉₉	—	2	1	no	—
25	C₂×C₄×C₁₉₉	—	2	1	no	—
26	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
30	C₂×D₁₉₉	—	4	2	no	—
31	C₂×D₁₉₉	—	4	2	no	—
32	C₂×D₁₉₉	—	4	2	no	—
33	C₄×C₁₉₉	—	2	2	no	—
34	C₄×C₁₉₉	—	2	1	no	—
35	C₄×C₁₉₉	—	4	1	no	—
36	C₄×C₁₉₉	SD₁₆	1	2	yes	complemented
37	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₉₉	#41	1	1	yes	characteristic
39	C₂×C₁₉₉	#121	1	1	yes	characteristic
40	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₁₉₉	—	8	2	no	—
43	C₁₉₉	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	199	1	no	maximal, Sylow
45	C₂×SD₁₆	—	199	2	no	—
46	#127	—	199	1	no	—
47	C₄×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
48	C₄⋊C₈	—	199	1	no	—
49	#102	—	199	2	no	—
50	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
51	C₂×D₄	—	398	1	no	—
52	C₂×D₄	—	398	1	no	—
53	C₂×D₄	—	199	2	no	—
54	SD₁₆	—	398	4	no	—
55	C₂×C₈	—	199	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
60	C₂³	—	398	2	no	—
61	Q₈	—	2	1	no	—
62	Q₈	D₇₉₆	1	2	yes	complemented
63	D₄	—	796	2	no	—
64	D₄	—	796	2	no	—
65	D₄	—	398	2	no	—
66	D₄	—	398	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
72	C₈	—	398	2	no	—
73	C₂²	—	796	2	no	—
74	C₂²	—	796	2	no	—
75	C₂²	—	796	2	no	—
76	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	4	1	no	—
80	C₄	#15691873	1	2	yes	complemented
81	C₄	C₂×D₇₉₆	1	1	yes	characteristic
82	C₄	#15691886	1	1	yes	characteristic
83	C₂	—	1592	2	no	—
84	C₂	#20263929	1	1	yes	characteristic
85	C₂	C₂×#15691873	1	1	yes	characteristic
86	C₂	#20263980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×Q₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	#20263952	C₂
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#460
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15691873	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. #20263875	C₂	semidirect
5. C₄×Q₈×C₁₉₉	C₂	semidirect
23. Q₈×C₁₉₉	D₄	semidirect
36. C₄×C₁₉₉	SD₁₆	semidirect
43. C₁₉₉	#460	semidirect
47. C₄×Q₈	D₁₉₉	semidirect
62. Q₈	D₇₉₆	semidirect
80. C₄	#15691873	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691873	2
#20263952	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263875	2
C₄×Q₈×C₁₉₉	2
#460	199