Subgroups — #27164295

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₇₉₆	C₂	1	1	yes	maximal, characteristic
2	#20263888	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263875	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₉₉	—	2	1	no	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
10	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₉₉	—	2	1	no	—
12	C₄⋊C₄×C₁₉₉	—	2	1	no	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	—	2	1	no	—
16	D₇₉₆	—	2	1	no	—
17	D₇₉₆	Q₈	1	2	yes	complemented
18	C₂×Dic₁₉₉	—	2	1	no	—
19	C₄×D₁₉₉	—	4	2	no	—
20	C₁₉₉⋊C₈	—	2	2	no	—
21	Q₈×C₁₉₉	—	4	2	no	—
22	C₂×C₄×C₁₉₉	—	2	2	no	—
23	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	2	no	—
28	C₂×D₁₉₉	—	4	1	no	—
29	C₂×D₁₉₉	—	4	1	no	—
30	C₄×C₁₉₉	—	2	1	no	—
31	C₄×C₁₉₉	—	4	2	no	—
32	C₄×C₁₉₉	D₈	1	2	yes	—
33	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₉₉	#41	1	1	yes	characteristic
35	Dic₁₉₉	—	4	1	no	—
36	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₉₉	#137	1	1	yes	characteristic
39	D₁₉₉	—	4	2	no	—
40	C₁₉₉	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	199	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	199	1	no	—
44	#107	—	199	2	no	—
45	C₄⋊C₈	—	199	1	no	—
46	#102	—	199	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	199	1	no	—
49	C₂²×C₄	—	398	1	no	—
50	C₂×C₈	—	199	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	199	1	no	—
53	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
54	C₂²⋊C₄	—	398	1	no	—
55	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
56	C₂³	—	398	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	398	1	no	—
59	D₄	—	199	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	796	2	no	—
62	C₂×C₄	—	398	1	no	—
63	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
66	C₈	—	398	2	no	—
67	C₂²	—	796	1	no	—
68	C₂²	—	796	1	no	—
69	C₂²	—	398	2	no	—
70	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	796	1	no	—
74	C₄	#15691871	1	2	yes	—
75	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
76	C₄	#15691887	1	1	yes	characteristic
77	C₂	—	796	2	no	—
78	C₂	#20263931	1	1	yes	characteristic
79	C₂	C₂×#15691871	1	1	yes	characteristic
80	C₂	#20263986	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	C₂×D₇₉₆	C₂²
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#474
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
2. #20263888	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₁₉₉	C₂	semidirect
17. D₇₉₆	Q₈	semidirect
40. C₁₉₉	#474	semidirect
42. C₄⋊Q₈	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₇₉₆	2
#20263888	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263875	2
C₄⋊Q₈×C₁₉₉	2
#474	199