Subgroups — #27164410

Order 12736.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691876	C₂	1	2	yes	maximal, complemented
2	C₂×#15691873	C₂	1	2	yes	maximal
3	#20263943	C₂	1	1	yes	maximal, characteristic
4	#20263866	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
7	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
8	#15691876	—	2	4	no	—
9	#15691873	—	2	4	no	—
10	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	(C₂×D₁₉₉)⋊C₄	—	2	2	no	—
12	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₉₉	—	2	2	no	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	—	2	1	no	—
16	Dic₃₉₈	—	2	2	no	—
17	D₇₉₆	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	1	no	—
19	C₁₉₉⋊C₈	D₄	1	4	yes	—
20	Q₈×C₁₉₉	—	2	2	no	—
21	Q₈×C₁₉₉	—	2	2	no	—
22	C₂×C₄×C₁₉₉	—	2	2	no	—
23	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₉₉	—	4	1	no	—
27	C₂×D₁₉₉	—	4	2	no	—
28	C₄×C₁₉₉	—	2	2	no	—
29	C₄×C₁₉₉	—	4	2	no	—
30	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
31	Dic₁₉₉	—	4	1	no	—
32	C₂×C₁₉₉	#127	1	1	yes	characteristic
33	C₂×C₁₉₉	#137	1	2	yes	—
34	D₁₉₉	—	8	1	no	—
35	C₁₉₉	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	199	1	no	maximal, Sylow
37	C₂×Q₁₆	—	199	2	no	—
38	C₂×SD₁₆	—	199	2	no	—
39	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic, complemented
40	#124	—	199	1	no	—
41	#97	—	199	1	no	—
42	C₂×Q₈	—	199	1	no	—
43	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
44	C₂×D₄	—	199	1	no	—
45	Q₁₆	—	398	4	no	—
46	SD₁₆	—	398	4	no	—
47	C₂×C₈	—	199	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	398	2	no	—
50	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
51	C₂³	—	398	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	398	2	no	—
55	D₄	—	398	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	398	1	no	—
58	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
59	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
60	C₈	—	199	4	no	—
61	C₂²	—	796	1	no	—
62	C₂²	—	796	2	no	—
63	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	796	1	no	—
67	C₄	D₄×D₁₉₉	1	2	yes	—
68	C₂	—	1592	1	no	—
69	C₂	#20263951	1	1	yes	characteristic
70	C₂	#20263986	1	2	yes	—

	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	#497
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×#15691876	C₂	semidirect
4. #20263866	C₂	semidirect
5. C₄⋊Q₈×C₁₉₉	C₂	semidirect
35. C₁₉₉	#497	semidirect
39. 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
C₂×#15691876	2
C₂×#15691873	2
#20263943	2
#20263866	2
C₄⋊Q₈×C₁₉₉	2
#497	199