Subgroups — #27133011

Order 12224.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259319	C₂	1	1	yes	maximal, characteristic
2	#20259242	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₆₄	—	2	1	no	—
6	C₂×D₇₆₄	—	2	1	no	—
7	C₂×D₇₆₄	C₂²	1	2	yes	—
8	C₂×C₁₉₁⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₁	—	2	1	no	—
10	C₄⋊C₄×C₁₉₁	—	2	1	no	—
11	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
12	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₂²×D₁₉₁	—	2	2	no	—
14	D₇₆₄	—	2	4	no	—
15	D₇₆₄	—	4	2	no	—
16	D₇₆₄	—	4	2	no	—
17	C₁₉₁⋊C₈	—	2	2	no	—
18	Q₈×C₁₉₁	—	4	2	no	—
19	C₂×C₄×C₁₉₁	—	2	2	no	—
20	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₉₁	—	4	2	no	—
25	C₂×D₁₉₁	—	4	2	no	—
26	C₂×D₁₉₁	—	4	2	no	—
27	C₄×C₁₉₁	—	4	2	no	—
28	C₄×C₁₉₁	D₈	1	2	yes	—
29	C₄×C₁₉₁	SD₁₆	1	2	yes	—
30	C₄×C₁₉₁	#41	1	2	yes	—
31	C₂×C₁₉₁	#124	1	1	yes	characteristic
32	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
33	C₂×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
34	D₁₉₁	—	8	2	no	—
35	C₁₉₁	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	191	1	no	maximal, Sylow
37	C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic, complemented
38	#127	—	191	1	no	—
39	#102	—	191	4	no	—
40	C₄×C₈	—	191	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	382	1	no	—
43	C₂×D₄	—	382	1	no	—
44	C₂×D₄	—	191	2	no	—
45	C₂×C₈	—	191	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
48	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
49	C₂³	—	382	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	382	4	no	—
52	D₄	—	764	2	no	—
53	D₄	—	764	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
58	C₈	—	382	2	no	—
59	C₂²	—	764	2	no	—
60	C₂²	—	764	2	no	—
61	C₂²	—	764	2	no	—
62	C₂²	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#15690272	1	2	yes	—
65	C₄	#15690274	1	2	yes	—
66	C₄	#15690288	1	2	yes	—
67	C₂	—	1528	2	no	—
68	C₂	#20259311	1	1	yes	characteristic
69	C₂	C₂×#15690272	1	1	yes	characteristic
70	C₂	C₂×#15690274	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₄⋊Q₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
prime radical	#20259319	C₂
hypercenter	C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#486
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20259242	C₂	semidirect
3. C₄×C₁₉₁⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₁₉₁	C₂	semidirect
35. C₁₉₁	#486	semidirect
37. C₄⋊Q₈	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#20259319	2
#20259242	2
C₄×C₁₉₁⋊C₈	2
C₄⋊Q₈×C₁₉₁	2
#486	191