Subgroups — #44470385

Order 29888.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28417950	C₂	1	1	yes	maximal, characteristic
2	#28417873	C₂	1	4	yes	maximal, complemented
3	C₄×C₄₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₄₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₈₆₈	—	2	1	no	—
6	C₂×D₁₈₆₈	—	2	1	no	—
7	C₂×D₁₈₆₈	C₂²	1	2	yes	—
8	C₂×C₄₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₄₆₇	—	2	1	no	—
10	C₄⋊C₄×C₄₆₇	—	2	1	no	—
11	C₄⋊C₄×C₄₆₇	C₂²	1	2	yes	—
12	C₄²×C₄₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₄₆₇	—	2	2	no	—
14	D₁₈₆₈	—	2	4	no	—
15	D₁₈₆₈	—	4	2	no	—
16	D₁₈₆₈	—	4	2	no	—
17	C₄₆₇⋊C₈	—	2	2	no	—
18	Q₈×C₄₆₇	—	4	2	no	—
19	C₂×C₄×C₄₆₇	—	2	2	no	—
20	C₂×C₄×C₄₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄₆₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₆₇	—	4	2	no	—
25	C₂×D₄₆₇	—	4	2	no	—
26	C₂×D₄₆₇	—	4	2	no	—
27	C₄×C₄₆₇	—	4	2	no	—
28	C₄×C₄₆₇	D₈	1	2	yes	—
29	C₄×C₄₆₇	SD₁₆	1	2	yes	—
30	C₄×C₄₆₇	#41	1	2	yes	—
31	C₂×C₄₆₇	#124	1	1	yes	characteristic
32	C₂×C₄₆₇	C₂×D₈	1	1	yes	characteristic
33	C₂×C₄₆₇	C₂×SD₁₆	1	1	yes	characteristic
34	D₄₆₇	—	8	2	no	—
35	C₄₆₇	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	467	1	no	maximal, Sylow
37	C₄⋊Q₈	D₄₆₇	1	1	yes	characteristic, complemented
38	#127	—	467	1	no	—
39	#102	—	467	4	no	—
40	C₄×C₈	—	467	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	934	1	no	—
43	C₂×D₄	—	934	1	no	—
44	C₂×D₄	—	467	2	no	—
45	C₂×C₈	—	467	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₄₆₇	1	2	yes	—
48	C₄²	C₂×D₄₆₇	1	1	yes	characteristic
49	C₂³	—	934	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	1868	2	no	—
52	D₄	—	1868	2	no	—
53	D₄	—	934	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₄₆₇	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₆₇	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₄₆₇	1	1	yes	characteristic
58	C₈	—	934	2	no	—
59	C₂²	—	1868	2	no	—
60	C₂²	—	1868	2	no	—
61	C₂²	—	1868	2	no	—
62	C₂²	C₂×C₂²⋊D₄₆₇	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#21418728	1	2	yes	—
65	C₄	#21418730	1	2	yes	—
66	C₄	#21418744	1	2	yes	—
67	C₂	—	3736	2	no	—
68	C₂	#28417942	1	1	yes	characteristic
69	C₂	C₂×#21418728	1	1	yes	characteristic
70	C₂	C₂×#21418730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₆₇
commutator, cosocle	C₂×C₄×C₄₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₆₇
Fitting	C₄⋊Q₈×C₄₆₇	C₂
socle	C₂²×C₄₆₇	C₂×D₄
prime radical	#28417950	C₂
hypercenter	C₄⋊Q₈	D₄₆₇
hypocenter	C₄₆₇	#486
LCS	G ▷ C₂×C₄×C₄₆₇ ▷ C₂×C₄₆₇ ▷ C₄₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #28417873	C₂	semidirect
3. C₄×C₄₆₇⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₄₆₇	C₂	semidirect
35. C₄₆₇	#486	semidirect
37. C₄⋊Q₈	D₄₆₇	semidirect

Maximal subgroups

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

subgroup	index
#28417950	2
#28417873	2
C₄×C₄₆₇⋊C₈	2
C₄⋊Q₈×C₄₆₇	2
#486	467