Subgroups — #18009729

Order 4736.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558438	C₂	1	2	yes	maximal
2	#14557907	C₂	1	1	yes	maximal, characteristic
3	#11924595	—	2	2	no	—
4	C₂×#10596941	—	2	2	no	—
5	#11924533	C₂²	1	1	yes	characteristic
6	C₄×#64436	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈×C₃₇	—	2	1	no	—
9	C₂×D₁₄₈	—	4	1	no	—
10	C₂×D₁₄₈	D₄	1	2	yes	—
11	C₂²⋊#3866	—	4	2	no	—
12	C₄⋊C₄×C₃₇	—	4	1	no	—
13	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
14	#10596941	—	4	4	no	—
15	C₂×#64436	—	2	1	no	—
16	C₂²×D₃₇	—	2	2	no	—
17	C₂×#3866	—	8	2	no	—
18	Q₈×C₃₇	—	4	2	no	—
19	C₂×C₄×C₃₇	—	2	1	no	—
20	C₂×C₄×C₃₇	—	4	1	no	—
21	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	D₁₄₈	—	4	2	no	—
23	D₁₄₈	—	8	2	no	—
24	#64436	—	2	2	no	—
25	C₂²×C₃₇	#99	1	1	yes	characteristic
26	C₂×D₃₇	—	4	2	no	—
27	C₂×D₃₇	—	8	2	no	—
28	#3866	—	16	2	no	—
29	C₄×C₃₇	—	2	2	no	—
30	C₄×C₃₇	—	2	1	no	—
31	C₄×C₃₇	—	8	1	no	—
32	#1409	—	37	1	no	maximal, Sylow
33	C₂×C₃₇	—	2	1	no	—
34	C₂×C₃₇	#353	1	1	yes	characteristic
35	D₃₇	—	8	2	no	—
36	#492	—	37	1	no	—
37	#353	—	37	2	no	—
38	C₃₇	#1409	1	1	yes	characteristic, complemented, Sylow
39	C₂×SD₁₆	—	74	2	no	—
40	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
41	#127	—	37	1	no	—
42	#99	—	74	2	no	—
43	C₄×C₈	—	37	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	37	2	no	—
46	C₂×D₄	—	148	1	no	—
47	SD₁₆	—	148	4	no	—
48	C₂×C₈	—	74	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	148	2	no	—
51	C₄²	C₂×#3866	1	1	yes	characteristic
52	C₂³	—	74	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	296	2	no	—
55	D₄	—	148	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	296	2	no	—
59	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
60	C₈	—	74	2	no	—
61	C₂²	—	296	2	no	—
62	C₂²	—	148	2	no	—
63	C₂²	#11924594	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	592	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	296	2	no	—
70	C₂	#14558437	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14558437
commutator	C₄²×C₃₇	C₂×C₄
Frattini	C₄²	C₂×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂×C₃₇	#353
prime radical, cosocle	#11924533	C₂²
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1409
LCS	G ▷ C₄²×C₃₇ ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#64436	C₄	semidirect
7. C₄⋊Q₈×C₃₇	C₄	semidirect
38. C₃₇	#1409	semidirect
40. C₄⋊Q₈	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558438	2
#14557907	2
#1409	37