Subgroups — #31795923

Order 17536.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578308	C₂	1	2	yes	maximal
2	#22577776	C₂	1	1	yes	maximal, characteristic
3	#17984147	—	2	2	no	—
4	C₂×#14555499	—	2	2	no	—
5	#17984088	C₄	1	1	yes	characteristic, complemented
6	#17984085	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10587739	—	4	2	no	—
9	C₂×Dic₂₇₄	—	2	1	no	—
10	C₂×D₅₄₈	—	4	1	no	—
11	C₂×D₅₄₈	D₄	1	2	yes	—
12	#14555499	—	4	4	no	—
13	C₄⋊Dic₁₃₇	—	4	1	no	—
14	C₂×C₈×C₁₃₇	—	2	1	no	—
15	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#10587739	—	8	2	no	—
17	C₂²×D₁₃₇	—	2	2	no	—
18	Dic₂₇₄	—	4	2	no	—
19	D₅₄₈	—	4	2	no	—
20	D₅₄₈	—	8	2	no	—
21	C₂×Dic₁₃₇	—	4	1	no	—
22	C₂×C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₁₃₇	#99	1	1	yes	characteristic
26	C₂×D₁₃₇	—	4	2	no	—
27	C₂×D₁₃₇	—	8	2	no	—
28	#10587739	—	16	2	no	—
29	C₄×C₁₃₇	—	2	2	no	—
30	C₄×C₁₃₇	—	2	1	no	—
31	Dic₁₃₇	—	8	1	no	—
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	#353	1	1	yes	characteristic
34	D₁₃₇	—	8	2	no	—
35	C₁₃₇	#1409	1	1	yes	characteristic, complemented, Sylow
36	#1409	—	137	1	no	maximal, Sylow
37	#492	—	137	1	no	—
38	#353	—	137	2	no	—
39	C₂×SD₁₆	—	274	2	no	—
40	C₄⋊Q₈	—	137	1	no	—
41	#127	—	137	1	no	—
42	#99	—	274	2	no	—
43	C₄×C₈	#10587739	1	1	yes	characteristic, complemented
44	C₂×Q₈	—	274	1	no	—
45	C₂×D₄	—	548	1	no	—
46	C₂×D₄	—	137	2	no	—
47	SD₁₆	—	548	4	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	548	1	no	—
50	C₂²⋊C₄	—	548	2	no	—
51	C₄²	C₂×#10587739	1	1	yes	characteristic
52	C₂³	—	274	2	no	—
53	Q₈	—	548	2	no	—
54	D₄	—	548	2	no	—
55	D₄	—	1096	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	548	1	no	—
58	C₂×C₄	—	1096	2	no	—
59	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₂²	—	548	2	no	—
62	C₂²	—	1096	2	no	—
63	C₂²	#17984147	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	1096	1	no	—
67	C₄	—	2192	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	1096	2	no	—
70	C₂	#22578308	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578308
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#353
prime radical, cosocle	#17984085	C₂²
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1409
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #17984088	C₄	semidirect
7. C₄×C₈×C₁₃₇	C₄	semidirect
35. C₁₃₇	#1409	semidirect
43. C₄×C₈	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578308	2
#22577776	2
#1409	137