Subgroups — #31795926

Order 17536.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578314	C₂	1	2	yes	maximal
2	#22577782	C₂	1	1	yes	maximal, characteristic
3	#17984149	—	2	2	no	—
4	C₂×#14555500	—	2	2	no	—
5	#17984088	C₂²	1	1	yes	characteristic
6	C₄×#11764040	C₄	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555519	—	4	2	no	—
9	C₂×Dic₂₇₄	D₄	1	2	yes	—
10	#14555500	—	4	4	no	—
11	C₂×#11764040	—	2	1	no	—
12	C₄⋊Dic₁₃₇	—	4	1	no	—
13	C₂×Q₈×C₁₃₇	—	2	1	no	—
14	C₄⋊C₄×C₁₃₇	—	4	1	no	—
15	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
16	#11764047	—	8	2	no	—
17	Dic₂₇₄	—	4	2	no	—
18	C₂×Dic₁₃₇	—	2	2	no	—
19	#11764040	—	2	2	no	—
20	Q₈×C₁₃₇	—	4	2	no	—
21	C₂×C₄×C₁₃₇	—	2	1	no	—
22	C₂×C₄×C₁₃₇	—	4	1	no	—
23	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₁₃₇	#99	1	1	yes	characteristic
25	C₄×C₁₃₇	—	2	2	no	—
26	C₄×C₁₃₇	—	2	1	no	—
27	C₄×C₁₃₇	—	8	1	no	—
28	Dic₁₃₇	—	4	2	no	—
29	C₂×C₁₃₇	—	2	1	no	—
30	C₂×C₁₃₇	#353	1	1	yes	characteristic
31	C₁₃₇	#1411	1	1	yes	characteristic, complemented, Sylow
32	#1411	—	137	1	no	maximal, Sylow
33	#494	—	137	1	no	—
34	#356	—	137	2	no	—
35	C₂×Q₁₆	—	274	2	no	—
36	C₄⋊Q₈	—	137	1	no	—
37	C₄⋊Q₈	#10587739	1	1	yes	characteristic
38	#101	—	274	2	no	—
39	C₄×C₈	—	137	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	—	137	2	no	—
42	Q₁₆	—	548	4	no	—
43	M₁₆	—	548	2	no	—
44	C₂×C₈	—	274	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	548	1	no	—
47	C₄²	C₂×#10587739	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	548	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	274	2	no	—
53	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
54	C₈	—	274	2	no	—
55	C₈	—	1096	2	no	—
56	C₂²	#17984146	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	1	no	—
59	C₄	—	8	1	no	—
60	C₄	—	548	2	no	—
61	C₂	—	2	1	no	—
62	C₂	#22578307	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578307
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#353
prime radical	C₂²×C₁₃₇	#99
cosocle	#17984088	C₂²
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1411
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
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
31. C₁₃₇	#1411	semidirect

Maximal subgroups

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

subgroup	index
#22578314	2
#22577782	2
#1411	137