Subgroups — #31795798

Order 17536.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578326	C₂	1	2	yes	maximal
2	#22577782	C₂	1	1	yes	maximal, characteristic
3	#17984162	—	2	2	no	—
4	C₂×#14555500	—	2	2	no	—
5	#17984088	C₄	1	1	yes	characteristic
6	C₄×#11764040	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555513	—	4	2	no	—
9	C₂×Dic₂₇₄	—	2	1	no	—
10	#14555500	—	4	2	no	—
11	#14555500	—	4	2	no	—
12	C₂×#11764040	D₄	1	2	yes	—
13	C₄⋊Dic₁₃₇	—	4	1	no	—
14	C₂×Q₈×C₁₃₇	—	2	1	no	—
15	C₄⋊C₄×C₁₃₇	—	4	1	no	—
16	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	Dic₂₇₄	—	4	2	no	—
18	C₂×Dic₁₃₇	—	4	1	no	—
19	#11764040	D₈	1	2	yes	—
20	#11764040	SD₁₆	1	2	yes	—
21	Q₈×C₁₃₇	—	4	2	no	—
22	C₂×C₄×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₁₃₇	—	4	1	no	—
24	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₃₇	#100	1	1	yes	characteristic
26	C₄×C₁₃₇	—	2	2	no	—
27	C₄×C₁₃₇	—	8	1	no	—
28	C₄×C₁₃₇	#102	1	2	yes	—
29	Dic₁₃₇	—	8	1	no	—
30	C₂×C₁₃₇	—	2	1	no	—
31	C₂×C₁₃₇	#331	1	1	yes	characteristic
32	C₁₃₇	#1358	1	1	yes	characteristic, complemented, Sylow
33	#1358	—	137	1	no	maximal, Sylow
34	#494	—	137	1	no	—
35	#364	—	137	2	no	—
36	C₂×Q₁₆	—	274	2	no	—
37	C₄⋊Q₈	—	137	1	no	—
38	C₄⋊Q₈	#10587739	1	1	yes	characteristic
39	M₃₂	—	274	2	no	—
40	C₄×C₈	—	137	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×Q₈	—	274	1	no	—
43	Q₁₆	—	548	2	no	—
44	Q₁₆	—	548	2	no	—
45	C₂×C₈	—	137	2	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	548	1	no	—
48	C₄²	C₂×#10587739	1	1	yes	characteristic
49	C₁₆	—	548	2	no	—
50	Q₈	—	4	2	no	—
51	Q₈	—	548	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	548	1	no	—
55	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
56	C₈	—	137	2	no	—
57	C₈	—	137	2	no	—
58	C₂²	#17984148	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	8	1	no	—
61	C₄	—	1096	1	no	—
62	C₄	#17984150	1	2	yes	—
63	C₂	—	2	1	no	—
64	C₂	#22578261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578261
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#331
prime radical	C₂²×C₁₃₇	#100
cosocle	C₄×#11764040	C₂²
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1358
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
32. C₁₃₇	#1358	semidirect

Maximal subgroups

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

subgroup	index
#22578326	2
#22577782	2
#1358	137