Subgroups — #31795931

Order 17536.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578314	C₂	1	2	yes	maximal
2	#22577773	C₂	1	1	yes	maximal, characteristic
3	#17984149	—	2	2	no	—
4	#17984088	C₂²	1	1	yes	characteristic
5	#17984011	—	2	2	no	—
6	#17983999	C₄	1	1	yes	characteristic
7	#125×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555519	—	4	2	no	—
9	C₂×Dic₂₇₄	D₄	1	2	yes	—
10	C₂×#11764040	—	2	1	no	—
11	C₄⋊Dic₁₃₇	—	4	1	no	—
12	C₄⋊C₄×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	—	4	1	no	—
14	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	#11764047	—	8	2	no	—
16	Dic₂₇₄	—	4	2	no	—
17	C₂×Dic₁₃₇	—	2	2	no	—
18	#11764040	—	4	1	no	—
19	C₂×C₄×C₁₃₇	—	2	1	no	—
20	C₂×C₄×C₁₃₇	—	4	1	no	—
21	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₁₃₇	#99	1	1	yes	characteristic
23	C₄×C₁₃₇	—	2	1	no	—
24	C₄×C₁₃₇	—	4	1	no	—
25	C₄×C₁₃₇	—	8	1	no	—
26	Dic₁₃₇	—	4	2	no	—
27	C₂×C₁₃₇	—	2	1	no	—
28	C₂×C₁₃₇	#354	1	1	yes	characteristic
29	C₁₃₇	#1413	1	1	yes	characteristic, complemented, Sylow
30	#1413	—	137	1	no	maximal, Sylow
31	#491	—	137	1	no	—
32	#356	—	137	2	no	—
33	C₄⋊Q₈	—	137	1	no	—
34	#125	#10587739	1	1	yes	characteristic
35	#103	—	274	2	no	—
36	#101	—	274	2	no	—
37	#97	—	137	1	no	—
38	C₂×Q₈	—	137	2	no	—
39	M₁₆	—	548	2	no	—
40	C₂×C₈	—	274	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	548	1	no	—
44	C₄²	C₂×#10587739	1	1	yes	characteristic
45	Q₈	—	548	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	274	2	no	—
49	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
50	C₈	—	548	1	no	—
51	C₈	—	1096	2	no	—
52	C₂²	#17984146	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	8	1	no	—
56	C₄	—	548	2	no	—
57	C₂	—	2	1	no	—
58	C₂	#22578309	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578309
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	#125×C₁₃₇	C₄
socle	C₂×C₁₃₇	#354
prime radical	C₂²×C₁₃₇	#99
cosocle	#17984088	C₂²
hypercenter	#125	#10587739
hypocenter	C₁₃₇	#1413
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
29. C₁₃₇	#1413	semidirect

Maximal subgroups

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

subgroup	index
#22578314	2
#22577773	2
#1413	137