Subgroups — #31795928

Order 17536.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578308	C₂	1	2	yes	maximal
2	#22577769	C₂	1	1	yes	maximal, characteristic
3	#17984147	—	2	2	no	—
4	#17984085	C₂²	1	1	yes	characteristic
5	#17984008	—	2	2	no	—
6	#17983999	C₄	1	1	yes	characteristic, complemented
7	#125×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10587739	—	4	2	no	—
9	C₂×D₅₄₈	—	4	1	no	—
10	C₂×D₅₄₈	D₄	1	2	yes	—
11	C₂×#11764040	—	2	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	C₂×#10587739	—	8	2	no	—
16	C₂²×D₁₃₇	—	2	2	no	—
17	D₅₄₈	—	4	2	no	—
18	D₅₄₈	—	8	2	no	—
19	#11764040	—	4	1	no	—
20	C₂×C₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×C₁₃₇	—	4	1	no	—
22	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₁₃₇	#99	1	1	yes	characteristic
24	C₂×D₁₃₇	—	4	2	no	—
25	C₂×D₁₃₇	—	8	2	no	—
26	#10587739	—	16	2	no	—
27	C₄×C₁₃₇	—	2	1	no	—
28	C₄×C₁₃₇	—	4	1	no	—
29	C₄×C₁₃₇	—	8	1	no	—
30	C₂×C₁₃₇	—	2	1	no	—
31	C₂×C₁₃₇	#354	1	1	yes	characteristic
32	D₁₃₇	—	8	2	no	—
33	C₁₃₇	#1412	1	1	yes	characteristic, complemented, Sylow
34	#1412	—	137	1	no	maximal, Sylow
35	#490	—	137	1	no	—
36	#353	—	137	2	no	—
37	#127	—	137	1	no	—
38	#125	#10587739	1	1	yes	characteristic, complemented
39	#102	—	274	2	no	—
40	#99	—	274	2	no	—
41	#97	—	137	1	no	—
42	C₂×D₄	—	548	1	no	—
43	C₂×D₄	—	137	2	no	—
44	C₂×C₈	—	274	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	548	2	no	—
48	C₄²	C₂×#10587739	1	1	yes	characteristic
49	C₂³	—	274	2	no	—
50	D₄	—	548	2	no	—
51	D₄	—	1096	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	1096	2	no	—
55	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
56	C₈	—	548	1	no	—
57	C₂²	—	548	2	no	—
58	C₂²	—	1096	2	no	—
59	C₂²	#17984146	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	2192	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	1096	2	no	—
66	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, cosocle	#17984085	C₂²
hypercenter	#125	#10587739
hypocenter	C₁₃₇	#1412
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
6. #17983999	C₄	semidirect
7. #125×C₁₃₇	C₄	semidirect
33. C₁₃₇	#1412	semidirect
38. #125	#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
#22577769	2
#1412	137