Subgroups — #31797933

Order 17536.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11764049	C₂	1	6	yes	maximal
2	C₄²×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764049	C₂²	1	4	yes	—
4	C₄⋊#11764049	—	2	12	no	—
5	C₂×C₄×#10587739	C₄	1	3	yes	—
6	C₂×C₄×#10587739	C₂²	1	3	yes	—
7	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#11764049	—	2	12	no	—
9	C₂²×#10587739	C₂×C₄	1	3	yes	—
10	C₂²×#10587739	C₂³	1	1	yes	characteristic
11	C₄×#10587739	—	2	6	no	—
12	C₄×#10587739	D₄	1	6	yes	—
13	C₄×#10587739	Q₈	1	6	yes	—
14	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
15	C₄×Dic₁₃₇	C₈	1	3	yes	complemented
16	C₄²×C₁₃₇	C₈	1	1	yes	characteristic, complemented
17	#11764049	—	4	4	no	—
18	C₂×#10587739	—	2	6	no	—
19	C₂×#10587739	C₄⋊C₄	1	6	yes	—
20	C₂×#10587739	C₂×D₄	1	3	yes	—
21	C₂×#10587739	C₂×Q₈	1	3	yes	—
22	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×Dic₁₃₇	C₂×C₈	1	3	yes	—
24	C₄×D₁₃₇	C₄⋊C₄	1	6	yes	—
25	C₄×D₁₃₇	M₁₆	1	6	yes	—
26	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
27	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
28	C₂×D₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×D₁₃₇	C₂×M₁₆	1	3	yes	—
30	#10587739	—	2	6	no	—
31	#10587739	#127	1	1	yes	characteristic
32	#10587739	C₄⋊Q₈	1	3	yes	—
33	C₄×C₁₃₇	C₄⋊C₈	1	6	yes	complemented
34	Dic₁₃₇	C₄⋊C₈	1	6	yes	complemented
35	C₂×C₁₃₇	C₂×C₄⋊C₈	1	3	yes	—
36	D₁₃₇	#384	1	1	yes	characteristic
37	D₁₃₇	#423	1	3	yes	—
38	C₁₃₇	#1842	1	1	yes	characteristic, complemented, Sylow
39	#1842	—	137	1	no	maximal, Sylow
40	C₂×C₄⋊C₈	—	137	6	no	—
41	C₄³	—	137	1	no	—
42	C₂²×C₈	—	137	4	no	—
43	C₂×C₄²	—	137	3	no	—
44	C₂×C₄²	—	137	3	no	—
45	C₂×C₄²	—	137	1	no	—
46	C₄⋊C₈	—	274	12	no	—
47	C₂²×C₄	—	137	3	no	—
48	C₂²×C₄	—	137	1	no	—
49	C₂²×C₄	—	137	3	no	—
50	C₂×C₈	—	274	12	no	—
51	C₄²	—	274	6	no	—
52	C₄²	—	137	6	no	—
53	C₄²	—	137	6	no	—
54	C₄²	—	137	3	no	—
55	C₄²	#11764049	1	1	yes	characteristic, complemented
56	C₂³	—	137	1	no	—
57	C₂×C₄	—	274	6	no	—
58	C₂×C₄	—	137	6	no	—
59	C₂×C₄	—	137	3	no	—
60	C₂×C₄	—	137	3	no	—
61	C₂×C₄	—	137	3	no	—
62	C₂×C₄	—	137	6	no	—
63	C₂×C₄	—	137	6	no	—
64	C₂×C₄	C₂×#11764049	1	3	yes	—
65	C₈	—	548	4	no	—
66	C₂²	—	137	3	no	—
67	C₂²	—	137	3	no	—
68	C₂²	C₂²×#11764049	1	1	yes	characteristic
69	C₄	—	274	6	no	—
70	C₄	—	137	1	no	—
71	C₄	—	137	3	no	—
72	C₄	—	137	6	no	—
73	C₄	C₄⋊#11764049	1	6	yes	complemented
74	C₂	—	137	3	no	—
75	C₂	—	137	1	no	—
76	C₂	C₂×C₄⋊#11764049	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₄²×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂²×#10587739	C₂³
hypercenter	C₄²	#11764049
hypocenter	C₁₃₇	#1842
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
15. C₄×Dic₁₃₇	C₈	semidirect
16. C₄²×C₁₃₇	C₈	semidirect
33. C₄×C₁₃₇	C₄⋊C₈	semidirect
34. Dic₁₃₇	C₄⋊C₈	semidirect
38. C₁₃₇	#1842	semidirect
55. C₄²	#11764049	semidirect
73. C₄	C₄⋊#11764049	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11764049	2
C₄²×#10587739	2
#1842	137