Subgroups — #31780964

Order 17536.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984024	C₂	1	4	yes	maximal
2	#22577427	C₂	1	2	yes	maximal
3	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555494	C₂²	1	4	yes	—
5	#17984024	—	2	8	no	—
6	C₄⋊#11764040	—	2	4	no	—
7	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#97×C₁₃₇	C₄	1	4	yes	—
10	#14555494	—	2	8	no	—
11	C₂×#11764040	—	2	4	no	—
12	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
13	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
15	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
16	C₂×C₈×C₁₃₇	Q₈	1	4	yes	—
17	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
18	C₄²×C₁₃₇	D₄	1	2	yes	—
19	#11764040	—	4	4	no	—
20	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	—	4	2	no	—
22	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
23	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
27	C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
28	C₂²×C₁₃₇	—	2	2	no	—
29	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
31	C₄×C₁₃₇	—	2	4	no	—
32	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
33	C₄×C₁₃₇	#127	1	1	yes	characteristic
34	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
35	C₂×C₁₃₇	—	2	2	no	—
36	C₂×C₁₃₇	#384	1	1	yes	characteristic
37	C₂×C₁₃₇	#430	1	2	yes	—
38	C₁₃₇	#1849	1	1	yes	characteristic, complemented, Sylow
39	#1849	—	137	1	no	maximal, Sylow
40	C₂×#108	—	137	4	no	—
41	#423	—	137	2	no	—
42	C₂×#97	D₁₃₇	1	1	yes	characteristic
43	C₂×M₁₆	—	137	4	no	—
44	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
45	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
46	#108	—	274	8	no	—
47	C₄⋊C₈	—	274	4	no	—
48	#97	Dic₁₃₇	1	4	yes	—
49	C₂²×C₄	Dic₂₇₄	1	2	yes	—
50	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
51	M₁₆	—	274	8	no	—
52	C₂×C₈	—	274	4	no	—
53	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
54	C₂×C₈	D₅₄₈	1	4	yes	—
55	C₂×C₈	Dic₂₇₄	1	4	yes	—
56	C₄²	C₂×Dic₁₃₇	1	2	yes	—
57	C₄²	D₅₄₈	1	2	yes	—
58	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
61	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
62	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
63	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
64	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
65	C₈	—	548	4	no	—
66	C₈	C₄⋊Dic₁₃₇	1	8	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
69	C₂²	#17984088	1	2	yes	—
70	C₄	—	2	4	no	—
71	C₄	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
72	C₄	#17984085	1	1	yes	characteristic
73	C₄	#17984088	1	1	yes	characteristic
74	C₂	—	2	2	no	—
75	C₂	#22577273	1	1	yes	characteristic
76	C₂	#22577452	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#97×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#1849
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
38. C₁₃₇	#1849	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984024	2
#22577427	2
C₂×#97×C₁₃₇	2
#1849	137