Subgroups — #31779081

Order 17536.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577456	C₂	1	2	yes	maximal
2	#22577125	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11764040	C₄	1	4	yes	—
5	C₂²⋊#11764040	C₄	1	4	yes	—
6	C₄×#11764040	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₃₇	C₄	1	2	yes	—
8	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₃₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#11764040	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	—	2	4	no	—
15	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
16	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
19	#11764040	—	2	8	no	—
20	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	—	2	4	no	—
22	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
23	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	M₁₆	1	4	yes	—
25	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
26	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₈×C₁₃₇	—	2	4	no	—
28	C₂²×C₁₃₇	—	2	2	no	—
29	C₂²×C₁₃₇	#97	1	2	yes	—
30	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
31	C₄×C₁₃₇	—	2	2	no	—
32	C₄×C₁₃₇	#97	1	2	yes	—
33	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
34	C₄×C₁₃₇	#131	1	4	yes	—
35	C₂×C₁₃₇	—	2	2	no	—
36	C₂×C₁₃₇	C₂×#97	1	1	yes	characteristic
37	C₂×C₁₃₇	#405	1	2	yes	—
38	C₁₃₇	#1452	1	1	yes	characteristic, complemented, Sylow
39	#1452	—	137	1	no	maximal, Sylow
40	#431	—	137	2	no	—
41	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
42	#322	—	137	4	no	—
43	C₂²×C₈	Dic₁₃₇	1	2	yes	—
44	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
45	C₄⋊C₈	—	137	4	no	—
46	C₂²⋊C₈	—	137	4	no	—
47	C₄×C₈	—	137	4	no	—
48	C₄×C₈	Dic₁₃₇	1	2	yes	—
49	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
50	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
51	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	137	8	no	—
54	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
55	C₄²	C₄×D₁₃₇	1	2	yes	—
56	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
57	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
58	C₂³	C₄×Dic₁₃₇	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄×Dic₁₃₇	1	2	yes	—
61	C₂×C₄	C₄×Dic₁₃₇	1	1	yes	characteristic
62	C₂×C₄	#14555494	1	4	yes	—
63	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
64	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
65	C₈	—	2	4	no	—
66	C₈	—	274	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#17983999	1	2	yes	—
69	C₂²	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	#17983999	1	2	yes	—
72	C₄	C₂×#14555494	1	2	yes	—
73	C₄	#17984094	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#17983999	1	1	yes	characteristic
76	C₂	#22577358	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
prime radical	C₂³×C₁₃₇	C₄²
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#1452
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22577125	C₂	semidirect
38. C₁₃₇	#1452	semidirect

Maximal subgroups

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

subgroup	index
#22577456	2
#22577125	2
C₂×C₄×C₈×C₁₃₇	2
#1452	137