Subgroups — #31782547

Order 17536.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984028	C₂	1	1	yes	maximal, characteristic
2	#22577358	C₂	1	1	yes	maximal, characteristic
3	#22577221	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	C₄	1	1	yes	characteristic
6	#17984047	C₄	1	1	yes	characteristic
7	#17984028	—	2	2	no	—
8	C₂×#14555485	C₂²	1	2	yes	—
9	#17983998	C₄	1	2	yes	—
10	C₈×Dic₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₃₇	—	2	2	no	—
13	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
14	#14555494	—	2	2	no	—
15	C₂×#11764040	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₃₇	C₈	1	2	yes	—
17	C₂²⋊Dic₁₃₇	C₈	1	2	yes	—
18	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
19	#14555485	—	2	2	no	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
23	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
24	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₃₇	—	2	2	no	—
26	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	—
27	#11764040	—	2	2	no	—
28	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₃₇	—	2	2	no	—
37	C₂²×C₁₃₇	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
42	Dic₁₃₇	—	2	4	no	—
43	C₂×C₁₃₇	—	2	2	no	—
44	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₃₇	#504	1	2	yes	—
46	C₁₃₇	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	137	1	no	maximal, Sylow
48	C₂×M₃₂	—	137	1	no	—
49	C₂²×C₁₆	D₁₃₇	1	1	yes	characteristic
50	#405	—	137	1	no	—
51	C₄⋊C₁₆	—	137	4	no	—
52	C₂×M₁₆	—	137	1	no	—
53	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
54	#117	—	137	1	no	—
55	M₃₂	—	274	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	137	2	no	—
58	C₂×C₁₆	C₂×D₁₃₇	1	2	yes	—
59	#97	—	137	2	no	—
60	C₄×C₈	—	137	2	no	—
61	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
62	M₁₆	—	274	2	no	—
63	C₂×C₈	—	137	2	no	—
64	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
66	C₂×C₈	Dic₂₇₄	1	2	yes	—
67	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
68	C₄⋊C₄	—	137	2	no	—
69	C₂²⋊C₄	—	137	2	no	—
70	C₄²	—	137	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	274	2	no	—
73	C₂³	C₈×D₁₃₇	1	1	yes	characteristic
74	C₂×C₄	—	137	4	no	—
75	C₂×C₄	#14555489	1	2	yes	—
76	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
77	C₂×C₄	#14555493	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
79	C₈	—	274	2	no	—
80	C₈	#14555489	1	2	yes	—
81	C₈	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
82	C₈	C₂×Dic₂₇₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#17984017	1	2	yes	—
85	C₂²	C₂×C₈×D₁₃₇	1	1	yes	characteristic
86	C₄	—	274	4	no	—
87	C₄	#17984017	1	2	yes	—
88	C₄	C₂×#14555489	1	1	yes	characteristic
89	C₄	C₂×#14555493	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#17984017	1	1	yes	characteristic
92	C₂	#22577819	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂²×C₁₆×C₁₃₇	C₂
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂²×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2151
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22577221	C₂	semidirect
46. C₁₃₇	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984028	2
#22577358	2
#22577221	2
C₂²×C₁₆×C₁₃₇	2
#2151	137