Subgroups — #31795712

Order 17536.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14555513	C₂	1	2	yes	maximal, complemented
2	#22577371	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555513	—	2	2	no	—
4	C₂×#14555510	C₄	1	1	yes	characteristic
5	C₂²×#11764040	C₂²	1	1	yes	characteristic
6	#17984000	—	2	2	no	—
7	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555513	—	4	2	no	—
9	#14555510	—	4	1	no	—
10	#14555510	—	4	1	no	—
11	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
12	C₂×D₅₄₈	C₈	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
14	C₂×C₄×D₁₃₇	—	2	1	no	—
15	C₂×#11764040	—	4	1	no	—
16	C₂×#11764040	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₃₇	—	4	1	no	—
19	C₂×C₈×C₁₃₇	—	2	1	no	—
20	C₂²×D₁₃₇	—	2	1	no	—
21	Dic₂₇₄	—	4	1	no	—
22	D₅₄₈	—	4	1	no	—
23	C₂²⋊D₁₃₇	—	4	2	no	—
24	C₂×Dic₁₃₇	—	2	1	no	—
25	C₄×D₁₃₇	—	4	1	no	—
26	C₄×D₁₃₇	—	4	1	no	—
27	#11764040	—	2	2	no	—
28	C₂³×C₁₃₇	M₁₆	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₃₇	—	4	1	no	—
34	C₂²×C₁₃₇	—	2	1	no	—
35	C₂²×C₁₃₇	—	2	2	no	—
36	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×D₁₃₇	—	4	1	no	—
38	C₂×D₁₃₇	—	4	2	no	—
39	C₄×C₁₃₇	—	2	1	no	—
40	C₄×C₁₃₇	#99	1	1	yes	characteristic
41	C₄×C₁₃₇	#100	1	1	yes	characteristic
42	Dic₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	—	4	1	no	—
44	C₂×C₁₃₇	#323	1	1	yes	characteristic
45	C₂×C₁₃₇	#350	1	2	yes	—
46	D₁₃₇	—	8	1	no	—
47	C₁₃₇	#1315	1	1	yes	characteristic, complemented, Sylow
48	#1315	—	137	1	no	maximal, Sylow
49	#408	—	137	1	no	—
50	C₂²⋊C₁₆	—	137	2	no	—
51	C₂×#41	—	137	1	no	—
52	C₂×M₁₆	#10587739	1	1	yes	characteristic
53	C₂²×C₈	—	137	1	no	—
54	C₂×C₁₆	—	274	2	no	—
55	C₂²⋊C₈	—	274	2	no	—
56	#41	—	548	1	no	—
57	#41	—	548	1	no	—
58	C₂×Q₈	—	137	1	no	—
59	C₂×D₄	—	274	1	no	—
60	C₂×D₄	—	137	1	no	—
61	C₂²×C₄	—	274	1	no	—
62	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	548	1	no	—
66	C₂×C₈	—	137	2	no	—
67	C₁₆	—	548	2	no	—
68	C₂³	—	274	1	no	—
69	C₂³	#14555520	1	1	yes	characteristic
70	Q₈	—	548	1	no	—
71	D₄	—	548	1	no	—
72	D₄	—	548	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	274	1	no	—
76	C₂×C₄	—	548	1	no	—
77	C₂×C₄	—	548	1	no	—
78	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
79	C₂×C₄	#14555518	1	1	yes	characteristic
80	C₈	—	4	1	no	—
81	C₈	—	274	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	548	1	no	—
85	C₂²	—	548	2	no	—
86	C₂²	#17984145	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	548	1	no	—
89	C₄	#17984147	1	1	yes	characteristic
90	C₄	#17984148	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	1096	1	no	—
93	C₂	#22578247	1	1	yes	characteristic
94	C₂	#22578302	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²⋊C₈
prime radical	C₂×#14555510	C₄
cosocle	C₂²×#11764040	C₂²
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#1315
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#14555513	C₂	semidirect
47. C₁₃₇	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14555513	2
#22577371	2
#1315	137