Subgroups — #31795697

Order 17536.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578406	C₂	1	2	yes	maximal
2	#22577426	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555519	C₄	1	4	yes	—
4	#17984155	—	2	2	no	—
5	C₄⋊#11764047	—	2	2	no	—
6	C₂×#14555494	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	#17984017	—	2	2	no	—
9	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
10	#14555519	—	4	4	no	—
11	C₂×#11764047	—	2	4	no	—
12	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
13	#14555494	—	4	1	no	—
14	C₂×#11764040	—	2	1	no	—
15	C₄×Dic₁₃₇	D₄	1	2	yes	—
16	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
17	C₄×Dic₁₃₇	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₁₃₇	—	4	1	no	—
20	C₂×C₈×C₁₃₇	—	2	1	no	—
21	#11764047	—	4	4	no	—
22	C₂×Dic₁₃₇	—	2	2	no	—
23	C₂×Dic₁₃₇	—	2	2	no	—
24	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
25	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
26	#11764040	—	4	1	no	—
27	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
32	C₈×C₁₃₇	—	4	1	no	—
33	C₂²×C₁₃₇	—	2	1	no	—
34	C₂²×C₁₃₇	—	2	2	no	—
35	C₂²×C₁₃₇	#95	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	#100	1	1	yes	characteristic
38	C₄×C₁₃₇	#101	1	1	yes	characteristic
39	Dic₁₃₇	—	2	2	no	—
40	Dic₁₃₇	#100	1	2	yes	—
41	Dic₁₃₇	#101	1	2	yes	—
42	C₂×C₁₃₇	—	4	1	no	—
43	C₂×C₁₃₇	#343	1	2	yes	—
44	C₂×C₁₃₇	#343	1	1	yes	characteristic
45	C₁₃₇	#1309	1	1	yes	characteristic, complemented, Sylow
46	#1309	—	137	1	no	maximal, Sylow
47	#423	—	137	2	no	—
48	#423	—	137	1	no	—
49	C₂×M₁₆	—	137	4	no	—
50	C₂×M₁₆	—	137	1	no	—
51	C₂×M₁₆	#10587739	1	1	yes	characteristic
52	C₂×C₄²	—	137	1	no	—
53	C₄⋊C₈	—	274	2	no	—
54	C₄⋊C₈	—	274	2	no	—
55	C₄⋊C₈	—	274	2	no	—
56	C₂²×C₄	—	137	2	no	—
57	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	548	1	no	—
60	M₁₆	—	548	4	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	274	4	no	—
63	C₂×C₈	—	274	1	no	—
64	C₄²	—	137	2	no	—
65	C₄²	—	137	1	no	—
66	C₄²	—	137	1	no	—
67	C₂³	C₄×#10587739	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	274	2	no	—
71	C₂×C₄	—	274	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	—	137	2	no	—
74	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
75	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
76	C₈	—	4	1	no	—
77	C₈	—	548	4	no	—
78	C₈	—	548	1	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	1	no	—
81	C₂²	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	274	2	no	—
84	C₄	—	137	2	no	—
85	C₄	—	137	2	no	—
86	C₄	#17984148	1	1	yes	characteristic
87	C₄	#17984149	1	1	yes	characteristic
88	C₂	—	4	1	no	—
89	C₂	#22578282	1	2	yes	—
90	C₂	#22578283	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#10587739)⋊C₄
commutator, prime radical	C₂³×C₁₃₇	C₄²
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#95
cosocle	C₂×C₄×Dic₁₃₇	C₂²
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#1309
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
45. C₁₃₇	#1309	semidirect

Maximal subgroups

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

subgroup	index
#22578406	2
#22577426	2
#1309	137