Subgroups — #31797130

Order 17536.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984162	C₂	1	2	yes	maximal
2	#22578325	C₂	1	4	yes	maximal, complemented
3	#22577361	C₂	1	1	yes	maximal, characteristic
4	#17984162	—	2	2	no	—
5	#17984162	C₂²	1	4	yes	—
6	C₂×#14555513	—	2	2	no	—
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	#17984047	C₄	1	1	yes	characteristic
9	#17983998	C₄	1	2	yes	—
10	C₈×Dic₁₃₇	C₂²	1	2	yes	—
11	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555513	—	2	4	no	—
13	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
14	C₂×#11764040	D₄	1	2	yes	—
15	C₂×#11764040	Q₈	1	2	yes	—
16	C₂×#11764040	C₂³	1	1	yes	characteristic
17	C₄⋊Dic₁₃₇	—	2	1	no	—
18	C₂²⋊Dic₁₃₇	—	2	1	no	—
19	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	M₁₆×C₁₃₇	C₈	1	4	yes	—
22	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₃₇	—	2	2	no	—
24	#11764040	C₄⋊C₄	1	2	yes	—
25	#11764040	C₂×D₄	1	1	yes	characteristic
26	#11764040	C₂×Q₈	1	1	yes	characteristic
27	C₂³×C₁₃₇	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₃₇	C₂×C₈	1	4	yes	—
33	C₂²×C₁₃₇	—	4	1	no	—
34	C₂²×C₁₃₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
37	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
39	Dic₁₃₇	—	4	2	no	—
40	C₂×C₁₃₇	—	2	1	no	—
41	C₂×C₁₃₇	—	2	2	no	—
42	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
43	C₁₃₇	#2153	1	1	yes	characteristic, complemented, Sylow
44	#2153	—	137	1	no	maximal, Sylow
45	C₂×M₃₂	—	137	2	no	—
46	#405	—	137	1	no	—
47	#364	—	137	4	no	—
48	C₂×M₁₆	#10587739	1	1	yes	characteristic
49	C₂²×C₈	—	137	1	no	—
50	#117	—	137	1	no	—
51	M₃₂	—	274	2	no	—
52	M₃₂	—	137	4	no	—
53	C₂×C₁₆	—	274	2	no	—
54	#97	—	137	2	no	—
55	C₄×C₈	—	137	2	no	—
56	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
57	M₁₆	#11764047	1	4	yes	—
58	C₂×C₈	—	137	2	no	—
59	C₂×C₈	—	137	2	no	—
60	C₂×C₈	—	137	1	no	—
61	C₂×C₈	—	137	1	no	—
62	C₂×C₈	C₂×#10587739	1	2	yes	—
63	C₄⋊C₄	—	274	1	no	—
64	C₂²⋊C₄	—	274	1	no	—
65	C₄²	—	137	2	no	—
66	C₁₆	—	274	4	no	—
67	C₂³	#14555520	1	1	yes	characteristic
68	C₂×C₄	—	274	2	no	—
69	C₂×C₄	C₄⋊#10587739	1	2	yes	—
70	C₂×C₄	C₂×#11764047	1	2	yes	—
71	C₂×C₄	#14555520	1	1	yes	characteristic
72	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
73	C₈	—	137	1	no	—
74	C₈	—	137	2	no	—
75	C₈	—	137	1	no	—
76	C₈	C₂×#11764047	1	4	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊#11764047	1	2	yes	—
79	C₂²	C₂×#14555520	1	1	yes	characteristic
80	C₄	—	548	2	no	—
81	C₄	C₄⋊#11764047	1	2	yes	—
82	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
83	C₄	C₂²×#11764047	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊#11764047	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊#10587739
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃₇	M₁₆
cosocle	C₂×#11764040	C₂³
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#2153
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22578325	C₂	semidirect
43. C₁₃₇	#2153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984162	2
#22578325	2
#22577361	2
#2153	137