Subgroups — #31798015

Order 17536.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578506	C₂	1	2	yes	maximal
2	C₄⋊#14555522	C₂	1	1	yes	maximal, characteristic
3	#22578497	C₂	1	2	yes	maximal
4	C₄×#14555522	C₂	1	1	yes	maximal, characteristic
5	Q₈×#11764047	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555522	C₂²	1	1	yes	characteristic
7	C₂×#14555522	C₂²	1	2	yes	—
8	C₂×#14555522	C₂²	1	1	yes	characteristic
9	C₄⋊#11764047	C₄	1	2	yes	—
10	C₄⋊#11764047	C₄	1	1	yes	characteristic
11	C₄×#11764047	C₂²	1	1	yes	characteristic
12	C₄×#11764047	C₂²	1	2	yes	—
13	Q₈×Dic₁₃₇	C₄	1	1	yes	characteristic
14	#14555522	—	2	1	no	—
15	#14555522	—	2	2	no	—
16	#14555522	Q₈	1	2	yes	complemented
17	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
18	C₂×#11764047	C₂×C₄	1	2	yes	—
19	C₂×#11764047	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₃₇	C₈	1	2	yes	—
21	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
22	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
25	#11764047	—	2	1	no	—
26	#11764047	—	2	2	no	—
27	#11764047	C₂×Q₈	1	1	yes	characteristic
28	#11764047	#41	1	1	yes	characteristic
29	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	Q₈×C₁₃₇	—	2	2	no	—
33	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	2	no	—
37	C₄×C₁₃₇	M₃₂	1	2	yes	—
38	Dic₁₃₇	—	2	2	no	—
39	Dic₁₃₇	—	2	1	no	—
40	Dic₁₃₇	C₄×Q₈	1	1	yes	characteristic
41	Dic₁₃₇	#131	1	1	yes	characteristic
42	C₂×C₁₃₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₁₃₇	#504	1	1	yes	characteristic
45	C₁₃₇	#2183	1	1	yes	characteristic, complemented, Sylow
46	#2183	—	137	1	no	maximal, Sylow
47	C₈×Q₈	—	137	1	no	—
48	C₄⋊C₁₆	—	137	1	no	—
49	C₄⋊C₁₆	—	137	2	no	—
50	#346	—	137	2	no	—
51	C₄×C₁₆	—	137	1	no	—
52	C₄×Q₈	—	137	1	no	—
53	C₂×C₁₆	—	137	1	no	—
54	C₂×C₁₆	—	137	2	no	—
55	C₂×C₁₆	—	137	1	no	—
56	C₄⋊C₈	—	137	2	no	—
57	C₄⋊C₈	—	137	1	no	—
58	C₄×C₈	—	137	1	no	—
59	C₄×C₈	—	137	2	no	—
60	C₂×Q₈	#11764049	1	1	yes	characteristic
61	C₂×C₈	—	137	1	no	—
62	C₂×C₈	—	137	2	no	—
63	C₂×C₈	—	137	1	no	—
64	C₄⋊C₄	—	137	2	no	—
65	C₄⋊C₄	—	137	1	no	—
66	C₄²	—	137	2	no	—
67	C₄²	—	137	1	no	—
68	C₁₆	—	274	1	no	—
69	C₁₆	—	274	2	no	—
70	C₁₆	—	137	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	—	137	1	no	—
74	C₂×C₄	—	137	1	no	—
75	C₂×C₄	C₂×#11764049	1	2	yes	—
76	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
77	C₈	—	274	1	no	—
78	C₈	—	274	2	no	—
79	C₈	—	137	1	no	—
80	C₈	—	137	1	no	—
81	C₂²	C₂²×#11764049	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	274	2	no	—
84	C₄	—	274	1	no	—
85	C₄	—	137	1	no	—
86	C₄	—	137	1	no	—
87	C₄	#17984184	1	2	yes	—
88	C₂	Q₈×#11764049	1	1	yes	characteristic
89	C₂	C₂×#17984184	1	1	yes	characteristic
90	C₂	#22578525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle, prime radical	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂×Q₈×C₁₃₇	C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	#2183
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
16. #14555522	Q₈	semidirect
45. C₁₃₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
#22578506	2
C₄⋊#14555522	2
#22578497	2
C₄×#14555522	2
Q₈×#11764047	2
#2183	137