Subgroups — #31797203

Order 17536.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578324	C₂	1	2	yes	maximal
2	C₄⋊#14555513	C₂	1	1	yes	maximal, characteristic
3	#22578289	C₂	1	2	yes	maximal
4	C₄×#14555513	C₂	1	1	yes	maximal, characteristic
5	#22577527	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555513	C₂²	1	1	yes	characteristic
7	C₂×#14555513	C₂²	1	2	yes	—
8	C₂×#14555513	C₂²	1	1	yes	characteristic
9	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
10	#17984017	C₄	1	2	yes	—
11	C₄×#11764040	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₃₇	C₂²	1	2	yes	—
13	C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
14	#14555513	—	2	1	no	—
15	#14555513	—	2	2	no	—
16	#14555513	Q₈	1	2	yes	complemented
17	C₂×Dic₂₇₄	C₈	1	1	yes	characteristic
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₂×#11764040	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
21	#14555489	C₈	1	2	yes	—
22	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	Dic₂₇₄	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
28	#11764040	—	2	1	no	—
29	#11764040	C₂×Q₈	1	1	yes	characteristic
30	#11764040	#41	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₃₇	#131	1	1	yes	characteristic
40	Dic₁₃₇	—	2	2	no	—
41	Dic₁₃₇	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₃₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₃₇	#504	1	1	yes	characteristic
45	C₁₃₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	Q₈×C₁₆	—	137	1	no	maximal, Sylow
47	C₈×Q₈	—	137	1	no	—
48	C₄⋊C₁₆	—	137	1	no	—
49	C₄⋊C₁₆	—	137	2	no	—
50	C₄×C₁₆	—	137	1	no	—
51	C₄×C₁₆	—	137	2	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	#10587739	1	1	yes	characteristic
58	C₄×C₈	—	137	2	no	—
59	C₄×C₈	—	137	1	no	—
60	C₂×Q₈	—	137	1	no	—
61	C₂×C₈	—	137	1	no	—
62	C₂×C₈	—	137	1	no	—
63	C₂×C₈	C₂×#10587739	1	2	yes	—
64	C₄⋊C₄	—	137	2	no	—
65	C₄⋊C₄	—	137	1	no	—
66	C₄²	—	137	2	no	—
67	C₄²	C₂×#10587739	1	1	yes	characteristic
68	C₁₆	—	274	1	no	—
69	C₁₆	—	274	2	no	—
70	C₁₆	—	137	2	no	—
71	Q₈	—	137	4	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	—	137	2	no	—
74	C₂×C₄	#14555518	1	1	yes	characteristic
75	C₂×C₄	#14555518	1	1	yes	characteristic
76	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	274	1	no	—
79	C₈	—	137	1	no	—
80	C₈	—	137	1	no	—
81	C₂²	C₂×#14555518	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	274	2	no	—
84	C₄	—	137	4	no	—
85	C₄	#17984161	1	2	yes	—
86	C₄	Q₈×#10587739	1	1	yes	characteristic
87	C₄	#17984175	1	1	yes	characteristic
88	C₂	#22578452	1	1	yes	characteristic
89	C₂	C₂×#17984161	1	1	yes	characteristic
90	C₂	#22578462	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂²×C₁₆
Fitting	C₄⋊C₈×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₂²×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	C₄⋊C₈	#10587739
hypocenter	C₁₃₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #14555513	Q₈	semidirect
25. Dic₂₇₄	C₁₆	semidirect
45. C₁₃₇	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22578324	2
C₄⋊#14555513	2
#22578289	2
C₄×#14555513	2
#22577527	2
Q₈×C₁₆	137