Subgroups — #31797186

Order 17536.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#10587739	C₂	1	2	yes	maximal
2	#22578330	C₂	1	1	yes	maximal, characteristic
3	#22578316	C₂	1	2	yes	maximal
4	#22578290	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10587739	—	2	2	no	—
7	#17984161	C₂²	1	1	yes	characteristic
8	#17984157	C₂²	1	2	yes	—
9	#17984151	—	2	2	no	—
10	C₈×#10587739	C₂²	1	1	yes	characteristic
11	Q₁₆×D₁₃₇	C₂²	1	2	yes	—
12	Dic₁₀₉₆	C₄	1	1	yes	characteristic, complemented
13	#17984034	C₄	1	2	yes	complemented
14	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	#14555518	D₄	1	1	yes	characteristic
17	C₄⋊#10587739	—	2	2	no	—
18	C₄⋊#10587739	—	4	2	no	—
19	C₄×#10587739	—	4	2	no	—
20	C₄×#10587739	D₄	1	1	yes	characteristic
21	#14555513	—	2	1	no	—
22	Q₈×D₁₃₇	—	2	2	no	—
23	Dic₅₄₈	C₂×C₄	1	2	yes	—
24	#14555500	—	2	2	no	—
25	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
26	#14555485	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#10587739	—	4	2	no	—
30	C₂×#10587739	D₈	1	2	yes	—
31	#11764047	—	2	1	no	—
32	Dic₂₇₄	—	2	2	no	—
33	Dic₂₇₄	—	4	2	no	—
34	C₄×D₁₃₇	—	4	2	no	—
35	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
36	#11764040	#41	1	1	yes	characteristic
37	Q₈×C₁₃₇	—	2	2	no	—
38	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₃₇	C₂×D₈	1	1	yes	characteristic
40	#10587739	—	2	2	no	—
41	#10587739	—	8	2	no	—
42	C₄×C₁₃₇	—	4	2	no	—
43	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
44	Dic₁₃₇	—	4	2	no	—
45	Dic₁₃₇	#135	1	1	yes	characteristic
46	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
47	D₁₃₇	#510	1	2	yes	—
48	C₁₃₇	#2176	1	1	yes	characteristic, complemented, Sylow
49	#2176	—	137	1	no	maximal, Sylow
50	C₂×Q₃₂	—	137	1	no	—
51	C₄×Q₁₆	—	137	2	no	—
52	#367	—	137	1	no	—
53	#358	—	137	2	no	—
54	#346	—	137	1	no	—
55	C₂×Q₁₆	—	137	2	no	—
56	C₄×Q₈	—	274	2	no	—
57	Q₃₂	—	137	2	no	—
58	Q₃₂	—	137	1	no	—
59	Q₃₂	#10587739	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	137	1	no	—
61	C₂×C₁₆	—	137	1	no	—
62	#107	—	137	2	no	—
63	#103	—	274	2	no	—
64	C₄×C₈	—	137	1	no	—
65	C₂×Q₈	—	274	2	no	—
66	Q₁₆	—	274	2	no	—
67	Q₁₆	—	137	2	no	—
68	Q₁₆	C₂×#10587739	1	2	yes	—
69	C₂×C₈	—	137	1	no	—
70	C₂×C₈	—	137	1	no	—
71	C₄⋊C₄	—	274	2	no	—
72	C₄⋊C₄	—	548	2	no	—
73	C₄²	—	548	2	no	—
74	C₄²	—	137	1	no	—
75	C₁₆	—	274	1	no	—
76	C₁₆	—	137	1	no	—
77	C₁₆	C₂×#10587739	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	274	2	no	—
80	Q₈	—	548	2	no	—
81	C₂×C₄	—	548	2	no	—
82	C₂×C₄	—	548	2	no	—
83	C₂×C₄	—	137	2	no	—
84	C₂×C₄	—	137	1	no	—
85	C₈	—	274	1	no	—
86	C₈	—	137	1	no	—
87	C₈	C₂²×#10587739	1	1	yes	characteristic
88	C₂²	—	137	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	274	2	no	—
91	C₄	—	548	2	no	—
92	C₄	—	1096	2	no	—
93	C₄	—	137	1	no	—
94	C₄	D₄×#10587739	1	1	yes	characteristic
95	C₂	—	137	2	no	—
96	C₂	D₈×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#10587739
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	Q₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₄×D₈
prime radical	C₂×D₁₃₇	C₂×D₈
cosocle	C₈×D₁₃₇	C₂³
hypercenter	Q₃₂	#10587739
hypocenter	C₁₃₇	#2176
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₁₀₉₆	C₄	semidirect
13. #17984034	C₄	semidirect
15. Q₃₂×C₁₃₇	C₄	semidirect
48. C₁₃₇	#2176	semidirect
59. Q₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#10587739	2
#22578330	2
#22578316	2
#22578290	2
Q₃₂×D₁₃₇	2
#2176	137