Subgroups — #31797192

Order 17536.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578447	C₂	1	2	yes	maximal, complemented
2	#22578332	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578318	C₂	1	2	yes	maximal, complemented
4	#22578290	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577836	C₂	1	1	yes	maximal, characteristic
6	#17984176	—	2	2	no	—
7	#17984161	C₂²	1	1	yes	characteristic
8	#17984159	C₂²	1	2	yes	—
9	#17984153	—	2	2	no	—
10	C₈×#10587739	C₂²	1	1	yes	characteristic
11	#17984111	C₂²	1	2	yes	—
12	#17984031	C₄	1	2	yes	complemented
13	D₂₁₉₂	C₄	1	1	yes	characteristic, complemented
14	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	#14555520	—	2	2	no	—
17	#14555520	—	4	2	no	—
18	#14555518	—	4	2	no	—
19	#14555518	D₄	1	1	yes	characteristic
20	C₄×#10587739	D₄	1	1	yes	characteristic
21	#14555513	—	2	1	no	—
22	#14555511	—	2	2	no	—
23	#14555497	—	2	2	no	—
24	D₁₀₉₆	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
30	#11764047	—	4	2	no	—
31	#11764047	D₈	1	2	yes	—
32	D₅₄₈	—	2	2	no	—
33	D₅₄₈	—	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	—	4	2	no	—
40	C₂×D₁₃₇	#135	1	1	yes	characteristic
41	#10587739	—	4	1	no	—
42	C₄×C₁₃₇	—	4	2	no	—
43	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
44	Dic₁₃₇	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
46	D₁₃₇	—	2	1	no	—
47	D₁₃₇	—	8	2	no	—
48	C₁₃₇	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	137	1	no	maximal, Sylow
50	#508	—	137	1	no	—
51	#443	—	137	2	no	—
52	#368	—	137	1	no	—
53	#359	—	137	2	no	—
54	#346	—	137	1	no	—
55	#135	—	137	2	no	—
56	#131	—	274	2	no	—
57	Q₃₂	#10587739	1	1	yes	characteristic, complemented
58	SD₃₂	—	137	2	no	—
59	D₁₆	—	137	1	no	—
60	C₂×C₁₆	—	137	1	no	—
61	C₂×C₁₆	—	137	1	no	—
62	#108	—	137	2	no	—
63	#104	—	274	2	no	—
64	C₄×C₈	—	137	1	no	—
65	#41	—	274	2	no	—
66	Q₁₆	C₂×#10587739	1	2	yes	—
67	SD₁₆	—	274	2	no	—
68	D₈	—	137	2	no	—
69	M₁₆	—	274	2	no	—
70	M₁₆	—	548	2	no	—
71	C₂×C₈	—	548	2	no	—
72	C₂×C₈	—	137	1	no	—
73	C₂×C₈	—	137	1	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	D₄	—	274	2	no	—
80	D₄	—	548	2	no	—
81	C₂×C₄	—	274	1	no	—
82	C₂×C₄	—	548	2	no	—
83	C₂×C₄	—	137	1	no	—
84	C₈	—	548	2	no	—
85	C₈	—	137	2	no	—
86	C₈	—	137	1	no	—
87	C₈	C₂²×#10587739	1	1	yes	characteristic
88	C₂²	—	548	2	no	—
89	C₂²	—	137	1	no	—
90	C₄	—	4	2	no	—
91	C₄	—	548	1	no	—
92	C₄	—	137	1	no	—
93	C₄	D₄×#10587739	1	1	yes	characteristic
94	C₂	—	274	1	no	—
95	C₂	—	1096	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	#22577836	C₂
cosocle	C₈×D₁₃₇	C₂³
hypercenter	Q₃₂	#10587739
hypocenter	C₁₃₇	#2179
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
1. #22578447	C₂	semidirect
2. #22578332	C₂	semidirect
3. #22578318	C₂	semidirect
4. #22578290	C₂	semidirect
12. #17984031	C₄	semidirect
13. D₂₁₉₂	C₄	semidirect
15. Q₃₂×C₁₃₇	C₄	semidirect
48. C₁₃₇	#2179	semidirect
57. Q₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578447	2
#22578332	2
#22578318	2
#22578290	2
#22577836	2
#2179	137