Subgroups — #31795766

Order 17536.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578300	C₂	1	1	yes	maximal, characteristic
2	#22578272	C₂	1	1	yes	maximal, characteristic
3	#22577662	C₂	1	1	yes	maximal, characteristic
4	D₁₃₇⋊M₁₆	—	2	1	no	—
5	#17984162	—	2	1	no	—
6	C₂×Dic₅₄₈	C₄	1	1	yes	characteristic
7	C₂×#14555493	C₂²	1	1	yes	characteristic
8	#17984068	—	2	1	no	—
9	#17984011	—	2	1	no	—
10	#106×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555520	—	4	1	no	—
12	#14555519	—	4	1	no	—
13	#14555518	—	2	2	no	—
14	#14555513	—	4	1	no	—
15	C₂×Dic₂₇₄	—	2	1	no	—
16	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
17	Dic₅₄₈	—	4	1	no	—
18	#14555493	—	4	1	no	—
19	C₂×#11764040	D₄	1	1	yes	characteristic
20	C₄⋊Dic₁₃₇	—	4	1	no	—
21	#14555487	—	4	1	no	—
22	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₇	—	2	1	no	—
24	#11764047	—	4	2	no	—
25	C₂²×D₁₃₇	SD₁₆	1	1	yes	characteristic
26	Dic₂₇₄	—	4	1	no	—
27	Dic₂₇₄	—	4	1	no	—
28	C₂×Dic₁₃₇	—	4	1	no	—
29	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
30	C₄×D₁₃₇	—	2	1	no	—
31	C₄×D₁₃₇	—	2	1	no	—
32	#11764040	—	2	1	no	—
33	C₂×C₄×C₁₃₇	—	4	1	no	—
34	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₈×C₁₃₇	—	2	1	no	—
36	C₂²×C₁₃₇	#102	1	1	yes	characteristic
37	C₂×D₁₃₇	—	2	1	no	—
38	C₂×D₁₃₇	—	4	1	no	—
39	C₄×C₁₃₇	—	8	1	no	—
40	C₄×C₁₃₇	#99	1	1	yes	characteristic
41	C₄×C₁₃₇	#104	1	1	yes	characteristic
42	Dic₁₃₇	—	2	1	no	—
43	Dic₁₃₇	—	8	1	no	—
44	C₂×C₁₃₇	—	2	1	no	—
45	C₂×C₁₃₇	#327	1	1	yes	characteristic
46	D₁₃₇	—	4	1	no	—
47	C₁₃₇	#1342	1	1	yes	characteristic, complemented, Sylow
48	#1342	—	137	1	no	maximal, Sylow
49	#470	—	137	1	no	—
50	#349	—	137	1	no	—
51	#338	—	137	1	no	—
52	C₂×Q₁₆	—	137	1	no	—
53	C₂×M₁₆	—	274	1	no	—
54	C₂×M₁₆	—	137	1	no	—
55	C₂²⋊Q₈	—	274	1	no	—
56	M₃₂	—	274	1	no	—
57	#106	#10587739	1	1	yes	characteristic
58	#103	—	274	1	no	—
59	C₂×Q₈	—	274	1	no	—
60	C₂²×C₄	—	137	1	no	—
61	Q₁₆	—	548	1	no	—
62	M₁₆	—	548	1	no	—
63	M₁₆	—	548	1	no	—
64	M₁₆	—	548	1	no	—
65	C₂×C₈	—	274	2	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
68	C₄⋊C₄	—	2	1	no	—
69	C₄⋊C₄	—	548	1	no	—
70	C₂²⋊C₄	—	548	1	no	—
71	C₁₆	—	548	1	no	—
72	C₂³	—	137	1	no	—
73	Q₈	—	548	1	no	—
74	Q₈	—	548	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	274	1	no	—
77	C₂×C₄	—	274	1	no	—
78	C₂×C₄	—	548	1	no	—
79	C₂×C₄	—	137	1	no	—
80	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	274	1	no	—
83	C₈	—	548	2	no	—
84	C₂²	—	274	1	no	—
85	C₂²	—	548	1	no	—
86	C₂²	#17984150	1	1	yes	characteristic
87	C₄	—	8	1	no	—
88	C₄	—	274	1	no	—
89	C₄	—	1096	1	no	—
90	C₄	#17984147	1	1	yes	characteristic
91	C₄	#17984153	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	548	1	no	—
94	C₂	#22578255	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578255
commutator	C₂×C₈×C₁₃₇	C₂×C₄
Frattini	C₂×C₈	C₂×#10587739
Fitting	#106×C₁₃₇	C₄
socle	C₂×C₁₃₇	#327
prime radical	C₂²×D₁₃₇	SD₁₆
cosocle	C₂×#14555493	C₂²
hypercenter	#106	#10587739
hypocenter	C₁₃₇	#1342
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
47. C₁₃₇	#1342	semidirect

Maximal subgroups

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

subgroup	index
#22578300	2
#22578272	2
#22577662	2
#1342	137