Subgroups — #31797823

Order 17536.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14555522	C₂	1	2	yes	maximal
2	C₂×#17984155	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555522	—	2	2	no	—
4	C₂²×#11764047	C₄	1	1	yes	characteristic
5	C₂²×#11764047	C₂²	1	1	yes	characteristic
6	#17984155	—	2	2	no	—
7	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
8	#14555522	—	4	2	no	—
9	C₂×#11764047	—	2	1	no	—
10	C₂×#11764047	—	4	1	no	—
11	C₂×#11764047	—	4	1	no	—
12	C₂×#11764047	D₄	1	2	yes	—
13	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
14	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₄×Dic₁₃₇	—	2	1	no	—
16	C₄×Dic₁₃₇	—	2	1	no	—
17	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
18	#11764047	—	2	2	no	—
19	#11764047	—	4	1	no	—
20	C₂×Dic₁₃₇	—	2	1	no	—
21	C₂×Dic₁₃₇	—	2	1	no	—
22	C₂×Dic₁₃₇	—	2	1	no	—
23	C₂×Dic₁₃₇	—	2	1	no	—
24	C₂×Dic₁₃₇	C₁₆	1	2	yes	complemented
25	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
27	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	4	1	no	—
30	C₂²×C₁₃₇	—	2	1	no	—
31	C₂²×C₁₃₇	—	2	1	no	—
32	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
34	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
35	C₄×C₁₃₇	—	4	1	no	—
36	Dic₁₃₇	—	2	1	no	—
37	Dic₁₃₇	—	2	2	no	—
38	Dic₁₃₇	#99	1	1	yes	characteristic
39	Dic₁₃₇	#101	1	1	yes	characteristic
40	C₂×C₁₃₇	—	2	1	no	—
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	#324	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₂×C₁₃₇	#349	1	1	yes	characteristic
45	C₁₃₇	#1322	1	1	yes	characteristic, complemented, Sylow
46	#1322	—	137	1	no	maximal, Sylow
47	C₂×C₄⋊C₈	—	137	1	no	—
48	C₂²⋊C₁₆	—	137	2	no	—
49	C₂²×C₈	—	137	1	no	—
50	C₂²×C₈	—	137	1	no	—
51	C₂×C₄²	—	137	1	no	—
52	C₂×C₁₆	—	274	2	no	—
53	C₄⋊C₈	—	274	2	no	—
54	C₂²×C₄	—	137	1	no	—
55	C₂²×C₄	—	137	1	no	—
56	C₂²×C₄	#11764049	1	1	yes	characteristic
57	C₂×C₈	—	274	1	no	—
58	C₂×C₈	—	548	1	no	—
59	C₂×C₈	—	548	1	no	—
60	C₂×C₈	—	137	2	no	—
61	C₄²	—	274	1	no	—
62	C₄²	—	274	1	no	—
63	C₁₆	—	548	2	no	—
64	C₂³	C₂×#11764049	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	274	1	no	—
68	C₂×C₄	—	274	1	no	—
69	C₂×C₄	—	274	1	no	—
70	C₂×C₄	—	274	1	no	—
71	C₂×C₄	—	137	1	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	—	137	1	no	—
74	C₈	—	274	2	no	—
75	C₈	—	548	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
79	C₂²	#17984183	1	1	yes	characteristic
80	C₂²	#17984185	1	1	yes	characteristic
81	C₄	—	4	1	no	—
82	C₄	—	274	1	no	—
83	C₄	—	274	2	no	—
84	C₄	—	137	1	no	—
85	C₄	—	137	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#22578489	1	1	yes	characteristic
89	C₂	#22578500	1	1	yes	characteristic
90	C₂	#22578501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11764049
commutator	C₂²×C₁₃₇	C₂×C₁₆
Frattini	C₂³	C₂×#11764049
Fitting	C₂²×C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²⋊C₈
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂²×#11764047	C₂²
hypercenter	C₂²×C₄	#11764049
hypocenter	C₁₃₇	#1322
LCS	G ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
24. C₂×Dic₁₃₇	C₁₆	semidirect
45. C₁₃₇	#1322	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14555522	2
C₂×#17984155	2
#1322	137