Subgroups — #31796714

Order 17536.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578408	C₂	1	2	yes	maximal
2	#22578279	C₂	1	4	yes	maximal
3	C₄×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555520	C₂²	1	4	yes	—
5	#17984155	—	2	4	no	—
6	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
7	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
8	C₄×#11764040	C₄	1	1	yes	characteristic
9	C₈×Dic₁₃₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555520	—	2	8	no	—
12	C₂×#11764047	—	2	4	no	—
13	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
14	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
15	C₂×#11764040	C₂×C₄	1	2	yes	—
16	C₈×D₁₃₇	—	2	4	no	—
17	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₁₃₇	D₄	1	2	yes	—
19	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	#11764047	—	4	4	no	—
22	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
23	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
24	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
25	C₄×D₁₃₇	—	2	4	no	—
26	C₄×D₁₃₇	#41	1	4	yes	—
27	#11764040	—	2	2	no	—
28	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×D₁₃₇	—	2	2	no	—
33	C₂×D₁₃₇	#125	1	2	yes	—
34	C₄×C₁₃₇	—	2	2	no	—
35	C₄×C₁₃₇	#117	1	2	yes	—
36	Dic₁₃₇	#108	1	4	yes	—
37	Dic₁₃₇	#124	1	2	yes	—
38	C₂×C₁₃₇	#382	1	1	yes	characteristic
39	C₂×C₁₃₇	C₂×#108	1	2	yes	—
40	D₁₃₇	—	2	2	no	—
41	C₁₃₇	#1837	1	1	yes	characteristic, complemented, Sylow
42	#1837	—	137	1	no	maximal, Sylow
43	#423	—	137	2	no	—
44	C₂×C₄×C₈	—	137	1	no	—
45	#341	—	137	4	no	—
46	C₂×M₁₆	—	137	4	no	—
47	C₂²×C₈	—	137	2	no	—
48	C₂×C₄²	—	137	1	no	—
49	C₄⋊C₈	—	274	4	no	—
50	C₄×C₈	—	137	2	no	—
51	C₄×C₈	—	137	1	no	—
52	C₄×C₈	#10587739	1	1	yes	characteristic
53	C₂²×C₄	—	137	2	no	—
54	C₂²×C₄	—	137	1	no	—
55	M₁₆	—	274	8	no	—
56	C₂×C₈	—	274	4	no	—
57	C₂×C₈	—	274	4	no	—
58	C₂×C₈	—	137	2	no	—
59	C₂×C₈	C₂×#10587739	1	2	yes	—
60	C₄²	—	137	2	no	—
61	C₄²	—	137	1	no	—
62	C₄²	C₂×#10587739	1	1	yes	characteristic
63	C₂³	—	137	1	no	—
64	C₂×C₄	—	274	4	no	—
65	C₂×C₄	—	137	1	no	—
66	C₂×C₄	—	137	2	no	—
67	C₂×C₄	—	137	4	no	—
68	C₂×C₄	C₄⋊#10587739	1	2	yes	—
69	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	274	2	no	—
72	C₈	—	548	4	no	—
73	C₂²	—	274	2	no	—
74	C₂²	—	137	2	no	—
75	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	137	2	no	—
78	C₄	—	137	4	no	—
79	C₄	#17984167	1	2	yes	—
80	C₂	—	274	2	no	—
81	C₂	#22578347	1	1	yes	characteristic
82	C₂	D₁₃₇⋊#108	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1837
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
41. C₁₃₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#22578408	2
#22578279	2
C₄×C₈×D₁₃₇	2
#1837	137