Subgroups — #31780926

Order 17536.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
3	#22577167	C₂	1	4	yes	maximal
4	#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	C₂²	1	2	yes	—
6	C₂²×#11764040	C₂²	1	2	yes	—
7	C₄⋊#11764040	—	2	2	no	—
8	C₄×#11764040	C₄	1	2	yes	—
9	C₄×#11764040	C₄	1	2	yes	—
10	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	—	2	2	no	—
13	#14555494	—	2	4	no	—
14	C₂×#11764040	—	2	2	no	—
15	C₂×#11764040	—	2	2	no	—
16	C₂×#11764040	—	2	2	no	—
17	C₂×#11764040	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₃₇	—	2	4	no	—
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	D₄	1	2	yes	—
25	#11764040	—	2	4	no	—
26	#11764040	—	4	2	no	—
27	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	4	no	—
29	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	#41	1	2	yes	—
34	C₂×C₄×C₁₃₇	#41	1	2	yes	—
35	C₈×C₁₃₇	—	4	2	no	—
36	C₂²×C₁₃₇	—	2	2	no	—
37	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₃₇	#125	1	2	yes	—
39	C₄×C₁₃₇	—	2	2	no	—
40	C₄×C₁₃₇	#108	1	4	yes	—
41	C₄×C₁₃₇	#117	1	2	yes	—
42	C₄×C₁₃₇	#124	1	1	yes	characteristic
43	C₄×C₁₃₇	#124	1	1	yes	characteristic
44	C₂×C₁₃₇	—	2	2	no	—
45	C₂×C₁₃₇	#382	1	1	yes	characteristic
46	C₂×C₁₃₇	C₂×#108	1	2	yes	—
47	C₁₃₇	#1837	1	1	yes	characteristic, complemented, Sylow
48	#1837	—	137	1	no	maximal, Sylow
49	#423	—	137	1	no	—
50	#423	D₁₃₇	1	1	yes	characteristic
51	C₂×C₄×C₈	—	137	1	no	—
52	#341	—	137	4	no	—
53	C₂×M₁₆	—	137	2	no	—
54	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
55	C₂²×C₈	—	137	2	no	—
56	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	274	2	no	—
59	C₄×C₈	—	137	2	no	—
60	C₄×C₈	—	137	2	no	—
61	C₂²×C₄	Dic₂₇₄	1	2	yes	—
62	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	274	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	274	2	no	—
67	C₂×C₈	—	274	2	no	—
68	C₂×C₈	—	274	2	no	—
69	C₂×C₈	—	137	4	no	—
70	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
71	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
72	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
73	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	#14555489	1	2	yes	—
76	C₂×C₄	#14555489	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
79	C₂×C₄	#14555509	1	2	yes	—
80	C₂×C₄	#14555511	1	2	yes	—
81	C₈	—	4	2	no	—
82	C₈	—	274	4	no	—
83	C₈	—	548	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×#14555489	1	1	yes	characteristic
86	C₂²	#17984079	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#17984023	1	4	yes	—
89	C₄	#17984046	1	2	yes	—
90	C₄	#17984075	1	1	yes	characteristic
91	C₄	#17984077	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#22577268	1	1	yes	characteristic
94	C₂	C₂×#17984023	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#423×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#423	D₁₃₇
hypocenter	C₁₃₇	#1837
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

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

Maximal subgroups

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

subgroup	index
#22577427	2
C₂×C₄×#11764040	2
#22577167	2
#423×C₁₃₇	2
#1837	137