Subgroups — #31780919

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
2	#22577269	C₂	1	1	yes	maximal, characteristic
3	#22577165	C₂	1	4	yes	maximal
4	#382×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
7	C₄×#11764040	C₄	1	4	yes	complemented
8	#17983995	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#95×C₁₃₇	—	2	2	no	—
12	C₂²×Dic₁₃₇	—	2	2	no	—
13	C₂×#11764040	—	2	2	no	—
14	C₂×#11764040	—	2	2	no	—
15	C₂×#11764040	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₃₇	—	2	4	no	—
17	C₂²×C₄×C₁₃₇	—	2	2	no	—
18	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	—	2	4	no	—
21	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₃₇	Q₈	1	2	yes	—
23	C₂×Dic₁₃₇	—	4	2	no	—
24	C₂×Dic₁₃₇	—	4	4	no	—
25	#11764040	—	2	4	no	—
26	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	—	2	4	no	—
28	C₂×C₄×C₁₃₇	—	4	2	no	—
29	C₂×C₄×C₁₃₇	—	4	4	no	—
30	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	#41	1	2	yes	—
34	C₂×C₄×C₁₃₇	#41	1	2	yes	—
35	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	#124	1	1	yes	characteristic
37	C₂²×C₁₃₇	#124	1	1	yes	characteristic
38	C₂²×C₁₃₇	#135	1	4	yes	—
39	C₄×C₁₃₇	—	2	4	no	—
40	C₄×C₁₃₇	—	8	2	no	—
41	C₄×C₁₃₇	#117	1	2	yes	—
42	C₄×C₁₃₇	#125	1	2	yes	—
43	Dic₁₃₇	—	8	2	no	—
44	C₂×C₁₃₇	#382	1	1	yes	characteristic
45	C₂×C₁₃₇	#427	1	2	yes	—
46	C₂×C₁₃₇	#488	1	2	yes	—
47	C₂×C₁₃₇	#488	1	2	yes	—
48	C₁₃₇	#1835	1	1	yes	characteristic, complemented, Sylow
49	#1835	—	137	1	no	maximal, Sylow
50	C₂×C₄×C₈	—	137	1	no	—
51	#382	—	137	1	no	—
52	#382	D₁₃₇	1	1	yes	characteristic
53	#340	—	137	4	no	—
54	C₂²×C₈	—	137	2	no	—
55	C₂×C₄⋊C₄	—	137	2	no	—
56	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
57	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
58	C₄×C₈	—	137	4	no	—
59	#95	—	2	2	no	—
60	#95	—	274	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	274	2	no	—
63	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₂×C₈	—	274	2	no	—
66	C₂×C₈	—	274	2	no	—
67	C₂×C₈	—	137	4	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	274	4	no	—
70	C₄²	C₄×D₁₃₇	1	2	yes	—
71	C₄²	Dic₂₇₄	1	2	yes	—
72	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	548	2	no	—
77	C₂×C₄	—	548	4	no	—
78	C₂×C₄	#14555489	1	4	yes	—
79	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
81	C₂×C₄	#14555509	1	2	yes	—
82	C₂×C₄	#14555511	1	2	yes	—
83	C₈	—	274	4	no	—
84	C₂²	C₂×#14555489	1	1	yes	characteristic
85	C₂²	#17984075	1	1	yes	characteristic
86	C₂²	#17984077	1	1	yes	characteristic
87	C₂²	#17984108	1	4	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	—	1096	2	no	—
91	C₄	#17984046	1	2	yes	—
92	C₄	#17984079	1	2	yes	—
93	C₂	#22577268	1	1	yes	characteristic
94	C₂	#22577439	1	2	yes	—
95	C₂	#22577757	1	2	yes	—
96	C₂	#22577760	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₇
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#382×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#382	D₁₃₇
hypocenter	C₁₃₇	#1835
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
7. C₄×#11764040	C₄	semidirect
48. C₁₃₇	#1835	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11764040	2
#22577269	2
#22577165	2
#382×C₁₃₇	2
#1835	137