Finite Group Databasebeta

Subgroups — #31797931

Order 17536.

260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11764049	C₂	1	2	yes	maximal
2	(C₂×#11764049)⋊C₄	C₂	1	4	yes	maximal
3	C₄²×#10587739	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764049	C₂²	1	4	yes	—
5	C₄⋊#11764049	—	2	4	no	—
6	C₂×C₄×#10587739	C₄	1	2	yes	—
7	C₂×C₄×#10587739	C₄	1	1	yes	characteristic
8	C₂×C₄×#10587739	C₂²	1	2	yes	—
9	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
10	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#11764049	—	2	4	no	—
12	C₂×#11764049	—	2	8	no	—
13	C₂²×#10587739	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10587739	C₂×C₄	1	2	yes	—
15	C₂²×#10587739	C₂³	1	1	yes	characteristic
16	C₄×#10587739	—	2	4	no	—
17	C₄×#10587739	—	2	4	no	—
18	C₄×#10587739	—	2	2	no	—
19	C₄×#10587739	D₄	1	2	yes	—
20	C₄×#10587739	Q₈	1	2	yes	—
21	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₃₇	C₈	1	2	yes	complemented
24	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic, complemented
25	C₄²×C₁₃₇	C₈	1	1	yes	characteristic, complemented
26	#11764049	—	4	4	no	—
27	C₂×#10587739	—	2	2	no	—
28	C₂×#10587739	—	2	2	no	—
29	C₂×#10587739	—	2	2	no	—
30	C₂×#10587739	C₄⋊C₄	1	2	yes	—
31	C₂×#10587739	M₁₆	1	4	yes	—
32	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
33	C₂×#10587739	C₂×Q₈	1	1	yes	characteristic
34	C₂×#10587739	#41	1	4	yes	—
35	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
37	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
38	C₄×D₁₃₇	—	2	2	no	—
39	C₄×D₁₃₇	—	2	2	no	—
40	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
41	C₄×D₁₃₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	C₂×D₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₁₃₇	#117	1	2	yes	—
47	C₂×D₁₃₇	C₂×M₁₆	1	2	yes	—
48	C₂×D₁₃₇	C₂×M₁₆	1	1	yes	characteristic
49	#10587739	—	2	2	no	—
50	#10587739	—	2	4	no	—
51	#10587739	#124	1	2	yes	—
52	#10587739	#125	1	2	yes	—
53	C₄×C₁₃₇	—	2	2	no	—
54	C₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
55	Dic₁₃₇	—	2	2	no	—
56	Dic₁₃₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₁₃₇	#431	1	2	yes	—
59	D₁₃₇	#382	1	1	yes	characteristic
60	D₁₃₇	#423	1	1	yes	characteristic
61	D₁₃₇	#432	1	2	yes	—
62	C₁₃₇	#1831	1	1	yes	characteristic, complemented, Sylow
63	#1831	—	137	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	137	2	no	—
65	C₄³	—	137	1	no	—
66	#336	—	137	4	no	—
67	C₂²×C₈	—	137	4	no	—
68	C₂×C₄²	—	137	2	no	—
69	C₂×C₄²	—	137	2	no	—
70	C₂×C₄²	—	137	1	no	—
71	C₂×C₄²	—	137	1	no	—
72	C₂×C₄²	—	137	1	no	—
73	C₄⋊C₈	—	274	4	no	—
74	C₂²×C₄	—	137	2	no	—
75	C₂²×C₄	—	137	1	no	—
76	C₂²×C₄	—	137	2	no	—
77	C₂²×C₄	—	137	1	no	—
78	C₂²×C₄	—	137	1	no	—
79	C₂×C₈	—	274	4	no	—
80	C₂×C₈	—	274	8	no	—
81	C₄²	—	274	4	no	—
82	C₄²	—	274	4	no	—
83	C₄²	—	274	2	no	—
84	C₄²	—	137	2	no	—
85	C₄²	—	137	2	no	—
86	C₄²	—	137	1	no	—
87	C₄²	—	137	2	no	—
88	C₄²	#11764049	1	1	yes	characteristic, complemented
89	C₂³	—	137	1	no	—
90	C₂×C₄	—	274	2	no	—
91	C₂×C₄	—	274	2	no	—
92	C₂×C₄	—	274	2	no	—
93	C₂×C₄	—	274	2	no	—
94	C₂×C₄	—	274	2	no	—
95	C₂×C₄	—	137	4	no	—
96	C₂×C₄	—	137	2	no	—
97	C₂×C₄	—	137	2	no	—
98	C₂×C₄	—	137	4	no	—
99	C₂×C₄	—	137	1	no	—
100	C₂×C₄	—	137	1	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	—	137	2	no	—
103	C₂×C₄	—	137	2	no	—
104	C₂×C₄	C₂×#11764049	1	2	yes	—
105	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
106	C₈	—	548	4	no	—
107	C₂²	—	137	1	no	—
108	C₂²	—	137	1	no	—
109	C₂²	—	137	2	no	—
110	C₂²	—	137	2	no	—
111	C₂²	C₂²×#11764049	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	274	2	no	—
114	C₄	—	274	4	no	—
115	C₄	—	274	2	no	—
116	C₄	—	137	2	no	—
117	C₄	—	137	2	no	—
118	C₄	—	137	2	no	—
119	C₄	C₄⋊#11764049	1	2	yes	—
120	C₂	—	137	1	no	—
121	C₂	—	137	2	no	—
122	C₂	—	137	1	no	—
123	C₂	C₂×C₄⋊#11764049	1	1	yes	characteristic
124	C₂	#22578516	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₄²×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂²×#10587739	C₂³
hypercenter	C₄²	#11764049
hypocenter	C₁₃₇	#1831
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
23. C₄×Dic₁₃₇	C₈	semidirect
24. C₄×Dic₁₃₇	C₈	semidirect
25. C₄²×C₁₃₇	C₈	semidirect
62. C₁₃₇	#1831	semidirect
88. C₄²	#11764049	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11764049	2
(C₂×#11764049)⋊C₄	2
C₄²×#10587739	2
#1831	137