Finite Group Databasebeta

Subgroups — #31797141

Order 17536.

156 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22578413	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984157	C₂	1	1	yes	maximal, characteristic
3	#22578330	C₂	1	2	yes	maximal, complemented
4	#22578329	C₂	1	2	yes	maximal, complemented
5	M₃₂×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984167	—	2	1	no	—
7	C₂×C₄⋊#10587739	—	2	1	no	—
8	#17984157	—	2	1	no	—
9	#17984157	C₂²	1	1	yes	characteristic
10	#17984157	C₂²	1	2	yes	—
11	#17984157	C₂²	1	1	yes	characteristic
12	#17984156	—	2	1	no	—
13	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
14	#17984028	C₄	1	1	yes	characteristic, complemented
15	#17984027	C₄	1	2	yes	complemented
16	C₁₆×D₁₃₇	C₂²	1	2	yes	—
17	M₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×#10587739	—	4	1	no	—
19	C₄⋊#10587739	—	2	1	no	—
20	C₄⋊#10587739	—	2	1	no	—
21	C₄⋊#10587739	—	2	2	no	—
22	C₄⋊#10587739	—	4	1	no	—
23	C₂²⋊#10587739	—	4	1	no	—
24	C₄×#10587739	—	4	1	no	—
25	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
26	C₂×#11764040	Q₈	1	1	yes	characteristic
27	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₃₇	D₄	1	1	yes	characteristic
29	C₈×D₁₃₇	Q₈	1	1	yes	characteristic
30	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
31	#14555485	C₂×C₄	1	2	yes	complemented
32	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
34	C₂×#10587739	—	4	1	no	—
35	C₂×#10587739	—	4	1	no	—
36	C₂×#10587739	—	4	2	no	—
37	C₂×#10587739	—	8	2	no	—
38	C₂²×D₁₃₇	D₈	1	1	yes	characteristic
39	C₂×Dic₁₃₇	Q₁₆	1	1	yes	characteristic
40	C₄×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×D₁₃₇	D₈	1	1	yes	characteristic
42	C₄×D₁₃₇	Q₁₆	1	1	yes	characteristic
43	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
44	#11764040	C₄⋊C₄	1	1	yes	characteristic
45	#11764040	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₁₃₇	#107	1	1	yes	characteristic
50	C₂×D₁₃₇	—	2	1	no	—
51	C₂×D₁₃₇	—	2	1	no	—
52	C₂×D₁₃₇	#107	1	1	yes	characteristic
53	C₂×D₁₃₇	C₂×D₈	1	1	yes	characteristic
54	#10587739	—	8	1	no	—
55	#10587739	—	8	2	no	—
56	#10587739	—	8	1	no	—
57	C₄×C₁₃₇	#107	1	1	yes	characteristic
58	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	Dic₁₃₇	#107	1	1	yes	characteristic
60	Dic₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₁₃₇	—	2	1	no	—
62	C₂×C₁₃₇	C₂×#107	1	1	yes	characteristic
63	D₁₃₇	—	2	1	no	—
64	D₁₃₇	#509	1	1	yes	characteristic
65	D₁₃₇	#510	1	1	yes	characteristic
66	C₁₃₇	#2159	1	1	yes	characteristic, complemented, Sylow
67	#2159	—	137	1	no	maximal, Sylow
68	C₂×M₃₂	—	137	1	no	—
69	#427	—	137	1	no	—
70	C₂×#107	—	137	1	no	—
71	#367	—	137	2	no	—
72	#366	—	137	2	no	—
73	C₂²×C₈	—	137	1	no	—
74	#117	—	274	1	no	—
75	C₂×C₄⋊C₄	—	274	1	no	—
76	M₃₂	—	137	1	no	—
77	M₃₂	—	137	2	no	—
78	M₃₂	#10587739	1	1	yes	characteristic, complemented
79	C₂×C₁₆	—	137	2	no	—
80	#107	—	274	1	no	—
81	#107	—	137	1	no	—
82	#107	—	137	2	no	—
83	#107	—	137	1	no	—
84	#106	—	274	1	no	—
85	C₂²×C₄	—	548	1	no	—
86	C₂²×C₄	—	137	1	no	—
87	C₂×C₈	—	137	1	no	—
88	C₂×C₈	—	137	1	no	—
89	C₂×C₈	—	137	1	no	—
90	C₂×C₈	—	137	1	no	—
91	C₂×C₈	—	137	1	no	—
92	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
93	C₄⋊C₄	—	274	1	no	—
94	C₄⋊C₄	—	274	1	no	—
95	C₄⋊C₄	—	274	2	no	—
96	C₄⋊C₄	—	548	1	no	—
97	C₂²⋊C₄	—	548	1	no	—
98	C₄²	—	548	1	no	—
99	C₁₆	—	137	2	no	—
100	C₁₆	C₂×#10587739	1	2	yes	complemented
101	C₂³	—	137	1	no	—
102	C₂×C₄	—	548	1	no	—
103	C₂×C₄	—	548	1	no	—
104	C₂×C₄	—	548	2	no	—
105	C₂×C₄	—	1096	2	no	—
106	C₂×C₄	—	137	1	no	—
107	C₂×C₄	—	137	1	no	—
108	C₂×C₄	—	137	1	no	—
109	C₂×C₄	—	137	1	no	—
110	C₂×C₄	—	137	1	no	—
111	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
112	C₈	—	137	1	no	—
113	C₈	—	137	1	no	—
114	C₈	C₄⋊#10587739	1	1	yes	characteristic
115	C₈	C₂²×#10587739	1	1	yes	characteristic
116	C₂²	—	274	1	no	—
117	C₂²	—	274	1	no	—
118	C₂²	—	137	1	no	—
119	C₂²	—	137	1	no	—
120	C₂²	#17984157	1	1	yes	characteristic
121	C₄	—	1096	2	no	—
122	C₄	—	1096	1	no	—
123	C₄	—	1096	1	no	—
124	C₄	—	137	1	no	—
125	C₄	—	137	1	no	—
126	C₄	#17984157	1	1	yes	characteristic
127	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	274	1	no	—
130	C₂	—	137	1	no	—
131	C₂	—	137	1	no	—
132	C₂	C₂×#17984157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17984157
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	M₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×#107
prime radical	C₂²×D₁₃₇	D₈
cosocle	C₈×D₁₃₇	C₂³
hypercenter	M₃₂	#10587739
hypocenter	C₁₃₇	#2159
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
3. #22578330	C₂	semidirect
4. #22578329	C₂	semidirect
14. #17984028	C₄	semidirect
15. #17984027	C₄	semidirect
17. M₃₂×C₁₃₇	C₄	semidirect
31. #14555485	C₂×C₄	semidirect
33. C₁₆×C₁₃₇	C₂×C₄	semidirect
66. C₁₃₇	#2159	semidirect
78. M₃₂	#10587739	semidirect
100. C₁₆	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578413	2
C₂×#17984157	2
#22578330	2
#22578329	2
M₃₂×D₁₃₇	2
#2159	137