Finite Group Databasebeta

Subgroups — #31797136

Order 17536.

144 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₁₃₇⋊#108	C₂	1	1	yes	maximal, characteristic
2	#22578413	C₂	1	1	yes	maximal, characteristic
3	#22578328	C₂	1	2	yes	maximal, complemented
4	#22578327	C₂	1	2	yes	maximal, complemented
5	C₂×#17984027	C₂	1	1	yes	maximal, characteristic
6	D₁₃₇⋊M₁₆	—	2	1	no	—
7	#17984167	—	2	1	no	—
8	#17984159	—	2	1	no	—
9	#17984158	C₂²	1	2	yes	—
10	#17984157	—	2	1	no	—
11	#17984156	C₂²	1	2	yes	—
12	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
13	#17984027	C₄	1	1	yes	characteristic, complemented
14	#17984027	C₄	1	1	yes	characteristic, complemented
15	#17984027	C₂²	1	1	yes	characteristic
16	#17984027	C₂²	1	1	yes	characteristic
17	C₂×#14555485	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
19	#14555520	—	2	2	no	—
20	#14555519	—	4	1	no	—
21	#14555518	—	4	1	no	—
22	C₄⋊#10587739	—	2	2	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	1	yes	characteristic
32	#14555485	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×#10587739	—	4	2	no	—
37	#11764047	—	4	2	no	—
38	C₂²×D₁₃₇	SD₁₆	1	1	yes	characteristic
39	C₂×Dic₁₃₇	SD₁₆	1	1	yes	characteristic
40	C₄×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×D₁₃₇	SD₁₆	1	1	yes	characteristic
42	C₄×D₁₃₇	SD₁₆	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	#106	1	1	yes	characteristic
50	C₂×D₁₃₇	—	4	1	no	—
51	C₂×D₁₃₇	#106	1	1	yes	characteristic
52	C₂×D₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
53	#10587739	—	8	2	no	—
54	C₄×C₁₃₇	#106	1	1	yes	characteristic
55	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	Dic₁₃₇	#106	1	1	yes	characteristic
57	Dic₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	C₂×#106	1	1	yes	characteristic
60	D₁₃₇	—	2	1	no	—
61	D₁₃₇	—	2	1	no	—
62	C₁₃₇	#2155	1	1	yes	characteristic, complemented, Sylow
63	#2155	—	137	1	no	maximal, Sylow
64	C₂×M₃₂	—	137	1	no	—
65	C₂×#108	—	137	1	no	—
66	#427	—	137	1	no	—
67	#365	—	137	2	no	—
68	#365	—	137	2	no	—
69	C₂×M₁₆	—	274	1	no	—
70	C₂²×C₈	—	137	1	no	—
71	#117	—	274	1	no	—
72	M₃₂	—	137	1	no	—
73	M₃₂	—	137	1	no	—
74	M₃₂	—	137	1	no	—
75	M₃₂	—	137	1	no	—
76	C₂×C₁₆	—	137	1	no	—
77	C₂×C₁₆	#10587739	1	1	yes	characteristic, complemented
78	#108	—	274	1	no	—
79	#108	—	137	2	no	—
80	#107	—	274	1	no	—
81	#106	—	137	2	no	—
82	C₂²×C₄	—	137	1	no	—
83	M₁₆	—	274	2	no	—
84	M₁₆	—	548	1	no	—
85	C₂×C₈	—	548	1	no	—
86	C₂×C₈	—	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	C₂×#10587739	1	1	yes	characteristic
92	C₄⋊C₄	—	274	2	no	—
93	C₂²⋊C₄	—	548	1	no	—
94	C₄²	—	548	1	no	—
95	C₁₆	—	137	1	no	—
96	C₁₆	—	137	1	no	—
97	C₁₆	C₂×#10587739	1	1	yes	characteristic
98	C₁₆	C₂×#10587739	1	1	yes	characteristic
99	C₂³	—	137	1	no	—
100	C₂×C₄	—	548	2	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	—	137	1	no	—
103	C₂×C₄	—	137	1	no	—
104	C₂×C₄	—	137	1	no	—
105	C₂×C₄	—	137	1	no	—
106	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
107	C₈	—	548	2	no	—
108	C₈	—	137	1	no	—
109	C₈	—	137	1	no	—
110	C₈	C₄⋊#10587739	1	1	yes	characteristic
111	C₈	C₂²×#10587739	1	1	yes	characteristic
112	C₂²	—	548	1	no	—
113	C₂²	—	137	1	no	—
114	C₂²	—	137	1	no	—
115	C₂²	#17984156	1	1	yes	characteristic
116	C₄	—	1096	2	no	—
117	C₄	—	137	1	no	—
118	C₄	—	137	1	no	—
119	C₄	#17984156	1	1	yes	characteristic
120	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	274	1	no	—
123	C₂	—	274	1	no	—
124	C₂	C₂×#17984156	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984156
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×#106
prime radical	C₂²×D₁₃₇	SD₁₆
cosocle	C₈×D₁₃₇	C₂³
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#2155
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22578328	C₂	semidirect
4. #22578327	C₂	semidirect
13. #17984027	C₄	semidirect
14. #17984027	C₄	semidirect
17. C₂×#14555485	C₄	semidirect
18. C₂×C₁₆×C₁₃₇	C₄	semidirect
62. C₁₃₇	#2155	semidirect
77. C₂×C₁₆	#10587739	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇⋊#108	2
#22578413	2
#22578328	2
#22578327	2
C₂×#17984027	2
#2155	137