Finite Group Databasebeta

Subgroups — #31797995

Order 17536.

138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22578506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#14555522	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578500	C₂	1	1	yes	maximal, characteristic
4	#22578500	C₂	1	1	yes	maximal, characteristic
5	#22578497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555522	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578368	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555522	C₂²	1	1	yes	characteristic
9	C₂×#14555522	C₂²	1	1	yes	characteristic
10	C₂×#14555522	C₂²	1	1	yes	characteristic
11	C₂×#14555522	C₂²	1	1	yes	characteristic
12	C₂×#14555520	C₄	1	1	yes	characteristic
13	C₂×#14555518	C₂²	1	1	yes	characteristic
14	#17984143	C₄	1	1	yes	characteristic
15	#17984143	C₄	1	1	yes	characteristic
16	C₄×#11764047	C₂²	1	1	yes	characteristic
17	C₄×#11764047	C₂²	1	1	yes	characteristic
18	#17984046	C₄	1	1	yes	characteristic
19	#14555522	—	2	1	no	—
20	#14555522	—	2	1	no	—
21	#14555522	—	2	1	no	—
22	#14555522	—	2	1	no	—
23	#14555520	—	2	2	no	—
24	#14555518	—	2	1	no	—
25	#14555518	—	2	1	no	—
26	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
27	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
28	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
29	C₂×#11764047	C₂³	1	1	yes	characteristic
30	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
32	#14555487	C₈	1	1	yes	characteristic
33	#14555487	C₈	1	1	yes	characteristic
34	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
37	#11764047	—	2	1	no	—
38	#11764047	—	2	1	no	—
39	#11764047	M₁₆	1	2	yes	—
40	#11764047	#41	1	1	yes	characteristic
41	#11764047	#41	1	1	yes	characteristic
42	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
43	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₁₃₇	—	2	1	no	—
47	C₄×D₁₃₇	—	2	1	no	—
48	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×D₁₃₇	—	2	1	no	—
53	C₂×D₁₃₇	—	2	1	no	—
54	C₂×D₁₃₇	—	2	1	no	—
55	C₄×C₁₃₇	—	2	1	no	—
56	C₄×C₁₃₇	—	2	1	no	—
57	C₄×C₁₃₇	—	2	1	no	—
58	Dic₁₃₇	—	2	1	no	—
59	Dic₁₃₇	—	2	1	no	—
60	Dic₁₃₇	#117	1	1	yes	characteristic
61	Dic₁₃₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₁₃₇	#431	1	1	yes	characteristic
63	C₂×C₁₃₇	#504	1	1	yes	characteristic
64	C₂×C₁₃₇	#504	1	1	yes	characteristic
65	D₁₃₇	—	4	1	no	—
66	C₁₃₇	#2164	1	1	yes	characteristic, complemented, Sylow
67	#2164	—	137	1	no	maximal, Sylow
68	#405	—	137	1	no	—
69	C₄⋊C₁₆	—	137	1	no	—
70	C₄⋊C₁₆	—	137	1	no	—
71	C₂²⋊C₁₆	—	137	1	no	—
72	C₂²⋊C₁₆	—	137	1	no	—
73	#346	—	137	1	no	—
74	C₄×C₁₆	—	137	1	no	—
75	C₂×M₁₆	—	137	1	no	—
76	C₂²×C₈	—	137	1	no	—
77	#117	—	137	1	no	—
78	C₂×C₁₆	—	137	1	no	—
79	C₂×C₁₆	—	137	1	no	—
80	C₂×C₁₆	—	137	1	no	—
81	C₂×C₁₆	—	137	1	no	—
82	#97	—	137	1	no	—
83	#97	—	137	1	no	—
84	C₄×C₈	—	137	1	no	—
85	C₄×C₈	—	137	1	no	—
86	C₂²×C₄	—	137	1	no	—
87	M₁₆	—	274	2	no	—
88	C₂×C₈	—	274	1	no	—
89	C₂×C₈	—	274	1	no	—
90	C₂×C₈	—	137	1	no	—
91	C₂×C₈	—	137	1	no	—
92	C₂×C₈	—	137	1	no	—
93	C₂×C₈	—	137	1	no	—
94	C₄⋊C₄	—	137	1	no	—
95	C₄⋊C₄	#11764049	1	1	yes	characteristic
96	C₂²⋊C₄	—	137	1	no	—
97	C₂²⋊C₄	—	137	1	no	—
98	C₄²	—	137	1	no	—
99	C₄²	—	137	1	no	—
100	C₁₆	—	274	1	no	—
101	C₁₆	—	274	1	no	—
102	C₁₆	—	274	1	no	—
103	C₁₆	—	274	1	no	—
104	C₂³	—	137	1	no	—
105	C₂×C₄	—	274	1	no	—
106	C₂×C₄	—	274	1	no	—
107	C₂×C₄	—	137	1	no	—
108	C₂×C₄	—	137	1	no	—
109	C₂×C₄	—	137	1	no	—
110	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
111	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
112	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
113	C₈	—	274	1	no	—
114	C₈	—	274	1	no	—
115	C₈	—	137	1	no	—
116	C₈	—	137	1	no	—
117	C₈	—	137	2	no	—
118	C₂²	—	274	1	no	—
119	C₂²	—	274	1	no	—
120	C₂²	—	274	1	no	—
121	C₂²	C₂²×#11764049	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	274	1	no	—
126	C₄	—	274	1	no	—
127	C₄	—	137	1	no	—
128	C₄	—	137	1	no	—
129	C₂	—	548	1	no	—
130	C₂	#22578516	1	1	yes	characteristic
131	C₂	#22578524	1	1	yes	characteristic
132	C₂	#22578525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₄⋊C₄×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₄⋊C₄	#11764049
hypocenter	C₁₃₇	#2164
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22578506	C₂	semidirect
2. C₄⋊#14555522	C₂	semidirect
5. #22578497	C₂	semidirect
6. C₄×#14555522	C₂	semidirect
66. C₁₃₇	#2164	semidirect

Maximal subgroups

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

subgroup	index
#22578506	2
C₄⋊#14555522	2
#22578500	2
#22578497	2
C₄×#14555522	2
#22578368	2
#2164	137