Finite Group Databasebeta

Subgroups — #31797952

Order 17536.

184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22578522	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984183	C₂	1	1	yes	maximal, characteristic
3	#22578498	C₂	1	2	yes	maximal, complemented
4	#10587739⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	#22578369	C₂	1	1	yes	maximal, characteristic
6	#17984185	C₄	1	2	yes	complemented
7	#17984184	C₄	1	2	yes	complemented
8	#17984183	C₄	1	2	yes	complemented
9	#17984183	C₂²	1	1	yes	characteristic
10	#17984183	C₂²	1	2	yes	—
11	#17984183	C₂²	1	1	yes	characteristic
12	C₂×#14555522	C₄	1	2	yes	complemented
13	C₂×#14555518	C₂²	1	1	yes	characteristic
14	#17984167	C₄	1	1	yes	characteristic
15	#17984142	C₄	1	2	yes	—
16	C₈×#10587739	C₂²	1	2	yes	—
17	M₁₆×D₁₃₇	C₄	1	1	yes	characteristic
18	#14555522	C₂×C₄	1	2	yes	—
19	#14555522	C₂×C₄	1	2	yes	—
20	#14555522	C₂×C₄	1	4	yes	—
21	#14555518	C₂×C₄	1	1	yes	characteristic
22	#14555518	C₂×C₄	1	1	yes	characteristic
23	#14555518	C₂×C₄	1	1	yes	characteristic
24	#14555518	C₂³	1	1	yes	characteristic
25	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
26	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
27	C₄⋊#10587739	C₈	1	2	yes	—
28	C₂²⋊#10587739	C₈	1	2	yes	—
29	C₄×#10587739	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
31	#14555494	C₈	1	1	yes	characteristic
32	#14555493	C₈	1	2	yes	—
33	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
34	M₁₆×C₁₃₇	C₈	1	1	yes	characteristic
35	C₂×#10587739	C₂×C₈	1	4	yes	—
36	#11764047	C₄²	1	1	yes	characteristic
37	#11764047	C₄²	1	1	yes	characteristic
38	#11764047	C₂²×C₄	1	1	yes	characteristic
39	#11764047	C₂²×C₄	1	1	yes	characteristic
40	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
42	C₄×D₁₃₇	C₄²	1	1	yes	characteristic
43	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
44	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
45	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	#11764040	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
48	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
49	C₂²×C₁₃₇	C₄×C₈	1	1	yes	characteristic
50	C₂×D₁₃₇	—	2	2	no	—
51	C₂×D₁₃₇	C₄×C₈	1	1	yes	characteristic
52	C₂×D₁₃₇	C₂²×C₈	1	1	yes	characteristic
53	#10587739	—	2	4	no	—
54	C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
55	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
56	Dic₁₃₇	C₄×C₈	1	1	yes	characteristic
57	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
60	D₁₃₇	—	2	1	no	—
61	D₁₃₇	#504	1	2	yes	—
62	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
63	#2108	—	137	1	no	maximal, Sylow
64	C₂×M₃₂	—	137	1	no	—
65	C₂²×C₁₆	—	137	1	no	—
66	#405	—	137	1	no	—
67	#346	—	137	2	no	—
68	C₄×C₁₆	—	137	2	no	—
69	C₂×M₁₆	—	137	1	no	—
70	C₂²×C₈	—	137	1	no	—
71	#117	—	137	1	no	—
72	M₃₂	—	137	2	no	—
73	M₃₂	—	137	2	no	—
74	C₂×C₁₆	—	137	1	no	—
75	C₂×C₁₆	—	137	2	no	—
76	C₂×C₁₆	—	137	1	no	—
77	C₂×C₁₆	—	137	2	no	—
78	C₂×C₁₆	—	137	2	no	—
79	#97	—	137	2	no	—
80	C₄×C₈	—	137	2	no	—
81	C₂²×C₄	—	137	1	no	—
82	M₁₆	—	137	1	no	—
83	M₁₆	—	137	2	no	—
84	M₁₆	#11764049	1	1	yes	characteristic
85	C₂×C₈	—	137	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	—	137	2	no	—
92	C₄⋊C₄	—	137	2	no	—
93	C₂²⋊C₄	—	137	2	no	—
94	C₄²	—	137	2	no	—
95	C₁₆	—	137	2	no	—
96	C₁₆	—	137	2	no	—
97	C₁₆	—	137	4	no	—
98	C₂³	—	137	1	no	—
99	C₂×C₄	—	137	4	no	—
100	C₂×C₄	—	137	1	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	C₂×#11764049	1	1	yes	characteristic
106	C₈	—	137	1	no	—
107	C₈	—	137	1	no	—
108	C₈	—	137	1	no	—
109	C₈	—	137	1	no	—
110	C₈	—	137	2	no	—
111	C₈	C₂×#11764049	1	2	yes	—
112	C₂²	—	274	2	no	—
113	C₂²	—	137	1	no	—
114	C₂²	—	137	1	no	—
115	C₂²	C₄×#11764049	1	1	yes	characteristic
116	C₄	—	274	4	no	—
117	C₄	—	137	1	no	—
118	C₄	—	137	1	no	—
119	C₄	C₄×#11764049	1	1	yes	characteristic
120	C₄	C₂²×#11764049	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	274	1	no	—
123	C₂	—	137	2	no	—
124	C₂	C₂×C₄×#11764049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11764049
commutator, socle	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	M₁₆×C₁₃₇	C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	#14555518	C₂³
hypercenter	M₁₆	#11764049
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #22578498	C₂	semidirect
4. #10587739⋊C₁₆	C₂	semidirect
6. #17984185	C₄	semidirect
7. #17984184	C₄	semidirect
8. #17984183	C₄	semidirect
12. C₂×#14555522	C₄	semidirect
62. C₁₃₇	#2108	semidirect

Maximal subgroups

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

subgroup	index
#22578522	2
C₂×#17984183	2
#22578498	2
#10587739⋊C₁₆	2
#22578369	2
#2108	137