Finite Group Databasebeta

Subgroups — #31797189

Order 17536.

146 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22578447	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578445	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578332	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578318	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578317	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577834	C₂	1	1	yes	maximal, characteristic
8	#17984176	—	2	1	no	—
9	#17984175	—	2	1	no	—
10	#17984161	C₂²	1	1	yes	characteristic
11	#17984159	C₂²	1	1	yes	characteristic
12	#17984159	C₂²	1	1	yes	characteristic
13	#17984153	—	2	1	no	—
14	#17984152	—	2	1	no	—
15	C₈×#10587739	C₂²	1	1	yes	characteristic
16	#17984111	C₂²	1	1	yes	characteristic
17	#17984107	C₂²	1	1	yes	characteristic
18	#17984034	C₄	1	1	yes	characteristic, complemented
19	#17984033	C₄	1	1	yes	characteristic, complemented
20	#17984029	C₄	1	1	yes	characteristic, complemented
21	C₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
22	SD₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
23	#14555520	—	2	1	no	—
24	#14555520	—	2	1	no	—
25	#14555520	—	4	1	no	—
26	#14555519	—	4	1	no	—
27	#14555518	—	4	1	no	—
28	#14555518	D₄	1	1	yes	characteristic
29	C₂×#11764047	—	4	1	no	—
30	C₄×#10587739	D₄	1	1	yes	characteristic
31	#14555513	—	2	1	no	—
32	#14555511	—	2	1	no	—
33	#14555509	—	2	1	no	—
34	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
35	#14555498	—	2	1	no	—
36	#14555497	—	2	1	no	—
37	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
39	#14555485	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×#10587739	—	2	1	no	—
44	#11764047	—	4	1	no	—
45	#11764047	—	4	1	no	—
46	#11764047	D₈	1	1	yes	characteristic
47	#11764047	D₈	1	1	yes	characteristic
48	Dic₂₇₄	—	2	1	no	—
49	D₅₄₈	—	2	1	no	—
50	D₅₄₈	—	4	1	no	—
51	C₂²⋊D₁₃₇	—	4	1	no	—
52	C₂×Dic₁₃₇	—	4	1	no	—
53	C₄×D₁₃₇	—	4	1	no	—
54	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
55	#11764040	#41	1	1	yes	characteristic
56	Q₈×C₁₃₇	—	2	1	no	—
57	D₄×C₁₃₇	—	2	1	no	—
58	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₁₃₇	—	4	1	no	—
60	C₂×D₁₃₇	—	4	1	no	—
61	C₂×D₁₃₇	#135	1	1	yes	characteristic
62	#10587739	—	2	2	no	—
63	C₄×C₁₃₇	—	4	1	no	—
64	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
65	Dic₁₃₇	—	4	1	no	—
66	Dic₁₃₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₁₃₇	—	8	1	no	—
68	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
69	D₁₃₇	—	2	1	no	—
70	D₁₃₇	—	8	1	no	—
71	C₁₃₇	#2178	1	1	yes	characteristic, complemented, Sylow
72	#2178	—	137	1	no	maximal, Sylow
73	#508	—	137	1	no	—
74	#443	—	137	1	no	—
75	#443	—	137	1	no	—
76	#368	—	137	1	no	—
77	#359	—	137	1	no	—
78	#359	—	137	1	no	—
79	C₄×C₁₆	—	137	1	no	—
80	#135	—	137	1	no	—
81	#135	—	137	1	no	—
82	#131	—	274	1	no	—
83	#131	—	274	1	no	—
84	Q₃₂	—	137	1	no	—
85	SD₃₂	—	137	1	no	—
86	SD₃₂	#10587739	1	1	yes	characteristic, complemented
87	D₁₆	—	137	1	no	—
88	C₂×C₁₆	—	137	1	no	—
89	C₂×C₁₆	—	137	1	no	—
90	#108	—	137	1	no	—
91	#108	—	137	1	no	—
92	#104	—	274	1	no	—
93	#104	—	274	1	no	—
94	C₄×C₈	—	137	1	no	—
95	#41	—	274	1	no	—
96	#41	—	274	1	no	—
97	Q₁₆	—	137	1	no	—
98	Q₁₆	C₂×#10587739	1	1	yes	characteristic
99	SD₁₆	—	274	1	no	—
100	SD₁₆	—	274	1	no	—
101	D₈	—	137	1	no	—
102	D₈	C₂×#10587739	1	1	yes	characteristic
103	M₁₆	—	274	1	no	—
104	M₁₆	—	274	1	no	—
105	M₁₆	—	548	1	no	—
106	M₁₆	—	548	1	no	—
107	C₂×C₈	—	548	1	no	—
108	C₂×C₈	—	548	1	no	—
109	C₂×C₈	—	137	1	no	—
110	C₂×C₈	—	137	1	no	—
111	C₄²	—	137	1	no	—
112	C₁₆	—	274	1	no	—
113	C₁₆	—	137	1	no	—
114	C₁₆	C₂×#10587739	1	1	yes	characteristic
115	Q₈	—	2	1	no	—
116	Q₈	—	274	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	274	1	no	—
119	D₄	—	548	1	no	—
120	D₄	—	548	1	no	—
121	C₂×C₄	—	274	1	no	—
122	C₂×C₄	—	548	1	no	—
123	C₂×C₄	—	548	1	no	—
124	C₂×C₄	—	137	1	no	—
125	C₈	—	548	1	no	—
126	C₈	—	548	1	no	—
127	C₈	—	137	1	no	—
128	C₈	—	137	1	no	—
129	C₈	—	137	1	no	—
130	C₈	C₂²×#10587739	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	548	1	no	—
133	C₂²	—	137	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	274	2	no	—
136	C₄	—	548	1	no	—
137	C₄	—	137	1	no	—
138	C₄	D₄×#10587739	1	1	yes	characteristic
139	C₂	—	8	1	no	—
140	C₂	—	274	1	no	—
141	C₂	—	1096	1	no	—
142	C₂	D₈×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#10587739
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	SD₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₄×D₈
prime radical	#22577834	C₂
cosocle	C₈×D₁₃₇	C₂³
hypercenter	SD₃₂	#10587739
hypocenter	C₁₃₇	#2178
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22578447	C₂	semidirect
2. #22578445	C₂	semidirect
3. #22578332	C₂	semidirect
4. #22578318	C₂	semidirect
5. #22578317	C₂	semidirect
6. C₁₆×#10587739	C₂	semidirect
18. #17984034	C₄	semidirect
19. #17984033	C₄	semidirect
20. #17984029	C₄	semidirect
22. SD₃₂×C₁₃₇	C₄	semidirect
71. C₁₃₇	#2178	semidirect
86. SD₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578447	2
#22578445	2
#22578332	2
#22578318	2
#22578317	2
C₁₆×#10587739	2
#22577834	2
#2178	137