Finite Group Databasebeta

← #31796083

Subgroups — #31796083

Order 17536.

276 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578422C211yesmaximal, characteristic, complemented
2C4×#14555518C211yesmaximal, characteristic, complemented
3Q8⋊#11764047C212yesmaximal, complemented
4#22578251C212yesmaximal, complemented
5C4×#14555511C211yesmaximal, characteristic
6C2×#14555518—21no—
7C4⋊#11764047C2212yes—
8#17984145—21no—
9C4×#11764047—21no—
10C4×#11764047—21no—
11C4×#11764047C2212yes—
12C2×#14555511C411yescharacteristic
13Q8×Dic137C2211yescharacteristic
14C4×D548—21no—
15C4×D548C411yescharacteristic
16#17984050—21no—
17#17984050C2211yescharacteristic
18#17984046—21no—
19#17984046C411yescharacteristic
20C42×D137—21no—
21C42×D137C2211yescharacteristic
22C4×Q8×C137C411yescharacteristic
23#14555518—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26#14555511—22no—
27#14555511C814yescomplemented
28C2×D548—21no—
29C2×D548C2×C411yescharacteristic
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C4⋊Dic137—21no—
37C4⋊Dic137C2×C411yescharacteristic
38#14555487—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137—21no—
42C4×Dic137—21no—
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C42×C137—21no—
50C42×C137C2×C411yescharacteristic
51#11764047—24no—
52#11764047—42no—
53C22×D137—21no—
54C22×D137C22⋊C411yescharacteristic
55D548—24no—
56D548—21no—
57D548C2×C812yescomplemented
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C4×D137—22no—
63C4×D137—24no—
64C4×D137—22no—
65C4×D137—42no—
66C4×D137—42no—
67C4×D137C2×C812yes—
68C4×D137M1612yes—
69Q8×C137—21no—
70Q8×C137C2×C812yescomplemented
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C22×C411yescharacteristic
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—22no—
80C2×D137—41no—
81C2×D137—41no—
82C2×D137C22⋊C812yes—
83C4×C137—22no—
84C4×C137—21no—
85C4×C137—41no—
86C4×C137#10412yes—
87C4×C137C22×C811yescharacteristic
88C4×C137C2×M1611yescharacteristic
89Dic137—21no—
90Dic137—41no—
91Dic137C22⋊C812yes—
92Dic137#13512yes—
93C2×C137C2×C22⋊C811yescharacteristic
94C2×C137#41611yescharacteristic
95C2×C137C2×#10411yescharacteristic
96D137—22no—
97D137—42no—
98C137#147611yescharacteristic, complemented, Sylow
99#1476—1371nomaximal, Sylow
100C4×#41—1371no—
101#431—1371no—
102C2×C4×C8—1371no—
103Q8⋊C8—1372no—
104D4⋊C8—1372no—
105C2×#41—1371no—
106C22×C8—2741no—
107C4×Q8—1371no—
108C4×Q8#1058773911yescharacteristic
109C4×D4—2741no—
110C4×D4—2741no—
111C4×D4—1371no—
112C4×D4—1371no—
113#117—2741no—
114#117—1371no—
115C2×C42—2741no—
116C2×C42—1371no—
117C4⋊C8—1372no—
118C22⋊C8—2741no—
119C4×C8—2741no—
120C4×C8—2741no—
121C4×C8—1372no—
122#41—2742no—
123#41—1374no—
124C2×Q8C2×#1058773911yescharacteristic
125C2×D4—2741no—
126C2×D4—1371no—
127C22×C4—2741no—
128C22×C4—2741no—
129C22×C4—2741no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C2×C8—2744no—
134C2×C8—2742no—
135C2×C8—2742no—
136C4⋊C4—21no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4C2×#1058773911yescharacteristic
140C22⋊C4—2741no—
141C22⋊C4—2741no—
142C22⋊C4—2741no—
143C42—21no—
144C42—2741no—
145C42—2741no—
146C42—1371no—
147C42—1371no—
148C42—1371no—
149C42C2×#1058773911yescharacteristic
150C23—2741no—
151C23—1371no—
152Q8—21no—
153Q8#1455551812yescomplemented
154D4—2744no—
155D4—2741no—
156D4—1372no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—2744no—
160C2×C4—2742no—
161C2×C4—2742no—
162C2×C4—2741no—
163C2×C4—5482no—
164C2×C4—5482no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4—1372no—
169C2×C4—1372no—
170C2×C4C22⋊#1058773911yescharacteristic
171C2×C4C22⋊#1058773911yescharacteristic
172C2×C4C22×#1058773911yescharacteristic
173C8—2744no—
174C8—5482no—
175C22—2741no—
176C22—2741no—
177C22—2742no—
178C22—5481no—
179C22—5481no—
180C22—1372no—
181C22C2×C22⋊#1058773911yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—2741no—
186C4—5481no—
187C4—1372no—
188C4—1372no—
189C4#1798415312yes—
190C4C2×#1455551811yescharacteristic
191C4D137⋊M1611yescharacteristic
192C2—2742no—
193C2—5482no—
194C2#2257837111yescharacteristic
195C2#2257839311yescharacteristic
196C2C2×#1798415311yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10587739
commutatorC4×C137C22×C8
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#14555511C4
cosocleC4×Dic137C23
hypercenterC4×Q8#10587739
hypocenterC137#1476
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. C4×#14555518C2semidirect
3. Q8⋊#11764047C2semidirect
4. #22578251C2semidirect
27. #14555511C8semidirect
57. D548C2×C8semidirect
70. Q8×C137C2×C8semidirect
98. C137#1476semidirect
153. Q8#14555518semidirect

Maximal subgroups

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

subgroupindex
#225784222
C4×#145555182
Q8⋊#117640472
#225782512
C4×#145555112
#1476137