Finite Group Databasebeta

← #31796794

Subgroups — #31796794

Order 17536.

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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#22578470C211yesmaximal, characteristic
2#22578396C212yesmaximal, complemented
3D137⋊#100C211yesmaximal, characteristic
4#22578274C212yesmaximal, complemented
5#124×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7D4×#10587739—24no—
8#17984167C2212yes—
9C2×C22⋊#10587739—22no—
10#17984151—24no—
11#17984148—22no—
12C2×Q8×D137C2211yescharacteristic
13C2×D4×D137C2211yescharacteristic
14#17984077C411yescharacteristic, complemented
15#17984076C411yescharacteristic, complemented
16#17984075C411yescharacteristic, complemented
17#17984074—22no—
18C22⋊C4×D137—22no—
19C42×D137C2211yescharacteristic
20#124×C137C411yescharacteristic, complemented
21C22×#10587739—24no—
22#14555520—22no—
23#14555519—22no—
24#14555518—22no—
25C4⋊#10587739D414yes—
26C22⋊#10587739—24no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29C23×D137D412yes—
30C2×Dic274—21no—
31C2×Dic274C2×C411yescharacteristic, complemented
32Q8×D137—22no—
33Q8×D137—22no—
34C2×D548C2×C411yescharacteristic
35C2×C22⋊D137D412yes—
36D4×D137—22no—
37D4×D137—22no—
38C2×C4×D137—21no—
39C2×C4×D137—21no—
40C2×C4×D137C2311yescharacteristic
41C22⋊Dic137—22no—
42#14555487—22no—
43#14555487—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic, complemented
46C2×Q8×C137C2×C411yescharacteristic, complemented
47C2×D4×C137C2×C411yescharacteristic
48C22⋊C4×C137—22no—
49C42×C137C2×C411yescharacteristic, complemented
50C2×#10587739—24no—
51C2×#10587739—84no—
52#11764047—42no—
53C22×D137—22no—
54C22×D137—22no—
55C22×D137—44no—
56C22×D137C22⋊C412yes—
57C22×D137C2×D411yescharacteristic
58Dic274—22no—
59Dic274—42no—
60D548—22no—
61C22⋊D137—24no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137C2×D411yescharacteristic
65C4×D137—42no—
66C4×D137—41no—
67C4×D137—41no—
68C4×D137C2×D412yes—
69C4×D137#4112yes—
70C23×C137C22⋊C412yes—
71Q8×C137—22no—
72D4×C137—22no—
73C2×C4×C137—21no—
74C2×C4×C137—21no—
75C2×C4×C137C22×C411yescharacteristic
76C22×C137—22no—
77C22×C137—42no—
78C22×C137C2×C22⋊C411yescharacteristic
79C2×D137—22no—
80C2×D137—22no—
81C2×D137—44no—
82C2×D137—42no—
83C2×D137—82no—
84C2×D137#12011yescharacteristic
85C2×D137#12311yescharacteristic
86#10587739—44no—
87C4×C137—41no—
88C4×C137—41no—
89C4×C137C4×D412yes—
90Dic137—41no—
91Dic137—41no—
92Dic137#12112yes—
93C2×C137—21no—
94C2×C137—42no—
95C2×C137#38611yescharacteristic
96D137—21no—
97D137—42no—
98D137#45512yes—
99C137#188511yescharacteristic, complemented, Sylow
100#1885—1371nomaximal, Sylow
101C2×#124—1371no—
102#518—1371no—
103#419—1372no—
104C2×#100—1371no—
105#339—1372no—
106C22×Q8—1371no—
107C22×D4—1371no—
108C2×M16—1372no—
109#124—2742no—
110#124—1371no—
111#124—1371no—
112#124—1371no—
113#124#1058773911yescharacteristic, complemented
114C4×D4—2744no—
115#117—1372no—
116C2×C22⋊C4—2742no—
117C2×C22⋊C4—2742no—
118C2×C42—1371no—
119#103—2744no—
120#100—2742no—
121C24—1372no—
122C2×Q8—2741no—
123C2×Q8—2742no—
124C2×Q8—2742no—
125C2×Q8—1371no—
126C2×Q8C2×#1058773911yescharacteristic, complemented
127C2×D4—2742no—
128C2×D4—2742no—
129C2×D4—1372no—
130C2×D4—1371no—
131C2×D4C2×#1058773911yescharacteristic
132C22×C4—2741no—
133C22×C4—2741no—
134C22×C4—2744no—
135C22×C4—1371no—
136M16—2742no—
137M16—2742no—
138C2×C8—2742no—
139C4⋊C4—1374no—
140C22⋊C4—22no—
141C22⋊C4—2742no—
142C22⋊C4—2742no—
143C22⋊C4—2744no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C42—2741no—
147C42—2742no—
148C42—1371no—
149C42C2×#1058773911yescharacteristic, complemented
150C23—2742no—
151C23—2742no—
152C23—5484no—
153C23—1371no—
154C23—1372no—
155C23C22⋊#1058773912yes—
156Q8—22no—
157Q8—2742no—
158Q8—5482no—
159D4—22no—
160D4—2744no—
161D4—2742no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—2744no—
165C2×C4—2741no—
166C2×C4—2741no—
167C2×C4—5482no—
168C2×C4—5481no—
169C2×C4—5481no—
170C2×C4—10964no—
171C2×C4—1371no—
172C2×C4—1372no—
173C2×C4—1372no—
174C2×C4C22×#1058773911yescharacteristic
175C8—5482no—
176C22—22no—
177C22—42no—
178C22—2742no—
179C22—2742no—
180C22—5482no—
181C22—5484no—
182C22—10962no—
183C22—1371no—
184C22—1371no—
185C22C2×C22⋊#1058773911yescharacteristic
186C4—41no—
187C4—41no—
188C4—5481no—
189C4—5484no—
190C4—5481no—
191C4—1372no—
192C4D4×#1058773912yes—
193C2—21no—
194C2—42no—
195C2—2741no—
196C2—5482no—
197C2—1372no—
198C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137#386
prime radicalC2×D4×D137C22
cosocleC2×C4×D137C23
hypercenter#124#10587739
hypocenterC137#1885
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #22578396C2semidirect
4. #22578274C2semidirect
14. #17984077C4semidirect
15. #17984076C4semidirect
16. #17984075C4semidirect
20. #124×C137C4semidirect
31. C2×Dic274C2×C4semidirect
45. C4×Dic137C2×C4semidirect
46. C2×Q8×C137C2×C4semidirect
49. C42×C137C2×C4semidirect
99. C137#1885semidirect
113. #124#10587739semidirect
126. C2×Q8C2×#10587739semidirect
149. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
#225783962
D137⋊#1002
#225782742
#124×D1372
#1885137