Finite Group Databasebeta

← #31796797

Subgroups — #31796797

Order 17536.

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

normal subgroups only

This group has 344 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#22578471C211yesmaximal, characteristic, complemented
2#22578395C212yesmaximal
3D137⋊#101C211yesmaximal, characteristic, complemented
4#22578274C212yesmaximal, complemented
5#124×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7Q8×#10587739—24no—
8#17984167—22no—
9#17984167C2212yes—
10#17984150—24no—
11#17984149—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
21#14555520—22no—
22#14555519—22no—
23#14555518—22no—
24C4⋊#10587739—24no—
25C4⋊#10587739D414yes—
26C22⋊#10587739—22no—
27C4×#10587739—24no—
28C4×#10587739—22no—
29C23×D137—21no—
30C2×Dic274C2×C411yescharacteristic
31C2×Dic274D412yes—
32Q8×D137—22no—
33Q8×D137—22no—
34C2×D548C2×C411yescharacteristic
35C2×C22⋊D137—21no—
36D4×D137—22no—
37D4×D137—22no—
38C2×C4×D137—21no—
39C2×C4×D137D412yes—
40C2×C4×D137C2311yescharacteristic
41C22⋊Dic137—22no—
42#14555487—22no—
43#14555487—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C2×D4×C137C2×C411yescharacteristic
48C22⋊C4×C137—22no—
49C42×C137C2×C411yescharacteristic
50C2×#10587739—24no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137—44no—
54C22×D137—41no—
55C22×D137—41no—
56C22×D137C2×D411yescharacteristic
57Dic274—22no—
58Dic274—24no—
59D548—22no—
60C22⋊D137—42no—
61C2×Dic137—21no—
62C2×Dic137C22⋊C412yes—
63C2×Dic137C2×D411yescharacteristic
64C4×D137—22no—
65C4×D137—22no—
66C4×D137—42no—
67C4×D137C2×D412yes—
68C4×D137#4112yes—
69C23×C137—21no—
70Q8×C137—22no—
71D4×C137—22no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C412yes—
74C2×C4×C137C22×C411yescharacteristic
75C22×C137—42no—
76C22×C137—41no—
77C22×C137C2×C22⋊C411yescharacteristic
78C2×D137—22no—
79C2×D137—42no—
80C2×D137—41no—
81C2×D137—82no—
82C2×D137—82no—
83C2×D137#12011yescharacteristic
84C2×D137#12311yescharacteristic
85#10587739—44no—
86C4×C137—22no—
87C4×C137—41no—
88C4×C137C4×D412yes—
89Dic137—22no—
90Dic137—41no—
91Dic137#12112yes—
92C2×C137—21no—
93C2×C137—81no—
94C2×C137#38611yescharacteristic
95D137—21no—
96D137—81no—
97D137#45412yes—
98C137#188811yescharacteristic, complemented, Sylow
99#1888—1371nomaximal, Sylow
100C2×#124—1371no—
101#519—1371no—
102#418—1372no—
103C2×#101—1371no—
104#339—1372no—
105C22×Q8—1371no—
106C22×D4—1371no—
107C2×M16—1372no—
108#124—2742no—
109#124—1371no—
110#124—1371no—
111#124—1371no—
112#124#1058773911yescharacteristic, complemented
113C4×Q8—2744no—
114#117—2742no—
115#117—1372no—
116C2×C22⋊C4—2742no—
117C2×C42—1371no—
118#102—2744no—
119#101—2742no—
120C24—2741no—
121C2×Q8—2742no—
122C2×Q8—2742no—
123C2×Q8—1372no—
124C2×Q8—1371no—
125C2×Q8C2×#1058773911yescharacteristic
126C2×D4—2742no—
127C2×D4—2742no—
128C2×D4—2741no—
129C2×D4—1371no—
130C2×D4C2×#1058773911yescharacteristic
131C22×C4—2741no—
132C22×C4—1372no—
133C22×C4—1371no—
134M16—2742no—
135M16—2742no—
136C2×C8—2742no—
137C4⋊C4—2744no—
138C4⋊C4—1374no—
139C22⋊C4—22no—
140C22⋊C4—2742no—
141C22⋊C4—2742no—
142C22⋊C4—2742no—
143C22⋊C4—2742no—
144C42—2741no—
145C42—2744no—
146C42—2742no—
147C42—1371no—
148C42C2×#1058773911yescharacteristic
149C23—21no—
150C23—2741no—
151C23—5484no—
152C23—5481no—
153C23—5481no—
154C23—1371no—
155Q8—22no—
156Q8—2744no—
157Q8—2742no—
158D4—22no—
159D4—2742no—
160D4—5482no—
161C2×C4—21no—
162C2×C4—2742no—
163C2×C4—2742no—
164C2×C4—2744no—
165C2×C4—2741no—
166C2×C4—5482no—
167C2×C4—1372no—
168C2×C4—1371no—
169C2×C4—1372no—
170C2×C4—1372no—
171C2×C4C22⋊#1058773912yes—
172C2×C4C22×#1058773911yescharacteristic
173C8—5482no—
174C22—42no—
175C22—41no—
176C22—2742no—
177C22—5482no—
178C22—5481no—
179C22—10962no—
180C22—10962no—
181C22—1371no—
182C22—1371no—
183C22C2×C22⋊#1058773911yescharacteristic
184C4—22no—
185C4—41no—
186C4—2742no—
187C4—5481no—
188C4—5484no—
189C4—1372no—
190C4D4×#1058773912yes—
191C2—21no—
192C2—81no—
193C2—2741no—
194C2—10961no—
195C2—1372no—
196C2#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#1888
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
1. #22578471C2semidirect
3. D137⋊#101C2semidirect
4. #22578274C2semidirect
14. #17984077C4semidirect
15. #17984076C4semidirect
16. #17984075C4semidirect
20. #124×C137C4semidirect
98. C137#1888semidirect
112. #124#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784712
#225783952
D137⋊#1012
#225782742
#124×D1372
#1888137