Finite Group Databasebeta

← #31796249

Subgroups — #31796249

Order 17536.

264 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic, complemented
2C4×#14555520C211yesmaximal, characteristic, complemented
3#22578259C212yesmaximal, complemented
4#22578258C212yesmaximal, complemented
5#124×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7#17984155—21no—
8#17984145—21no—
9#17984143C2212yescomplemented
10#17984143C2212yescomplemented
11C4×#11764047—21no—
12C2×Q8×D137C411yescharacteristic
13C2×D4×D137C411yescharacteristic
14#17984077C2211yescharacteristic
15#17984076C411yescharacteristic
16#17984075C2211yescharacteristic
17#17984074—21no—
18#17984074—21no—
19C22⋊C4×D137—21no—
20C22⋊C4×D137—21no—
21C42×D137C2211yescharacteristic
22#124×C137C411yescharacteristic
23#14555520—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C23×D137—21no—
27C2×Dic274—21no—
28C2×Dic274C2×C411yescharacteristic
29Q8×D137—42no—
30C2×D548C2×C411yescharacteristic
31C2×C22⋊D137—21no—
32D4×D137—42no—
33C2×C4×D137—21no—
34C2×C4×D137C2×C411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C2×D4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49C22⋊C4×C137—21no—
50C42×C137C2×C411yescharacteristic
51#11764047—24no—
52#11764047—42no—
53C22×D137—21no—
54C22×D137—42no—
55C22×D137—42no—
56C22×D137—42no—
57C22×D137C22⋊C411yescharacteristic
58Dic274—42no—
59Dic274—41no—
60D548—41no—
61C22⋊D137—42no—
62C2×Dic137—21no—
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C4×D137—22no—
67C4×D137—22no—
68C4×D137—41no—
69C4×D137—42no—
70C23×C137—21no—
71Q8×C137—41no—
72D4×C137—41no—
73C2×C4×C137—21no—
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C22⋊C411yescharacteristic
76C2×C4×C137C22×C411yescharacteristic
77C22×C137—41no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137C2×C22⋊C411yescharacteristic
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—41no—
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—84no—
87C2×D137#10412yes—
88C4×C137—21no—
89C4×C137—21no—
90C4×C137—21no—
91C4×C137—41no—
92Dic137—21no—
93Dic137—21no—
94Dic137—41no—
95Dic137#10012yes—
96C2×C137—81no—
97C2×C137C2×#10011yescharacteristic
98C2×C137C2×#10411yescharacteristic
99C2×C137#42111yescharacteristic
100D137—22no—
101D137—81no—
102C137#152711yescharacteristic, complemented, Sylow
103#1527—1371nomaximal, Sylow
104C2×#124—1371no—
105#432—1371no—
106C4×M16—1371no—
107#329—1372no—
108#329—1372no—
109C22×Q8—1371no—
110C22×D4—1371no—
111C2×M16—2741no—
112#124—2741no—
113#124—2741no—
114#124—1371no—
115#124—1371no—
116#124—1371no—
117#124#1058773911yescharacteristic
118C2×C22⋊C4—2741no—
119C2×C22⋊C4—2741no—
120C2×C42—1371no—
121C4⋊C8—2741no—
122C22⋊C8—2741no—
123#97—1372no—
124#97—1372no—
125C4×C8—2741no—
126C24—2741no—
127C2×Q8—2741no—
128C2×Q8—5482no—
129C2×Q8—1371no—
130C2×Q8C2×#1058773911yescharacteristic
131C2×D4—2741no—
132C2×D4—5482no—
133C2×D4—1371no—
134C2×D4C2×#1058773911yescharacteristic
135C22×C4—2741no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139M16—2744no—
140C2×C8—2742no—
141C2×C8—2742no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—2741no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2741no—
149C22⋊C4—2741no—
150C42—1371no—
151C42—1371no—
152C42—1371no—
153C42C2×#1058773911yescharacteristic
154C23—21no—
155C23—2741no—
156C23—5482no—
157C23—5482no—
158C23—5482no—
159C23—1371no—
160Q8—41no—
161Q8—5481no—
162Q8—5482no—
163D4—41no—
164D4—5481no—
165D4—5482no—
166C2×C4—21no—
167C2×C4—2742no—
168C2×C4—2742no—
169C2×C4—2741no—
170C2×C4—5481no—
171C2×C4—5482no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4C22⋊#1058773911yescharacteristic
176C2×C4C22⋊#1058773911yescharacteristic
177C2×C4C22×#1058773911yescharacteristic
178C8—2744no—
179C8—5482no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—2741no—
184C22—2741no—
185C22—5481no—
186C22—5481no—
187C22—5481no—
188C22—10964no—
189C22—1372no—
190C22C2×C22⋊#1058773911yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—41no—
195C4—2741no—
196C4—2741no—
197C4—5481no—
198C4—1372no—
199C2—81no—
200C2—2742no—
201C2—10961no—
202C2D137⋊#10011yescharacteristic
203C2D137⋊#10411yescharacteristic
204C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C4
cosocleC4×Dic137C23
hypercenter#124#10587739
hypocenterC137#1527
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #22578426C2semidirect
2. C4×#14555520C2semidirect
3. #22578259C2semidirect
4. #22578258C2semidirect
9. #17984143C22semidirect
10. #17984143C22semidirect
102. C137#1527semidirect

Maximal subgroups

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

subgroupindex
#225784262
C4×#145555202
#225782592
#225782582
#124×D1372
#1527137