Finite Group Databasebeta

← #31796224

Subgroups — #31796224

Order 17536.

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

normal subgroups only

This group has 340 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
1C2×#17984145C211yesmaximal, characteristic, complemented
2#22578345C211yesmaximal, characteristic, complemented
3#22578255C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#121×D137C211yesmaximal, characteristic
6C22×#11764047—21no—
7C2×C4⋊#10587739—21no—
8C2×C22⋊#10587739—21no—
9#17984145—21no—
10#17984145C2212yescomplemented
11(C2×#10587739)⋊C4C2212yescomplemented
12C2×D4×D137—21no—
13C2×D4×D137C411yescharacteristic, complemented
14C22×C4×D137C2211yescharacteristic
15#17984063C2211yescharacteristic
16#17984062C411yescharacteristic, complemented
17#17984061—21no—
18#17984060—21no—
19#17984059C2211yescharacteristic
20C4⋊C4×D137C411yescharacteristic, complemented
21C22⋊C4×D137—21no—
22#121×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C4⋊#10587739—42no—
27C22⋊#10587739—42no—
28C23×D137—21no—
29C23×D137D411yescharacteristic
30C2×D548—21no—
31C2×D548C2×C411yescharacteristic
32C2×C22⋊D137—21no—
33C2×C22⋊D137—21no—
34C2×C22⋊D137—21no—
35D4×D137—42no—
36D4×D137—42no—
37D4×D137—42no—
38C22×Dic137D411yescharacteristic
39C2×C4×D137—21no—
40C2×C4×D137—42no—
41C2×C4×D137C2×C411yescharacteristic
42C2×C4×D137D411yescharacteristic
43C2×C4×D137D411yescharacteristic
44C2×C4×D137C2311yescharacteristic
45C4⋊Dic137C2×C411yescharacteristic
46#14555489—21no—
47C22⋊Dic137—21no—
48#14555487—21no—
49#14555487—21no—
50C2×D4×C137—21no—
51C2×D4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic
53C4⋊C4×C137C2×C411yescharacteristic
54C22⋊C4×C137—21no—
55C2×#10587739—42no—
56C2×#10587739—42no—
57C2×#10587739—42no—
58#11764047—42no—
59C22×D137—21no—
60C22×D137—41no—
61C22×D137—41no—
62C22×D137—42no—
63C22×D137—42no—
64C22×D137—42no—
65C22×D137C22⋊C411yescharacteristic
66C22×D137D812yes—
67C22×D137SD1612yes—
68C22×D137C2×D411yescharacteristic
69D548—41no—
70D548—41no—
71D548—41no—
72C22⋊D137—42no—
73C22⋊D137—41no—
74C22⋊D137—41no—
75C22⋊D137—42no—
76C2×Dic137—22no—
77C2×Dic137—21no—
78C2×Dic137C22⋊C411yescharacteristic
79C2×Dic137C2×D411yescharacteristic
80C4×D137—22no—
81C4×D137—22no—
82C4×D137—41no—
83C4×D137—41no—
84C4×D137—42no—
85C23×C137—21no—
86C23×C137C22⋊C411yescharacteristic
87D4×C137—41no—
88D4×C137—42no—
89C2×C4×C137—21no—
90C2×C4×C137—41no—
91C2×C4×C137C22⋊C411yescharacteristic
92C2×C4×C137C22×C411yescharacteristic
93C22×C137—21no—
94C22×C137—21no—
95C22×C137—41no—
96C22×C137—41no—
97C22×C137—41no—
98C22×C137#10212yes—
99C22×C137C2×C22⋊C411yescharacteristic
100C2×D137—21no—
101C2×D137—21no—
102C2×D137—21no—
103C2×D137—21no—
104C2×D137—42no—
105C2×D137—42no—
106C2×D137—41no—
107C2×D137—41no—
108C2×D137—41no—
109C2×D137—84no—
110C2×D137#10212yes—
111C2×D137C2×D811yescharacteristic
112C2×D137C2×SD1611yescharacteristic
113#10587739—82no—
114C4×C137—21no—
115C4×C137—21no—
116C4×C137—41no—
117Dic137—21no—
118Dic137—41no—
119Dic137#9912yes—
120C2×C137—22no—
121C2×C137—81no—
122C2×C137C2×#9911yescharacteristic
123C2×C137C2×#10211yescharacteristic
124C2×C137#42111yescharacteristic
125D137—21no—
126D137—22no—
127D137—21no—
128D137—81no—
129C137#151911yescharacteristic, complemented, Sylow
130#1519—1371nomaximal, Sylow
131C2×#121—1371no—
132C2×C22⋊C8—1371no—
133#380—1371no—
134#327—1372no—
135#327—1372no—
136C22×D4—2741no—
137C22×D4—1371no—
138C23×C4—1371no—
139C22×C8—2741no—
140#121—2741no—
141#121—2741no—
142#121—1371no—
143#121—1371no—
144#121—1371no—
145#121#1058773911yescharacteristic, complemented
146C2×C4⋊C4—2741no—
147C2×C4⋊C4—1371no—
148C2×C22⋊C4—2741no—
149C2×C22⋊C4—2741no—
150C22⋊C8—2741no—
151C22⋊C8—1372no—
152#95—1372no—
153C24—2741no—
154C24—1371no—
155C2×D4—21no—
156C2×D4—2741no—
157C2×D4—2741no—
158C2×D4—2741no—
159C2×D4—2741no—
160C2×D4—5482no—
161C2×D4—5482no—
162C2×D4—5482no—
163C2×D4—1371no—
164C2×D4C2×#1058773911yescharacteristic
165C22×C4—2741no—
166C22×C4—2742no—
167C22×C4—5482no—
168C22×C4—1371no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4—1371no—
172C22×C4—1371no—
173C22×C4C2×#1058773911yescharacteristic
174C2×C8—2742no—
175C2×C8—5482no—
176C4⋊C4—2741no—
177C4⋊C4—5482no—
178C4⋊C4—1371no—
179C4⋊C4C2×#1058773911yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—2741no—
182C22⋊C4—2741no—
183C22⋊C4—2741no—
184C22⋊C4—5482no—
185C23—21no—
186C23—2741no—
187C23—5481no—
188C23—5481no—
189C23—5482no—
190C23—5482no—
191C23—5482no—
192C23—1371no—
193C23—1371no—
194C23—1372no—
195C23—1372no—
196C23C22⋊#1058773911yescharacteristic
197D4—41no—
198D4—42no—
199D4—5481no—
200D4—5481no—
201D4—5481no—
202D4—5481no—
203D4—5481no—
204D4—5482no—
205D4—5482no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—2742no—
209C2×C4—2742no—
210C2×C4—2742no—
211C2×C4—2741no—
212C2×C4—5481no—
213C2×C4—5481no—
214C2×C4—5482no—
215C2×C4—5482no—
216C2×C4—5482no—
217C2×C4—5482no—
218C2×C4—1371no—
219C2×C4—1371no—
220C2×C4C22⋊#1058773911yescharacteristic
221C2×C4C22×#1058773911yescharacteristic
222C8—5482no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—2741no—
229C22—2741no—
230C22—2741no—
231C22—2741no—
232C22—5482no—
233C22—5482no—
234C22—5481no—
235C22—5481no—
236C22—5481no—
237C22—10964no—
238C22—1371no—
239C22—1372no—
240C22—1371no—
241C22#1798415012yes—
242C22C2×C22⋊#1058773911yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—2741no—
247C4—5481no—
248C4—10962no—
249C4—1372no—
250C2—22no—
251C2—81no—
252C2—2741no—
253C2—2742no—
254C2—2741no—
255C2—10961no—
256C2D137⋊#9911yescharacteristic
257C2C2×#1798415011yescharacteristic
258C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#121×D137C2
cosocleC2×C4×D137C23
hypercenter#121#10587739
hypocenterC137#1519
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#17984145C2semidirect
2. #22578345C2semidirect
3. #22578255C2semidirect
4. #22578254C2semidirect
10. #17984145C22semidirect
11. (C2×#10587739)⋊C4C22semidirect
13. C2×D4×D137C4semidirect
16. #17984062C4semidirect
20. C4⋊C4×D137C4semidirect
22. #121×C137C4semidirect
129. C137#1519semidirect
145. #121#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841452
#225783452
#225782552
#225782542
#121×D1372
#1519137