Finite Group Databasebeta

← #31784078

Subgroups — #31784078

Order 17536.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#17984053C211yesmaximal, characteristic, complemented
2C2×#17984050C211yesmaximal, characteristic
3#22577341C211yesmaximal, characteristic, complemented
4#22577332C212yesmaximal, complemented
5#22577321C211yesmaximal, characteristic
6#22577320C211yesmaximal, characteristic
7#22577316C212yesmaximal, complemented
8#22577250C212yesmaximal, complemented
9#22577250C212yesmaximal, complemented
10#22577248C211yesmaximal, characteristic, complemented
11#400×C137C211yesmaximal, characteristic, complemented
12C22×Dic274C2211yescharacteristic
13C22×D548C2211yescharacteristic
14C22×C4×D137C2212yes—
15#17984053—24no—
16#17984050—24no—
17C2×#14555489C2211yescharacteristic
18C2×#14555489C2211yescharacteristic
19C2×#14555487C2212yes—
20C2×#14555487C2212yes—
21C2×#14555487C2212yes—
22C2×#14555487C2212yes—
23C2×#14555487C2212yes—
24C2×C4×Dic137C2212yes—
25C2×C4×Dic137C2211yescharacteristic
26C2×C4×Dic137C2211yescharacteristic
27C2×C4×Dic137C2211yescharacteristic
28#17983995C2212yes—
29#17983995C2211yescharacteristic
30#17983995C2212yes—
31#17983995C2212yes—
32C2×C4⋊C4×C137C2211yescharacteristic
33C2×C4⋊C4×C137C2211yescharacteristic
34#95×C137C2211yescharacteristic
35#95×C137C2212yes—
36#95×C137C2212yes—
37C23×D137C2312yes—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C2×Dic274—22no—
41C2×D548—24no—
42C2×D548—22no—
43C22×Dic137C2311yescharacteristic
44C22×Dic137C2312yes—
45C22×Dic137C2311yescharacteristic
46C22×Dic137C2311yescharacteristic
47C22×Dic137C2311yescharacteristic
48C2×C4×D137—24no—
49C2×C4×D137—24no—
50C2×C4×D137—24no—
51#14555489—22no—
52#14555489—22no—
53#14555487—24no—
54#14555487—24no—
55#14555487—24no—
56#14555487—24no—
57#14555487—24no—
58C4×Dic137—22no—
59C4×Dic137—22no—
60C4×Dic137—24no—
61C4×Dic137D414yes—
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2312yes—
64C22×C4×C137C2312yes—
65C22×C4×C137C2311yescharacteristic
66C22×C4×C137C2311yescharacteristic
67C4⋊C4×C137—22no—
68C4⋊C4×C137—22no—
69C22×D137—22no—
70C22×D137—22no—
71C22×D137—22no—
72C22×D137—22no—
73C22×D137—22no—
74C22×D137—22no—
75C22×D137#4114yes—
76Dic274—28no—
77D548—28no—
78C2×Dic137—21no—
79C2×Dic137—21no—
80C2×Dic137—21no—
81C2×Dic137—21no—
82C2×Dic137—21no—
83C2×Dic137—22no—
84C2×Dic137—22no—
85C2×Dic137—22no—
86C2×Dic137—22no—
87C2×Dic137C2×D412yes—
88C2×Dic137C2×D412yes—
89C2×Dic137#4112yes—
90C2×Dic137#4114yes—
91C4×D137—48no—
92C23×C137C2411yescharacteristic
93C2×C4×C137—21no—
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—21no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137—22no—
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137—22no—
104C2×C4×C137C2×D412yes—
105C2×C4×C137#4112yes—
106C22×C137C22×D411yescharacteristic
107C22×C137C2×#4112yes—
108C22×C137C2×#4111yescharacteristic
109C22×C137C2×#4112yes—
110C22×C137C2×#4111yescharacteristic
111C2×D137—24no—
112C2×D137—24no—
113C2×D137—24no—
114C2×D137—42no—
115C2×D137—42no—
116C2×D137—42no—
117C2×D137—42no—
118C4×C137—22no—
119C4×C137—22no—
120C4×C137—41no—
121C4×C137—42no—
122C4×C137—42no—
123Dic137—22no—
124Dic137—22no—
125Dic137—41no—
126Dic137—42no—
127Dic137#12414yes—
128C2×C137#52511yescharacteristic
129C2×C137#52511yescharacteristic
130C2×C137C2×#12411yescharacteristic
131C2×C137#52912yes—
132C2×C137#52912yes—
133D137—44no—
134C137#239811yescharacteristic, complemented, Sylow
135#2398—1371nomaximal, Sylow
136C2×C4×Q8—1371no—
137C2×C4×D4—1371no—
138#400—1371no—
139#400D13711yescharacteristic, complemented
140#397—1372no—
141#394—1371no—
142#394—1371no—
143#393—1372no—
144C4×C22⋊C4—1372no—
145C4×C22⋊C4—1372no—
146#376—1371no—
147C22×Q8—1371no—
148C22×D4—1371no—
149C23×C4—1372no—
150C4×Q8—2744no—
151C4×D4—2744no—
152C2×C4⋊C4—1371no—
153C2×C4⋊C4—1371no—
154C2×C4⋊C4C2×D13711yescharacteristic
155C2×C4⋊C4C2×D13711yescharacteristic
156C2×C22⋊C4—1372no—
157C2×C22⋊C4—1372no—
158C2×C22⋊C4—1372no—
159C2×C22⋊C4—1372no—
160C2×C22⋊C4—1372no—
161C2×C42—1371no—
162C2×C42—1371no—
163C2×C42—1372no—
164C2×C42—1371no—
165#95—1372no—
166#95—1371no—
167#95—1372no—
168#95—1372no—
169#95C2×D13711yescharacteristic
170#95C2×D13712yes—
171#95C2×D13712yes—
172C24—1372no—
173C2×Q8—2742no—
174C2×Q8—2742no—
175C2×Q8—2742no—
176C2×D4—2744no—
177C2×D4—2742no—
178C22×C4—2744no—
179C22×C4—2744no—
180C22×C4—2744no—
181C22×C4—1372no—
182C22×C4—1371no—
183C22×C4—1371no—
184C22×C4—1371no—
185C22×C4—1371no—
186C22×C4C22×D13712yes—
187C22×C4C22×D13712yes—
188C22×C4C22×D13711yescharacteristic
189C22×C4C22×D13711yescharacteristic
190C22×C4C22×D13711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—2742no—
194C4⋊C4—2742no—
195C22⋊C4—2744no—
196C22⋊C4—2744no—
197C22⋊C4—2744no—
198C22⋊C4—2744no—
199C22⋊C4—2744no—
200C42—2744no—
201C42—2742no—
202C42—2742no—
203C42—1374no—
204C23—2742no—
205C23—2742no—
206C23—2742no—
207C23—2742no—
208C23—2742no—
209C23—2742no—
210C23—1374no—
211C23C23×D13711yescharacteristic
212Q8—2748no—
213D4—2748no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—2741no—
226C2×C4—2741no—
227C2×C4—2741no—
228C2×C4—2741no—
229C2×C4—2741no—
230C2×C4—2742no—
231C2×C4—2742no—
232C2×C4—2742no—
233C2×C4—2742no—
234C2×C4—5488no—
235C2×C4—1372no—
236C2×C4—1372no—
237C2×C4—1372no—
238C2×C4—1374no—
239C2×C4D4×D13712yes—
240C2×C4#1455551112yes—
241C22—2744no—
242C22—2744no—
243C22—2744no—
244C22—5482no—
245C22—5482no—
246C22—5482no—
247C22—5482no—
248C22C2×D4×D13711yescharacteristic
249C22#41×D13711yescharacteristic
250C22#41×D13712yes—
251C22#41×D13712yes—
252C22C2×#1455551111yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—2742no—
259C4—2742no—
260C4—5482no—
261C4—5481no—
262C4—1374no—
263C2—5484no—
264C2#2257791111yescharacteristic
265C2#2257791611yescharacteristic
266C2#124×D13711yescharacteristic
267C2#2257793612yes—
268C2#2257793812yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#400×C137C2
prime radicalC22×D548C22
hypercenter#400D137
hypocenterC137#2398
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#17984053C2semidirect
3. #22577341C2semidirect
4. #22577332C2semidirect
7. #22577316C2semidirect
8. #22577250C2semidirect
9. #22577250C2semidirect
10. #22577248C2semidirect
11. #400×C137C2semidirect
134. C137#2398semidirect
139. #400D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840532
C2×#179840502
#225773412
#225773322
#225773212
#225773202
#225773162
#225772502
#225772482
#400×C1372
#2398137