Finite Group Databasebeta

← #31784053

Subgroups — #31784053

Order 17536.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17984050C211yesmaximal, characteristic
2C2×#17984046C211yesmaximal, characteristic, complemented
3#22577342C211yesmaximal, characteristic, complemented
4#22577340C211yesmaximal, characteristic, complemented
5#22577330C212yesmaximal, complemented
6#22577329C211yesmaximal, characteristic, complemented
7#22577320C212yesmaximal
8C4⋊C4×Dic137C212yesmaximal, complemented
9#22577250C211yesmaximal, characteristic, complemented
10#22577250C212yesmaximal, complemented
11#398×C137C211yesmaximal, characteristic, complemented
12C22×D548C2211yescharacteristic
13C22×C4×D137C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15#17984050—24no—
16#17984046—24no—
17C2×C4⋊Dic137C2212yes—
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555487C2212yes—
20C2×#14555487C2212yes—
21C2×#14555487C2212yes—
22C2×#14555487C2212yes—
23C2×#14555487C2211yescharacteristic
24C2×#14555487C2211yescharacteristic
25C2×C4×Dic137C2212yes—
26C2×C4×Dic137C2212yes—
27C2×C4×Dic137C2211yescharacteristic
28#17983995C2211yescharacteristic
29#17983995C2212yes—
30#17983995C2212yes—
31#17983995C2212yes—
32C2×C4⋊C4×C137C2212yes—
33C2×C4⋊C4×C137C2211yescharacteristic
34C2×C4⋊C4×C137C2211yescharacteristic
35#95×C137C2212yes—
36#95×C137C2211yescharacteristic
37C23×D137C2311yescharacteristic
38C23×D137C2311yescharacteristic
39C2×D548—22no—
40C2×D548—22no—
41C2×D548—22no—
42C22×Dic137C2311yescharacteristic
43C22×Dic137C2312yes—
44C22×Dic137C2312yes—
45C22×Dic137C2311yescharacteristic
46C2×C4×D137—24no—
47C2×C4×D137—22no—
48C2×C4×D137—24no—
49C2×C4×D137D414yes—
50C4⋊Dic137—22no—
51C4⋊Dic137—24no—
52#14555487—24no—
53#14555487—22no—
54#14555487—24no—
55#14555487—22no—
56#14555487—24no—
57#14555487—24no—
58C4×Dic137—24no—
59C4×Dic137—22no—
60C4×Dic137—24no—
61C22×C4×C137C2312yes—
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2311yescharacteristic
64C22×C4×C137C2312yes—
65C22×C4×C137C2311yescharacteristic
66C4⋊C4×C137—22no—
67C4⋊C4×C137—22no—
68C4⋊C4×C137—24no—
69C22×D137—21no—
70C22×D137—22no—
71C22×D137—22no—
72C22×D137—22no—
73C22×D137—22no—
74C22×D137—21no—
75C22×D137—21no—
76C22×D137—21no—
77C22×D137C2×D412yes—
78C22×D137#4112yes—
79D548—28no—
80C2×Dic137—21no—
81C2×Dic137—22no—
82C2×Dic137—22no—
83C2×Dic137—22no—
84C2×Dic137—22no—
85C2×Dic137—22no—
86C2×Dic137—22no—
87C2×Dic137—22no—
88C2×Dic137C2×D412yes—
89C2×Dic137#4114yes—
90C4×D137—28no—
91C4×D137—44no—
92C23×C137C2411yescharacteristic
93C2×C4×C137—21no—
94C2×C4×C137—22no—
95C2×C4×C137—22no—
96C2×C4×C137—22no—
97C2×C4×C137—21no—
98C2×C4×C137—22no—
99C2×C4×C137—21no—
100C2×C4×C137—22no—
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137C2×D412yes—
104C2×C4×C137#4112yes—
105C2×C4×C137#4114yes—
106C22×C137C22×D411yescharacteristic
107C22×C137C2×#4112yes—
108C22×C137C2×#4111yescharacteristic
109C22×C137C2×#4111yescharacteristic
110C22×C137C2×#4112yes—
111C2×D137—22no—
112C2×D137—22no—
113C2×D137—22no—
114C2×D137—24no—
115C2×D137—22no—
116C2×D137—41no—
117C2×D137—42no—
118C2×D137—42no—
119C2×D137—42no—
120C2×D137—41no—
121C4×C137—22no—
122C4×C137—22no—
123C4×C137—24no—
124C4×C137—41no—
125C4×C137—42no—
126Dic137—42no—
127Dic137—41no—
128Dic137—42no—
129Dic137#12314yes—
130C2×C137C2×#12311yescharacteristic
131C2×C137#52511yescharacteristic
132C2×C137#52912yes—
133C2×C137#52912yes—
134C2×C137#53211yescharacteristic
135D137—42no—
136D137—42no—
137C137#239511yescharacteristic, complemented, Sylow
138#2395—1371nomaximal, Sylow
139C2×C4×D4—1371no—
140C2×#117—1371no—
141#400—1371no—
142#399—1371no—
143#398D13711yescharacteristic, complemented
144#397—1371no—
145#397—1372no—
146#394—1372no—
147C4×C4⋊C4—1372no—
148C4×C22⋊C4—1372no—
149C4×C22⋊C4—1371no—
150C22×D4—1371no—
151C23×C4—1371no—
152C23×C4—1371no—
153C4×D4—2744no—
154#117—2744no—
155C2×C4⋊C4—1371no—
156C2×C4⋊C4—1372no—
157C2×C4⋊C4C2×D13711yescharacteristic
158C2×C4⋊C4C2×D13711yescharacteristic
159C2×C4⋊C4C2×D13712yes—
160C2×C22⋊C4—1372no—
161C2×C22⋊C4—1371no—
162C2×C22⋊C4—1371no—
163C2×C22⋊C4—1372no—
164C2×C22⋊C4—1372no—
165C2×C22⋊C4—1372no—
166C2×C42—1372no—
167C2×C42—1371no—
168C2×C42—1372no—
169#95—1371no—
170#95—1372no—
171#95—1372no—
172#95—1372no—
173#95C2×D13712yes—
174#95C2×D13711yescharacteristic
175C24—1371no—
176C24—1371no—
177C2×D4—2742no—
178C2×D4—2742no—
179C2×D4—2742no—
180C22×C4—2744no—
181C22×C4—2742no—
182C22×C4—2744no—
183C22×C4—1372no—
184C22×C4—1371no—
185C22×C4—1372no—
186C22×C4—1374no—
187C22×C4—1371no—
188C22×C4C22×D13711yescharacteristic
189C22×C4C22×D13712yes—
190C22×C4C22×D13711yescharacteristic
191C22×C4C22×D13711yescharacteristic
192C22×C4C22×D13712yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—2742no—
197C4⋊C4—2744no—
198C22⋊C4—2744no—
199C22⋊C4—2744no—
200C22⋊C4—2742no—
201C22⋊C4—2742no—
202C22⋊C4—2744no—
203C22⋊C4—2744no—
204C42—2744no—
205C42—2744no—
206C42—2742no—
207C23—2741no—
208C23—2742no—
209C23—2742no—
210C23—2741no—
211C23—2741no—
212C23—2741no—
213C23—2742no—
214C23—2742no—
215C23—1372no—
216C23—1372no—
217C23C23×D13711yescharacteristic
218D4—2748no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—2748no—
230C2×C4—2741no—
231C2×C4—2742no—
232C2×C4—2742no—
233C2×C4—2742no—
234C2×C4—2742no—
235C2×C4—2742no—
236C2×C4—2742no—
237C2×C4—2742no—
238C2×C4—5484no—
239C2×C4—1372no—
240C2×C4—1374no—
241C2×C4D4×D13712yes—
242C2×C4#1455550912yes—
243C2×C4#1455551114yes—
244C22—2744no—
245C22—2742no—
246C22—2742no—
247C22—2742no—
248C22—2742no—
249C22—5482no—
250C22—5482no—
251C22—5482no—
252C22—5481no—
253C22—5481no—
254C22C2×D4×D13711yescharacteristic
255C22#41×D13711yescharacteristic
256C22#41×D13712yes—
257C22C2×#1455550911yescharacteristic
258C22C2×#1455551112yes—
259C4—24no—
260C4—22no—
261C4—22no—
262C4—41no—
263C4—42no—
264C4—5482no—
265C4—5481no—
266C4—5482no—
267C4—1374no—
268C2—5482no—
269C2—5482no—
270C2#123×D13711yescharacteristic
271C2#2257791311yescharacteristic
272C2#2257793812yes—
273C2#2257793912yes—
274C2#2257795411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#398×C137C2
prime radicalC22×D548C22
hypercenter#398D137
hypocenterC137#2395
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#17984046C2semidirect
3. #22577342C2semidirect
4. #22577340C2semidirect
5. #22577330C2semidirect
6. #22577329C2semidirect
8. C4⋊C4×Dic137C2semidirect
9. #22577250C2semidirect
10. #22577250C2semidirect
11. #398×C137C2semidirect
137. C137#2395semidirect
143. #398D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840502
C2×#179840462
#225773422
#225773402
#225773302
#225773292
#225773202
C4⋊C4×Dic1372
#225772502
#398×C1372
#2395137