Finite Group Databasebeta

← #31784074

Subgroups — #31784074

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×#17984048C211yesmaximal, characteristic
3#22577344C212yesmaximal, complemented
4#22577329C211yesmaximal, characteristic, complemented
5#22577318C212yesmaximal
6#22577316C211yesmaximal, characteristic, complemented
7#22577315C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic137C212yesmaximal, complemented
9#22577250C212yesmaximal, complemented
10#22577248C211yesmaximal, characteristic, complemented
11#397×C137C211yesmaximal, characteristic, complemented
12C22×Dic274C2211yescharacteristic
13C22×C22⋊D137C2211yescharacteristic
14C23×Dic137C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984053—24no—
17#17984048—24no—
18C2×#14555489C2212yes—
19C2×#14555489C2211yescharacteristic
20C2×C22⋊Dic137C2212yes—
21C2×#14555487C2212yes—
22C2×#14555487C2212yes—
23C2×#14555487C2211yescharacteristic
24C2×C4×Dic137C2212yes—
25C2×C4×Dic137C2212yes—
26C2×C4×Dic137C2211yescharacteristic
27#17983995C2211yescharacteristic
28#17983995C2212yes—
29#17983995C2212yes—
30#17983995C2212yes—
31#17983995C2212yes—
32C2×C4⋊C4×C137C2211yescharacteristic
33C2×C22⋊C4×C137C2212yes—
34C2×C22⋊C4×C137C2211yescharacteristic
35#95×C137C2211yescharacteristic
36#95×C137C2212yes—
37C23×D137C2311yescharacteristic
38C2×Dic274—24no—
39C2×Dic274—22no—
40C2×C22⋊D137—22no—
41C2×C22⋊D137—22no—
42C2×C22⋊D137—22no—
43C22×Dic137—22no—
44C22×Dic137—24no—
45C22×Dic137C2312yes—
46C22×Dic137C2311yescharacteristic
47C22×Dic137C2311yescharacteristic
48C22×Dic137C2312yes—
49C22×Dic137C2311yescharacteristic
50C2×C4×D137—22no—
51C2×C4×D137—24no—
52#14555489—22no—
53#14555489—24no—
54C22⋊Dic137—24no—
55#14555487—24no—
56#14555487—24no—
57#14555487—22no—
58C4×Dic137—24no—
59C4×Dic137—24no—
60C4×Dic137D414yes—
61C24×C137C2311yescharacteristic
62C22×C4×C137C2312yes—
63C22×C4×C137C2312yes—
64C22×C4×C137C2311yescharacteristic
65C22×C4×C137C2311yescharacteristic
66C4⋊C4×C137—22no—
67C22⋊C4×C137—24no—
68C22⋊C4×C137—22no—
69C22×D137—21no—
70C22×D137—22no—
71C22×D137—22no—
72C22×D137—21no—
73C22×D137#4112yes—
74Dic274—28no—
75C22⋊D137—28no—
76C2×Dic137—22no—
77C2×Dic137—21no—
78C2×Dic137—21no—
79C2×Dic137—22no—
80C2×Dic137—21no—
81C2×Dic137—22no—
82C2×Dic137—22no—
83C2×Dic137—22no—
84C2×Dic137—22no—
85C2×Dic137—44no—
86C2×Dic137C2×D412yes—
87C2×Dic137C2×D412yes—
88C2×Dic137#4114yes—
89C2×Dic137#4114yes—
90C4×D137—44no—
91C23×C137—21no—
92C23×C137—22no—
93C23×C137—22no—
94C23×C137—21no—
95C23×C137#4112yes—
96C23×C137C2411yescharacteristic
97C2×C4×C137—22no—
98C2×C4×C137—21no—
99C2×C4×C137—22no—
100C2×C4×C137—22no—
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137—22no—
104C2×C4×C137—22no—
105C2×C4×C137—22no—
106C2×C4×C137C2×D412yes—
107C22×C137—22no—
108C22×C137—24no—
109C22×C137—41no—
110C22×C137—41no—
111C22×C137—42no—
112C22×C137C22×D411yescharacteristic
113C22×C137C2×#4111yescharacteristic
114C22×C137C2×#4112yes—
115C22×C137C2×#4112yes—
116C22×C137C2×#4111yescharacteristic
117C2×D137—22no—
118C2×D137—24no—
119C2×D137—41no—
120C2×D137—41no—
121C2×D137—42no—
122C4×C137—22no—
123C4×C137—42no—
124C4×C137—41no—
125C4×C137—42no—
126Dic137—22no—
127Dic137—24no—
128Dic137—41no—
129Dic137—42no—
130Dic137#12414yes—
131C2×C137—42no—
132C2×C137#52511yescharacteristic
133C2×C137#52511yescharacteristic
134C2×C137C2×#12411yescharacteristic
135C2×C137#52912yes—
136C2×C137#52912yes—
137D137—42no—
138C137#239811yescharacteristic, complemented, Sylow
139#2398—1371nomaximal, Sylow
140C2×C4×Q8—1371no—
141C2×C4×D4—1371no—
142#400—1372no—
143#397—1371no—
144#397D13711yescharacteristic, complemented
145#394—1372no—
146#393—1371no—
147#393—1371no—
148C4×C22⋊C4—1372no—
149C4×C22⋊C4—1372no—
150#376—1371no—
151C22×Q8—1371no—
152C22×D4—1371no—
153C23×C4—1371no—
154C23×C4—1371no—
155C4×Q8—2744no—
156C4×D4—2744no—
157C2×C4⋊C4—1371no—
158C2×C4⋊C4—1372no—
159C2×C4⋊C4C2×D13711yescharacteristic
160C2×C22⋊C4—1372no—
161C2×C22⋊C4—1371no—
162C2×C22⋊C4—1372no—
163C2×C22⋊C4—1372no—
164C2×C22⋊C4C2×D13711yescharacteristic
165C2×C22⋊C4C2×D13712yes—
166C2×C42—1372no—
167C2×C42—1372no—
168C2×C42—1371no—
169#95—1371no—
170#95—1372no—
171#95—1372no—
172#95—1372no—
173#95—1372no—
174#95C2×D13711yescharacteristic
175#95C2×D13712yes—
176C24—1371no—
177C24C22×D13711yescharacteristic
178C2×Q8—2744no—
179C2×Q8—2742no—
180C2×D4—2742no—
181C2×D4—2742no—
182C2×D4—2742no—
183C22×C4—2742no—
184C22×C4—2744no—
185C22×C4—2742no—
186C22×C4—2744no—
187C22×C4—1372no—
188C22×C4—1371no—
189C22×C4—1372no—
190C22×C4—1371no—
191C22×C4—1371no—
192C22×C4C22×D13712yes—
193C22×C4C22×D13711yescharacteristic
194C22×C4C22×D13712yes—
195C22×C4C22×D13711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—2742no—
198C4⋊C4—2744no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—2744no—
202C22⋊C4—2744no—
203C22⋊C4—2744no—
204C22⋊C4—2742no—
205C42—2744no—
206C42—2744no—
207C42—1374no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—2741no—
213C23—2742no—
214C23—2742no—
215C23—2741no—
216C23—1372no—
217C23#1455550912yes—
218C23C23×D13711yescharacteristic
219Q8—2748no—
220D4—2748no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—2742no—
231C2×C4—2741no—
232C2×C4—2742no—
233C2×C4—2741no—
234C2×C4—2741no—
235C2×C4—2742no—
236C2×C4—2742no—
237C2×C4—2742no—
238C2×C4—2742no—
239C2×C4—5484no—
240C2×C4—5484no—
241C2×C4—1372no—
242C2×C4—1374no—
243C2×C4—1372no—
244C2×C4—1374no—
245C2×C4D4×D13712yes—
246C22—24no—
247C22—22no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—2744no—
252C22—2742no—
253C22—5482no—
254C22—5481no—
255C22—5481no—
256C22C2×D4×D13711yescharacteristic
257C22#41×D13712yes—
258C22#41×D13712yes—
259C22#41×D13711yescharacteristic
260C22C2×#1455550911yescharacteristic
261C4—22no—
262C4—42no—
263C4—42no—
264C4—41no—
265C4—2744no—
266C4—2742no—
267C4—5482no—
268C4—5481no—
269C4—1374no—
270C2—42no—
271C2—5482no—
272C2#2257791111yescharacteristic
273C2#2257791311yescharacteristic
274C2#124×D13711yescharacteristic
275C2#2257793012yes—
276C2#2257793612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#397×C137C2
prime radicalC22×C22⋊D137C22
hypercenter#397D137
hypocenterC137#2398
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#17984053C2semidirect
3. #22577344C2semidirect
4. #22577329C2semidirect
6. #22577316C2semidirect
7. #22577315C2semidirect
8. C22⋊C4×Dic137C2semidirect
9. #22577250C2semidirect
10. #22577248C2semidirect
11. #397×C137C2semidirect
138. C137#2398semidirect
144. #397D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840532
C2×#179840482
#225773442
#225773292
#225773182
#225773162
#225773152
C22⋊C4×Dic1372
#225772502
#225772482
#397×C1372
#2398137