Finite Group Databasebeta

← #31784051

Subgroups — #31784051

Order 17536.

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

normal subgroups only

This group has 560 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×#17984048C211yesmaximal, characteristic
2C2×#17984046C211yesmaximal, characteristic, complemented
3#22577343C211yesmaximal, characteristic, complemented
4#22577339C211yesmaximal, characteristic, complemented
5#22577334C211yesmaximal, characteristic, complemented
6#22577332C212yesmaximal, complemented
7#22577318C212yesmaximal
8#22577254C212yesmaximal, complemented
9C22⋊C4×Dic137C212yesmaximal, complemented
10#22577250C211yesmaximal, characteristic, complemented
11#397×C137C211yesmaximal, characteristic, complemented
12C22×C22⋊D137C2211yescharacteristic
13C23×Dic137C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15#17984048—24no—
16#17984046—24no—
17C2×C4⋊Dic137C2211yescharacteristic
18C2×#14555489C2212yes—
19C2×#14555489C2212yes—
20C2×#14555489C2211yescharacteristic
21C2×C22⋊Dic137C2212yes—
22C2×#14555487C2212yes—
23C2×#14555487C2212yes—
24C2×#14555487C2211yescharacteristic
25C2×C4×Dic137C2212yes—
26C2×C4×Dic137C2212yes—
27C2×C4×Dic137C2211yescharacteristic
28#17983995C2211yescharacteristic
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×C22⋊D137—22no—
39C2×C22⋊D137—22no—
40C2×C22⋊D137—22no—
41C22×Dic137—22no—
42C22×Dic137—24no—
43C22×Dic137C2312yes—
44C22×Dic137C2311yescharacteristic
45C22×Dic137C2312yes—
46C22×Dic137C2311yescharacteristic
47C22×Dic137C2311yescharacteristic
48C2×C4×D137—24no—
49C2×C4×D137D414yes—
50C4⋊Dic137—22no—
51#14555489—24no—
52#14555489—24no—
53#14555489—22no—
54C22⋊Dic137—24no—
55#14555487—24no—
56#14555487—22no—
57#14555487—24no—
58C4×Dic137—24no—
59C4×Dic137—22no—
60C4×Dic137—24no—
61C24×C137C2311yescharacteristic
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2312yes—
64C22×C4×C137C2312yes—
65C22×C4×C137C2311yescharacteristic
66C4⋊C4×C137—22no—
67C22⋊C4×C137—24no—
68C22⋊C4×C137—22no—
69C22×D137—22no—
70C22×D137—22no—
71C22×D137—21no—
72C22×D137—21no—
73C22×D137C2×D412yes—
74C22⋊D137—28no—
75C2×Dic137—21no—
76C2×Dic137—22no—
77C2×Dic137—22no—
78C2×Dic137—21no—
79C2×Dic137—22no—
80C2×Dic137—21no—
81C2×Dic137—22no—
82C2×Dic137—22no—
83C2×Dic137—22no—
84C2×Dic137—44no—
85C2×Dic137C2×D412yes—
86C2×Dic137#4114yes—
87C2×Dic137#4112yes—
88C2×Dic137#4114yes—
89C4×D137—28no—
90C23×C137—21no—
91C23×C137—22no—
92C23×C137—22no—
93C23×C137—21no—
94C23×C137#4112yes—
95C23×C137C2411yescharacteristic
96C2×C4×C137—22no—
97C2×C4×C137—21no—
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×C137—22no—
105C2×C4×C137C2×D412yes—
106C22×C137—22no—
107C22×C137—24no—
108C22×C137—41no—
109C22×C137—42no—
110C22×C137—41no—
111C22×C137C22×D411yescharacteristic
112C22×C137C2×#4112yes—
113C22×C137C2×#4111yescharacteristic
114C22×C137C2×#4112yes—
115C22×C137C2×#4111yescharacteristic
116C2×D137—22no—
117C2×D137—22no—
118C2×D137—22no—
119C2×D137—42no—
120C2×D137—42no—
121C4×C137—22no—
122C4×C137—42no—
123C4×C137—42no—
124C4×C137—41no—
125Dic137—22no—
126Dic137—24no—
127Dic137—41no—
128Dic137—42no—
129Dic137#12314yes—
130C2×C137—42no—
131C2×C137C2×#12311yescharacteristic
132C2×C137#52511yescharacteristic
133C2×C137#52912yes—
134C2×C137#52912yes—
135C2×C137#53211yescharacteristic
136D137—42no—
137C137#239511yescharacteristic, complemented, Sylow
138#2395—1371nomaximal, Sylow
139C2×C4×D4—1371no—
140C2×#117—1371no—
141#400—1371no—
142#399—1371no—
143#398—1371no—
144#397—1372no—
145#397D13711yescharacteristic, complemented
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—1371no—
157C2×C4⋊C4—1372no—
158C2×C4⋊C4—1372no—
159C2×C4⋊C4C2×D13711yescharacteristic
160C2×C22⋊C4—1372no—
161C2×C22⋊C4—1372no—
162C2×C22⋊C4—1371no—
163C2×C22⋊C4—1372no—
164C2×C22⋊C4C2×D13711yescharacteristic
165C2×C22⋊C4C2×D13712yes—
166C2×C42—1372no—
167C2×C42—1371no—
168C2×C42—1372no—
169#95—1372no—
170#95—1372no—
171#95—1371no—
172#95—1372no—
173#95C2×D13711yescharacteristic
174#95C2×D13712yes—
175C24—1371no—
176C24C22×D13711yescharacteristic
177C2×D4—2742no—
178C2×D4—2742no—
179C2×D4—2742no—
180C22×C4—2744no—
181C22×C4—2742no—
182C22×C4—2744no—
183C22×C4—1371no—
184C22×C4—1372no—
185C22×C4—1372no—
186C22×C4—1371no—
187C22×C4—1374no—
188C22×C4—1371no—
189C22×C4C22×D13712yes—
190C22×C4C22×D13712yes—
191C22×C4C22×D13711yescharacteristic
192C22×C4C22×D13711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2744no—
195C4⋊C4—2744no—
196C4⋊C4—2742no—
197C4⋊C4—2742no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—2744no—
201C22⋊C4—2742no—
202C22⋊C4—2744no—
203C22⋊C4—2744no—
204C42—2744no—
205C42—2744no—
206C42—2742no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—2742no—
212C23—2742no—
213C23—2741no—
214C23—2741no—
215C23—1372no—
216C23#1455550912yes—
217C23C23×D13711yescharacteristic
218D4—2748no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—2748no—
229C2×C4—2741no—
230C2×C4—2742no—
231C2×C4—2742no—
232C2×C4—2741no—
233C2×C4—2742no—
234C2×C4—2741no—
235C2×C4—2742no—
236C2×C4—2742no—
237C2×C4—2742no—
238C2×C4—5484no—
239C2×C4—1374no—
240C2×C4—1372no—
241C2×C4—1372no—
242C2×C4—1374no—
243C2×C4D4×D13712yes—
244C22—24no—
245C22—22no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—2742no—
250C22—2742no—
251C22—2742no—
252C22—5482no—
253C22—5482no—
254C22C2×D4×D13711yescharacteristic
255C22#41×D13712yes—
256C22#41×D13712yes—
257C22#41×D13711yescharacteristic
258C22C2×#1455550911yescharacteristic
259C4—22no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—2744no—
264C4—2742no—
265C4—5482no—
266C4—5481no—
267C4—1374no—
268C2—42no—
269C2—5482no—
270C2#123×D13711yescharacteristic
271C2#2257791311yescharacteristic
272C2#2257793012yes—
273C2#2257793612yes—
274C2#2257795211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#397×C137C2
prime radicalC22×C22⋊D137C22
hypercenter#397D137
hypocenterC137#2395
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#17984046C2semidirect
3. #22577343C2semidirect
4. #22577339C2semidirect
5. #22577334C2semidirect
6. #22577332C2semidirect
8. #22577254C2semidirect
9. C22⋊C4×Dic137C2semidirect
10. #22577250C2semidirect
11. #397×C137C2semidirect
137. C137#2395semidirect
145. #397D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840482
C2×#179840462
#225773432
#225773392
#225773342
#225773322
#225773182
#225772542
C22⋊C4×Dic1372
#225772502
#397×C1372
#2395137