Finite Group Databasebeta

← #28340091

Subgroups — #28340091

Order 13696.

836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#13133817C211yesmaximal, characteristic
2C2×#16812109C211yesmaximal, characteristic, complemented
3C2×#16812096C211yesmaximal, characteristic, complemented
4C2×#16812089C211yesmaximal, characteristic, complemented
5#20292543C214yesmaximal, complemented
6#20292532C211yesmaximal, characteristic, complemented
7#20292502C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9#390×C107C211yesmaximal, characteristic, complemented
10C2×#13133817—24no—
11C2×#13133817—24no—
12C2×#13133817—24no—
13C22×Dic214C2211yescharacteristic
14C22×Dic214C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C23×Dic107C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812109—24no—
19#16812096—24no—
20#16812089—24no—
21C2×C4⋊Dic107C2212yes—
22C2×C4⋊Dic107C2214yes—
23C2×C22⋊Dic107C2214yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2212yes—
28C107⋊#95C2214yes—
29C22×D4×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C22⋊C4×C107C2214yes—
32C2×C42×C107C2211yescharacteristic
33C23×D107C2311yescharacteristic
34#13133817—232no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—22no—
40C2×Dic214D414yes—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—24no—
44C22×Dic107—24no—
45C22×Dic107—24no—
46C22×Dic107—24no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2312yes—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107D414yes—
54C4⋊Dic107—24no—
55C4⋊Dic107—28no—
56C22⋊Dic107—28no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—22no—
60C24×C107C2312yes—
61C2×D4×C107—22no—
62C2×D4×C107—22no—
63C2×D4×C107D414yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—22no—
68C22⋊C4×C107—28no—
69C42×C107—22no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—24no—
73C23×C107—24no—
74C23×C107C2×D414yes—
75C23×C107C2411yescharacteristic
76C22×D107—21no—
77C22×D107—21no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107C2×D412yes—
81D4×C107—28no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—24no—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107#4114yescomplemented
90C22⋊D107—216no—
91C2×Dic107—216no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107C2×D412yes—
103C2×Dic107C2×D414yes—
104C4×D107—28no—
105Dic214—28no—
106Dic214—44no—
107C22×C107—24no—
108C22×C107—28no—
109C22×C107—42no—
110C22×C107—42no—
111C22×C107—44no—
112C22×C107C22×D411yescharacteristic
113C22×C107C22×D411yescharacteristic
114C22×C107C22×D411yescharacteristic
115C22×C107C2×#4112yes—
116C22×C1072−1+412yes—
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—42no—
121C2×D107—42no—
122C4×C107—24no—
123C4×C107—42no—
124C4×C107#12014yes—
125Dic107—24no—
126Dic107—22no—
127Dic107—42no—
128Dic107—42no—
129C2×C107—44no—
130C2×C107C2×#12011yescharacteristic
131C2×C107#53211yescharacteristic
132C2×C107#53611yescharacteristic
133C2×C107#54712yes—
134C2×C107#54812yes—
135D107—42no—
136#2432—1071nomaximal, Sylow
137C107#243211yescharacteristic, complemented, Sylow
138C22×#41—1071no—
139C2×C4⋊Q8—1071no—
140C2×#124—1071no—
141C2×C22⋊Q8—1071no—
142#394—1074no—
143#391—1071no—
144#390D10711yescharacteristic, complemented
145#385—1074no—
146C4×C22⋊C4—1071no—
147C2×#41—2144no—
148C2×#41—2144no—
149C2×#41—2144no—
150C22×Q8—1071no—
151C22×Q8—1071no—
152C22×D4—1072no—
153C22×D4C2×D10711yescharacteristic
154C23×C4—1071no—
155C23×C4—1072no—
156C4⋊Q8—2144no—
157#124—2144no—
158C22⋊Q8—2144no—
159C2×C4⋊C4—1072no—
160C2×C4⋊C4—1074no—
161C2×C4⋊C4C2×D10711yescharacteristic
162C2×C22⋊C4—1074no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4C2×D10714yes—
166C2×C42—1071no—
167C2×C42C2×D10711yescharacteristic
168#95—1072no—
169#95—1074no—
170C24—1071no—
171C24C22×D10712yes—
172#41—21432no—
173C2×Q8—1074no—
174C2×Q8—2142no—
175C2×Q8—2142no—
176C2×Q8—2142no—
177C2×Q8—2142no—
178C2×Q8—2142no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—2144no—
182C2×D4—2144no—
183C2×D4—2144no—
184C2×D4D42814yes—
185C22×C4—1071no—
186C22×C4—1072no—
187C22×C4—1074no—
188C22×C4—1072no—
189C22×C4—1072no—
190C22×C4—2144no—
191C22×C4—2144no—
192C22×C4—2144no—
193C22×C4—2142no—
194C22×C4—2142no—
195C22×C4C22×D10712yes—
196C22×C4C22×D10712yes—
197C22×C4C22×D10711yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—2144no—
200C4⋊C4—2148no—
201C22⋊C4—28no—
202C22⋊C4—2144no—
203C22⋊C4—2148no—
204C22⋊C4—2144no—
205C42—22no—
206C42—2142no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—1072no—
212C23—2141no—
213C23—2141no—
214C23—2142no—
215C23—2142no—
216C23C2×D42814yes—
217C23C23×D10711yescharacteristic
218Q8—4284no—
219Q8—2148no—
220D4—28no—
221D4—21416no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—1072no—
227C2×C4—1074no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—21416no—
235C2×C4—2141no—
236C2×C4—2142no—
237C2×C4—2141no—
238C2×C4—2142no—
239C2×C4—2148no—
240C2×C4D4×D10712yes—
241C2×C4D4×D10712yes—
242C2×C4C2×D42812yes—
243C2×C4#1313381714yescomplemented
244C22—28no—
245C22—24no—
246C22—42no—
247C22—42no—
248C22—44no—
249C22—4282no—
250C22—4282no—
251C22—2142no—
252C22—2142no—
253C22—2142no—
254C22C2×D4×D10711yescharacteristic
255C22C2×D4×D10711yescharacteristic
256C22C22×D42811yescharacteristic
257C22C2×#1313381712yes—
258C22#1681215612yes—
259C4—24no—
260C4—42no—
261C4—4282no—
262C4—4282no—
263C4—2142no—
264C4—2144no—
265C4#1681207914yes—
266C2—44no—
267C2—4282no—
268C2C2×#1681207911yescharacteristic
269C2#2029317511yescharacteristic
270C2#2029319711yescharacteristic
271C2#2029328112yes—
272C2#2029328312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
prime radicalC22×#13133817C2
hypercenter#390D107
hypocenterC107#2432
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16812109C2semidirect
3. C2×#16812096C2semidirect
4. C2×#16812089C2semidirect
5. #20292543C2semidirect
6. #20292532C2semidirect
7. #20292502C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #390×C107C2semidirect
89. C2×C4×C107#41semidirect
137. C107#2432semidirect
144. #390D107semidirect
243. C2×C4#13133817semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168121092
C2×#168120962
C2×#168120892
#202925432
#202925322
#202925022
C4×(C2×D107)⋊C42
#390×C1072
#2432107