Finite Group Databasebeta

← #28340099

Subgroups — #28340099

Order 13696.

1348 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×D428C211yesmaximal, characteristic, complemented
2C2×#16812105C212yesmaximal, complemented
3C2×#16812077C214yesmaximal, complemented
4#20292536C214yesmaximal, complemented
5#20292528C212yesmaximal, complemented
6C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
7C2×#121×C107C211yesmaximal, characteristic, complemented
8C24×D107C2212yes—
9C22×D428—28no—
10C22×D428—24no—
11C22×D428C2212yes—
12C22×D428C2212yes—
13C22×C22⋊D107C2218yescomplemented
14#16812105—28no—
15#16812077—216no—
16C2×C22⋊Dic107C2212yes—
17C2×(C2×D107)⋊C4C2218yescomplemented
18C2×C4×Dic107C2212yescomplemented
19C107⋊#95C2212yescomplemented
20C22×D4×C107C2212yes—
21C22×D4×C107C2211yescharacteristic, complemented
22C23×C4×C107C2211yescharacteristic
23#121×C107—24no—
24C2×C4⋊C4×C107C2211yescharacteristic, complemented
25C2×C22⋊C4×C107C2212yescomplemented
26C23×D107—24no—
27C23×D107—28no—
28C23×D107—28no—
29C23×D107—44no—
30C23×D107C2314yes—
31C2×D428—232no—
32C2×D428—28no—
33C2×D428—24no—
34C2×D428—24no—
35C2×D428—24no—
36C2×D428D418yescomplemented
37C2×C22⋊D107—216no—
38C2×C22⋊D107—216no—
39C2×C22⋊D107—48no—
40C22×Dic107C2314yes—
41C22⋊Dic107D418yescomplemented
42(C2×D107)⋊C4—216no—
43C4×Dic107—24no—
44C24×C107C2312yes—
45C24×C107C2311yescharacteristic
46C2×D4×C107—28no—
47C2×D4×C107—24no—
48C2×D4×C107—24no—
49C2×D4×C107—22no—
50C22×C4×C107—24no—
51C22×C4×C107D414yescomplemented
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—22no—
56C22⋊C4×C107—24no—
57C23×C107—22no—
58C23×C107—22no—
59C23×C107—22no—
60C23×C107—24no—
61C23×C107—24no—
62C23×C107—24no—
63C23×C107C2×D414yes—
64C23×C107C2×D412yes—
65C23×C107C2411yescharacteristic
66C22×D107—216no—
67C22×D107—28no—
68C22×D107—24no—
69C22×D107—24no—
70C22×D107—28no—
71C22×D107—432no—
72C22×D107—48no—
73C22×D107—48no—
74C22×D107C2×D418yes—
75D4×C107—48no—
76D4×C107—44no—
77C2×C4×C107—22no—
78C2×C4×C107—28no—
79C2×C4×C107—22no—
80C2×C4×C107—24no—
81C2×C4×C107C2×D414yes—
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2×D412yes—
84C22⋊D107—432no—
85C2×Dic107—24no—
86C2×Dic107—24no—
87C2×Dic107C2×D418yescomplemented
88D428—216no—
89D428—216no—
90C22×C107—22no—
91C22×C107—22no—
92C22×C107—28no—
93C22×C107—42no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107—48no—
97C22×C107#12714yescomplemented
98C22×C107C22×D412yes—
99C22×C107C22×D412yes—
100C22×C107C22×D411yescharacteristic
101C22×C107H(5,𝔽2)12yes—
102C2×D107—28no—
103C2×D107—216no—
104C2×D107—432no—
105C2×D107—44no—
106C2×D107—44no—
107C2×D107—48no—
108C4×C107—22no—
109C4×C107—42no—
110C4×C107#12014yescomplemented
111Dic107—28no—
112C2×C107—24no—
113C2×C107—82no—
114C2×C107C2×#12011yescharacteristic
115C2×C107C2×#12711yescharacteristic
116C2×C107#53511yescharacteristic
117C2×C107D4214yes—
118D107—48no—
119#2433—1071nomaximal, Sylow
120C107#243311yescharacteristic, complemented, Sylow
121C23×D4—1071no—
122C2×#127—1072no—
123C2×#121D10711yescharacteristic, complemented
124C2×#120—1074no—
125#392—1074no—
126#390—1072no—
127C4×C22⋊C4—1071no—
128C25—1072no—
129C22×D4—1078no—
130C22×D4—1072no—
131C22×D4—1072no—
132C22×D4—2148no—
133C22×D4—2144no—
134C22×D4C2×D10712yes—
135C22×D4C2×D10711yescharacteristic, complemented
136C23×C4C2×D10711yescharacteristic
137#127—2148no—
138#121—24no—
139#120—21416no—
140C2×C4⋊C4C2×D10711yescharacteristic, complemented
141C2×C22⋊C4—1072no—
142C2×C22⋊C4—1078no—
143C2×C22⋊C4C2×D10712yescomplemented
144C2×C42—1072no—
145#95—1072no—
146C24—1074no—
147C24—4284no—
148C24—2148no—
149C24—2148no—
150C24—2144no—
151C24C22×D10712yes—
152C24C22×D10711yescharacteristic
153C2×D4—28no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—1078no—
158C2×D4—4288no—
159C2×D4—2148no—
160C2×D4—21432no—
161C2×D4—21416no—
162C2×D4—21416no—
163C2×D4—2144no—
164C2×D4—2144no—
165C2×D4—2144no—
166C22×C4—24no—
167C22×C4—1074no—
168C22×C4C22⋊D10714yescomplemented
169C22×C4C22×D10712yes—
170C22×C4C22×D10711yescharacteristic
171C22×C4C22×D10711yescharacteristic
172C4⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—1078no—
175C22⋊C4—21416no—
176C42—2144no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—1078no—
184C23—4288no—
185C23—42832no—
186C23—4288no—
187C23—21416no—
188C23—2148no—
189C23—2144no—
190C23—2144no—
191C23—2148no—
192C23D4×D10714yes—
193C23C2×C22⋊D10712yes—
194C23C23×D10711yescharacteristic
195D4—48no—
196D4—44no—
197D4—42832no—
198D4—21416no—
199D4—21416no—
200C2×C4—22no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—1078no—
205C2×C4—2144no—
206C2×C4—2144no—
207C2×C4D4×D10714yes—
208C2×C4C2×C22⋊D10712yes—
209C2×C4C2×C22⋊D10712yes—
210C22—22no—
211C22—28no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—48no—
216C22—42no—
217C22—42832no—
218C22—4284no—
219C22—4284no—
220C22—4288no—
221C22—21416no—
222C22—2148no—
223C22#1681210514yescomplemented
224C22C2×D4×D10712yes—
225C22C2×D4×D10712yes—
226C22C22×C22⋊D10711yescharacteristic
227C22#1681215512yes—
228C4—22no—
229C4—42no—
230C4—2148no—
231C4#1681207714yescomplemented
232C2—24no—
233C2—82no—
234C2—4288no—
235C2C2×#1681207711yescharacteristic
236C2C2×#1681210511yescharacteristic
237C2#2029319111yescharacteristic
238C2#2029326114yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#121×C107C2
hypercenterC2×#121D107
hypocenterC107#2433
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D428C2semidirect
2. C2×#16812105C2semidirect
3. C2×#16812077C2semidirect
4. #20292536C2semidirect
5. #20292528C2semidirect
6. C4×C22⋊Dic107C2semidirect
7. C2×#121×C107C2semidirect
13. C22×C22⋊D107C22semidirect
17. C2×(C2×D107)⋊C4C22semidirect
18. C2×C4×Dic107C22semidirect
19. C107⋊#95C22semidirect
21. C22×D4×C107C22semidirect
24. C2×C4⋊C4×C107C22semidirect
25. C2×C22⋊C4×C107C22semidirect
36. C2×D428D4semidirect
41. C22⋊Dic107D4semidirect
51. C22×C4×C107D4semidirect
87. C2×Dic107C2×D4semidirect
97. C22×C107#127semidirect
110. C4×C107#120semidirect
120. C107#2433semidirect
123. C2×#121D107semidirect
135. C22×D4C2×D107semidirect
140. C2×C4⋊C4C2×D107semidirect
143. C2×C22⋊C4C2×D107semidirect
168. C22×C4C22⋊D107semidirect
223. C22#16812105semidirect
231. C4#16812077semidirect

Maximal subgroups

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

subgroupindex
C23×D4282
C2×#168121052
C2×#168120772
#202925362
#202925282
C4×C22⋊Dic1072
C2×#121×C1072
#2433107