Finite Group Databasebeta

← #28334786

Subgroups — #28334786

Order 13696.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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×C22⋊C4×D107C211yesmaximal, characteristic
2#20292589C212yesmaximal, complemented
3#20292351C211yesmaximal, characteristic, complemented
4#20292350C212yesmaximal, complemented
5#323×C107C211yesmaximal, characteristic, complemented
6C24×D107C411yescharacteristic
7C22×C4×D107C411yescharacteristic
8C22×C4×D107C2211yescharacteristic
9C2×#13133801—22no—
10C2×C22⋊Dic107C411yescharacteristic
11C2×(C2×D107)⋊C4C411yescharacteristic
12C2×(C2×D107)⋊C4C2211yescharacteristic
13C22⋊C4×D107—22no—
14C22⋊C4×D107—22no—
15C22⋊(C107⋊C8)C2212yescomplemented
16(C2×D107)⋊C8—22no—
17C2×C22⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yescomplemented
19C23×D107—22no—
20C23×D107—21no—
21C23×D107—21no—
22C23×D107—22no—
23C23×D107—42no—
24C23×D107—42no—
25C23×D107C2×C411yescharacteristic
26C23×D107C2×C411yescharacteristic
27C22×Dic107C2×C411yescharacteristic
28C22×Dic107C2×C411yescharacteristic
29C2×C4×D107—21no—
30C2×C4×D107—21no—
31C2×C4×D107—42no—
32C2×C4×D107—42no—
33C2×C4×D107D412yes—
34C2×C4×D107D412yes—
35#13133801—44no—
36C2×C107⋊C8—22no—
37C22⋊Dic107—22no—
38(C2×D107)⋊C4—22no—
39(C2×D107)⋊C4—22no—
40C24×C107C2×C411yescharacteristic
41C22×C4×C107C2×C411yescharacteristic
42C22×C4×C107C2311yescharacteristic
43C2×C8×C107—22no—
44C22⋊C4×C107—22no—
45C23×C107—21no—
46C23×C107—21no—
47C23×C107—41no—
48C23×C107—41no—
49C23×C107M1612yescomplemented
50C23×C107C22×C411yescharacteristic
51C22×D107—21no—
52C22×D107—21no—
53C22×D107—22no—
54C22×D107—22no—
55C22×D107—24no—
56C22×D107—24no—
57C22×D107—21no—
58C22×D107—21no—
59C22×D107—48no—
60C22×D107—48no—
61C22×D107—42no—
62C22×D107—42no—
63C22×D107—42no—
64C22×D107—42no—
65C22×D107—41no—
66C22×D107—41no—
67C22×D107C22⋊C411yescharacteristic
68C22×D107C22⋊C411yescharacteristic
69C22×D107M1612yescomplemented
70C2×C4×C107—21no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107C2×D412yes—
74C2×Dic107—21no—
75C2×Dic107—41no—
76C2×Dic107—41no—
77C2×Dic107C22⋊C412yes—
78C4×D107—24no—
79C4×D107—42no—
80C4×D107—42no—
81C4×D107—42no—
82C8×C107—42no—
83C107⋊C8—42no—
84C22×C107—21no—
85C22×C107—21no—
86C22×C107—22no—
87C22×C107—42no—
88C22×C107—42no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107C2×C22⋊C411yescharacteristic
92C22×C107C2×M1611yescharacteristic
93C22×C107C2×M1611yescharacteristic
94C2×D107—24no—
95C2×D107—24no—
96C2×D107—21no—
97C2×D107—21no—
98C2×D107—21no—
99C2×D107—21no—
100C2×D107—22no—
101C2×D107—48no—
102C2×D107—48no—
103C2×D107—42no—
104C2×D107—41no—
105C2×D107—41no—
106C2×D107—42no—
107C2×D107#9912yes—
108C2×D107#10012yes—
109C4×C107—22no—
110C4×C107—41no—
111Dic107—22no—
112Dic107—41no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—42no—
116C2×C107#40711yescharacteristic
117C2×C107C2×#9911yescharacteristic
118C2×C107C2×#10011yescharacteristic
119D107—22no—
120D107—22no—
121D107—42no—
122#1462—1071nomaximal, Sylow
123C107#146211yescharacteristic, complemented, Sylow
124C22×C22⋊C4—1071no—
125#407—1072no—
126#323—1071no—
127#323—1072no—
128#323D10711yescharacteristic, complemented
129C25—1071no—
130C23×C4—1071no—
131C23×C4—1071no—
132C2×M16—2142no—
133C2×C22⋊C4—1071no—
134C2×C22⋊C4—1071no—
135C2×C22⋊C4—1071no—
136C2×C22⋊C4—2142no—
137C2×C22⋊C4—2142no—
138C2×C22⋊C4C2×D10711yescharacteristic
139C22⋊C8—1072no—
140C22⋊C8—2142no—
141C22⋊C8C2×D10712yescomplemented
142C24—1071no—
143C24—1071no—
144C24—4282no—
145C24—4282no—
146C24—2141no—
147C24—2141no—
148C24—2142no—
149C24—2142no—
150C24C4×D10711yescharacteristic
151C22×C4—1071no—
152C22×C4—1072no—
153C22×C4—1072no—
154C22×C4—1071no—
155C22×C4—4282no—
156C22×C4—4282no—
157C22×C4—2141no—
158C22×C4—2141no—
159C22×C4C4×D10711yescharacteristic
160C22×C4C22×D10711yescharacteristic
161M16—4284no—
162C2×C8—22no—
163C2×C8—2142no—
164C22⋊C4—22no—
165C22⋊C4—2142no—
166C22⋊C4—2142no—
167C22⋊C4—2142no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—1071no—
173C23—1071no—
174C23—1072no—
175C23—4281no—
176C23—4281no—
177C23—4282no—
178C23—4282no—
179C23—4288no—
180C23—4288no—
181C23—4282no—
182C23—4282no—
183C23—2142no—
184C23—2142no—
185C23—2144no—
186C23—2144no—
187C23—2141no—
188C23—2141no—
189C23—2141no—
190C23—2141no—
191C23#1313380112yescomplemented
192C23C2×C4×D10711yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1072no—
197C2×C4—4282no—
198C2×C4—4282no—
199C2×C4—4282no—
200C2×C4—4281no—
201C2×C4—4281no—
202C2×C4—2144no—
203C2×C4—2141no—
204C2×C4D4×D10712yes—
205C8—42no—
206C8—4282no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—1072no—
215C22—1072no—
216C22—4282no—
217C22—4282no—
218C22—4288no—
219C22—4288no—
220C22—4281no—
221C22—4281no—
222C22—2144no—
223C22—2144no—
224C22—2142no—
225C22—2141no—
226C22—2141no—
227C22—2141no—
228C22—2141no—
229C22C22⋊C4×D10711yescharacteristic
230C22M16×D10711yescharacteristic
231C22C2×#1313380111yescharacteristic
232C4—22no—
233C4—41no—
234C4—4281no—
235C4—2142no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—4282no—
240C2—2142no—
241C2—2142no—
242C2#2029258911yescharacteristic
243C2#99×D10711yescharacteristic
244C2#100×D10711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC24×D107C4
cosocleC22×C4×C107C23
hypercenter#323D107
hypocenterC107#1462
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292589C2semidirect
3. #20292351C2semidirect
4. #20292350C2semidirect
5. #323×C107C2semidirect
15. C22⋊(C107⋊C8)C22semidirect
18. C22⋊C8×C107C22semidirect
49. C23×C107M16semidirect
69. C22×D107M16semidirect
123. C107#1462semidirect
128. #323D107semidirect
141. C22⋊C8C2×D107semidirect
191. C23#13133801semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1072
#202925892
#202923512
#202923502
#323×C1072
#1462107