Finite Group Databasebeta

← #28339274

Subgroups — #28339274

Order 13696.

540 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812097C211yesmaximal, characteristic
2#20292531C212yesmaximal, complemented
3#20292518C212yesmaximal, complemented
4#20292517C212yesmaximal, complemented
5#20292505C212yesmaximal
6#20292498C212yesmaximal, complemented
7#20292495C211yesmaximal, characteristic, complemented
8C4×(C2×D107)⋊C4C212yesmaximal, complemented
9C4×C4⋊C4×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C22×C4×D107C2212yes—
13#16812097C418yescomplemented
14C2×Dic107⋊C4C2212yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×C4×Dic107C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2211yescharacteristic
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C107⋊#95C2212yes—
27C2×C4⋊C4×C107C2212yes—
28C2×C42×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30C2×C42×C107C2211yescharacteristic
31#95×C107C2212yes—
32C23×D107C2312yes—
33C2×Dic214—42no—
34C2×Dic214C2×C414yescomplemented
35C2×D428—42no—
36C2×D428C2×C414yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C2×C4×D107—24no—
41C2×C4×D107—42no—
42C2×C4×D107—42no—
43Dic107⋊C4—42no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—42no—
46(C2×D107)⋊C4—42no—
47(C2×D107)⋊C4C2×C418yescomplemented
48(C2×D107)⋊C4C2×C418yescomplemented
49C4×Dic107—42no—
50C22×C4×C107C2312yes—
51C22×C4×C107C2311yescharacteristic
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2312yes—
55C4⋊C4×C107—24no—
56C42×C107—41no—
57C42×C107—41no—
58C42×C107C2×C414yescomplemented
59C23×C107C2411yescharacteristic
60C22×D107—22no—
61C22×D107—22no—
62C22×D107—22no—
63C22×D107—22no—
64C22×D107—22no—
65C22×D107—22no—
66C22×D107C22×C414yes—
67C2×C4×C107—22no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—21no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107C22×C412yes—
77C2×C4×C107C22×C412yes—
78C2×C4×C107C22×C412yes—
79C2×C4×C107#4114yes—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107C22×C414yes—
89D428—44no—
90C4×D107—48no—
91Dic214—44no—
92C22×C107C23×C411yescharacteristic
93C22×C107C2×#4112yes—
94C22×C107H(5,𝔽2)11yescharacteristic
95C22×C107H(5,𝔽2)12yes—
96C22×C1072−1+411yescharacteristic
97C2×D107—24no—
98C2×D107—24no—
99C2×D107—24no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—42no—
103C2×D107—42no—
104C4×C107—22no—
105C4×C107—22no—
106C4×C107—22no—
107C4×C107—24no—
108C4×C107—42no—
109Dic107—24no—
110Dic107—42no—
111Dic107—42no—
112C2×C107C4×#4111yescharacteristic
113C2×C107#51811yescharacteristic
114C2×C107#52011yescharacteristic
115C2×C107#53812yes—
116C2×C107#54212yes—
117D107—44no—
118#2337—1071nomaximal, Sylow
119C107#233711yescharacteristic, complemented, Sylow
120C2×#124—1071no—
121#391—1072no—
122#388—1072no—
123#388—1072no—
124#386—1072no—
125#385—1072no—
126#383—1071no—
127C4×C4⋊C4D10711yescharacteristic, complemented
128C4×C22⋊C4—1072no—
129C22×Q8—1071no—
130C22×D4—1071no—
131C23×C4—1072no—
132#124—1078no—
133C2×C4⋊C4—1072no—
134C2×C4⋊C4C2×D10712yes—
135C2×C22⋊C4—1072no—
136C2×C22⋊C4—1072no—
137C2×C22⋊C4—1072no—
138C2×C22⋊C4—1072no—
139C2×C22⋊C4—1072no—
140C2×C42—1072no—
141C2×C42C2×D10711yescharacteristic
142C2×C42C2×D10711yescharacteristic
143C2×C42C2×D10711yescharacteristic
144#95—1072no—
145#95—1072no—
146#95—1071no—
147#95—1072no—
148#95—1071no—
149#95—1072no—
150#95C2×D10712yes—
151C24—1072no—
152C2×Q8—1074no—
153C2×Q8—4282no—
154C2×D4—1074no—
155C2×D4—4282no—
156C22×C4—1072no—
157C22×C4—1072no—
158C22×C4—1072no—
159C22×C4—4282no—
160C22×C4—4282no—
161C22×C4—2144no—
162C22×C4C22×D10712yes—
163C22×C4C22×D10711yescharacteristic
164C22×C4C22×D10711yescharacteristic
165C22×C4C22×D10711yescharacteristic
166C22×C4C22×D10712yes—
167C4⋊C4—24no—
168C4⋊C4—4282no—
169C22⋊C4—1078no—
170C22⋊C4—1078no—
171C22⋊C4—4282no—
172C22⋊C4—4282no—
173C22⋊C4—2144no—
174C42—41no—
175C42—41no—
176C42—4282no—
177C42C4×D10714yescomplemented
178C23—1074no—
179C23—2142no—
180C23—2142no—
181C23—2142no—
182C23—2142no—
183C23—2142no—
184C23—2142no—
185C23C23×D10711yescharacteristic
186Q8—4284no—
187D4—4284no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—1074no—
198C2×C4—4288no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4C2×C4×D10712yes—
208C2×C4C2×C4×D10712yes—
209C2×C4C2×C4×D10712yes—
210C2×C4#1313381814yes—
211C22—4282no—
212C22—4282no—
213C22—4282no—
214C22—4282no—
215C22—2144no—
216C22—2144no—
217C22—2144no—
218C22C22×C4×D10711yescharacteristic
219C22C2×#1313381812yes—
220C22#1681215411yescharacteristic
221C22#1681215512yes—
222C22#1681215711yescharacteristic
223C4—24no—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—42no—
228C4—4282no—
229C4—4282no—
230C4—2144no—
231C2—4284no—
232C2C4×#1313381811yescharacteristic
233C2#2029310111yescharacteristic
234C2#2029310811yescharacteristic
235C2#2029321012yes—
236C2#2029324912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC4×C4⋊C4×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D428C22
hypercenterC4×C4⋊C4D107
hypocenterC107#2337
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20292531C2semidirect
3. #20292518C2semidirect
4. #20292517C2semidirect
6. #20292498C2semidirect
7. #20292495C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. C4×C4⋊C4×C107C2semidirect
13. #16812097C4semidirect
34. C2×Dic214C2×C4semidirect
47. (C2×D107)⋊C4C2×C4semidirect
48. (C2×D107)⋊C4C2×C4semidirect
58. C42×C107C2×C4semidirect
119. C107#2337semidirect
127. C4×C4⋊C4D107semidirect
177. C42C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120972
#202925312
#202925182
#202925172
#202925052
#202924982
#202924952
C4×(C2×D107)⋊C42
C4×C4⋊C4×C1072
#2337107