Finite Group Databasebeta

← #28344764

Subgroups — #28344764

Order 13696.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#16812098C211yesmaximal, characteristic, complemented
2C2×#16812083C211yesmaximal, characteristic
3C22×C4×Dic107C211yesmaximal, characteristic, complemented
4#20292534C211yesmaximal, characteristic, complemented
5#20292528C211yesmaximal, characteristic, complemented
6#20292517C214yesmaximal, complemented
7#20292504C214yesmaximal, complemented
8#20292492C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C23×Dic107C2212yes—
13#16812098—24no—
14#16812083—24no—
15C2×C4⋊Dic107C2211yescharacteristic
16C2×(C2×D107)⋊C4C2214yes—
17C2×(C2×D107)⋊C4C2214yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×C4×Dic107—24no—
20C2×C4×Dic107C2212yes—
21C2×C4×Dic107C2211yescharacteristic
22C2×C4×Dic107C2211yescharacteristic
23C107⋊#95C2214yes—
24C107⋊#95C2214yes—
25C22×Q8×C107C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27C22⋊Q8×C107—24no—
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2212yes—
30C2×C22⋊C4×C107C2212yes—
31C23×D107C2312yes—
32C2×D428—22no—
33C2×D428—22no—
34C2×D428—22no—
35C2×C22⋊D107—24no—
36C2×C22⋊D107—28no—
37C22×Dic107—24no—
38C22×Dic107—24no—
39C22×Dic107—24no—
40C22×Dic107C2312yes—
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C4⋊Dic107—22no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—28no—
46(C2×D107)⋊C4—28no—
47C4×Dic107—22no—
48C4×Dic107—24no—
49C4×Dic107D414yescomplemented
50C24×C107C2311yescharacteristic
51C2×Q8×C107—22no—
52C2×Q8×C107—22no—
53C2×Q8×C107—22no—
54C22×C4×C107—22no—
55C22×C4×C107—22no—
56C22×C4×C107D414yes—
57C22×C4×C107C2312yes—
58C22×C4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107—24no—
63C22⋊C4×C107—24no—
64C23×C107—21no—
65C23×C107—21no—
66C23×C107—22no—
67C23×C107C2×D412yes—
68C23×C107#4114yes—
69C23×C107C2411yescharacteristic
70C22×D107—22no—
71C22×D107—22no—
72C22×D107—24no—
73C22×D107—24no—
74C22×D107—22no—
75Q8×C107—44no—
76C2×C4×C107—21no—
77C2×C4×C107—21no—
78C2×C4×C107—28no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—24no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D412yes—
87C2×C4×C107#4112yes—
88C22⋊D107—48no—
89C2×Dic107—216no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—24no—
96C2×Dic107C2×D414yes—
97C2×Dic107#4114yes—
98D428—44no—
99C22×C107—22no—
100C22×C107—24no—
101C22×C107—24no—
102C22×C107—22no—
103C22×C107#12414yescomplemented
104C22×C107C22×D411yescharacteristic
105C22×C107C22×D411yescharacteristic
106C22×C107C2×#4112yes—
107C22×C107C2×#4111yescharacteristic
108C22×C107C2×#4112yes—
109C2×D107—42no—
110C2×D107—44no—
111C2×D107—42no—
112C2×D107—44no—
113C2×D107—42no—
114C4×C107—22no—
115C4×C107—42no—
116C4×C107—42no—
117C4×C107#12114yes—
118Dic107—24no—
119Dic107—24no—
120Dic107—42no—
121C2×C107—24no—
122C2×C107C2×#12111yescharacteristic
123C2×C107#52512yes—
124C2×C107C2×#12411yescharacteristic
125C2×C107#52912yes—
126C2×C107#53211yescharacteristic
127D107—82no—
128#2850—1071nomaximal, Sylow
129C107#285011yescharacteristic, complemented, Sylow
130C2×#124—1071no—
131C2×C22⋊Q8D10711yescharacteristic, complemented
132C2×#121—1071no—
133C22×C42—1071no—
134#391—1071no—
135#390—1071no—
136#388—1074no—
137#386—1074no—
138#382—1071no—
139C22×Q8C2×D10711yescharacteristic
140C22×D4—1072no—
141C22×D4—1071no—
142C23×C4—1072no—
143C23×C4C2×D10711yescharacteristic
144#124—2144no—
145C22⋊Q8—24no—
146#121—2144no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4C2×D10711yescharacteristic
149C2×C4⋊C4C2×D10712yes—
150C2×C22⋊C4—1072no—
151C2×C22⋊C4—1074no—
152C2×C22⋊C4—1074no—
153C2×C22⋊C4C2×D10712yes—
154C2×C42—1071no—
155C2×C42—1072no—
156C2×C42—1071no—
157C2×C42—2144no—
158#95—1074no—
159#95—1074no—
160C24—1072no—
161C24C22×D10711yescharacteristic
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—2144no—
166C2×D4—2148no—
167C2×D4—2142no—
168C2×D4—2142no—
169C2×D4—2142no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—1072no—
173C22×C4—1072no—
174C22×C4—1072no—
175C22×C4—2144no—
176C22×C4—2144no—
177C22×C4—2144no—
178C22×C4C22⋊D10714yes—
179C22×C4C22×D10712yes—
180C22×C4C22×D10712yes—
181C22×C4C22×D10711yescharacteristic
182C22×C4C22×D10711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—2142no—
186C22⋊C4—24no—
187C22⋊C4—2148no—
188C22⋊C4—2148no—
189C22⋊C4—2144no—
190C42—1074no—
191C42—2144no—
192C42—2142no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—2142no—
197C23—2142no—
198C23—2144no—
199C23—2142no—
200C23—2144no—
201C23C2×C22⋊D10712yes—
202C23#1313381914yes—
203C23C23×D10711yescharacteristic
204Q8—44no—
205D4—4288no—
206D4—4284no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—1074no—
216C2×C4—1074no—
217C2×C4—2142no—
218C2×C4—2144no—
219C2×C4—21416no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10712yes—
225C2×C4C2×C22⋊D10712yes—
226C2×C4C2×C22⋊D10712yes—
227C2×C4#1313381712yes—
228C22—24no—
229C22—24no—
230C22—22no—
231C22—22no—
232C22—4284no—
233C22—4282no—
234C22—4284no—
235C22—4282no—
236C22—4282no—
237C22#1681209814yescomplemented
238C22C2×D4×D10711yescharacteristic
239C22C22×C22⋊D10711yescharacteristic
240C22#41×D10712yes—
241C22C2×#1313381711yescharacteristic
242C22C2×#1313381912yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—4282no—
247C4—2144no—
248C4—2144no—
249C4#1681208114yes—
250C2—24no—
251C2—8562no—
252C2C2×#1681208111yescharacteristic
253C2#2029314012yes—
254C2C2×#1681209811yescharacteristic
255C2#2029316312yes—
256C2#2029317311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC2×#16812083C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#2850
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812098C2semidirect
3. C22×C4×Dic107C2semidirect
4. #20292534C2semidirect
5. #20292528C2semidirect
6. #20292517C2semidirect
7. #20292504C2semidirect
8. #20292492C2semidirect
9. C2×C22⋊Q8×C107C2semidirect
49. C4×Dic107D4semidirect
103. C22×C107#124semidirect
129. C107#2850semidirect
131. C2×C22⋊Q8D107semidirect
237. C22#16812098semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
C2×#168120832
C22×C4×Dic1072
#202925342
#202925282
#202925172
#202925042
#202924922
C2×C22⋊Q8×C1072
#2850107