Finite Group Databasebeta

← #28340090

Subgroups — #28340090

Order 13696.

836 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133818C211yesmaximal, characteristic
2C2×#16812107C211yesmaximal, characteristic, complemented
3C2×#16812098C211yesmaximal, characteristic, complemented
4#20292542C214yesmaximal, complemented
5#20292532C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292498C214yesmaximal, complemented
8C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
10C2×#13133818—24no—
11C2×#13133818—24no—
12C2×#13133818—24no—
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C22×C4×D107C2212yes—
17#16812107—24no—
18#16812098—24no—
19C2×Dic107⋊C4C2214yes—
20C2×C22⋊Dic107C2212yes—
21C2×(C2×D107)⋊C4C2214yes—
22C2×(C2×D107)⋊C4C2214yes—
23C2×C4×Dic107C2211yescharacteristic
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2214yes—
27C22×Q8×C107C2211yescharacteristic
28C23×C4×C107C2211yescharacteristic
29C22⋊Q8×C107—24no—
30C2×C4⋊C4×C107C2212yes—
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107C2212yes—
33C23×D107C2312yes—
34#13133818—232no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214D414yes—
38C2×D428—22no—
39C2×D428—22no—
40C2×D428D414yes—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—24no—
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2312yes—
47C2×C4×D107—24no—
48C2×C4×D107—24no—
49C2×C4×D107—24no—
50Dic107⋊C4—28no—
51C22⋊Dic107—24no—
52(C2×D107)⋊C4—28no—
53(C2×D107)⋊C4—28no—
54C4×Dic107—22no—
55C4×Dic107—22no—
56C24×C107C2311yescharacteristic
57C2×Q8×C107—22no—
58C2×Q8×C107—22no—
59C2×Q8×C107—22no—
60C22×C4×C107—22no—
61C22×C4×C107—22no—
62C22×C4×C107D414yes—
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107—22no—
68C4⋊C4×C107—24no—
69C22⋊C4×C107—24no—
70C23×C107—21no—
71C23×C107—21no—
72C23×C107—22no—
73C23×C107—22no—
74C23×C107C2×D412yes—
75C23×C107C2411yescharacteristic
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—24no—
79C22×D107—24no—
80C22×D107C2×D414yes—
81Q8×C107—44no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—28no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—24no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D412yes—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×D412yes—
94C22⋊D107—216no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107C2×D414yes—
103C2×Dic107#4114yescomplemented
104D428—28no—
105C4×D107—216no—
106Dic214—28no—
107C22×C107—22no—
108C22×C107—22no—
109C22×C107—22no—
110C22×C107—42no—
111C22×C107—42no—
112C22×C107C22×D411yescharacteristic
113C22×C107C22×D411yescharacteristic
114C22×C107C22×D411yescharacteristic
115C22×C107C2×#4112yes—
116C22×C1072−1+412yes—
117C2×D107—24no—
118C2×D107—28no—
119C2×D107—42no—
120C2×D107—42no—
121C2×D107—44no—
122C4×C107—22no—
123C4×C107—42no—
124C4×C107—42no—
125C4×C107#12014yes—
126Dic107—24no—
127Dic107—24no—
128Dic107—42no—
129C2×C107—42no—
130C2×C107C2×#12011yescharacteristic
131C2×C107#53211yescharacteristic
132C2×C107#53611yescharacteristic
133C2×C107#54712yes—
134C2×C107#54812yes—
135D107—44no—
136#2432—1071nomaximal, Sylow
137C107#243211yescharacteristic, complemented, Sylow
138C22×#41—1071no—
139C2×C4⋊Q8—1071no—
140C2×#124—1071no—
141C2×C22⋊Q8D10711yescharacteristic, complemented
142#394—1074no—
143#391—1071no—
144#390—1071no—
145#385—1074no—
146C4×C22⋊C4—1071no—
147C2×#41—2144no—
148C2×#41—2144no—
149C2×#41—2144no—
150C22×Q8—1071no—
151C22×Q8C2×D10711yescharacteristic
152C22×D4—1072no—
153C22×D4—1071no—
154C23×C4—1072no—
155C23×C4C2×D10711yescharacteristic
156C4⋊Q8—2144no—
157#124—2144no—
158C22⋊Q8—24no—
159C2×C4⋊C4—1074no—
160C2×C4⋊C4C2×D10712yes—
161C2×C4⋊C4C2×D10711yescharacteristic
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1074no—
164C2×C22⋊C4—1074no—
165C2×C22⋊C4C2×D10712yes—
166C2×C42—1071no—
167C2×C42—1071no—
168#95—1072no—
169#95—1074no—
170C24—1072no—
171C24C22×D10711yescharacteristic
172#41—21432no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—1074no—
177C2×Q8—2142no—
178C2×Q8—2142no—
179C2×D4—1074no—
180C2×D4—2144no—
181C2×D4—2144no—
182C2×D4—2144no—
183C2×D4—2142no—
184C2×D4—2142no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—1072no—
188C22×C4—1072no—
189C22×C4—1072no—
190C22×C4—2144no—
191C22×C4—2144no—
192C22×C4—2144no—
193C22×C4C22⋊D10714yes—
194C22×C4C22×D10712yes—
195C22×C4C22×D10712yes—
196C22×C4C22×D10711yescharacteristic
197C22×C4C22×D10711yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—2148no—
201C22⋊C4—24no—
202C22⋊C4—2148no—
203C22⋊C4—2148no—
204C22⋊C4—2144no—
205C42—2142no—
206C42—2142no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—1074no—
212C23—2142no—
213C23—2142no—
214C23—2144no—
215C23—2144no—
216C23C2×C22⋊D10712yes—
217C23C23×D10711yescharacteristic
218Q8—44no—
219Q8—2148no—
220D4—21416no—
221D4—2148no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—1074no—
231C2×C4—1074no—
232C2×C4—21416no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4D4×D10712yes—
241C2×C4D4×D10712yes—
242C2×C4C2×C22⋊D10712yes—
243C2×C4C2×C22⋊D10712yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—4282no—
250C22—4282no—
251C22—4284no—
252C22—2148no—
253C22—2144no—
254C22C2×D4×D10711yescharacteristic
255C22C2×D4×D10711yescharacteristic
256C22C22×C22⋊D10711yescharacteristic
257C22#41×D10712yes—
258C22#1681215712yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—4282no—
263C4—2144no—
264C4—2144no—
265C4#1681207714yes—
266C2—42no—
267C2—4284no—
268C2C2×#1681207711yescharacteristic
269C2#2029317311yescharacteristic
270C2#2029320111yescharacteristic
271C2#2029327912yes—
272C2#2029328712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
prime radicalC22×#13133818C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#2432
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16812107C2semidirect
3. C2×#16812098C2semidirect
4. #20292542C2semidirect
5. #20292532C2semidirect
6. #20292528C2semidirect
7. #20292498C2semidirect
8. C4×C22⋊Dic107C2semidirect
9. C2×C22⋊Q8×C107C2semidirect
103. C2×Dic107#41semidirect
137. C107#2432semidirect
141. C2×C22⋊Q8D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121072
C2×#168120982
#202925422
#202925322
#202925282
#202924982
C4×C22⋊Dic1072
C2×C22⋊Q8×C1072
#2432107