Finite Group Databasebeta

← #28340085

Subgroups — #28340085

Order 13696.

836 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 185 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×#16812107C211yesmaximal, characteristic, complemented
2C2×#16812096C211yesmaximal, characteristic, complemented
3C2×#16812088C211yesmaximal, characteristic, complemented
4#20292541C214yesmaximal, complemented
5#20292534C211yesmaximal, characteristic, complemented
6#20292530C211yesmaximal, characteristic, complemented
7#20292500C214yesmaximal, complemented
8C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
9C22×#41×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11#16812107—24no—
12#16812096—24no—
13#16812088—24no—
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4C2212yes—
16C2×Dic107⋊C4C2214yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2214yes—
19C2×C22⋊Dic107C2212yes—
20C2×C22⋊Dic107C2214yes—
21C2×C4×Dic107C2211yescharacteristic
22C2×C4×Dic107C2211yescharacteristic
23C107⋊#95C2212yes—
24C107⋊#95C2214yes—
25C2×#41×C107—24no—
26C2×#41×C107—24no—
27C2×#41×C107—24no—
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107C2212yes—
30C22×D4×C107C2211yescharacteristic
31C23×C4×C107C2212yes—
32C23×C4×C107C2211yescharacteristic
33C2×Dic214—22no—
34C2×Dic214—22no—
35C2×Dic214—22no—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C4⋊Dic107—22no—
41Dic107⋊C4—24no—
42Dic107⋊C4—28no—
43C22⋊Dic107—24no—
44C22⋊Dic107—28no—
45C22⋊Dic107—24no—
46C22⋊Dic107—28no—
47C4×Dic107—22no—
48C4×Dic107—22no—
49C24×C107C2312yes—
50C24×C107C2311yescharacteristic
51#41×C107—232no—
52C2×Q8×C107—22no—
53C2×Q8×C107—22no—
54C2×Q8×C107D414yes—
55C2×D4×C107—24no—
56C2×D4×C107—24no—
57C2×D4×C107—24no—
58C2×D4×C107—22no—
59C2×D4×C107—22no—
60C2×D4×C107D414yes—
61C22×C4×C107—24no—
62C22×C4×C107—24no—
63C22×C4×C107—24no—
64C22×C4×C107—22no—
65C22×C4×C107—22no—
66C22×C4×C107D414yes—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C23×C107—22no—
71C23×C107—21no—
72C23×C107—22no—
73C23×C107—21no—
74C23×C107—22no—
75C23×C107—24no—
76C23×C107—22no—
77C23×C107—24no—
78C23×C107C2×D414yes—
79C23×C107C2×D412yes—
80C23×C107C2411yescharacteristic
81Q8×C107—28no—
82D4×C107—216no—
83D4×C107—28no—
84C2×C4×C107—216no—
85C2×C4×C107—21no—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—28no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D414yes—
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D412yes—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—24no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107#4114yescomplemented
106Dic214—44no—
107C22×C107—24no—
108C22×C107—22no—
109C22×C107—22no—
110C22×C107—22no—
111C22×C107—28no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—44no—
117C22×C107C22×D411yescharacteristic
118C22×C107C22×D411yescharacteristic
119C22×C107C22×D411yescharacteristic
120C22×C107C2×#4112yes—
121C22×C1072−1+412yes—
122C4×C107—22no—
123C4×C107—24no—
124C4×C107#12014yes—
125Dic107—24no—
126Dic107—42no—
127Dic107—42no—
128Dic107—42no—
129C2×C107—42no—
130C2×C107—44no—
131C2×C107C2×#12011yescharacteristic
132C2×C107#53211yescharacteristic
133C2×C107#53611yescharacteristic
134C2×C107#54712yes—
135C2×C107#54812yes—
136#2432—1071nomaximal, Sylow
137C107#243211yescharacteristic, complemented, Sylow
138C22×#41D10711yescharacteristic
139C2×C4⋊Q8—1071no—
140C2×#124—1071no—
141C2×C22⋊Q8—1071no—
142#394—1074no—
143#391—1071no—
144#390—1071no—
145#385—1074no—
146C4×C22⋊C4—1071no—
147C2×#41—24no—
148C2×#41—24no—
149C2×#41—24no—
150C22×Q8—1071no—
151C22×Q8C2×D10711yescharacteristic
152C22×D4C2×D10712yes—
153C22×D4C2×D10711yescharacteristic
154C23×C4C2×D10712yes—
155C23×C4C2×D10711yescharacteristic
156C4⋊Q8—2144no—
157#124—2144no—
158C22⋊Q8—2144no—
159C2×C4⋊C4—1072no—
160C2×C4⋊C4—1071no—
161C2×C4⋊C4—1074no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1074no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4—1074no—
166C2×C42—1071no—
167C2×C42—1071no—
168#95—1072no—
169#95—1074no—
170C24C22×D10712yes—
171C24C22×D10711yescharacteristic
172#41—232no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—2142no—
176C2×Q8—2142no—
177C2×Q8—2142no—
178C2×Q8C22⋊D10714yes—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4C22⋊D10714yes—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1072no—
191C22×C4—1072no—
192C22×C4—1072no—
193C22×C4—1072no—
194C22×C4C22⋊D10714yes—
195C22×C4C22×D10711yescharacteristic
196C22×C4C22×D10712yes—
197C22×C4C22×D10711yescharacteristic
198C4⋊C4—2144no—
199C4⋊C4—2142no—
200C4⋊C4—2148no—
201C22⋊C4—2144no—
202C22⋊C4—2148no—
203C22⋊C4—2144no—
204C22⋊C4—2148no—
205C42—2142no—
206C42—2142no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—24no—
213C23—22no—
214C23—24no—
215C23C2×C22⋊D10714yes—
216C23C2×C22⋊D10712yes—
217C23C23×D10711yescharacteristic
218Q8—28no—
219Q8—4284no—
220D4—216no—
221D4—28no—
222C2×C4—216no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—28no—
228C2×C4—1074no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—2144no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4C2×C22⋊D10712yes—
240C2×C4C2×C22⋊D10714yes—
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10712yes—
243C2×C4C2×C22⋊D10712yes—
244C22—22no—
245C22—22no—
246C22—28no—
247C22—24no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—44no—
254C22C22×C22⋊D10711yescharacteristic
255C22C22×C22⋊D10711yescharacteristic
256C22C22×C22⋊D10711yescharacteristic
257C22#41×D10712yes—
258C22#1681215612yes—
259C4—22no—
260C4—24no—
261C4—4282no—
262C4—4282no—
263C4—4282no—
264C4—2144no—
265C4#1681207814yes—
266C2—42no—
267C2—44no—
268C2C2×#1681207811yescharacteristic
269C2#2029317311yescharacteristic
270C2#2029319811yescharacteristic
271C2#2029327612yes—
272C2#2029328412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting, prime radicalC22×#41×C107C2
hypercenterC22×#41D107
hypocenterC107#2432
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812107C2semidirect
2. C2×#16812096C2semidirect
3. C2×#16812088C2semidirect
4. #20292541C2semidirect
5. #20292534C2semidirect
6. #20292530C2semidirect
7. #20292500C2semidirect
8. C4×C22⋊Dic107C2semidirect
105. C2×Dic107#41semidirect
137. C107#2432semidirect

Maximal subgroups

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

subgroupindex
C2×#168121072
C2×#168120962
C2×#168120882
#202925412
#202925342
#202925302
#202925002
C4×C22⋊Dic1072
C22×#41×C1072
#2432107