Finite Group Databasebeta

← #28337540

Subgroups — #28337540

Order 13696.

426 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133807C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic
3C2×#16812096C211yesmaximal, characteristic, complemented
4C2×#16812032C211yesmaximal, characteristic, complemented
5#20292528C211yesmaximal, characteristic, complemented
6#20292381C211yesmaximal, characteristic, complemented
7C2×#102×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10C2×#13133807—22no—
11C2×#13133807—22no—
12C2×#13133807—22no—
13C2×#13133803—22no—
14C2×#13133803—22no—
15C2×#13133803—22no—
16C22×C107⋊C8C2211yescharacteristic
17#16812096—22no—
18#16812096—22no—
19C2×C22⋊Dic107—22no—
20C2×(C2×D107)⋊C4—22no—
21C2×C4×Dic107C2211yescharacteristic
22#16812032—22no—
23C22×D4×C107C2211yescharacteristic
24C22×C8×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2211yescharacteristic
26#102×C107—22no—
27C23×D107—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214D412yes—
34C2×D428—21no—
35C2×D428—21no—
36C2×D428—21no—
37C2×D428—21no—
38C2×D428—21no—
39C2×D428D412yes—
40C22×Dic107—21no—
41C22×Dic107D412yes—
42#13133807—28no—
43#13133803—28no—
44C2×C107⋊C8—21no—
45C2×C107⋊C8—21no—
46C2×C107⋊C8D412yes—
47C22⋊Dic107—28no—
48(C2×D107)⋊C4—44no—
49C4×Dic107—21no—
50C4×Dic107—21no—
51C24×C107—21no—
52C2×D4×C107—21no—
53C2×D4×C107—21no—
54C2×D4×C107—21no—
55C2×D4×C107—21no—
56C2×D4×C107—21no—
57C2×D4×C107D412yes—
58C22×C4×C107—21no—
59C22×C4×C107C2311yescharacteristic
60C2×C8×C107—21no—
61C2×C8×C107—21no—
62C2×C8×C107D412yes—
63C4⋊C4×C107—21no—
64C4⋊C4×C107—21no—
65C23×C107—22no—
66C23×C107—41no—
67C23×C107—41no—
68C23×C107—42no—
69C23×C107—42no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—41no—
73C22×D107—41no—
74C22×D107—42no—
75C22×D107—42no—
76D4×C107—22no—
77D4×C107—22no—
78D4×C107—41no—
79D4×C107—41no—
80C2×C4×C107—41no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4111yescharacteristic
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—41no—
93C2×Dic107—41no—
94D428—22no—
95D428—22no—
96D428—41no—
97D428—41no—
98Dic214—22no—
99Dic214—22no—
100Dic214—41no—
101Dic214—41no—
102C8×C107—22no—
103C107⋊C8—22no—
104C22×C107—42no—
105C22×C107—44no—
106C22×C107—82no—
107C22×C107—81no—
108C22×C107—81no—
109C22×C107#12011yescharacteristic
110C22×C107#12111yescharacteristic
111C22×C107#12711yescharacteristic
112C22×C107#13512yes—
113C22×C107AGL(1,ℤ/8)12yes—
114C2×D107—42no—
115C2×D107—44no—
116C2×D107—82no—
117C2×D107—81no—
118C2×D107—81no—
119C4×C107—81no—
120C4×C107#12011yescharacteristic
121C4×C107#12011yescharacteristic
122C4×C107#12111yescharacteristic
123C4×C107#12111yescharacteristic
124Dic107—42no—
125Dic107—42no—
126C2×C107—82no—
127C2×C107#39211yescharacteristic
128C2×C107#44912yes—
129C2×C107#46312yes—
130C2×C107#49511yescharacteristic
131C2×C107#49611yescharacteristic
132D107—82no—
133#2012—1071nomaximal, Sylow
134C107#201211yescharacteristic, complemented, Sylow
135C22×SD16—1071no—
136C22×D8—1071no—
137C2×#124—1071no—
138C2×#103—1071no—
139C2×#102D10711yescharacteristic, complemented
140#390—1071no—
141#336—1071no—
142C22×Q8—1071no—
143C22×D4—1071no—
144C22×D4C2×D10711yescharacteristic
145C2×SD16—2142no—
146C2×SD16—2142no—
147C2×SD16—2142no—
148C2×D8—2142no—
149C2×D8—2142no—
150C2×D8—2142no—
151C22×C8—1071no—
152C22×C8C2×D10711yescharacteristic
153#124—2142no—
154#124—2142no—
155C2×C4⋊C4C2×D10711yescharacteristic
156C2×C22⋊C4—2142no—
157C2×C22⋊C4—2142no—
158C2×C42—1071no—
159#103—2142no—
160#102—22no—
161C24—21no—
162C24—2141no—
163C2×Q8—1072no—
164C2×Q8—2141no—
165C2×Q8—2141no—
166C2×Q8—2141no—
167C2×Q8—2141no—
168C2×Q8—2141no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1072no—
175C2×D4—2141no—
176C2×D4—2141no—
177C2×D4—2141no—
178C2×D4—2141no—
179C2×D4—2141no—
180C2×D4D42812yes—
181C22×C4—21no—
182C22×C4—1072no—
183C22×C4—2141no—
184C22×C4C22×D10711yescharacteristic
185SD16—2148no—
186D8—2148no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—1072no—
190C2×C8—2141no—
191C2×C8—2141no—
192C2×C8C22⋊D10712yes—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C22⋊C4—4284no—
196C22⋊C4—2148no—
197C42—2141no—
198C42—2141no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—4281no—
205C23—4281no—
206C23—4282no—
207C23—4282no—
208C23—2142no—
209C23D4×D10711yescharacteristic
210Q8—4281no—
211Q8—4281no—
212Q8—2142no—
213Q8—2142no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—4281no—
219D4—4281no—
220D4—2142no—
221D4—2142no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—4281no—
225C2×C4—4281no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4D4×D10711yescharacteristic
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4C2×C22⋊D10711yescharacteristic
234C2×C4C2×D42811yescharacteristic
235C2×C4#1313381811yescharacteristic
236C8—22no—
237C8—2142no—
238C22—44no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22—8561no—
244C22—8561no—
245C22—8562no—
246C22—4282no—
247C22—4284no—
248C22#1681207911yescharacteristic
249C22#1681208211yescharacteristic
250C22#1681210511yescharacteristic
251C22#1681213012yes—
252C22#1681213312yes—
253C4—81no—
254C4—4282no—
255C4—4282no—
256C4#1681207711yescharacteristic
257C4#1681207911yescharacteristic
258C4#1681208211yescharacteristic
259C4#1681208311yescharacteristic
260C2—82no—
261C2—8562no—
262C2#2029253611yescharacteristic
263C2#2029277312yes—
264C2#2029284112yes—
265C2#2029300811yescharacteristic
266C2#2029301511yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#102×C107C2
socleC23×C107C2×D4
prime radicalC22×#13133803C2
hypercenterC2×#102D107
hypocenterC107#2012
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
3. C2×#16812096C2semidirect
4. C2×#16812032C2semidirect
5. #20292528C2semidirect
6. #20292381C2semidirect
7. C2×#102×C107C2semidirect
134. C107#2012semidirect
139. C2×#102D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
C22×#131338032
C2×#168120962
C2×#168120322
#202925282
#202923812
C2×#102×C1072
#2012107