Finite Group Databasebeta

← #28344474

Subgroups — #28344474

Order 13696.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812088C211yesmaximal, characteristic, complemented
2#20292560C212yesmaximal, complemented
3#20292557C211yesmaximal, characteristic, complemented
4#20292555C212yesmaximal, complemented
5#20292552C211yesmaximal, characteristic, complemented
6#20292550C212yesmaximal, complemented
7#20292547C211yesmaximal, characteristic, complemented
8#20292541C212yesmaximal
9#20292539C211yesmaximal, characteristic, complemented
10#20292487C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic
12C22×Dic214C2211yescharacteristic
13#16812088—24no—
14C2×C4⋊Dic107C2211yescharacteristic
15C2×C4⋊Dic107C2212yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×Dic107⋊C4C2212yes—
19C2×C22⋊Dic107C2212yes—
20C2×C22⋊Dic107C2212yes—
21C2×C22⋊Dic107C2212yes—
22C2×C22⋊Dic107C2212yes—
23C107⋊#95C2211yescharacteristic
24C107⋊#95C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C22×D4×C107C2211yescharacteristic
30C23×C4×C107C2211yescharacteristic
31#123×C107—24no—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×Dic214—22no—
38C2×Dic214—42no—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C4⋊Dic107—22no—
44C4⋊Dic107—42no—
45Dic107⋊C4—24no—
46Dic107⋊C4—42no—
47Dic107⋊C4—42no—
48C22⋊Dic107—24no—
49C22⋊Dic107—42no—
50C22⋊Dic107—42no—
51C22⋊Dic107—42no—
52C24×C107C2311yescharacteristic
53C24×C107C2311yescharacteristic
54C2×D4×C107—22no—
55C2×D4×C107—42no—
56C22×C4×C107—42no—
57C22×C4×C107D414yes—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C22×C4×C107C2311yescharacteristic
62C22×C4×C107C2311yescharacteristic
63C4⋊C4×C107—22no—
64C4⋊C4×C107—22no—
65C22⋊C4×C107—22no—
66C22⋊C4×C107—22no—
67C22⋊C4×C107—22no—
68C23×C107—22no—
69C23×C107—21no—
70C23×C107—22no—
71C23×C107—21no—
72C23×C107—21no—
73C23×C107—22no—
74C23×C107—21no—
75C23×C107—22no—
76C23×C107—21no—
77C23×C107C2×D412yes—
78C23×C107C2411yescharacteristic
79D4×C107—44no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—22no—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—44no—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D412yes—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104Dic214—44no—
105C22×C107—22no—
106C22×C107—22no—
107C22×C107—22no—
108C22×C107—42no—
109C22×C107—42no—
110C22×C107—41no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107—42no—
114C22×C107—42no—
115C22×C107C22×D411yescharacteristic
116C22×C107H(5,𝔽2)12yes—
117C22×C107H(5,𝔽2)11yescharacteristic
118C22×C1072−1+412yes—
119C22×C1072−1+411yescharacteristic
120C4×C107—22no—
121C4×C107—22no—
122C4×C107—41no—
123C4×C107—41no—
124C4×C107—41no—
125Dic107—42no—
126Dic107—42no—
127Dic107—42no—
128Dic107—42no—
129C2×C107—42no—
130C2×C107—81no—
131C2×C107#53411yescharacteristic
132C2×C107#53611yescharacteristic
133C2×C107#53711yescharacteristic
134C2×C107#56212yes—
135C2×C107#56312yes—
136#2814—1071nomaximal, Sylow
137C107#281411yescharacteristic, complemented, Sylow
138C2×#123D10711yescharacteristic
139C2×C22⋊Q8—1071no—
140#398—1071no—
141#398—1072no—
142#397—1071no—
143#397—1072no—
144#396—1072no—
145#395—1071no—
146#394—1072no—
147#393—1071no—
148#380—1071no—
149C22×Q8—1071no—
150C22×D4C2×D10711yescharacteristic
151C23×C4C2×D10711yescharacteristic
152#123—24no—
153C22⋊Q8—2144no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4—1072no—
156C2×C4⋊C4—1072no—
157C2×C4⋊C4—1072no—
158C2×C4⋊C4—1071no—
159C2×C4⋊C4C2×D10711yescharacteristic
160C2×C4⋊C4C2×D10711yescharacteristic
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4C2×D10711yescharacteristic
166C2×C22⋊C4C2×D10711yescharacteristic
167C2×C22⋊C4C2×D10711yescharacteristic
168#95—1071no—
169#95—1071no—
170#95—1072no—
171#95—1072no—
172#95—1072no—
173#95—1072no—
174C24C22×D10711yescharacteristic
175C24C22×D10711yescharacteristic
176C2×Q8—4282no—
177C2×Q8—2142no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—42no—
181C22×C4—1072no—
182C22×C4—1072no—
183C22×C4—1072no—
184C22×C4—1072no—
185C22×C4C22⋊D10714yes—
186C22×C4C22×D10711yescharacteristic
187C22×C4C22×D10711yescharacteristic
188C22×C4C22×D10711yescharacteristic
189C22×C4C22×D10711yescharacteristic
190C22×C4C22×D10711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—4282no—
194C4⋊C4—4282no—
195C4⋊C4—4282no—
196C4⋊C4—2144no—
197C4⋊C4—2142no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—4282no—
202C22⋊C4—4282no—
203C22⋊C4—4282no—
204C22⋊C4—2144no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23C2×C22⋊D10712yes—
215C23C23×D10711yescharacteristic
216Q8—4284no—
217D4—44no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4C2×C22⋊D10712yes—
241C2×C4C2×C22⋊D10712yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22C22×C22⋊D10711yescharacteristic
253C22#1681215412yes—
254C22#1681215511yescharacteristic
255C22#1681215611yescharacteristic
256C22#1681215612yes—
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—4282no—
263C4—4282no—
264C4—4282no—
265C4—4282no—
266C2—42no—
267C2—81no—
268C2#2029318611yescharacteristic
269C2#2029319811yescharacteristic
270C2#2029320511yescharacteristic
271C2#2029337412yes—
272C2#2029338012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
prime radicalC22×D4×C107C22
hypercenterC2×#123D107
hypocenterC107#2814
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16812088C2semidirect
2. #20292560C2semidirect
3. #20292557C2semidirect
4. #20292555C2semidirect
5. #20292552C2semidirect
6. #20292550C2semidirect
7. #20292547C2semidirect
9. #20292539C2semidirect
10. #20292487C2semidirect
137. C107#2814semidirect

Maximal subgroups

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

subgroupindex
C2×#168120882
#202925602
#202925572
#202925552
#202925522
#202925502
#202925472
#202925412
#202925392
#202924872
C2×#123×C1072
#2814107