Finite Group Databasebeta

← #28336893

Subgroups — #28336893

Order 13696.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13133801C211yesmaximal, characteristic
2C2×#16812089C211yesmaximal, characteristic
3C2×#16812034C211yesmaximal, characteristic, complemented
4C2×#16812032C211yesmaximal, characteristic, complemented
5#20292484C211yesmaximal, characteristic
6#20292389C211yesmaximal, characteristic, complemented
7#340×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×C4×D107C2211yescharacteristic
10C2×#13133801—22no—
11C2×#13133801—22no—
12C2×#13133801—22no—
13C22×C107⋊C8C2211yescharacteristic
14#16812089—22no—
15#16812089C414yescomplemented
16C2×C4⋊Dic107—21no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×(C2×D107)⋊C4—21no—
19C2×(C2×D107)⋊C4—21no—
20#16812034—22no—
21#16812032—22no—
22C107⋊#95—21no—
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2211yescharacteristic
26C23×D107D411yescharacteristic
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214C2×C412yes—
33C22×Dic107—21no—
34C22×Dic107—21no—
35C22×Dic107D411yescharacteristic
36C2×C4×D107—21no—
37C2×C4×D107—21no—
38C2×C4×D107—21no—
39C2×C4×D107—21no—
40C2×C4×D107C2×C412yes—
41C2×C4×D107D412yes—
42#13133801—28no—
43C2×C107⋊C8—21no—
44C2×C107⋊C8—21no—
45C2×C107⋊C8D412yes—
46C4⋊Dic107—24no—
47C4⋊Dic107—21no—
48C4⋊Dic107—21no—
49(C2×D107)⋊C4—24no—
50(C2×D107)⋊C4—42no—
51C22×C4×C107—21no—
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—21no—
55C2×C8×C107—21no—
56C2×C8×C107D412yes—
57C4⋊C4×C107—21no—
58C4⋊C4×C107—21no—
59C4⋊C4×C107—21no—
60C4⋊C4×C107C2×C412yes—
61C23×C107C2×D411yescharacteristic
62C22×D107—21no—
63C22×D107—21no—
64C22×D107—21no—
65C22×D107—21no—
66C22×D107—21no—
67C22×D107—21no—
68C22×D107C22⋊C412yes—
69C2×C4×C107—22no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107C22×C411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107#4111yescharacteristic
80C2×C4×C107#4111yescharacteristic
81C2×Dic107—21no—
82C2×Dic107—21no—
83C2×Dic107—22no—
84C2×Dic107—41no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88C2×Dic107—41no—
89C2×Dic107C22⋊C412yes—
90C4×D107—22no—
91C4×D107—22no—
92C4×D107—22no—
93C4×D107—22no—
94Dic214—22no—
95Dic214—22no—
96Dic214—41no—
97Dic214—41no—
98C8×C107—22no—
99C107⋊C8—22no—
100C22×C107C2×C22⋊C411yescharacteristic
101C22×C107#12111yescharacteristic
102C22×C107#12111yescharacteristic
103C22×C107AGL(1,ℤ/8)11yescharacteristic
104C22×C107#13711yescharacteristic
105C22×C107#13711yescharacteristic
106C22×C107#13711yescharacteristic
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—22no—
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C2×D107—41no—
114C4×C107—42no—
115C4×C107—81no—
116C4×C107C4×D411yescharacteristic
117C4×C107C4×D411yescharacteristic
118C4×C107#12011yescharacteristic
119C4×C107#12311yescharacteristic
120Dic107—22no—
121Dic107—42no—
122Dic107—81no—
123C2×C107#38611yescharacteristic
124C2×C107#41711yescharacteristic
125C2×C107#41911yescharacteristic
126C2×C107#46811yescharacteristic
127C2×C107#46811yescharacteristic
128C2×C107#47011yescharacteristic
129C2×C107#47011yescharacteristic
130D107—42no—
131#1894—1071nomaximal, Sylow
132C107#189411yescharacteristic, complemented, Sylow
133C22×M16—1071no—
134C2×C22⋊Q8—1071no—
135C2×#103—1071no—
136C2×#103—1071no—
137#380—1071no—
138#340—1071no—
139#340D10711yescharacteristic, complemented
140C22×Q8—1071no—
141C23×C4—1071no—
142C2×M16—2142no—
143C2×M16—2142no—
144C2×M16—2142no—
145C22×C8—1071no—
146C22×C8C2×D10711yescharacteristic
147C22⋊Q8—1074no—
148C22⋊Q8—2142no—
149C2×C4⋊C4—1071no—
150C2×C4⋊C4—2141no—
151C2×C4⋊C4C2×D10711yescharacteristic
152C2×C4⋊C4C2×D10711yescharacteristic
153C2×C22⋊C4—2141no—
154C2×C22⋊C4—2141no—
155#103—2142no—
156#103—2142no—
157#95—2141no—
158C24—1071no—
159C2×Q8—1072no—
160C2×Q8—2141no—
161C2×Q8—2141no—
162C2×Q8—2141no—
163C2×Q8—2141no—
164C2×Q8—2141no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—1071no—
168C22×C4—1072no—
169C22×C4—1072no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4—2141no—
174C22×C4—2141no—
175C22×C4—2141no—
176C22×C4C22×D10711yescharacteristic
177M16—2148no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—1072no—
181C2×C8—2141no—
182C2×C8—2141no—
183C2×C8D42812yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—2144no—
188C4⋊C4—2141no—
189C4⋊C4—2141no—
190C4⋊C4C4×D10712yes—
191C22⋊C4—4282no—
192C22⋊C4—2144no—
193C23—1072no—
194C23—2141no—
195C23—2141no—
196C23—2141no—
197C23—2141no—
198C23—2141no—
199C23—2141no—
200C23D4×D10711yescharacteristic
201Q8—4281no—
202Q8—4281no—
203Q8—2142no—
204Q8—2142no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1072no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—4281no—
216C2×C4—4281no—
217C2×C4—2142no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4C2×C4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4D4×D10711yescharacteristic
227C2×C4C2×D42811yescharacteristic
228C2×C4#1313381811yescharacteristic
229C2×C4#1313381911yescharacteristic
230C8—22no—
231C8—2142no—
232C22—4281no—
233C22—4281no—
234C22—4281no—
235C22—4281no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C22⋊C4×D10711yescharacteristic
240C22#1681208211yescharacteristic
241C22#1681208411yescharacteristic
242C22#1681213311yescharacteristic
243C22#1681213811yescharacteristic
244C22#1681213811yescharacteristic
245C22#1681214111yescharacteristic
246C4—42no—
247C4—81no—
248C4—8561no—
249C4—4282no—
250C4—2142no—
251C4#1681207111yescharacteristic
252C4C4×D42811yescharacteristic
253C4#1681207911yescharacteristic
254C4#1681209411yescharacteristic
255C2—4282no—
256C2#2029250511yescharacteristic
257C2#2029262511yescharacteristic
258C2#2029263411yescharacteristic
259C2#2029287211yescharacteristic
260C2#2029287411yescharacteristic
261C2#2029288411yescharacteristic
262C2#2029288611yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
cosocleC22×C4×C107C23
hypercenter#340D107
hypocenterC107#1894
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
3. C2×#16812034C2semidirect
4. C2×#16812032C2semidirect
6. #20292389C2semidirect
7. #340×C107C2semidirect
15. #16812089C4semidirect
132. C107#1894semidirect
139. #340D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338012
C2×#168120892
C2×#168120342
C2×#168120322
#202924842
#202923892
#340×C1072
#1894107