Finite Group Databasebeta

← #28336891

Subgroups — #28336891

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×#16812085C211yesmaximal, characteristic
3C2×#16812033C211yesmaximal, characteristic, complemented
4#20292484C211yesmaximal, characteristic
5#20292390C211yesmaximal, characteristic, complemented
6#20292389C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107C2211yescharacteristic
9C2×#13133801—22no—
10C2×#13133801—22no—
11C2×#13133801—22no—
12C22×C107⋊C8C2211yescharacteristic
13#16812085—22no—
14#16812085C414yescomplemented
15C2×C4⋊Dic107C2211yescharacteristic
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4—21no—
18C2×(C2×D107)⋊C4—21no—
19C2×(C2×D107)⋊C4—21no—
20#16812033—22no—
21C107⋊#95—21no—
22C22×Q8×C107C2211yescharacteristic
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25#103×C107—22no—
26C23×D107D411yescharacteristic
27C22×Dic107—21no—
28C22×Dic107—21no—
29C22×Dic107D411yescharacteristic
30C2×C4×D107—21no—
31C2×C4×D107—21no—
32C2×C4×D107—21no—
33C2×C4×D107—21no—
34C2×C4×D107C2×C412yes—
35C2×C4×D107D412yes—
36#13133801—28no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C2×C107⋊C8D412yes—
40C4⋊Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107—21no—
43C4⋊Dic107C2×C412yes—
44Dic107⋊C4—24no—
45(C2×D107)⋊C4—24no—
46(C2×D107)⋊C4—42no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107—21no—
52C2×Q8×C107C2×C412yes—
53C22×C4×C107—21no—
54C22×C4×C107—21no—
55C22×C4×C107C2311yescharacteristic
56C2×C8×C107—21no—
57C2×C8×C107—21no—
58C2×C8×C107D412yes—
59C4⋊C4×C107—21no—
60C4⋊C4×C107—21no—
61C23×C107C2×D411yescharacteristic
62C22×D107—21no—
63C22×D107—21no—
64C22×D107—21no—
65C22×D107—21no—
66C22×D107—21no—
67C22×D107—21no—
68C22×D107C22⋊C412yes—
69Q8×C107—22no—
70Q8×C107—22no—
71Q8×C107—41no—
72Q8×C107—41no—
73C2×C4×C107—22no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107—41no—
78C2×C4×C107—41no—
79C2×C4×C107C22×C411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107#4111yescharacteristic
84C2×C4×C107#4111yescharacteristic
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—22no—
88C2×Dic107—41no—
89C2×Dic107—41no—
90C2×Dic107—41no—
91C2×Dic107—41no—
92C2×Dic107—41no—
93C2×Dic107C22⋊C412yes—
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—22no—
97C4×D107—22no—
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×#103D10711yescharacteristic, complemented
137#380—1071no—
138#340—1071no—
139#340—1071no—
140C22×Q8C2×D10711yescharacteristic
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—1071no—
151C2×C4⋊C4—2141no—
152C2×C4⋊C4C2×D10711yescharacteristic
153C2×C22⋊C4—2141no—
154C2×C22⋊C4—2141no—
155#103—22no—
156#103—2142no—
157#95—2141no—
158C24—1071no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C4×D10712yes—
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×C8C22⋊D10712yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—1072no—
187C4⋊C4—2144no—
188C4⋊C4—2141no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C22⋊C4—4282no—
192C22⋊C4—2144no—
193C23—1072no—
194C23—2141no—
195C23—2141no—
196C23—2141no—
197C23—2141no—
198C23—2141no—
199C23—2141no—
200C23D4×D10711yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—41no—
204Q8—41no—
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×C22⋊D10711yescharacteristic
228C2×C4#1313381711yescharacteristic
229C2×C4#1313381811yescharacteristic
230C8—22no—
231C8—2142no—
232C22—4281no—
233C22—4281no—
234C22—4281no—
235C22—4281no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C22⋊C4×D10711yescharacteristic
240C22#1681208011yescharacteristic
241C22#1681208211yescharacteristic
242C22#1681213611yescharacteristic
243C22#1681213811yescharacteristic
244C22#1681214111yescharacteristic
245C22#1681214111yescharacteristic
246C4—42no—
247C4—81no—
248C4—8561no—
249C4—4282no—
250C4—2142no—
251C4(C22⋊D107)⋊C411yescharacteristic
252C4C4×C22⋊D10711yescharacteristic
253C4#1681207711yescharacteristic
254C4#1681209111yescharacteristic
255C2—4282no—
256C2#2029250311yescharacteristic
257C2#2029262711yescharacteristic
258C2#2029263411yescharacteristic
259C2#2029287011yescharacteristic
260C2#2029287611yescharacteristic
261C2#2029288211yescharacteristic
262C2#2029288411yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
cosocleC22×C4×C107C23
hypercenterC2×#103D107
hypocenterC107#1894
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16812033C2semidirect
5. #20292390C2semidirect
6. #20292389C2semidirect
7. C2×#103×C107C2semidirect
14. #16812085C4semidirect
132. C107#1894semidirect
136. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338012
C2×#168120852
C2×#168120332
#202924842
#202923902
#202923892
C2×#103×C1072
#1894107