Finite Group Databasebeta

← #28337662

Subgroups — #28337662

Order 13696.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16812085C211yesmaximal, characteristic
2C2×#16812042C211yesmaximal, characteristic, complemented
3C2×#16812033C211yesmaximal, characteristic, complemented
4C2×(C2×D107)⋊C8C211yesmaximal, characteristic
5#20292484C211yesmaximal, characteristic
6#20292390C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107C2211yescharacteristic
9C22×C107⋊C8C2211yescharacteristic
10#16812085—22no—
11#16812085—22no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4—21no—
15C2×(C2×D107)⋊C4—21no—
16C2×(C2×D107)⋊C4—21no—
17#16812042—22no—
18#16812033—22no—
19(C2×D107)⋊C8—22no—
20C107⋊#95—21no—
21C22×Q8×C107C2211yescharacteristic
22C22×C8×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24#103×C107—22no—
25C23×D107D411yescharacteristic
26C22×Dic107—21no—
27C22×Dic107—21no—
28C22×Dic107D411yescharacteristic
29C2×C4×D107—21no—
30C2×C4×D107—21no—
31C2×C4×D107—21no—
32C2×C4×D107—21no—
33C2×C4×D107—21no—
34C2×C4×D107—21no—
35C2×C107⋊C8—21no—
36C2×C107⋊C8—21no—
37C2×C107⋊C8D412yes—
38C4⋊Dic107—21no—
39C4⋊Dic107—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107D412yes—
42Dic107⋊C4—24no—
43(C2×D107)⋊C4—24no—
44(C2×D107)⋊C4—42no—
45C2×Q8×C107—21no—
46C2×Q8×C107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107D412yes—
51C22×C4×C107—21no—
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—21no—
55C2×C8×C107—21no—
56C2×C8×C107—21no—
57C4⋊C4×C107—21no—
58C4⋊C4×C107—21no—
59C23×C107C2×D411yescharacteristic
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—21no—
63C22×D107—21no—
64C22×D107—21no—
65C22×D107—21no—
66C22×D107Q1612yes—
67Q8×C107—22no—
68Q8×C107—22no—
69Q8×C107—41no—
70Q8×C107—41no—
71C2×C4×C107—22no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107#4111yescharacteristic
81C2×C4×C107#4111yescharacteristic
82C2×C4×C107#4111yescharacteristic
83C2×Dic107—22no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—41no—
88C2×Dic107—41no—
89C2×Dic107—41no—
90C2×Dic107—41no—
91C2×Dic107—41no—
92C4×D107—41no—
93C4×D107—41no—
94C4×D107—41no—
95C4×D107—41no—
96C8×C107—41no—
97C107⋊C8—22no—
98C22×C107#12011yescharacteristic
99C22×C107#12111yescharacteristic
100C22×C107#12311yescharacteristic
101C22×C107C2×Q1611yescharacteristic
102C22×C107#13511yescharacteristic
103C22×C107AGL(1,ℤ/8)11yescharacteristic
104C22×C107#13711yescharacteristic
105C2×D107—22no—
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—41no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C4×C107—42no—
113C4×C107—81no—
114C4×C107#12111yescharacteristic
115C4×C107#12111yescharacteristic
116C4×C107#12311yescharacteristic
117C4×C107#12411yescharacteristic
118Dic107—41no—
119Dic107—42no—
120Dic107—81no—
121C2×C107#39411yescharacteristic
122C2×C107#44911yescharacteristic
123C2×C107#45111yescharacteristic
124C2×C107#46711yescharacteristic
125C2×C107#46811yescharacteristic
126C2×C107#48411yescharacteristic
127C2×C107#48511yescharacteristic
128D107—42no—
129#2036—1071nomaximal, Sylow
130C107#203611yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—1071no—
132C2×#107—1071no—
133C2×#103—1071no—
134C2×#103D10711yescharacteristic, complemented
135C2×C22⋊C8—1071no—
136#380—1071no—
137#340—1071no—
138C22×Q8C2×D10711yescharacteristic
139C23×C4—1071no—
140C22×C8—1071no—
141C22×C8C2×D10711yescharacteristic
142C22⋊Q8—2142no—
143C22⋊Q8—2142no—
144C2×C4⋊C4—1071no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4—2141no—
147C2×C4⋊C4C2×D10711yescharacteristic
148C2×C22⋊C4—2141no—
149C2×C22⋊C4—2141no—
150#107—2142no—
151#103—22no—
152#103—2142no—
153C22⋊C8—2142no—
154#95—2141no—
155C24—1071no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D10712yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1071no—
165C22×C4—2141no—
166C22×C4—2141no—
167C22×C4—2141no—
168C22×C4—2141no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1072no—
178C2×C8—2141no—
179C2×C8—2141no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1072no—
183C4⋊C4—2144no—
184C4⋊C4—2141no—
185C4⋊C4—2141no—
186C4⋊C4—2141no—
187C22⋊C4—4282no—
188C22⋊C4—2144no—
189C23—1072no—
190C23—2141no—
191C23—2141no—
192C23—2141no—
193C23—2141no—
194C23—2141no—
195C23—2141no—
196C23D4×D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—4281no—
208C2×C4—4281no—
209C2×C4—4281no—
210C2×C4—4281no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—4281no—
216C2×C4—2142no—
217C2×C4—2141no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4D4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C2×C4C2×C22⋊D10711yescharacteristic
223C2×C4#1313381711yescharacteristic
224C2×C4#1313381811yescharacteristic
225C2×C4#1313381811yescharacteristic
226C8—41no—
227C8—2142no—
228C22—4281no—
229C22—4281no—
230C22—4281no—
231C22—4281no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22#1681207711yescharacteristic
236C22#1681208211yescharacteristic
237C22#1681209111yescharacteristic
238C22Q16×D10711yescharacteristic
239C22#1681213011yescharacteristic
240C22#1681213611yescharacteristic
241C22#1681214111yescharacteristic
242C4—42no—
243C4—81no—
244C4—8561no—
245C4—4282no—
246C4—4281no—
247C4#1681208111yescharacteristic
248C4#1681208211yescharacteristic
249C4#1681209211yescharacteristic
250C4#1681209511yescharacteristic
251C2—4282no—
252C2#2029254211yescharacteristic
253C2#2029277111yescharacteristic
254C2#2029278111yescharacteristic
255C2#2029286711yescharacteristic
256C2#2029287611yescharacteristic
257C2#2029296111yescharacteristic
258C2#2029296611yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
hypercenterC2×#103D107
hypocenterC107#2036
LCSG ▷ C22×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
2. C2×#16812042C2semidirect
3. C2×#16812033C2semidirect
6. #20292390C2semidirect
7. C2×#103×C107C2semidirect
130. C107#2036semidirect
134. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C2×#168120422
C2×#168120332
C2×(C2×D107)⋊C82
#202924842
#202923902
C2×#103×C1072
#2036107