Finite Group Databasebeta

← #28337698

Subgroups — #28337698

Order 13696.

318 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16812098C211yesmaximal, characteristic
2C2×#16812033C211yesmaximal, characteristic, complemented
3C2×#16812031C211yesmaximal, characteristic
4C2×#16812029C211yesmaximal, characteristic
5#20292523C211yesmaximal, characteristic, complemented
6#20292381C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C22×C107⋊C8C2211yescharacteristic
10#16812098—22no—
11#16812098—22no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4—21no—
14C2×(C2×D107)⋊C4—21no—
15C2×(C2×D107)⋊C4—21no—
16C2×C4×Dic107C2211yescharacteristic
17#16812033—22no—
18#16812031—22no—
19#16812029—22no—
20C107⋊#95—21no—
21C22×Q8×C107C2211yescharacteristic
22C22×C8×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24#103×C107—22no—
25C23×D107—21no—
26C2×D428—21no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428D412yes—
32C22×Dic107—21no—
33C22×Dic107D411yescharacteristic
34C22×Dic107D411yescharacteristic
35C2×C107⋊C8—21no—
36C2×C107⋊C8—21no—
37C2×C107⋊C8—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107D412yes—
40Dic107⋊C4—42no—
41(C2×D107)⋊C4—24no—
42(C2×D107)⋊C4—24no—
43C4×Dic107—21no—
44C4×Dic107—21no—
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—22no—
61C22×D107—41no—
62C22×D107—41no—
63C22×D107—41no—
64C22×D107—41no—
65C22×D107—41no—
66C22×D107—41no—
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—21no—
84C2×Dic107—21no—
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—41no—
91C2×Dic107—41no—
92D428—22no—
93D428—22no—
94D428—41no—
95D428—41no—
96C8×C107—41no—
97C107⋊C8—41no—
98C22×C107#12011yescharacteristic
99C22×C107#12111yescharacteristic
100C22×C107#12411yescharacteristic
101C22×C107#13511yescharacteristic
102C22×C107#13511yescharacteristic
103C22×C107AGL(1,ℤ/8)11yescharacteristic
104C22×C107#13711yescharacteristic
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—81no—
109C2×D107—81no—
110C2×D107—81no—
111C2×D107—81no—
112C4×C107—42no—
113C4×C107—81no—
114C4×C107#12111yescharacteristic
115C4×C107#12111yescharacteristic
116C4×C107#12311yescharacteristic
117C4×C107#12311yescharacteristic
118Dic107—41no—
119Dic107—41no—
120Dic107—42no—
121C2×C107#39411yescharacteristic
122C2×C107#44911yescharacteristic
123C2×C107#45211yescharacteristic
124C2×C107#46311yescharacteristic
125C2×C107#46411yescharacteristic
126C2×C107#48811yescharacteristic
127C2×C107#48911yescharacteristic
128D107—82no—
129#2042—1071nomaximal, Sylow
130C107#204211yescharacteristic, complemented, Sylow
131C2×#124—1071no—
132C2×#103—1071no—
133C2×#103D10711yescharacteristic, complemented
134C2×#102—1071no—
135C2×#102—1071no—
136#389—1071no—
137#336—1071no—
138C22×Q8C2×D10711yescharacteristic
139C22×D4—1071no—
140C22×C8—1071no—
141C22×C8C2×D10711yescharacteristic
142#124—2142no—
143#124—2142no—
144C2×C4⋊C4—1071no—
145C2×C4⋊C4—2141no—
146C2×C4⋊C4C2×D10711yescharacteristic
147C2×C22⋊C4—2141no—
148C2×C22⋊C4—2141no—
149C2×C42—1071no—
150#103—22no—
151#103—2142no—
152#102—2142no—
153#102—2142no—
154#95—2141no—
155C24—2141no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D10712yes—
162C2×D4—1072no—
163C2×D4—2141no—
164C2×D4—2141no—
165C2×D4—2141no—
166C2×D4—2141no—
167C2×D4—2141no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1071no—
171C22×C4—1071no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2141no—
178C2×C8—2141no—
179C2×C8—2141no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1072no—
183C4⋊C4—4282no—
184C4⋊C4—2141no—
185C22⋊C4—2144no—
186C22⋊C4—2144no—
187C42—2141no—
188C42—2141no—
189C23—4281no—
190C23—4281no—
191C23—4281no—
192C23—4281no—
193C23—4281no—
194C23—4281no—
195C23—2142no—
196C23D4×D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—4281no—
202D4—4281no—
203D4—2142no—
204D4—2142no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—2141no—
214C2×C4—2141no—
215C2×C4—2142no—
216C2×C4—2141no—
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—4281no—
228C22—8561no—
229C22—8561no—
230C22—8561no—
231C22—8561no—
232C22—4282no—
233C22—4282no—
234C22—4282no—
235C22#1681207711yescharacteristic
236C22#1681208211yescharacteristic
237C22#1681209511yescharacteristic
238C22#1681213011yescharacteristic
239C22#1681213211yescharacteristic
240C22#1681213611yescharacteristic
241C22#1681214111yescharacteristic
242C4—42no—
243C4—81no—
244C4—4281no—
245C4—4281no—
246C4—4282no—
247C4#1681208111yescharacteristic
248C4#1681208211yescharacteristic
249C4#1681209111yescharacteristic
250C4#1681209211yescharacteristic
251C2—8562no—
252C2#2029254211yescharacteristic
253C2#2029277111yescharacteristic
254C2#2029278811yescharacteristic
255C2#2029284511yescharacteristic
256C2#2029285211yescharacteristic
257C2#2029298311yescharacteristic
258C2#2029298811yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC22×D428C22
hypercenterC2×#103D107
hypocenterC107#2042
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×#16812033C2semidirect
5. #20292523C2semidirect
6. #20292381C2semidirect
7. C2×#103×C107C2semidirect
130. C107#2042semidirect
133. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
C2×#168120332
C2×#168120312
C2×#168120292
#202925232
#202923812
C2×#103×C1072
#2042107