Finite Group Databasebeta

← #28335448

Subgroups — #28335448

Order 13696.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#20293185C211yesmaximal, characteristic
2#20292941C211yesmaximal, characteristic, complemented
3#20292938C211yesmaximal, characteristic, complemented
4#20292554C211yesmaximal, characteristic, complemented
5#20292360C211yesmaximal, characteristic, complemented
6#20292350C211yesmaximal, characteristic, complemented
7#327×C107C211yesmaximal, characteristic, complemented
8C2×D4×D107—21no—
9#16812094—21no—
10#16812091—21no—
11#16812083C2211yescharacteristic
12#16812081—21no—
13#16812079—21no—
14#16812077—21no—
15C2×C4⋊Dic107C2211yescharacteristic, complemented
16C2×(C2×D107)⋊C4—21no—
17C2×(C2×D107)⋊C4C2211yescharacteristic
18#16812043—21no—
19#16812042—21no—
20#16812031—21no—
21#16812030—21no—
22C22⋊(C107⋊C8)C2211yescharacteristic, complemented
23C107⋊#95—21no—
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic, complemented
26#95×C107C2211yescharacteristic, complemented
27C23×D107—21no—
28C23×D107D411yescharacteristic
29C2×D428—21no—
30C2×D428D411yescharacteristic
31C2×C22⋊D107—21no—
32C2×C22⋊D107—21no—
33C2×C22⋊D107—21no—
34D4×D107—42no—
35D4×D107—42no—
36C22×Dic107—21no—
37C22×Dic107D411yescharacteristic
38C2×C4×D107—21no—
39C2×C107⋊C8—21no—
40C4⋊Dic107—21no—
41C4⋊Dic107—21no—
42C4⋊Dic107D411yescharacteristic
43Dic107⋊C4—21no—
44C22⋊Dic107—21no—
45(C2×D107)⋊C4—21no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—21no—
48(C2×D107)⋊C4—21no—
49(C2×D107)⋊C4—42no—
50C2×D4×C107—21no—
51C2×D4×C107D411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—21no—
55C4⋊C4×C107D411yescharacteristic
56C22⋊C4×C107—21no—
57C23×C107—21no—
58C23×C107C2×D411yescharacteristic
59C22×D107—21no—
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—42no—
63C22×D107—41no—
64C22×D107—41no—
65C22×D107—81no—
66C22×D107—81no—
67C22×D107D812yes—
68D4×C107—41no—
69D4×C107—41no—
70D4×C107—41no—
71C2×C4×C107—21no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C22⋊D107—42no—
79C22⋊D107—41no—
80C22⋊D107—41no—
81C22⋊D107—81no—
82C2×Dic107—21no—
83C2×Dic107—21no—
84C2×Dic107—41no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88D428—41no—
89D428—42no—
90C4×D107—42no—
91C8×C107—41no—
92C107⋊C8—41no—
93C22×C107—21no—
94C22×C107—21no—
95C22×C107—41no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107#12011yescharacteristic
99C22×C107C2×D811yescharacteristic
100C22×C107AGL(1,ℤ/8)11yescharacteristic
101C2×D107—22no—
102C2×D107—41no—
103C2×D107—41no—
104C2×D107—42no—
105C2×D107—41no—
106C2×D107—41no—
107C2×D107—41no—
108C2×D107—81no—
109C2×D107—84no—
110C4×C107—21no—
111C4×C107—21no—
112C4×C107—41no—
113C4×C107—81no—
114Dic107—41no—
115Dic107—41no—
116Dic107—81no—
117C2×C107—21no—
118C2×C107—21no—
119C2×C107—81no—
120C2×C107#44711yescharacteristic
121C2×C107#45511yescharacteristic
122C2×C107#45811yescharacteristic
123D107—42no—
124D107—81no—
125#1607—1071nomaximal, Sylow
126C107#160711yescharacteristic, complemented, Sylow
127#534—1071no—
128#480—1071no—
129#480—1071no—
130#397—1071no—
131#327—1071no—
132#327D10711yescharacteristic, complemented
133#323—1071no—
134C22×D4—2141no—
135#123—2141no—
136#123—2141no—
137#121—1071no—
138#121—2141no—
139#121C2×D10711yescharacteristic
140#120—2141no—
141#120—2141no—
142C2×C4⋊C4—1071no—
143C2×C22⋊C4—1071no—
144C2×C22⋊C4—2141no—
145#107—2141no—
146#107—2141no—
147#102—2141no—
148#102—2141no—
149C22⋊C8—1071no—
150C22⋊C8C2×D10711yescharacteristic, complemented
151#95—2141no—
152#95C2×D10711yescharacteristic, complemented
153C24—1071no—
154C24—2141no—
155C2×D4—21no—
156C2×D4—1071no—
157C2×D4—4282no—
158C2×D4—4282no—
159C2×D4—2141no—
160C2×D4—2141no—
161C2×D4—2141no—
162C2×D4—2141no—
163C2×D4D42811yescharacteristic
164C22×C4—21no—
165C22×C4—1071no—
166C22×C4—2141no—
167C22×C4—2141no—
168C22×C4C22×D10711yescharacteristic
169C2×C8—21no—
170C2×C8—2141no—
171C4⋊C4—1071no—
172C4⋊C4—2141no—
173C4⋊C4—2141no—
174C4⋊C4—2141no—
175C4⋊C4D42811yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—4282no—
178C22⋊C4—2141no—
179C22⋊C4—2141no—
180C22⋊C4—2141no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C23—21no—
184C23—8561no—
185C23—8561no—
186C23—1072no—
187C23—4281no—
188C23—4281no—
189C23—4282no—
190C23—2141no—
191C23—2141no—
192C23—2141no—
193C23D4×D10711yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—8561no—
198D4—4281no—
199D4—4282no—
200D4—4282no—
201D4—4281no—
202D4—4281no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—4282no—
209C2×C4—4281no—
210C2×C4—4281no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—2141no—
214C2×C4—2141no—
215C2×C4D4×D10711yescharacteristic
216C2×C4C2×D42811yescharacteristic
217C8—41no—
218C8—4281no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—8561no—
225C22—8564no—
226C22—4282no—
227C22—4281no—
228C22—4281no—
229C22—4281no—
230C22—4281no—
231C22—4281no—
232C22—2142no—
233C22#1681207911yescharacteristic
234C22D8×D10711yescharacteristic
235C22#1681213611yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—8561no—
241C4—4281no—
242C4—4281no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—8561no—
247C2—4282no—
248C2#2029276211yescharacteristic
249C2#2029280611yescharacteristic
250C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107#120
prime radical#20293185C2
hypercenter#327D107
hypocenterC107#1607
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20292941C2semidirect
3. #20292938C2semidirect
4. #20292554C2semidirect
5. #20292360C2semidirect
6. #20292350C2semidirect
7. #327×C107C2semidirect
15. C2×C4⋊Dic107C22semidirect
22. C22⋊(C107⋊C8)C22semidirect
25. C22⋊C8×C107C22semidirect
26. #95×C107C22semidirect
126. C107#1607semidirect
132. #327D107semidirect
150. C22⋊C8C2×D107semidirect
152. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931852
#202929412
#202929382
#202925542
#202923602
#202923502
#327×C1072
#1607107