Finite Group Databasebeta

← #28335482

Subgroups — #28335482

Order 13696.

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

normal subgroups only

This group has 288 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#20292946C211yesmaximal, characteristic, complemented
3#20292943C211yesmaximal, characteristic, complemented
4#20292564C211yesmaximal, 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⋊Dic107—21no—
16C2×C4⋊Dic107C2211yescharacteristic, complemented
17C2×(C2×D107)⋊C4—21no—
18C2×(C2×D107)⋊C4C2211yescharacteristic
19#16812041—21no—
20#16812040—21no—
21#16812031—21no—
22#16812030—21no—
23C22⋊(C107⋊C8)C2211yescharacteristic, complemented
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⋊Dic107—42no—
43C4⋊Dic107D411yescharacteristic
44Dic107⋊C4—21no—
45C22⋊Dic107—21no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—21no—
48(C2×D107)⋊C4—21no—
49(C2×D107)⋊C4—21no—
50(C2×D107)⋊C4—42no—
51C2×D4×C107—21no—
52C2×D4×C107D411yescharacteristic
53C22×C4×C107—21no—
54C22×C4×C107C2311yescharacteristic
55C2×C8×C107—21no—
56C4⋊C4×C107D411yescharacteristic
57C22⋊C4×C107—21no—
58C23×C107—21no—
59C23×C107C2×D411yescharacteristic
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—21no—
63C22×D107—42no—
64C22×D107—41no—
65C22×D107—41no—
66C22×D107—81no—
67C22×D107—81no—
68C22×D107SD1612yes—
69D4×C107—41no—
70D4×C107—41no—
71D4×C107—41no—
72C2×C4×C107—22no—
73C2×C4×C107—21no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C22⋊D107—42no—
80C22⋊D107—41no—
81C22⋊D107—41no—
82C22⋊D107—81no—
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88C2×Dic107—41no—
89D428—41no—
90D428—42no—
91C4×D107—42no—
92C8×C107—41no—
93C107⋊C8—41no—
94C22×C107—21no—
95C22×C107—21no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107—41no—
99C22×C107#12011yescharacteristic
100C22×C107C2×SD1611yescharacteristic
101C22×C107AGL(1,ℤ/8)11yescharacteristic
102C2×D107—22no—
103C2×D107—41no—
104C2×D107—41no—
105C2×D107—42no—
106C2×D107—41no—
107C2×D107—41no—
108C2×D107—41no—
109C2×D107—81no—
110C2×D107—84no—
111C4×C107—21no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107—41no—
115Dic107—41no—
116Dic107—41no—
117Dic107—81no—
118C2×C107—21no—
119C2×C107—21no—
120C2×C107—81no—
121C2×C107#45011yescharacteristic
122C2×C107#45311yescharacteristic
123C2×C107#45811yescharacteristic
124D107—42no—
125D107—81no—
126#1613—1071nomaximal, Sylow
127C107#161311yescharacteristic, complemented, Sylow
128#534—1071no—
129#481—1071no—
130#481—1071no—
131#399—1071no—
132#327—1071no—
133#327D10711yescharacteristic, complemented
134#323—1071no—
135C22×D4—2141no—
136#123—2141no—
137#123—2141no—
138#121—1071no—
139#121—2141no—
140#121C2×D10711yescharacteristic
141#120—2141no—
142#120—2141no—
143C2×C4⋊C4—1071no—
144C2×C4⋊C4—2141no—
145C2×C22⋊C4—1071no—
146C2×C22⋊C4—2141no—
147#106—2141no—
148#106—2141no—
149#102—2141no—
150#102—2141no—
151C22⋊C8—1071no—
152C22⋊C8C2×D10711yescharacteristic, complemented
153#95C2×D10711yescharacteristic, complemented
154C24—1071no—
155C24—2141no—
156C2×D4—21no—
157C2×D4—1071no—
158C2×D4—4282no—
159C2×D4—4282no—
160C2×D4—2141no—
161C2×D4—2141no—
162C2×D4—2141no—
163C2×D4—2141no—
164C2×D4D42811yescharacteristic
165C22×C4—21no—
166C22×C4—1071no—
167C22×C4—2141no—
168C22×C4—2141no—
169C22×C4C22×D10711yescharacteristic
170C2×C8—21no—
171C2×C8—2141no—
172C4⋊C4—1071no—
173C4⋊C4—4282no—
174C4⋊C4—2141no—
175C4⋊C4—2141no—
176C4⋊C4—2141no—
177C4⋊C4D42811yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4282no—
180C22⋊C4—2141no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C22⋊C4—2141no—
184C22⋊C4—2141no—
185C23—21no—
186C23—8561no—
187C23—8561no—
188C23—1072no—
189C23—4281no—
190C23—4281no—
191C23—4282no—
192C23—2141no—
193C23—2141no—
194C23—2141no—
195C23D4×D10711yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—8561no—
200D4—4281no—
201D4—4282no—
202D4—4282no—
203D4—4281no—
204D4—4281no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—4282no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—2141no—
216C2×C4—2141no—
217C2×C4D4×D10711yescharacteristic
218C2×C4C2×D42811yescharacteristic
219C8—41no—
220C8—4281no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—8561no—
227C22—8564no—
228C22—4282no—
229C22—4281no—
230C22—4281no—
231C22—4281no—
232C22—4281no—
233C22—4281no—
234C22—2142no—
235C22#1681207911yescharacteristic
236C22SD16×D10711yescharacteristic
237C22#1681213311yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—8561no—
243C4—4281no—
244C4—4281no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—8561no—
249C2—4282no—
250C2#2029277911yescharacteristic
251C2#2029279611yescharacteristic
252C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107#120
prime radical#20293185C2
hypercenter#327D107
hypocenterC107#1613
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. #20292946C2semidirect
3. #20292943C2semidirect
4. #20292564C2semidirect
5. #20292360C2semidirect
6. #20292350C2semidirect
7. #327×C107C2semidirect
16. C2×C4⋊Dic107C22semidirect
23. C22⋊(C107⋊C8)C22semidirect
25. C22⋊C8×C107C22semidirect
26. #95×C107C22semidirect
127. C107#1613semidirect
133. #327D107semidirect
152. C22⋊C8C2×D107semidirect
153. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931852
#202929462
#202929432
#202925642
#202923602
#202923502
#327×C1072
#1613107