Finite Group Databasebeta

← #28334949

Subgroups — #28334949

Order 13696.

304 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 65 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#20293137C211yesmaximal, characteristic
2C22⋊C8×D107C211yesmaximal, characteristic, complemented
3#20292474C211yesmaximal, characteristic, complemented
4#20292365C211yesmaximal, characteristic, complemented
5#20292363C211yesmaximal, characteristic, complemented
6#20292361C211yesmaximal, characteristic, complemented
7#327×C107C211yesmaximal, characteristic, complemented
8C2×#13133817C411yescharacteristic, complemented
9C22×C4×D107C2211yescharacteristic
10C2×C8×D107—21no—
11#16812091—21no—
12#16812089C411yescharacteristic, complemented
13#16812088C2211yescharacteristic
14#16812080C411yescharacteristic, complemented
15#16812077—21no—
16D4×Dic107—21no—
17(C22⋊D107)⋊C4—21no—
18#16812067C411yescharacteristic, complemented
19C2×(C2×D107)⋊C4—21no—
20C2×C4×Dic107—21no—
21C22⋊(C107⋊C8)C2211yescharacteristic
22(C2×D107)⋊C8—21no—
23C107⋊#95C2211yescharacteristic
24#121×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28#13133817—42no—
29C2×Dic214C2×C411yescharacteristic
30C2×C22⋊D107—21no—
31C2×C22⋊D107—21no—
32C22×Dic107—21no—
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C2×C4×D107—42no—
36C2×C4×D107C2×C411yescharacteristic
37C2×C4×D107C2×C411yescharacteristic
38C2×C4×D107D411yescharacteristic
39C2×C4×D107D411yescharacteristic
40C2×C107⋊C8—21no—
41C8×D107—42no—
42C4⋊Dic107—21no—
43C4⋊Dic107C2×C411yescharacteristic
44Dic107⋊C4—21no—
45C22⋊Dic107—21no—
46C22⋊Dic107—21no—
47(C2×D107)⋊C4—24no—
48(C2×D107)⋊C4—21no—
49(C2×D107)⋊C4—21no—
50C4×Dic107—21no—
51C4×Dic107—42no—
52C2×D4×C107—21no—
53C2×D4×C107C2×C411yescharacteristic
54C22×C4×C107—21no—
55C22×C4×C107C2311yescharacteristic
56C2×C8×C107—21no—
57C4⋊C4×C107C2×C411yescharacteristic
58C22⋊C4×C107—21no—
59C23×C107—21no—
60C23×C107C2×D411yescharacteristic
61C22×D107—21no—
62C22×D107—21no—
63C22×D107—22no—
64C22×D107—41no—
65C22×D107C22⋊C411yescharacteristic
66C22×D107C22⋊C411yescharacteristic
67D4×C107—41no—
68D4×C107—41no—
69D4×C107—41no—
70C2×C4×C107—22no—
71C2×C4×C107—21no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107C22×C411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C22⋊D107—42no—
78C22⋊D107—42no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—21no—
82C2×Dic107—21no—
83C2×Dic107—41no—
84C2×Dic107—41no—
85C2×Dic107—42no—
86C2×Dic107C22⋊C411yescharacteristic
87C2×Dic107C22⋊C411yescharacteristic
88C4×D107—22no—
89C4×D107—22no—
90C4×D107—41no—
91C4×D107—41no—
92Dic214—41no—
93C8×C107—41no—
94C107⋊C8—41no—
95C22×C107—21no—
96C22×C107—21no—
97C22×C107—41no—
98C22×C107—41no—
99C22×C107—41no—
100C22×C107C2×C22⋊C411yescharacteristic
101C22×C107#13511yescharacteristic
102C22×C107#13511yescharacteristic
103C2×D107—21no—
104C2×D107—21no—
105C2×D107—21no—
106C2×D107—21no—
107C2×D107—21no—
108C2×D107—42no—
109C2×D107—42no—
110C2×D107#10412yes—
111C4×C107—21no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107—41no—
115Dic107—21no—
116Dic107—42no—
117Dic107—41no—
118Dic107—41no—
119Dic107#9912yes—
120C2×C107—21no—
121C2×C107—21no—
122C2×C107—81no—
123C2×C107C2×#9911yescharacteristic
124C2×C107#41611yescharacteristic
125C2×C107C2×#10411yescharacteristic
126D107—22no—
127D107—41no—
128#1501—1071nomaximal, Sylow
129C107#150111yescharacteristic, complemented, Sylow
130#525—1071no—
131C2×C22⋊C8—1071no—
132C4×C22⋊C4—1071no—
133#328—1071no—
134#328—1071no—
135#327—1071no—
136#327D10711yescharacteristic, complemented
137C2×#41—1071no—
138C23×C4—1071no—
139C22×C8—2141no—
140#123—2141no—
141C22⋊Q8—1071no—
142C22⋊Q8—1071no—
143#121—1071no—
144#121C2×D10711yescharacteristic
145#120—2141no—
146C4×D4—2141no—
147C4×D4—2141no—
148#117—1071no—
149C2×C22⋊C4—2141no—
150C2×C42—2141no—
151C22⋊C8—1071no—
152C22⋊C8—2141no—
153C22⋊C8C2×D10711yescharacteristic
154#95—1071no—
155#95C2×D10711yescharacteristic
156C24—1071no—
157#41—4282no—
158C2×Q8—1071no—
159C2×D4—21no—
160C2×D4—2141no—
161C2×D4—2141no—
162C2×D4C4×D10711yescharacteristic
163C22×C4—21no—
164C22×C4—1071no—
165C22×C4—1071no—
166C22×C4—1071no—
167C22×C4—1071no—
168C22×C4—1071no—
169C22×C4—4282no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4C22×D10711yescharacteristic
173C2×C8—21no—
174C2×C8—4282no—
175C2×C8—2141no—
176C4⋊C4—1071no—
177C4⋊C4—2141no—
178C4⋊C4—2141no—
179C4⋊C4C4×D10711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C22⋊C4—2144no—
184C22⋊C4—2141no—
185C22⋊C4—2141no—
186C42—4282no—
187C42—2141no—
188C23—21no—
189C23—1071no—
190C23—1071no—
191C23—4281no—
192C23—2142no—
193C23—2141no—
194C23—2141no—
195C23D4×D10711yescharacteristic
196Q8—4281no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—4282no—
201D4—4282no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—1071no—
208C2×C4—1071no—
209C2×C4—4281no—
210C2×C4—4281no—
211C2×C4—4282no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4C2×C4×D10711yescharacteristic
221C2×C4D4×D10711yescharacteristic
222C8—41no—
223C8—4281no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—1072no—
230C22—4282no—
231C22—4282no—
232C22—2141no—
233C22—2141no—
234C22—2141no—
235C22—2141no—
236C22—2141no—
237C22C22⋊C4×D10711yescharacteristic
238C22#1681212811yescharacteristic
239C22#1681213011yescharacteristic
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—1072no—
245C4—4282no—
246C4—4281no—
247C4—4281no—
248C4—2141no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—4281no—
253C2—2142no—
254C2#99×D10711yescharacteristic
255C2#2029262011yescharacteristic
256C2#104×D10711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#327×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293137C2
cosocleC22×C4×C107C23
hypercenter#327D107
hypocenterC107#1501
LCSG ▷ C2×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. C22⋊C8×D107C2semidirect
3. #20292474C2semidirect
4. #20292365C2semidirect
5. #20292363C2semidirect
6. #20292361C2semidirect
7. #327×C107C2semidirect
8. C2×#13133817C4semidirect
12. #16812089C4semidirect
14. #16812080C4semidirect
18. #16812067C4semidirect
129. C107#1501semidirect
136. #327D107semidirect

Maximal subgroups

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

subgroupindex
#202931372
C22⋊C8×D1072
#202924742
#202923652
#202923632
#202923612
#327×C1072
#1501107