Finite Group Databasebeta

← #28335016

Subgroups — #28335016

Order 13696.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#20293140C211yesmaximal, characteristic
2C22⋊C8×D107C211yesmaximal, characteristic, complemented
3#20292489C211yesmaximal, characteristic, complemented
4#20292364C211yesmaximal, characteristic, complemented
5#20292362C211yesmaximal, characteristic, complemented
6#20292360C211yesmaximal, characteristic, complemented
7#328×C107C211yesmaximal, characteristic, complemented
8C2×#13133819C411yescharacteristic, complemented
9C22×C4×D107C2211yescharacteristic
10C2×C8×D107—21no—
11#16812094—21no—
12#16812085C411yescharacteristic, complemented
13#16812084C411yescharacteristic, complemented
14#16812083C2211yescharacteristic
15#16812079—21no—
16#16812071—21no—
17(C22⋊D107)⋊C4—21no—
18#16812067C411yescharacteristic, complemented
19C2×(C2×D107)⋊C4—21no—
20C22⋊(C107⋊C8)C2211yescharacteristic
21(C2×D107)⋊C8—21no—
22C107⋊#95—21no—
23C107⋊#95C2211yescharacteristic
24C22⋊Q8×C107C2211yescharacteristic
25C22⋊C8×C107C2211yescharacteristic
26#95×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28#13133819—42no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428C2×C411yescharacteristic
32C2×C22⋊D107—21no—
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C2×C4×D107—21no—
36C2×C4×D107—42no—
37C2×C4×D107C2×C411yescharacteristic
38C2×C4×D107C2×C411yescharacteristic
39C2×C4×D107D411yescharacteristic
40C2×C4×D107D411yescharacteristic
41C2×C107⋊C8—21no—
42C8×D107—42no—
43C4⋊Dic107C2×C411yescharacteristic
44Dic107⋊C4—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—
50C4×Dic107—21no—
51C2×Q8×C107C2×C411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54C2×C8×C107—21no—
55C4⋊C4×C107—21no—
56C4⋊C4×C107C2×C411yescharacteristic
57C22⋊C4×C107—21no—
58C23×C107C2×D411yescharacteristic
59C22×D107—21no—
60C22×D107—21no—
61C22×D107—22no—
62C22×D107—21no—
63C22×D107—41no—
64C22×D107C22⋊C411yescharacteristic
65C22×D107C22⋊C411yescharacteristic
66Q8×C107—41no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C22⋊D107—41no—
76C22⋊D107—41no—
77C2×Dic107—21no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107—41no—
82C2×Dic107C22⋊C411yescharacteristic
83C2×Dic107C22⋊C411yescharacteristic
84D428—41no—
85D428—42no—
86D428—42no—
87C4×D107—22no—
88C4×D107—22no—
89C4×D107—41no—
90C4×D107—41no—
91C4×D107—42no—
92C8×C107—41no—
93C107⋊C8—41no—
94C22×C107—21no—
95C22×C107—21no—
96C22×C107C2×C22⋊C411yescharacteristic
97C22×C107#13511yescharacteristic
98C22×C107#13511yescharacteristic
99C2×D107—21no—
100C2×D107—21no—
101C2×D107—21no—
102C2×D107—21no—
103C2×D107—21no—
104C2×D107—42no—
105C2×D107—42no—
106C2×D107—41no—
107C2×D107—41no—
108C2×D107—41no—
109C2×D107#9912yes—
110C4×C107—21no—
111C4×C107—21no—
112C4×C107—41no—
113C4×C107—41no—
114C4×C107—81no—
115Dic107—21no—
116Dic107—21no—
117Dic107—41no—
118Dic107—81no—
119C2×C107—21no—
120C2×C107—21no—
121C2×C107C2×#9911yescharacteristic
122C2×C107#41611yescharacteristic
123C2×C107#42111yescharacteristic
124D107—22no—
125D107—41no—
126D107—81no—
127#1516—1071nomaximal, Sylow
128C107#151611yescharacteristic, complemented, Sylow
129#525—1071no—
130C2×C22⋊C8—1071no—
131#381—1071no—
132#328—1071no—
133#328D10711yescharacteristic, complemented
134#327—1071no—
135#327—1071no—
136C2×#41—1071no—
137C23×C4—1071no—
138C22×C8—2141no—
139#123—2141no—
140C22⋊Q8—1071no—
141C22⋊Q8C2×D10711yescharacteristic
142#121—1071no—
143#121—1071no—
144#120—2141no—
145C4×D4—2141no—
146C4×D4—2141no—
147#117—1071no—
148C2×C22⋊C4—2141no—
149C22⋊C8—1071no—
150C22⋊C8—2141no—
151C22⋊C8C2×D10711yescharacteristic
152#95—1071no—
153#95—2141no—
154#95C2×D10711yescharacteristic
155C24—1071no—
156#41—4282no—
157C2×Q8C4×D10711yescharacteristic
158C2×D4—1071no—
159C2×D4—2141no—
160C2×D4—2141no—
161C2×D4—2141no—
162C22×C4—21no—
163C22×C4—1071no—
164C22×C4—1071no—
165C22×C4—1071no—
166C22×C4—1071no—
167C22×C4—1071no—
168C22×C4—4282no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4C22×D10711yescharacteristic
172C2×C8—21no—
173C2×C8—4282no—
174C2×C8—2141no—
175C4⋊C4—21no—
176C4⋊C4—1071no—
177C4⋊C4—2141no—
178C4⋊C4C4×D10711yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—4282no—
181C22⋊C4—2141no—
182C22⋊C4—2141no—
183C22⋊C4—2141no—
184C22⋊C4—2141no—
185C42—2141no—
186C23—1071no—
187C23—1071no—
188C23—4281no—
189C23—2142no—
190C23—2141no—
191C23—2141no—
192C23—2141no—
193C23D4×D10711yescharacteristic
194Q8—41no—
195D4—4281no—
196D4—4282no—
197D4—4282no—
198D4—4281no—
199D4—4281no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—1071no—
207C2×C4—1071no—
208C2×C4—4281no—
209C2×C4—4281no—
210C2×C4—4282no—
211C2×C4—4281no—
212C2×C4—4281no—
213C2×C4—4281no—
214C2×C4—4281no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2141no—
218C2×C4C2×C4×D10711yescharacteristic
219C2×C4D4×D10711yescharacteristic
220C8—41no—
221C8—4281no—
222C22—21no—
223C22—21no—
224C22—1072no—
225C22—4282no—
226C22—4282no—
227C22—4281no—
228C22—4281no—
229C22—4281no—
230C22—2141no—
231C22—2141no—
232C22—2141no—
233C22—2141no—
234C22—2141no—
235C22C22⋊C4×D10711yescharacteristic
236C22#1681213011yescharacteristic
237C22#1681213211yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—81no—
243C4—8561no—
244C4—4281no—
245C4—2141no—
246C4—2141no—
247C2—21no—
248C2—21no—
249C2—8561no—
250C2—4281no—
251C2—2142no—
252C2#99×D10711yescharacteristic
253C2#2029262211yescharacteristic
254C2#2029264211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#328×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293140C2
cosocleC22×C4×C107C23
hypercenter#328D107
hypocenterC107#1516
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D107C2semidirect
3. #20292489C2semidirect
4. #20292364C2semidirect
5. #20292362C2semidirect
6. #20292360C2semidirect
7. #328×C107C2semidirect
8. C2×#13133819C4semidirect
12. #16812085C4semidirect
13. #16812084C4semidirect
18. #16812067C4semidirect
128. C107#1516semidirect
133. #328D107semidirect

Maximal subgroups

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

subgroupindex
#202931402
C22⋊C8×D1072
#202924892
#202923642
#202923622
#202923602
#328×C1072
#1516107