Finite Group Databasebeta

← #28340100

Subgroups — #28340100

Order 13696.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×Q8×D107C211yesmaximal, characteristic
2C2×#16812108C212yesmaximal, complemented
3C2×#16812087C211yesmaximal, characteristic
4#20292548C211yesmaximal, characteristic, complemented
5#20292547C211yesmaximal, characteristic, complemented
6#20292546C211yesmaximal, characteristic, complemented
7#20292531C212yesmaximal, complemented
8#20292501C212yesmaximal
9#20292498C212yesmaximal
10#20292474C211yesmaximal, characteristic
11#395×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C22×Dic214C2212yes—
15C2×Q8×D107—28no—
16C2×Q8×D107—24no—
17C22×C4×D107C2212yes—
18C22×C4×D107C2211yescharacteristic
19#16812108—28no—
20#16812087—24no—
21C2×C4⋊Dic107C2212yes—
22C2×Dic107⋊C4C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2212yes—
25C2×Dic107⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2212yes—
29C2×C4×Dic107C2212yes—
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107C2212yes—
37#95×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×Dic214—22no—
41C2×Dic214—24no—
42C2×Dic214—24no—
43C2×Dic214—24no—
44C2×Dic214—22no—
45C2×Dic214—24no—
46C2×Dic214D418yes—
47Q8×D107—232no—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2312yes—
51C22×Dic107C2312yes—
52C22×Dic107C2311yescharacteristic
53C2×C4×D107—24no—
54C2×C4×D107—24no—
55C2×C4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107D414yes—
58C4⋊Dic107—24no—
59Dic107⋊C4—22no—
60Dic107⋊C4—24no—
61Dic107⋊C4—22no—
62Dic107⋊C4—24no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4Q818yescomplemented
66C4×Dic107—24no—
67C2×Q8×C107—24no—
68C2×Q8×C107—22no—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2312yes—
74C4⋊C4×C107—22no—
75C4⋊C4×C107—24no—
76C23×C107C2411yescharacteristic
77C22×D107—22no—
78C22×D107—21no—
79C22×D107—21no—
80C22×D107C2×D412yes—
81C22×D107C2×Q814yes—
82Q8×C107—28no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
93C2×C4×C107C2×D414yes—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×Q814yescomplemented
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107C2×D414yes—
106C2×Dic107C2×D414yes—
107C2×Dic107C2×D412yes—
108C2×Dic107C2×Q814yescomplemented
109C4×D107—216no—
110C4×D107—28no—
111Dic214—28no—
112Dic214—216no—
113Dic214—44no—
114C22×C107C22×D411yescharacteristic
115C22×C107C22×D412yes—
116C22×C107C22×Q812yes—
117C22×C1072−1+411yescharacteristic
118C22×C1072−1+411yescharacteristic
119C2×D107—22no—
120C2×D107—22no—
121C2×D107—24no—
122C2×D107—24no—
123C2×D107C4⋊Q814yescomplemented
124C4×C107—22no—
125C4×C107—24no—
126C4×C107—24no—
127C4×C107—41no—
128C4×C107—41no—
129Dic107—24no—
130Dic107—24no—
131Dic107—41no—
132Dic107—41no—
133Dic107#12014yes—
134C2×C107C2×#12011yescharacteristic
135C2×C107C2×C4⋊Q811yescharacteristic
136C2×C107#53611yescharacteristic
137C2×C107D4×Q812yes—
138C2×C107D4×Q812yes—
139D107—24no—
140#2434—1071nomaximal, Sylow
141C107#243411yescharacteristic, complemented, Sylow
142C23×Q8—1071no—
143C2×C4⋊Q8—1072no—
144C2×C22⋊Q8—1071no—
145#395—1071no—
146#395—1071no—
147#395—1071no—
148#395D10711yescharacteristic, complemented
149#391—1072no—
150#385—1072no—
151#385—1072no—
152C4×C22⋊C4—1071no—
153C22×Q8—1072no—
154C22×Q8—1071no—
155C22×Q8—1071no—
156C22×Q8—2144no—
157C22×Q8—2148no—
158C22×Q8C2×D10711yescharacteristic
159C23×C4—1071no—
160C23×C4—1072no—
161C4⋊Q8—2148no—
162C22⋊Q8—2144no—
163C2×C4⋊C4—1071no—
164C2×C4⋊C4—1071no—
165C2×C4⋊C4—1072no—
166C2×C4⋊C4—1072no—
167C2×C4⋊C4—1072no—
168C2×C4⋊C4C2×D10711yescharacteristic
169C2×C4⋊C4C2×D10712yes—
170C2×C22⋊C4—1072no—
171C2×C22⋊C4—1071no—
172C2×C22⋊C4—1071no—
173C2×C42—1072no—
174#95—1071no—
175#95—1072no—
176#95—1072no—
177#95—1072no—
178#95C2×D10711yescharacteristic
179#95C2×D10712yes—
180C24—1071no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—1078no—
184C2×Q8—2142no—
185C2×Q8—2144no—
186C2×Q8—2144no—
187C2×Q8—2144no—
188C2×Q8—2144no—
189C2×Q8—21432no—
190C2×Q8—2142no—
191C22×C4—1071no—
192C22×C4—1072no—
193C22×C4—1072no—
194C22×C4—1074no—
195C22×C4—1071no—
196C22×C4—1071no—
197C22×C4—2144no—
198C22×C4—2144no—
199C22×C4—2144no—
200C22×C4—2144no—
201C22×C4C22×D10711yescharacteristic
202C22×C4C22×D10711yescharacteristic
203C22×C4C22×D10711yescharacteristic
204C22×C4C22×D10712yes—
205C22×C4C22×D10712yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—2142no—
209C4⋊C4—2144no—
210C4⋊C4—2144no—
211C4⋊C4—2142no—
212C4⋊C4—2144no—
213C22⋊C4—1078no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C42—2144no—
217C23—1072no—
218C23—1074no—
219C23—2142no—
220C23—2141no—
221C23—2141no—
222C23C23×D10711yescharacteristic
223Q8—28no—
224Q8—4284no—
225Q8—21416no—
226Q8—2148no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—1074no—
238C2×C4—1072no—
239C2×C4—1074no—
240C2×C4—1074no—
241C2×C4—2148no—
242C2×C4—21416no—
243C2×C4—2141no—
244C2×C4—2141no—
245C2×C4—2142no—
246C2×C4—2141no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4D4×D10712yes—
253C2×C4D4×D10714yes—
254C2×C4Q8×D10714yescomplemented
255C22—1074no—
256C22—2142no—
257C22—2144no—
258C22—2144no—
259C22—2142no—
260C22C2×D4×D10711yescharacteristic
261C22C2×D4×D10712yes—
262C22C2×Q8×D10712yes—
263C22#1681215611yescharacteristic
264C22#1681215711yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—1074no—
271C4—4281no—
272C4—4281no—
273C4—2144no—
274C4—2144no—
275C2—2144no—
276C2#120×D10711yescharacteristic
277C2C4⋊Q8×D10711yescharacteristic
278C2#2029320011yescharacteristic
279C2#2029329012yes—
280C2#2029329212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#395×C107C2
prime radicalC23×D107C23
hypercenter#395D107
hypocenterC107#2434
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#16812108C2semidirect
4. #20292548C2semidirect
5. #20292547C2semidirect
6. #20292546C2semidirect
7. #20292531C2semidirect
11. #395×C107C2semidirect
65. (C2×D107)⋊C4Q8semidirect
95. C2×C4×C107C2×Q8semidirect
108. C2×Dic107C2×Q8semidirect
123. C2×D107C4⋊Q8semidirect
141. C107#2434semidirect
148. #395D107semidirect
254. C2×C4Q8×D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168121082
C2×#168120872
#202925482
#202925472
#202925462
#202925312
#202925012
#202924982
#202924742
#395×C1072
#2434107