Finite Group Databasebeta

← #28344786

Subgroups — #28344786

Order 13696.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#16812093C211yesmaximal, characteristic
2C22×C4×Dic107C211yesmaximal, characteristic, complemented
3#20292517C212yesmaximal, complemented
4#20292516C212yesmaximal, complemented
5#20292514C212yesmaximal, complemented
6#20292513C212yesmaximal, complemented
7#20292506C211yesmaximal, characteristic
8#20292504C211yesmaximal, characteristic
9#20292499C211yesmaximal, characteristic, complemented
10#20292488C211yesmaximal, characteristic, complemented
11#385×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C23×Dic107C2211yescharacteristic
15#16812093—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4C2211yescharacteristic
21C2×(C2×D107)⋊C4C2211yescharacteristic
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×C4×Dic107—24no—
25C2×C4×Dic107C2212yes—
26C2×C4×Dic107C2212yes—
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C23×C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C2×C22⋊D107—22no—
39C2×C22⋊D107—22no—
40C2×C22⋊D107—22no—
41C22×Dic107—22no—
42C22×Dic107—22no—
43C22×Dic107—22no—
44C22×Dic107—24no—
45C22×Dic107D414yes—
46C22×Dic107C2312yes—
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2312yes—
51C4⋊Dic107—22no—
52C4⋊Dic107—22no—
53Dic107⋊C4—24no—
54C22⋊Dic107—22no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—24no—
60C4×Dic107—24no—
61C24×C107C2311yescharacteristic
62C22×C4×C107—22no—
63C22×C4×C107—24no—
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2312yes—
67C22×C4×C107C2312yes—
68C4⋊C4×C107—24no—
69C22⋊C4×C107—22no—
70C22⋊C4×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107C2×D412yes—
74C23×C107#4112yes—
75C23×C107#4112yes—
76C23×C107C2411yescharacteristic
77C22×D107—21no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—21no—
81C22×D107—21no—
82C2×C4×C107—28no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—21no—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107#4114yes—
93C22⋊D107—44no—
94C2×Dic107—28no—
95C2×Dic107—28no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107C2×D412yes—
109C2×Dic107C2×D412yes—
110C2×Dic107#4114yes—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107—24no—
114C22×C107—24no—
115C22×C107#12314yes—
116C22×C107C22×D411yescharacteristic
117C22×C107C2×#4112yes—
118C22×C107C2×#4112yes—
119C22×C107C2×#4111yescharacteristic
120C22×C107C2×#4111yescharacteristic
121C2×D107—41no—
122C2×D107—41no—
123C2×D107—41no—
124C2×D107—42no—
125C2×D107—42no—
126C4×C107—24no—
127C4×C107—41no—
128C4×C107—41no—
129C4×C107—42no—
130Dic107—22no—
131Dic107—22no—
132Dic107—24no—
133Dic107—41no—
134Dic107—42no—
135C2×C107—24no—
136C2×C107C2×#12311yescharacteristic
137C2×C107#52511yescharacteristic
138C2×C107#52511yescharacteristic
139C2×C107#52912yes—
140C2×C107#52912yes—
141D107—81no—
142#2853—1071nomaximal, Sylow
143C107#285311yescharacteristic, complemented, Sylow
144C2×#123—1071no—
145C22×C42—1071no—
146#388—1072no—
147#388—1072no—
148#387—1072no—
149#387—1072no—
150#386—1071no—
151#386—1071no—
152#385—1071no—
153#385D10711yescharacteristic, complemented
154#381—1071no—
155C22×D4—1071no—
156C23×C4—1071no—
157C23×C4—1071no—
158C23×C4C2×D10711yescharacteristic
159#123—2144no—
160C2×C4⋊C4—1071no—
161C2×C4⋊C4—1071no—
162C2×C4⋊C4—1072no—
163C2×C4⋊C4C2×D10712yes—
164C2×C22⋊C4—1071no—
165C2×C22⋊C4—1071no—
166C2×C22⋊C4—1071no—
167C2×C22⋊C4—1072no—
168C2×C22⋊C4—1072no—
169C2×C22⋊C4C2×D10711yescharacteristic
170C2×C22⋊C4C2×D10711yescharacteristic
171C2×C42—1072no—
172C2×C42—1072no—
173C2×C42—2144no—
174#95—1072no—
175#95—1072no—
176#95—1072no—
177#95—1072no—
178#95—1072no—
179#95C2×D10712yes—
180C24—1071no—
181C24C22×D10711yescharacteristic
182C2×D4—2142no—
183C2×D4—2142no—
184C2×D4—2142no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—1074no—
188C22×C4—1071no—
189C22×C4—1071no—
190C22×C4—1072no—
191C22×C4—1072no—
192C22×C4—1071no—
193C22×C4—2144no—
194C22×C4—2142no—
195C22×C4—2142no—
196C22×C4—2142no—
197C22×C4C22×D10712yes—
198C22×C4C22×D10711yescharacteristic
199C22×C4C22×D10711yescharacteristic
200C22×C4C22×D10712yes—
201C4⋊C4—24no—
202C4⋊C4—2142no—
203C4⋊C4—2142no—
204C4⋊C4—2144no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C22⋊C4—2144no—
210C22⋊C4—2142no—
211C22⋊C4—2144no—
212C42—2144no—
213C42—2144no—
214C23—22no—
215C23—22no—
216C23—2141no—
217C23—2142no—
218C23—2142no—
219C23—2141no—
220C23—2141no—
221C23D4×D10712yes—
222C23#1313381812yes—
223C23#1313381912yes—
224C23C23×D10711yescharacteristic
225D4—4284no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—1072no—
237C2×C4—1072no—
238C2×C4—1074no—
239C2×C4—2141no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4—2142no—
244C2×C4—2141no—
245C2×C4—2148no—
246C2×C4—2148no—
247C2×C4—2142no—
248C2×C4—2142no—
249C2×C4—2141no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2141no—
253C2×C4#1313381714yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—4281no—
259C22—4281no—
260C22—4281no—
261C22—4282no—
262C22—4282no—
263C22#1681209414yes—
264C22C2×D4×D10711yescharacteristic
265C22#41×D10712yes—
266C22C2×#1313381712yes—
267C22C2×#1313381811yescharacteristic
268C22C2×#1313381911yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—4281no—
274C4—4282no—
275C4—2142no—
276C4—2142no—
277C4—2144no—
278C2—24no—
279C2—8561no—
280C2C2×#1681209411yescharacteristic
281C2#2029313311yescharacteristic
282C2#2029313511yescharacteristic
283C2#2029315712yes—
284C2#2029316312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#385D107
hypocenterC107#2853
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic107C2semidirect
3. #20292517C2semidirect
4. #20292516C2semidirect
5. #20292514C2semidirect
6. #20292513C2semidirect
9. #20292499C2semidirect
10. #20292488C2semidirect
11. #385×C107C2semidirect
143. C107#2853semidirect
153. #385D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120932
C22×C4×Dic1072
#202925172
#202925162
#202925142
#202925132
#202925062
#202925042
#202924992
#202924882
#385×C1072
#2853107