Finite Group Databasebeta

← #28339743

Subgroups — #28339743

Order 13696.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic107C211yesmaximal, characteristic, complemented
2C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
3#20292569C212yesmaximal, complemented
4#20292554C211yesmaximal, characteristic, complemented
5#20292552C211yesmaximal, characteristic, complemented
6#20292543C212yesmaximal
7#20292538C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic107C212yesmaximal, complemented
9#20292474C212yesmaximal, complemented
10#20292472C211yesmaximal, characteristic, complemented
11#393×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15Q8×Dic107—24no—
16(C22⋊D107)⋊C4—24no—
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4C2211yescharacteristic
20C2×C22⋊Dic107C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×C4×Dic107C2212yes—
25C2×C4×Dic107C2212yes—
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2211yescharacteristic
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×Q8×C107C2211yescharacteristic
33C2×C22⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35#95×C107C2211yescharacteristic
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C2×C22⋊D107—22no—
39C2×C22⋊D107—22no—
40C2×C22⋊D107—22no—
41C22×Dic107—22no—
42C22×Dic107—24no—
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C2×C4×D107—22no—
49C2×C4×D107—24no—
50C4⋊Dic107—22no—
51C4⋊Dic107—24no—
52Dic107⋊C4—22no—
53C22⋊Dic107—24no—
54(C2×D107)⋊C4—24no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—24no—
57C4×Dic107—24no—
58C4×Dic107—24no—
59C4×Dic107D414yes—
60C24×C107C2311yescharacteristic
61C2×Q8×C107—22no—
62C2×Q8×C107—24no—
63C22×C4×C107C2312yes—
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C22×C4×C107C2312yes—
67C22⋊C4×C107—22no—
68C22⋊C4×C107—24no—
69C23×C107—21no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—21no—
73C23×C107#4112yes—
74C23×C107C2411yescharacteristic
75C22×D107—21no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—21no—
79C22×D107#4112yes—
80Q8×C107—28no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—21no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107#4114yes—
91C22⋊D107—28no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—44no—
103C2×Dic107C2×D412yes—
104C2×Dic107C2×D412yes—
105C2×Dic107#4114yes—
106C4×D107—44no—
107C22×C107—22no—
108C22×C107—24no—
109C22×C107—41no—
110C22×C107—41no—
111C22×C107—42no—
112C22×C107C22×D411yescharacteristic
113C22×C107C2×#4111yescharacteristic
114C22×C107C2×#4112yes—
115C22×C107C2×#4112yes—
116C22×C107C2×#4111yescharacteristic
117C2×D107—22no—
118C2×D107—24no—
119C2×D107—41no—
120C2×D107—41no—
121C2×D107—42no—
122C4×C107—22no—
123C4×C107—24no—
124C4×C107—41no—
125C4×C107—42no—
126Dic107—22no—
127Dic107—42no—
128Dic107—41no—
129Dic107—42no—
130Dic107#12414yes—
131C2×C107—42no—
132C2×C107#52511yescharacteristic
133C2×C107#52511yescharacteristic
134C2×C107C2×#12411yescharacteristic
135C2×C107#52912yes—
136C2×C107#52912yes—
137D107—42no—
138#2398—1071nomaximal, Sylow
139C107#239811yescharacteristic, complemented, Sylow
140C2×C4×Q8—1071no—
141C2×C4×D4—1071no—
142#400—1072no—
143#397—1071no—
144#397—1071no—
145#394—1072no—
146#393—1071no—
147#393D10711yescharacteristic, complemented
148C4×C22⋊C4—1072no—
149C4×C22⋊C4—1072no—
150#376—1071no—
151C22×Q8C2×D10711yescharacteristic
152C22×D4—1071no—
153C23×C4—1071no—
154C23×C4—1071no—
155C4×Q8—2144no—
156C4×D4—2144no—
157C2×C4⋊C4—1071no—
158C2×C4⋊C4—1072no—
159C2×C4⋊C4—1071no—
160C2×C22⋊C4—1072no—
161C2×C22⋊C4—1071no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4C2×D10711yescharacteristic
165C2×C22⋊C4C2×D10712yes—
166C2×C42—1071no—
167C2×C42—1072no—
168C2×C42—1072no—
169#95—1071no—
170#95—1072no—
171#95—1072no—
172#95—1072no—
173#95—1072no—
174#95C2×D10711yescharacteristic
175#95C2×D10712yes—
176C24—1071no—
177C24C22×D10711yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—2142no—
181C2×D4—2142no—
182C2×D4—2142no—
183C22×C4—1071no—
184C22×C4—1071no—
185C22×C4—1072no—
186C22×C4—1071no—
187C22×C4—1072no—
188C22×C4—2142no—
189C22×C4—2144no—
190C22×C4—2142no—
191C22×C4—2144no—
192C22×C4C22×D10712yes—
193C22×C4C22×D10711yescharacteristic
194C22×C4C22×D10711yescharacteristic
195C22×C4C22×D10712yes—
196C4⋊C4—2142no—
197C4⋊C4—2144no—
198C4⋊C4—2142no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—2144no—
202C22⋊C4—2144no—
203C22⋊C4—2142no—
204C22⋊C4—2144no—
205C42—1074no—
206C42—2144no—
207C42—2144no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—1072no—
213C23—2141no—
214C23—2142no—
215C23—2142no—
216C23—2141no—
217C23#1313381912yes—
218C23C23×D10711yescharacteristic
219Q8—28no—
220D4—2148no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1072no—
230C2×C4—1074no—
231C2×C4—1072no—
232C2×C4—4284no—
233C2×C4—4284no—
234C2×C4—2142no—
235C2×C4—2141no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2141no—
243C2×C4—2141no—
244C2×C4D4×D10712yes—
245C2×C4#1313381714yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—4281no—
252C22—4281no—
253C22—4282no—
254C22—2142no—
255C22—2144no—
256C22C2×D4×D10711yescharacteristic
257C22#41×D10711yescharacteristic
258C22#41×D10712yes—
259C22C2×#1313381712yes—
260C22C2×#1313381911yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—1074no—
266C4—4282no—
267C4—4281no—
268C4—4282no—
269C4—2142no—
270C2—42no—
271C2—4282no—
272C2#2029313311yescharacteristic
273C2#2029314011yescharacteristic
274C2#124×D10711yescharacteristic
275C2#2029315612yes—
276C2#2029316312yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#393×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#393D107
hypocenterC107#2398
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic107C2semidirect
3. #20292569C2semidirect
4. #20292554C2semidirect
5. #20292552C2semidirect
7. #20292538C2semidirect
8. C22⋊C4×Dic107C2semidirect
9. #20292474C2semidirect
10. #20292472C2semidirect
11. #393×C107C2semidirect
139. C107#2398semidirect
147. #393D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1072
C2×(C22⋊D107)⋊C42
#202925692
#202925542
#202925522
#202925432
#202925382
C22⋊C4×Dic1072
#202924742
#202924722
#393×C1072
#2398107