Finite Group Databasebeta

← #28344765

Subgroups — #28344765

Order 13696.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812098C211yesmaximal, characteristic, complemented
2C2×#16812085C211yesmaximal, characteristic, complemented
3C2×#16812084C211yesmaximal, characteristic
4C2×C42×D107C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic, complemented
6#20292516C214yesmaximal, complemented
7#20292505C214yesmaximal, complemented
8#20292492C211yesmaximal, characteristic, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×D428C2212yes—
12C22×C4×D107C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812098—24no—
15#16812085—24no—
16#16812084—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×Dic107⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2214yes—
22C2×(C2×D107)⋊C4C2214yes—
23C2×C4×Dic107C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C42×D107—24no—
26C107⋊#95C2214yes—
27C22×Q8×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2214yes—
31C23×D107C2312yes—
32C23×D107C2311yescharacteristic
33C2×D428—24no—
34C2×D428—28no—
35C2×D428—22no—
36C2×D428—22no—
37C2×D428—22no—
38C22×Dic107C2312yes—
39C22×Dic107C2311yescharacteristic
40C22×Dic107C2312yes—
41C2×C4×D107—24no—
42C2×C4×D107—22no—
43C2×C4×D107—22no—
44C2×C4×D107—24no—
45C2×C4×D107—24no—
46C2×C4×D107D414yes—
47C4⋊Dic107—22no—
48Dic107⋊C4—24no—
49(C2×D107)⋊C4—24no—
50(C2×D107)⋊C4—28no—
51(C2×D107)⋊C4—24no—
52(C2×D107)⋊C4—28no—
53C4×Dic107—24no—
54C4×Dic107D414yescomplemented
55C2×Q8×C107—22no—
56C2×Q8×C107—22no—
57C2×Q8×C107—22no—
58C22×C4×C107C2312yes—
59C22×C4×C107C2312yes—
60C22×C4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62C4⋊C4×C107—22no—
63C42×C107—22no—
64C23×C107C2411yescharacteristic
65C22×D107—21no—
66C22×D107—21no—
67C22×D107—22no—
68C22×D107—22no—
69C22×D107—22no—
70C22×D107—24no—
71C22×D107—24no—
72C22×D107—22no—
73C22×D107C2×D412yes—
74C22×D107#4114yes—
75Q8×C107—44no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—24no—
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D412yes—
85C2×C4×C107#4112yes—
86C2×C4×C107#4114yes—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—24no—
93C2×Dic107C2×D414yes—
94C2×Dic107C2×D412yes—
95D428—48no—
96D428—44no—
97C4×D107—216no—
98C4×D107—28no—
99C22×C107C22×D411yescharacteristic
100C22×C107C22×D411yescharacteristic
101C22×C107C2×#4112yes—
102C22×C107C2×#4111yescharacteristic
103C22×C107C2×#4112yes—
104C2×D107—22no—
105C2×D107—24no—
106C2×D107—24no—
107C2×D107—22no—
108C2×D107—42no—
109C2×D107—44no—
110C2×D107—42no—
111C2×D107—44no—
112C2×D107—42no—
113C2×D107#12414yescomplemented
114C4×C107—24no—
115C4×C107—42no—
116C4×C107—42no—
117C4×C107#12114yes—
118Dic107—22no—
119Dic107—24no—
120Dic107—42no—
121C2×C107C2×#12111yescharacteristic
122C2×C107#52512yes—
123C2×C107C2×#12411yescharacteristic
124C2×C107#52912yes—
125C2×C107#53211yescharacteristic
126D107—24no—
127D107—82no—
128#2850—1071nomaximal, Sylow
129C107#285011yescharacteristic, complemented, Sylow
130C2×#124—1071no—
131C2×C22⋊Q8—1071no—
132C2×#121—1071no—
133C22×C42—1071no—
134#391D10711yescharacteristic, complemented
135#390—1071no—
136#388—1074no—
137#386—1074no—
138#382—1071no—
139C22×Q8C2×D10711yescharacteristic
140C22×D4—1071no—
141C22×D4—1072no—
142C23×C4—1071no—
143C23×C4—1072no—
144#124—2144no—
145C22⋊Q8—2144no—
146#121—2144no—
147C2×C4⋊C4—1072no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—1072no—
151C2×C22⋊C4—1072no—
152C2×C22⋊C4—1074no—
153C2×C22⋊C4—1074no—
154C2×C42—1071no—
155C2×C42—1072no—
156C2×C42—2144no—
157C2×C42C2×D10711yescharacteristic
158#95—1074no—
159#95C2×D10714yes—
160C24—1071no—
161C24—1072no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—2144no—
166C2×D4—2148no—
167C2×D4—2142no—
168C2×D4—2142no—
169C2×D4—2142no—
170C22×C4—1072no—
171C22×C4—1071no—
172C22×C4—1074no—
173C22×C4—1072no—
174C22×C4—2144no—
175C22×C4—2144no—
176C22×C4—2142no—
177C22×C4—2142no—
178C22×C4—2144no—
179C22×C4C22×D10712yes—
180C22×C4C22×D10712yes—
181C22×C4C22×D10712yes—
182C22×C4C22×D10711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—2144no—
185C4⋊C4—2142no—
186C22⋊C4—2144no—
187C22⋊C4—2148no—
188C22⋊C4—2144no—
189C22⋊C4—2148no—
190C42—22no—
191C42—1074no—
192C42—2144no—
193C23—1072no—
194C23—1074no—
195C23—2142no—
196C23—2142no—
197C23—2144no—
198C23—2142no—
199C23—2144no—
200C23—2141no—
201C23—2141no—
202C23—2142no—
203C23C23×D10711yescharacteristic
204Q8—44no—
205D4—4288no—
206D4—4284no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—1074no—
215C2×C4—1072no—
216C2×C4—21416no—
217C2×C4—2142no—
218C2×C4—2144no—
219C2×C4—2141no—
220C2×C4—2142no—
221C2×C4—2141no—
222C2×C4—2142no—
223C2×C4—2148no—
224C2×C4D4×D10712yes—
225C2×C4D4×D10712yes—
226C2×C4#1313381812yes—
227C2×C4#1313381914yes—
228C22—1074no—
229C22—4284no—
230C22—4282no—
231C22—4284no—
232C22—4282no—
233C22—4282no—
234C22—2144no—
235C22—2144no—
236C22—2142no—
237C22—2142no—
238C22C2×D4×D10711yescharacteristic
239C22C2×D4×D10711yescharacteristic
240C22#41×D10712yes—
241C22C2×#1313381811yescharacteristic
242C22C2×#1313381912yes—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—4282no—
247C4—2144no—
248C4—2142no—
249C4#1681208214yes—
250C2—8562no—
251C2—2144no—
252C2C2×#1681208211yescharacteristic
253C2#2029314012yes—
254C2#124×D10711yescharacteristic
255C2#2029316412yes—
256C2#2029317811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
prime radicalC2×#16812084C2
hypercenter#391D107
hypocenterC107#2850
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16812098C2semidirect
2. C2×#16812085C2semidirect
4. C2×C42×D107C2semidirect
5. #20292529C2semidirect
6. #20292516C2semidirect
7. #20292505C2semidirect
8. #20292492C2semidirect
9. #391×C107C2semidirect
54. C4×Dic107D4semidirect
113. C2×D107#124semidirect
129. C107#2850semidirect
134. #391D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120982
C2×#168120852
C2×#168120842
C2×C42×D1072
#202925292
#202925162
#202925052
#202924922
#391×C1072
#2850107