Finite Group Databasebeta

← #28340116

Subgroups — #28340116

Order 13696.

852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 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, complemented
2C2×#16812096C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic
4#20292539C214yesmaximal, complemented
5#20292534C211yesmaximal, characteristic, complemented
6#20292533C211yesmaximal, characteristic, complemented
7#20292503C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2212yes—
11C22×Dic214C2211yescharacteristic
12C2×Q8×D107—24no—
13C2×Q8×D107—24no—
14C2×Q8×D107—24no—
15C22×C22⋊D107C2212yes—
16C22×C4×D107C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812096—24no—
19#16812080—24no—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C22⋊Dic107C2214yes—
22C2×C22⋊Dic107C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×C4×Dic107C2211yescharacteristic
25C107⋊#95C2212yes—
26C107⋊#95C2214yes—
27C107⋊#95C2214yes—
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30#124×C107—24no—
31C2×C22⋊C4×C107C2214yes—
32C2×C42×C107C2211yescharacteristic
33C23×D107C2311yescharacteristic
34C2×Dic214—22no—
35C2×Dic214—24no—
36C2×Dic214—22no—
37C2×Dic214—24no—
38C2×Dic214—24no—
39C2×Dic214D414yescomplemented
40Q8×D107—232no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—28no—
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—24no—
48C2×C4×D107—22no—
49C2×C4×D107—24no—
50C2×C4×D107—22no—
51C2×C4×D107—24no—
52C2×C4×D107D414yes—
53C4⋊Dic107—22no—
54C22⋊Dic107—24no—
55C22⋊Dic107—28no—
56(C2×D107)⋊C4—24no—
57C4×Dic107—22no—
58C24×C107C2312yes—
59C2×Q8×C107—22no—
60C2×Q8×C107—22no—
61C2×Q8×C107D414yescomplemented
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107—22no—
65C22×C4×C107C2312yes—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C22⋊C4×C107—28no—
69C42×C107—22no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—24no—
74C23×C107—24no—
75C23×C107C2411yescharacteristic
76C22×D107—21no—
77C22×D107—21no—
78C22×D107—22no—
79C22×D107C2×D412yes—
80C22×D107#4114yes—
81Q8×C107—28no—
82D4×C107—44no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D414yes—
91C2×C4×C107C2×D412yes—
92C22⋊D107—48no—
93C2×Dic107—22no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—24no—
101C2×Dic107—22no—
102C2×Dic107C2×D414yes—
103C2×Dic107C2×D412yes—
104C4×D107—216no—
105C4×D107—28no—
106Dic214—216no—
107Dic214—28no—
108C22×C107—42no—
109C22×C107—42no—
110C22×C107—42no—
111C22×C107—44no—
112C22×C107—44no—
113C22×C107C22×D411yescharacteristic
114C22×C107C22×D411yescharacteristic
115C22×C107C22×D411yescharacteristic
116C22×C107C2×#4112yes—
117C22×C1072−1+412yes—
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—24no—
121C2×D107—24no—
122C2×D107#12414yescomplemented
123C4×C107—24no—
124C4×C107—42no—
125C4×C107#12014yes—
126Dic107—22no—
127Dic107—24no—
128Dic107—42no—
129Dic107—42no—
130C2×C107—82no—
131C2×C107C2×#12011yescharacteristic
132C2×C107C2×#12411yescharacteristic
133C2×C107#53611yescharacteristic
134C2×C107#54812yes—
135C2×C107#54812yes—
136D107—24no—
137#2436—1071nomaximal, Sylow
138C107#243611yescharacteristic, complemented, Sylow
139C23×Q8—1071no—
140C2×#124—1071no—
141C2×#124D10711yescharacteristic, complemented
142C2×#121—1071no—
143#393—1074no—
144#391—1071no—
145#391—1071no—
146#386—1074no—
147C4×C22⋊C4—1071no—
148C22×Q8—1071no—
149C22×Q8—1072no—
150C22×Q8—2144no—
151C22×Q8—2144no—
152C22×Q8—2144no—
153C22×Q8C2×D10711yescharacteristic
154C22×D4—1072no—
155C22×D4C2×D10711yescharacteristic
156C23×C4—1071no—
157C23×C4—1072no—
158#124—24no—
159#124—2144no—
160#121—2144no—
161C2×C4⋊C4—1071no—
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1074no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4C2×D10714yes—
166C2×C42—1071no—
167C2×C42C2×D10711yescharacteristic
168#95—1072no—
169#95—1074no—
170#95—1074no—
171C24—1071no—
172C24C22×D10712yes—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—1074no—
176C2×Q8—2144no—
177C2×Q8—2144no—
178C2×Q8—21432no—
179C2×Q8—2142no—
180C2×Q8—2144no—
181C2×Q8—2142no—
182C2×Q8C22⋊D10714yescomplemented
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—2144no—
187C2×D4—2148no—
188C22×C4—1071no—
189C22×C4—1072no—
190C22×C4—1074no—
191C22×C4—1072no—
192C22×C4—1072no—
193C22×C4—2144no—
194C22×C4—2144no—
195C22×C4—2142no—
196C22×C4—2144no—
197C22×C4—2142no—
198C22×C4C22×D10712yes—
199C22×C4C22×D10712yes—
200C22×C4C22×D10711yescharacteristic
201C4⋊C4—2142no—
202C22⋊C4—28no—
203C22⋊C4—2144no—
204C22⋊C4—2144no—
205C22⋊C4—2148no—
206C42—22no—
207C42—2142no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—24no—
212C23—24no—
213C23—1072no—
214C23—1074no—
215C23—2141no—
216C23—2141no—
217C23—2142no—
218C23C23×D10711yescharacteristic
219Q8—28no—
220Q8—21416no—
221Q8—2148no—
222D4—44no—
223D4—4288no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1072no—
230C2×C4—1074no—
231C2×C4—21416no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4—2142no—
235C2×C4—2144no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4—2142no—
241C2×C4—2148no—
242C2×C4D4×D10712yes—
243C2×C4D4×D10712yes—
244C2×C4C2×C22⋊D10714yes—
245C2×C4C2×C22⋊D10712yes—
246C22—42no—
247C22—42no—
248C22—44no—
249C22—44no—
250C22—42no—
251C22—1074no—
252C22—2142no—
253C22—2144no—
254C22—2144no—
255C22—2142no—
256C22C2×D4×D10711yescharacteristic
257C22C2×D4×D10711yescharacteristic
258C22C22×C22⋊D10711yescharacteristic
259C22#41×D10712yes—
260C22#1681215612yes—
261C4—24no—
262C4—42no—
263C4—4282no—
264C4—4282no—
265C4—2142no—
266C4—2144no—
267C4#1681207714yes—
268C2—82no—
269C2—2144no—
270C2C2×#1681207711yescharacteristic
271C2#124×D10711yescharacteristic
272C2#2029319711yescharacteristic
273C2#2029328412yes—
274C2#2029328512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#124×C107C2
prime radicalC2×#16812080C2
hypercenterC2×#124D107
hypocenterC107#2436
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812096C2semidirect
4. #20292539C2semidirect
5. #20292534C2semidirect
6. #20292533C2semidirect
7. #20292503C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. C2×#124×C107C2semidirect
39. C2×Dic214D4semidirect
61. C2×Q8×C107D4semidirect
122. C2×D107#124semidirect
138. C107#2436semidirect
141. C2×#124D107semidirect
182. C2×Q8C22⋊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×#168120962
C2×#168120802
#202925392
#202925342
#202925332
#202925032
C4×(C2×D107)⋊C42
C2×#124×C1072
#2436107