Finite Group Databasebeta

← #28340118

Subgroups — #28340118

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×#16812098C211yesmaximal, characteristic, complemented
3C2×#16812097C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic
5#20292540C214yesmaximal, complemented
6#20292532C211yesmaximal, characteristic, complemented
7#20292505C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2212yes—
11C22×Dic214C2211yescharacteristic
12C2×Q8×D107—24no—
13C2×Q8×D107—24no—
14C2×Q8×D107—24no—
15C22×D428C2211yescharacteristic
16C22×D428C2212yes—
17C22×C4×D107C2212yes—
18C22×C4×D107C2211yescharacteristic
19#16812098—24no—
20#16812097—24no—
21#16812084—24no—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2214yes—
24C2×(C2×D107)⋊C4C2214yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2212yes—
28C107⋊#95C2214yes—
29C22×Q8×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C42×C107C2211yescharacteristic
32#95×C107C2214yes—
33C23×D107C2312yes—
34C23×D107C2311yescharacteristic
35C2×Dic214—22no—
36C2×Dic214—24no—
37C2×Dic214—22no—
38C2×Dic214—24no—
39C2×Dic214—24no—
40C2×Dic214D414yescomplemented
41Q8×D107—232no—
42C2×D428—24no—
43C2×D428—28no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428—22no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C2×C4×D107—24no—
51C2×C4×D107—22no—
52C2×C4×D107—24no—
53C2×C4×D107—22no—
54C2×C4×D107—24no—
55C2×C4×D107D414yes—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—28no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—28no—
60C4×Dic107—22no—
61C2×Q8×C107—22no—
62C2×Q8×C107—22no—
63C2×Q8×C107D414yescomplemented
64C22×C4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C42×C107—22no—
70C23×C107C2411yescharacteristic
71C22×D107—21no—
72C22×D107—21no—
73C22×D107—22no—
74C22×D107—22no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—24no—
78C22×D107—24no—
79C22×D107C2×D412yes—
80C22×D107#4114yes—
81Q8×C107—28no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—24no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×C4×C107C2×D414yes—
92C2×C4×C107C2×D412yes—
93C2×Dic107—22no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107C2×D414yes—
101C2×Dic107C2×D412yes—
102D428—48no—
103D428—44no—
104C4×D107—216no—
105C4×D107—28no—
106Dic214—216no—
107Dic214—28no—
108C22×C107C22×D411yescharacteristic
109C22×C107C22×D411yescharacteristic
110C22×C107C22×D411yescharacteristic
111C22×C107C2×#4112yes—
112C22×C1072−1+412yes—
113C2×D107—22no—
114C2×D107—22no—
115C2×D107—24no—
116C2×D107—24no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—44no—
121C2×D107—44no—
122C2×D107#12414yescomplemented
123C4×C107—24no—
124C4×C107—42no—
125C4×C107—42no—
126C4×C107#12014yes—
127Dic107—22no—
128Dic107—24no—
129Dic107—42no—
130C2×C107C2×#12011yescharacteristic
131C2×C107C2×#12411yescharacteristic
132C2×C107#53611yescharacteristic
133C2×C107#54812yes—
134C2×C107#54812yes—
135D107—24no—
136D107—82no—
137#2436—1071nomaximal, Sylow
138C107#243611yescharacteristic, complemented, Sylow
139C23×Q8—1071no—
140C2×#124—1071no—
141C2×#124—1071no—
142C2×#121—1071no—
143#393—1074no—
144#391—1071no—
145#391D10711yescharacteristic, complemented
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—1071no—
155C22×D4—1072no—
156C23×C4—1071no—
157C23×C4—1072no—
158#124—2144no—
159#124—2144no—
160#121—2144no—
161C2×C4⋊C4C2×D10711yescharacteristic
162C2×C22⋊C4—1072no—
163C2×C22⋊C4—1074no—
164C2×C22⋊C4—1074no—
165C2×C22⋊C4—1072no—
166C2×C42—1071no—
167C2×C42C2×D10711yescharacteristic
168#95—1072no—
169#95—1074no—
170#95C2×D10714yes—
171C24—1071no—
172C24—1072no—
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×Q8D42814yescomplemented
183C2×D4—2144no—
184C2×D4—2148no—
185C2×D4—2142no—
186C2×D4—2142no—
187C2×D4—2142no—
188C22×C4—1071no—
189C22×C4—1072no—
190C22×C4—1074no—
191C22×C4—1072no—
192C22×C4—2144no—
193C22×C4—2144no—
194C22×C4—2142no—
195C22×C4—2144no—
196C22×C4—2142no—
197C22×C4C22×D10712yes—
198C22×C4C22×D10712yes—
199C22×C4C22×D10712yes—
200C22×C4C22×D10711yescharacteristic
201C4⋊C4—22no—
202C22⋊C4—2144no—
203C22⋊C4—2148no—
204C22⋊C4—2144no—
205C22⋊C4—2148no—
206C42—22no—
207C42—2142no—
208C23—1072no—
209C23—1074no—
210C23—2142no—
211C23—2142no—
212C23—2142no—
213C23—2144no—
214C23—2144no—
215C23—2141no—
216C23—2141no—
217C23—2142no—
218C23C23×D10711yescharacteristic
219Q8—28no—
220Q8—21416no—
221Q8—2148no—
222D4—4288no—
223D4—4284no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—1072no—
232C2×C4—1074no—
233C2×C4—21416no—
234C2×C4—2142no—
235C2×C4—2142no—
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×D42814yes—
245C2×C4C2×D42812yes—
246C22—1074no—
247C22—4282no—
248C22—4282no—
249C22—4284no—
250C22—4284no—
251C22—4282no—
252C22—2142no—
253C22—2144no—
254C22—2144no—
255C22—2142no—
256C22C2×D4×D10711yescharacteristic
257C22C2×D4×D10711yescharacteristic
258C22C22×D42811yescharacteristic
259C22#41×D10712yes—
260C22#1681215712yes—
261C4—24no—
262C4—42no—
263C4—42no—
264C4—4282no—
265C4—2142no—
266C4—2144no—
267C4#1681207914yes—
268C2—8562no—
269C2—2144no—
270C2C2×#1681207911yescharacteristic
271C2#124×D10711yescharacteristic
272C2#2029320211yescharacteristic
273C2#2029328712yes—
274C2#2029328912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
prime radicalC2×#16812084C2
hypercenter#391D107
hypocenterC107#2436
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812098C2semidirect
3. C2×#16812097C2semidirect
5. #20292540C2semidirect
6. #20292532C2semidirect
7. #20292505C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #391×C107C2semidirect
40. C2×Dic214D4semidirect
63. C2×Q8×C107D4semidirect
122. C2×D107#124semidirect
138. C107#2436semidirect
145. #391D107semidirect
182. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168120982
C2×#168120972
C2×#168120842
#202925402
#202925322
#202925052
C4×(C2×D107)⋊C42
#391×C1072
#2436107