Finite Group Databasebeta

← #28339052

Subgroups — #28339052

Order 13696.

880 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 229 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×#13133819C211yesmaximal, characteristic
2C2×#16812066C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292534C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292505C214yesmaximal, complemented
8#20292486C214yesmaximal, complemented
9#391×C107C211yesmaximal, characteristic, complemented
10C2×#13133819—24no—
11C2×#13133819—24no—
12C2×#13133819C418yescomplemented
13C22×D428C2212yes—
14C22×D428C2211yescharacteristic
15C22×C4×D107C2212yes—
16C22×C4×D107C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812066—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C4⋊C4×D107—24no—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2214yes—
25C2×(C2×D107)⋊C4C2214yes—
26C2×C4×Dic107C2211yescharacteristic
27C107⋊#95C2214yes—
28C22×Q8×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2211yescharacteristic
30C2×C42×C107C2211yescharacteristic
31#95×C107C2214yes—
32C23×D107C2312yes—
33C23×D107C2311yescharacteristic
34#13133819—232no—
35C2×D428—24no—
36C2×D428—24no—
37C2×D428—22no—
38C2×D428—22no—
39C2×D428C2×C418yes—
40C2×D428C2×C414yescomplemented
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C2×C4×D107—24no—
45C2×C4×D107—24no—
46C2×C4×D107—24no—
47C2×C4×D107—44no—
48C2×C4×D107C2×C414yes—
49C2×C4×D107C2×C418yescomplemented
50C2×C4×D107D414yes—
51C2×C4×D107D414yes—
52C4⋊Dic107—22no—
53Dic107⋊C4—24no—
54Dic107⋊C4—24no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—44no—
57(C2×D107)⋊C4—44no—
58C4×Dic107—22no—
59C2×Q8×C107—22no—
60C2×Q8×C107—22no—
61C2×Q8×C107C2×C414yescomplemented
62C22×C4×C107C2312yes—
63C22×C4×C107C2312yes—
64C22×C4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—22no—
67C42×C107—22no—
68C23×C107C2411yescharacteristic
69C22×D107—22no—
70C22×D107—22no—
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—24no—
74C22×D107—24no—
75C22×D107C22×C414yes—
76C22×D107C22×C412yes—
77C22×D107C2×D412yes—
78C22×D107C2×D412yes—
79Q8×C107—28no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—24no—
86C2×C4×C107—22no—
87C2×C4×C107C22×C414yes—
88C2×C4×C107C22×C412yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D412yes—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—24no—
95C2×Dic107—22no—
96C2×Dic107C22×C412yes—
97C2×Dic107C2×D412yes—
98C2×Dic107C2×D412yes—
99D428—216no—
100D428—28no—
101C4×D107—216no—
102C4×D107—48no—
103C4×D107C22⋊C4116yescomplemented
104C22×C107C23×C411yescharacteristic
105C22×C107C22×D411yescharacteristic
106C22×C107C22×D411yescharacteristic
107C22×C107H(5,𝔽2)12yes—
108C22×C1072−1+412yes—
109C2×D107—24no—
110C2×D107—22no—
111C2×D107—22no—
112C2×D107—24no—
113C2×D107—24no—
114C2×D107—28no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—44no—
118C2×D107C2×C22⋊C414yes—
119C4×C107—24no—
120C4×C107—42no—
121C4×C107—42no—
122C4×C107C2×C22⋊C414yes—
123Dic107—42no—
124Dic107—42no—
125Dic107C2×C22⋊C414yes—
126C2×C107C22×C22⋊C411yescharacteristic
127C2×C107#52012yes—
128C2×C107#53511yescharacteristic
129C2×C107#53611yescharacteristic
130C2×C107#53712yes—
131D107—24no—
132D107—44no—
133#2311—1071nomaximal, Sylow
134C107#231111yescharacteristic, complemented, Sylow
135C22×#41—1071no—
136C2×#117—1071no—
137C22×C4⋊C4—1071no—
138#391—1071no—
139#391D10711yescharacteristic, complemented
140#390—1071no—
141#390—1071no—
142#386—1074no—
143#380—1074no—
144C2×#41—1078no—
145C2×#41—2144no—
146C2×#41—2144no—
147C22×Q8C2×D10711yescharacteristic
148C22×D4—1072no—
149C22×D4—1071no—
150C23×C4—1071no—
151C23×C4—1072no—
152C23×C4—1072no—
153#117—2144no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4—1071no—
156C2×C4⋊C4—1072no—
157C2×C4⋊C4—2144no—
158C2×C4⋊C4C2×D10711yescharacteristic
159C2×C22⋊C4—1074no—
160C2×C22⋊C4—1074no—
161C2×C22⋊C4—1072no—
162C2×C42—1071no—
163C2×C42C2×D10711yescharacteristic
164#95—1074no—
165#95C2×D10714yes—
166C24—1071no—
167C24—1072no—
168#41—21432no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8C4×D10714yescomplemented
172C2×D4—1078no—
173C2×D4—1074no—
174C2×D4—2144no—
175C2×D4—2144no—
176C2×D4—2142no—
177C2×D4—2142no—
178C22×C4—1071no—
179C22×C4—1074no—
180C22×C4—1074no—
181C22×C4—1074no—
182C22×C4—1078no—
183C22×C4—1072no—
184C22×C4—1072no—
185C22×C4—4284no—
186C22×C4—2144no—
187C22×C4—2144no—
188C22×C4—2144no—
189C22×C4C22×D10712yes—
190C22×C4C22×D10712yes—
191C22×C4C22×D10712yes—
192C22×C4C22×D10711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—2144no—
195C4⋊C4—2144no—
196C4⋊C4—2142no—
197C22⋊C4—4284no—
198C22⋊C4—4284no—
199C22⋊C4—2144no—
200C42—22no—
201C42—2142no—
202C23—1072no—
203C23—1072no—
204C23—1072no—
205C23—1074no—
206C23—2142no—
207C23—2142no—
208C23—2144no—
209C23—2144no—
210C23—2142no—
211C23—2142no—
212C23C23×D10711yescharacteristic
213Q8—28no—
214D4—21416no—
215D4—2148no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—1072no—
224C2×C4—1072no—
225C2×C4—1072no—
226C2×C4—10716no—
227C2×C4—4288no—
228C2×C4—21416no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4—2144no—
233C2×C4—2142no—
234C2×C4C2×C4×D10714yes—
235C2×C4C2×C4×D10712yes—
236C2×C4D4×D10712yes—
237C2×C4D4×D10712yes—
238C22—1074no—
239C22—4282no—
240C22—4282no—
241C22—4284no—
242C22—2144no—
243C22—2148no—
244C22—2142no—
245C22—2142no—
246C22—2144no—
247C22—2144no—
248C22C22×C4×D10711yescharacteristic
249C22C2×D4×D10711yescharacteristic
250C22C2×D4×D10711yescharacteristic
251C22#1681215512yes—
252C22#1681215712yes—
253C4—24no—
254C4—42no—
255C4—42no—
256C4—1074no—
257C4—4282no—
258C4—4282no—
259C4C22⋊C4×D10714yes—
260C2—4284no—
261C2—2144no—
262C2C2×C22⋊C4×D10711yescharacteristic
263C2#2029310912yes—
264C2#2029319511yescharacteristic
265C2#2029320211yescharacteristic
266C2#2029320712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#391×C107C2
socle, cosocleC23×C107C24
prime radicalC22×#13133819C2
hypercenter#391D107
hypocenterC107#2311
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#16812066C2semidirect
3. C2×C4⋊C4×D107C2semidirect
4. #20292534C2semidirect
5. #20292529C2semidirect
6. #20292528C2semidirect
7. #20292505C2semidirect
8. #20292486C2semidirect
9. #391×C107C2semidirect
12. C2×#13133819C4semidirect
40. C2×D428C2×C4semidirect
49. C2×C4×D107C2×C4semidirect
61. C2×Q8×C107C2×C4semidirect
103. C4×D107C22⋊C4semidirect
134. C107#2311semidirect
139. #391D107semidirect
171. C2×Q8C4×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168120662
C2×C4⋊C4×D1072
#202925342
#202925292
#202925282
#202925052
#202924862
#391×C1072
#2311107