Finite Group Databasebeta

← #28340077

Subgroups — #28340077

Order 13696.

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

normal subgroups only

This group has 952 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×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812096C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic, complemented
4#20292535C214yesmaximal, complemented
5#20292533C211yesmaximal, characteristic, complemented
6#20292530C211yesmaximal, characteristic, complemented
7#20292507C214yesmaximal, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9C2×#127×C107C211yesmaximal, characteristic, complemented
10C2×#13133817—24no—
11C2×#13133817—24no—
12C2×#13133817—24no—
13C22×Dic214C2211yescharacteristic
14C22×C22⋊D107C2212yes—
15C22×C22⋊D107C2212yes—
16C23×Dic107C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812096—24no—
19#16812080—24no—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C22⋊Dic107C2214yes—
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2214yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×C4×Dic107C2211yescharacteristic
26C107⋊#95C2212yes—
27C107⋊#95C2214yes—
28C22×D4×C107C2211yescharacteristic
29C22×D4×C107C2214yes—
30C22×D4×C107C2211yescharacteristic
31#127×C107—24no—
32C2×C42×C107C2211yescharacteristic
33C23×D107C2311yescharacteristic
34#13133817—232no—
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214D414yescomplemented
38C2×C22⋊D107—24no—
39C2×C22⋊D107—24no—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—44no—
43C22×Dic107—24no—
44C22×Dic107—24no—
45C22×Dic107—24no—
46C22×Dic107C2312yes—
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2312yes—
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107D414yescomplemented
53C4⋊Dic107—22no—
54C22⋊Dic107—24no—
55C22⋊Dic107—28no—
56C22⋊Dic107—44no—
57(C2×D107)⋊C4—24no—
58C4×Dic107—22no—
59C24×C107C2312yes—
60C24×C107C2312yes—
61C2×D4×C107—28no—
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107—22no—
67C2×D4×C107—28no—
68C2×D4×C107—44no—
69C2×D4×C107D414yescomplemented
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C42×C107—22no—
73C23×C107—22no—
74C23×C107—22no—
75C23×C107—24no—
76C23×C107—24no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—24no—
81C23×C107—24no—
82C23×C107C2×D414yes—
83C23×C107C2411yescharacteristic
84C22×D107—21no—
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107C2×D412yes—
89D4×C107—28no—
90D4×C107—416no—
91D4×C107—44no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D412yes—
97C2×C4×C107#4114yescomplemented
98C22⋊D107—216no—
99C22⋊D107—48no—
100C2×Dic107—216no—
101C2×Dic107—21no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—24no—
110C2×Dic107C2×D412yes—
111C2×Dic107C2×D414yes—
112C4×D107—28no—
113Dic214—28no—
114C22×C107—24no—
115C22×C107—28no—
116C22×C107—42no—
117C22×C107—42no—
118C22×C107—44no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—44no—
123C22×C107—44no—
124C22×C107C22×D411yescharacteristic
125C22×C107C22×D411yescharacteristic
126C22×C107C22×D411yescharacteristic
127C22×C107C2×#4112yes—
128C22×C107H(5,𝔽2)12yes—
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—22no—
132C2×D107—42no—
133C2×D107—42no—
134C4×C107—24no—
135C4×C107#12014yes—
136Dic107—24no—
137Dic107—22no—
138Dic107—42no—
139Dic107—42no—
140C2×C107—44no—
141C2×C107—82no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#53211yescharacteristic
144C2×C107#53511yescharacteristic
145C2×C107#54612yes—
146C2×C107#55012yes—
147D107—42no—
148#2431—1071nomaximal, Sylow
149C107#243111yescharacteristic, complemented, Sylow
150C22×#41—1071no—
151C2×#127D10711yescharacteristic, complemented
152C2×#124—1071no—
153C2×#121—1071no—
154#392—1074no—
155#391—1071no—
156#390—1071no—
157#386—1074no—
158C4×C22⋊C4—1071no—
159C2×#41—2144no—
160C2×#41—2144no—
161C2×#41—2144no—
162C22×Q8—1071no—
163C22×D4—1072no—
164C22×D4—1072no—
165C22×D4C2×D10711yescharacteristic
166C22×D4C2×D10714yes—
167C22×D4C2×D10711yescharacteristic
168C23×C4—1071no—
169C23×C4—1072no—
170#127—24no—
171#124—2144no—
172#121—2144no—
173C2×C4⋊C4—1071no—
174C2×C22⋊C4—1074no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1074no—
177C2×C22⋊C4—1072no—
178C2×C42—1071no—
179C2×C42C2×D10711yescharacteristic
180#95—1072no—
181#95—1074no—
182C24—1071no—
183C24C22×D10712yes—
184C24C22×D10712yes—
185#41—21432no—
186C2×Q8—1074no—
187C2×Q8—2142no—
188C2×Q8—2142no—
189C2×D4—28no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—28no—
196C2×D4—44no—
197C2×D4—4284no—
198C2×D4—2144no—
199C2×D4—2144no—
200C2×D4—2144no—
201C2×D4—2144no—
202C2×D4C22⋊D10714yescomplemented
203C22×C4—1071no—
204C22×C4—1072no—
205C22×C4—1074no—
206C22×C4—1072no—
207C22×C4—1072no—
208C22×C4—2144no—
209C22×C4—2144no—
210C22×C4—2144no—
211C22×C4—2142no—
212C22×C4—2142no—
213C22×C4C22×D10712yes—
214C22×C4C22×D10711yescharacteristic
215C4⋊C4—2142no—
216C22⋊C4—4284no—
217C22⋊C4—2144no—
218C22⋊C4—2148no—
219C22⋊C4—2144no—
220C42—22no—
221C42—2142no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—1072no—
232C23—2141no—
233C23—2141no—
234C23—2142no—
235C23—2142no—
236C23C2×C22⋊D10714yes—
237C23C23×D10711yescharacteristic
238Q8—2148no—
239D4—28no—
240D4—416no—
241D4—44no—
242D4—4288no—
243D4—21416no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—1072no—
247C2×C4—1074no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2144no—
253C2×C4—21416no—
254C2×C4—2141no—
255C2×C4—2142no—
256C2×C4—2141no—
257C2×C4—2142no—
258C2×C4—2148no—
259C2×C4D4×D10712yes—
260C2×C4D4×D10712yes—
261C2×C4C2×C22⋊D10712yes—
262C2×C4#1313381714yescomplemented
263C22—28no—
264C22—24no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—44no—
271C22—44no—
272C22—42no—
273C22—4282no—
274C22—4282no—
275C22—2142no—
276C22—2142no—
277C22—2142no—
278C22C2×D4×D10711yescharacteristic
279C22C2×D4×D10711yescharacteristic
280C22C22×C22⋊D10711yescharacteristic
281C22C2×#1313381712yes—
282C22#1681215412yes—
283C4—24no—
284C4—4282no—
285C4—4282no—
286C4—2142no—
287C4—2144no—
288C4#1681207714yes—
289C2—44no—
290C2—82no—
291C2—4282no—
292C2C2×#1681207711yescharacteristic
293C2#2029317511yescharacteristic
294C2#2029319011yescharacteristic
295C2#2029327112yes—
296C2#2029329712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#127×C107C2
hypercenterC2×#127D107
hypocenterC107#2431
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812096C2semidirect
3. C2×#16812080C2semidirect
4. #20292535C2semidirect
5. #20292533C2semidirect
6. #20292530C2semidirect
7. #20292507C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. C2×#127×C107C2semidirect
37. C2×Dic214D4semidirect
52. C2×C4×D107D4semidirect
69. C2×D4×C107D4semidirect
97. C2×C4×C107#41semidirect
149. C107#2431semidirect
151. C2×#127D107semidirect
202. C2×D4C22⋊D107semidirect
262. C2×C4#13133817semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168120962
C2×#168120802
#202925352
#202925332
#202925302
#202925072
C4×(C2×D107)⋊C42
C2×#127×C1072
#2431107