Finite Group Databasebeta

← #28339594

Subgroups — #28339594

Order 13696.

552 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 185 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×#16812090C211yesmaximal, characteristic
2#20292530C211yesmaximal, characteristic
3#20292515C212yesmaximal, complemented
4#20292515C212yesmaximal, complemented
5#20292514C212yesmaximal, complemented
6#20292478C212yesmaximal, complemented
7C22⋊C4×Dic107C212yesmaximal, complemented
8C4×C22⋊Dic107C212yesmaximal, complemented
9#386×C107C211yesmaximal, characteristic
10C23×Dic107C2211yescharacteristic
11#16812090C418yescomplemented
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×C22⋊Dic107C2212yes—
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2211yescharacteristic
19C2×C4×Dic107C2212yes—
20C2×C4×Dic107C2212yes—
21C2×C4×Dic107C2212yes—
22C2×C4×Dic107C2211yescharacteristic
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C22×D4×C107C2211yescharacteristic
28C23×C4×C107C2211yescharacteristic
29C2×C22⋊C4×C107C2212yes—
30C2×C22⋊C4×C107C2211yescharacteristic
31#95×C107C2212yes—
32C22×Dic107—41no—
33C22×Dic107—41no—
34C22×Dic107C2×C414yescomplemented
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C4⋊Dic107—41no—
40Dic107⋊C4—24no—
41Dic107⋊C4C2×C418yescomplemented
42C22⋊Dic107—24no—
43C22⋊Dic107—24no—
44C22⋊Dic107C2×C418yes—
45C22⋊Dic107C2×C414yescomplemented
46C4×Dic107—24no—
47C4×Dic107—41no—
48C4×Dic107—42no—
49C4×Dic107—42no—
50C24×C107C2311yescharacteristic
51C24×C107C2311yescharacteristic
52C2×D4×C107—41no—
53C2×D4×C107—41no—
54C2×D4×C107C2×C414yes—
55C22×C4×C107—24no—
56C22×C4×C107—41no—
57C22×C4×C107C2312yes—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2312yes—
60C22⋊C4×C107—24no—
61C22⋊C4×C107—41no—
62C23×C107—22no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—21no—
67C23×C107—21no—
68C23×C107—21no—
69C23×C107—21no—
70C23×C107C22×C412yes—
71C23×C107C22×C412yes—
72C23×C107C2411yescharacteristic
73D4×C107—44no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—21no—
77C2×C4×C107—21no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—44no—
82C2×C4×C107C22×C412yes—
83C2×C4×C107#4114yes—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—44no—
94C2×Dic107C22×C414yes—
95C2×Dic107C22×C414yes—
96C2×Dic107#4114yes—
97C22×C107—24no—
98C22×C107—24no—
99C22×C107—22no—
100C22×C107—22no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107—42no—
104C22×C107—42no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107C23×C411yescharacteristic
108C22×C107C2×#4112yes—
109C22×C107C2×#4112yes—
110C22×C107H(5,𝔽2)11yescharacteristic
111C22×C107H(5,𝔽2)11yescharacteristic
112C4×C107—22no—
113C4×C107—24no—
114C4×C107—42no—
115Dic107—24no—
116Dic107—24no—
117Dic107—24no—
118Dic107—42no—
119C2×C107—42no—
120C2×C107—42no—
121C2×C107C4×#4112yes—
122C2×C107#51811yescharacteristic
123C2×C107#55111yescharacteristic
124C2×C107#55312yes—
125C2×C107#55911yescharacteristic
126#2374—1071nomaximal, Sylow
127C107#237411yescharacteristic, complemented, Sylow
128C2×#123—1071no—
129#390—1071no—
130#388—1072no—
131#388—1072no—
132#387—1072no—
133#386D10711yescharacteristic
134C4×C4⋊C4—1072no—
135C4×C22⋊C4—1072no—
136C4×C22⋊C4—1072no—
137C22×D4C2×D10711yescharacteristic
138C23×C4—1071no—
139C23×C4C2×D10711yescharacteristic
140#123—1078no—
141C2×C4⋊C4—1072no—
142C2×C4⋊C4—1071no—
143C2×C4⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4—1071no—
146C2×C22⋊C4—1072no—
147C2×C22⋊C4—1072no—
148C2×C22⋊C4C2×D10712yes—
149C2×C22⋊C4C2×D10711yescharacteristic
150C2×C42—1072no—
151C2×C42—1072no—
152C2×C42—1071no—
153C2×C42—1072no—
154#95—1072no—
155#95—1072no—
156#95—1072no—
157#95—1072no—
158#95C2×D10712yes—
159C24C22×D10711yescharacteristic
160C24C22×D10711yescharacteristic
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C4×D10714yes—
164C22×C4—24no—
165C22×C4—41no—
166C22×C4—1074no—
167C22×C4—1072no—
168C22×C4—1072no—
169C22×C4—1072no—
170C22×C4—1072no—
171C22×C4—4281no—
172C22×C4—4281no—
173C22×C4C22×D10712yes—
174C22×C4C22×D10711yescharacteristic
175C22×C4C22×D10712yes—
176C4⋊C4—1078no—
177C4⋊C4—4281no—
178C4⋊C4—2144no—
179C22⋊C4—24no—
180C22⋊C4—41no—
181C22⋊C4—1078no—
182C22⋊C4—1074no—
183C22⋊C4—2144no—
184C22⋊C4—2144no—
185C42—4282no—
186C42—4281no—
187C42—4282no—
188C42—2144no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23C2×C4×D10712yes—
198C23C2×C4×D10712yes—
199C23C23×D10711yescharacteristic
200D4—44no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—44no—
209C2×C4—1074no—
210C2×C4—1074no—
211C2×C4—1074no—
212C2×C4—4284no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4C2×C4×D10712yes—
223C2×C4#1313381714yes—
224C22—24no—
225C22—24no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C22×C4×D10711yescharacteristic
235C22#41×D10712yes—
236C22C2×#1313381712yes—
237C22#1681215411yescharacteristic
238C22#1681215411yescharacteristic
239C4—24no—
240C4—22no—
241C4—42no—
242C4—4282no—
243C4—2144no—
244C4—2144no—
245C4—2144no—
246C2—42no—
247C2—42no—
248C2C4×#1313381712yes—
249C2#2029310011yescharacteristic
250C2#2029330811yescharacteristic
251C2#2029332412yes—
252C2#2029336111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#386×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D4×C107C22
hypercenter#386D107
hypocenterC107#2374
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #20292515C2semidirect
4. #20292515C2semidirect
5. #20292514C2semidirect
6. #20292478C2semidirect
7. C22⋊C4×Dic107C2semidirect
8. C4×C22⋊Dic107C2semidirect
11. #16812090C4semidirect
34. C22×Dic107C2×C4semidirect
41. Dic107⋊C4C2×C4semidirect
45. C22⋊Dic107C2×C4semidirect
127. C107#2374semidirect

Maximal subgroups

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

subgroupindex
C2×#168120902
#202925302
#202925152
#202925142
#202924782
C22⋊C4×Dic1072
C4×C22⋊Dic1072
#386×C1072
#2374107