Finite Group Databasebeta

← #28339597

Subgroups — #28339597

Order 13696.

552 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 221 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
1#20292530C211yesmaximal, characteristic
2#20292520C212yesmaximal, complemented
3#20292520C212yesmaximal, complemented
4#20292513C212yesmaximal, complemented
5#20292507C211yesmaximal, characteristic
6C4⋊C4×Dic107C212yesmaximal, complemented
7C22⋊C4×Dic107C212yesmaximal, complemented
8C4×C22⋊Dic107C212yesmaximal, complemented
9C2×#123×C107C211yesmaximal, characteristic
10C23×Dic107C2211yescharacteristic
11C2×C4⋊Dic107C2212yes—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C22⋊Dic107C2212yes—
14C2×C22⋊Dic107C2212yes—
15C2×C22⋊Dic107C2212yes—
16C2×C22⋊Dic107C2211yescharacteristic
17C2×C4×Dic107C2212yes—
18C2×C4×Dic107C2212yes—
19C2×C4×Dic107C2212yes—
20C2×C4×Dic107C2211yescharacteristic
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×D4×C107C2211yescharacteristic
27C23×C4×C107C2211yescharacteristic
28#123×C107C418yescomplemented
29C2×C4⋊C4×C107C2212yes—
30C2×C22⋊C4×C107C2212yes—
31C2×C22⋊C4×C107C2211yescharacteristic
32C22×Dic107—24no—
33C22×Dic107—41no—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C4⋊Dic107—24no—
39C4⋊Dic107—41no—
40C22⋊Dic107—24no—
41C22⋊Dic107—24no—
42C22⋊Dic107—24no—
43C22⋊Dic107—41no—
44C4×Dic107—24no—
45C4×Dic107—41no—
46C4×Dic107—42no—
47C4×Dic107—42no—
48C24×C107C2311yescharacteristic
49C24×C107C2311yescharacteristic
50C2×D4×C107—41no—
51C2×D4×C107—41no—
52C2×D4×C107C2×C414yes—
53C22×C4×C107—41no—
54C22×C4×C107—41no—
55C22×C4×C107C2×C414yescomplemented
56C22×C4×C107C2312yes—
57C22×C4×C107C2312yes—
58C22×C4×C107C2311yescharacteristic
59C4⋊C4×C107C2×C418yescomplemented
60C22⋊C4×C107C2×C418yes—
61C22⋊C4×C107C2×C414yescomplemented
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—22no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—44no—
81C2×C4×C107C22×C414yes—
82C2×C4×C107C22×C414yes—
83C2×C4×C107C22×C412yes—
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—22no—
94C2×Dic107—44no—
95C2×Dic107#4114yes—
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—24no—
113C4×C107—24no—
114C4×C107—22no—
115Dic107—24no—
116Dic107—24no—
117Dic107—42no—
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×#123D10711yescharacteristic
129#390—1071no—
130#388—1072no—
131#388—1072no—
132#387—1072no—
133#386—1071no—
134C4×C4⋊C4—1072no—
135C4×C22⋊C4—1072no—
136C4×C22⋊C4—1072no—
137C22×D4C2×D10711yescharacteristic
138C23×C4—1071no—
139C23×C4C2×D10711yescharacteristic
140#123Dic10718yescomplemented
141C2×C4⋊C4—1071no—
142C2×C4⋊C4—1072no—
143C2×C4⋊C4C2×D10712yes—
144C2×C22⋊C4—1071no—
145C2×C22⋊C4—1072no—
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#95—1072no—
159C24C22×D10711yescharacteristic
160C24C22×D10711yescharacteristic
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C2×Dic10714yes—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—1072no—
167C22×C4—1072no—
168C22×C4—1072no—
169C22×C4—1072no—
170C22×C4—4281no—
171C22×C4—2144no—
172C22×C4C2×Dic10714yescomplemented
173C22×C4C22×D10712yes—
174C22×C4C22×D10712yes—
175C22×C4C22×D10711yescharacteristic
176C4⋊C4—4281no—
177C4⋊C4—2144no—
178C4⋊C4C2×Dic10718yescomplemented
179C22⋊C4—4281no—
180C22⋊C4—2144no—
181C22⋊C4—2144no—
182C22⋊C4—2144no—
183C22⋊C4C2×Dic10718yes—
184C22⋊C4C2×Dic10714yescomplemented
185C42—4282no—
186C42—4282no—
187C42—4281no—
188C42—2144no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23C22×Dic10712yes—
198C23C22×Dic10712yes—
199C23C23×D10711yescharacteristic
200D4—44no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—44no—
208C2×C4—1074no—
209C2×C4—1074no—
210C2×C4—4284no—
211C2×C4—2142no—
212C2×C4—2142no—
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×C4C22×Dic10714yes—
222C2×C4C22×Dic10714yes—
223C2×C4C22×Dic10712yes—
224C22—24no—
225C22—24no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C23×Dic10711yescharacteristic
235C22#41×D10712yes—
236C22#41×D10712yes—
237C22#1681215411yescharacteristic
238C22#1681215411yescharacteristic
239C4—24no—
240C4—24no—
241C4—22no—
242C4—4282no—
243C4—4282no—
244C4—2144no—
245C4—2144no—
246C2—42no—
247C2—42no—
248C2#41×Dic10712yes—
249C2#2029310211yescharacteristic
250C2#2029330511yescharacteristic
251C2#2029332212yes—
252C2#2029336111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC2×#123×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D4×C107C22
hypercenterC2×#123D107
hypocenterC107#2374
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #20292520C2semidirect
3. #20292520C2semidirect
4. #20292513C2semidirect
6. C4⋊C4×Dic107C2semidirect
7. C22⋊C4×Dic107C2semidirect
8. C4×C22⋊Dic107C2semidirect
28. #123×C107C4semidirect
55. C22×C4×C107C2×C4semidirect
59. C4⋊C4×C107C2×C4semidirect
61. C22⋊C4×C107C2×C4semidirect
127. C107#2374semidirect
140. #123Dic107semidirect
172. C22×C4C2×Dic107semidirect
178. C4⋊C4C2×Dic107semidirect
184. C22⋊C4C2×Dic107semidirect

Maximal subgroups

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

subgroupindex
#202925302
#202925202
#202925132
#202925072
C4⋊C4×Dic1072
C22⋊C4×Dic1072
C4×C22⋊Dic1072
C2×#123×C1072
#2374107