Finite Group Databasebeta

← #28344497

Subgroups — #28344497

Order 13696.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812089C211yesmaximal, characteristic
2C2×#16812087C211yesmaximal, characteristic
3#20292569C212yesmaximal, complemented
4#20292565C211yesmaximal, characteristic, complemented
5#20292556C212yesmaximal
6#20292554C212yesmaximal
7#20292548C211yesmaximal, characteristic, complemented
8#20292548C212yesmaximal, complemented
9#20292547C211yesmaximal, characteristic, complemented
10#20292489C211yesmaximal, characteristic
11#400×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15#16812089—24no—
16#16812087—24no—
17C2×C4⋊Dic107C2212yes—
18C2×C4⋊Dic107C2212yes—
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2211yescharacteristic
21C2×Dic107⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2211yescharacteristic
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C107⋊#95C2211yescharacteristic
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34#95×C107C2211yescharacteristic
35#95×C107C2212yes—
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C2×Dic214—22no—
39C2×Dic214—22no—
40C2×Dic214—42no—
41C2×Dic214—42no—
42C22×Dic107C2311yescharacteristic
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2312yes—
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—42no—
48C2×C4×D107D414yes—
49C4⋊Dic107—24no—
50C4⋊Dic107—42no—
51Dic107⋊C4—22no—
52Dic107⋊C4—22no—
53Dic107⋊C4—42no—
54(C2×D107)⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—42no—
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2312yes—
61C22×C4×C107C2312yes—
62C22×C4×C107C2311yescharacteristic
63C4⋊C4×C107—22no—
64C4⋊C4×C107—22no—
65C23×C107C2411yescharacteristic
66C22×D107—22no—
67C22×D107—21no—
68C22×D107—22no—
69C22×D107—21no—
70C22×D107C2×D412yes—
71C2×C4×C107—22no—
72C2×C4×C107—21no—
73C2×C4×C107—22no—
74C2×C4×C107—21no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—21no—
78C2×C4×C107—24no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107C2×D412yes—
83C2×Dic107—22no—
84C2×Dic107—21no—
85C2×Dic107—21no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107C2×D412yes—
96C4×D107—44no—
97Dic214—44no—
98Dic214—44no—
99C22×C107C22×D411yescharacteristic
100C22×C107H(5,𝔽2)12yes—
101C22×C1072−1+411yescharacteristic
102C22×C1072−1+412yes—
103C22×C1072−1+411yescharacteristic
104C2×D107—22no—
105C2×D107—22no—
106C2×D107—22no—
107C2×D107—42no—
108C2×D107—42no—
109C4×C107—22no—
110C4×C107—41no—
111C4×C107—42no—
112C4×C107—41no—
113C4×C107—42no—
114Dic107—22no—
115Dic107—41no—
116Dic107—42no—
117Dic107—41no—
118Dic107—42no—
119C2×C107#53411yescharacteristic
120C2×C107#53611yescharacteristic
121C2×C107#53611yescharacteristic
122C2×C107#56312yes—
123C2×C107#56312yes—
124D107—42no—
125#2817—1071nomaximal, Sylow
126C107#281711yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—1071no—
128C2×C22⋊Q8—1071no—
129#400—1071no—
130#400—1072no—
131#400D10711yescharacteristic, complemented
132#397—1072no—
133#397—1072no—
134#395—1071no—
135#395—1072no—
136#395—1071no—
137#381—1071no—
138C22×Q8—1071no—
139C22×Q8—1071no—
140C23×C4—1071no—
141C22⋊Q8—2144no—
142C22⋊Q8—2144no—
143C2×C4⋊C4—1072no—
144C2×C4⋊C4—1072no—
145C2×C4⋊C4—1071no—
146C2×C4⋊C4—1071no—
147C2×C4⋊C4—1072no—
148C2×C4⋊C4C2×D10711yescharacteristic
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—1072no—
151C2×C22⋊C4—1071no—
152C2×C22⋊C4—1071no—
153C2×C22⋊C4—1072no—
154#95—1072no—
155#95—1071no—
156#95—1072no—
157#95—1072no—
158#95—1072no—
159#95—1072no—
160#95C2×D10711yescharacteristic
161#95C2×D10712yes—
162#95C2×D10712yes—
163C24—1071no—
164C2×Q8—4282no—
165C2×Q8—4282no—
166C2×Q8—2142no—
167C2×Q8—2142no—
168C22×C4—1071no—
169C22×C4—1074no—
170C22×C4—1072no—
171C22×C4—1071no—
172C22×C4—1072no—
173C22×C4—1071no—
174C22×C4—4282no—
175C22×C4C22×D10712yes—
176C22×C4C22×D10712yes—
177C22×C4C22×D10711yescharacteristic
178C22×C4C22×D10711yescharacteristic
179C22×C4C22×D10711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—4282no—
183C4⋊C4—4282no—
184C4⋊C4—2142no—
185C4⋊C4—2144no—
186C4⋊C4—2142no—
187C22⋊C4—4282no—
188C22⋊C4—2142no—
189C22⋊C4—2144no—
190C22⋊C4—2142no—
191C23—1072no—
192C23—2142no—
193C23—2141no—
194C23—2142no—
195C23—2141no—
196C23C23×D10711yescharacteristic
197Q8—4284no—
198Q8—4284no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—1072no—
211C2×C4—4284no—
212C2×C4—2141no—
213C2×C4—2142no—
214C2×C4—2141no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4—2141no—
223C2×C4—2142no—
224C2×C4D4×D10712yes—
225C22—4282no—
226C22—4282no—
227C22—2142no—
228C22—2142no—
229C22—2142no—
230C22C2×D4×D10711yescharacteristic
231C22#1681215512yes—
232C22#1681215611yescharacteristic
233C22#1681215612yes—
234C22#1681215711yescharacteristic
235C4—22no—
236C4—42no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—4282no—
241C4—4282no—
242C4—4281no—
243C4—4281no—
244C4—2142no—
245C2—4282no—
246C2#2029318811yescharacteristic
247C2#2029319711yescharacteristic
248C2#2029320011yescharacteristic
249C2#2029337912yes—
250C2#2029338112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#400×C107C2
prime radicalC23×D107C23
hypercenter#400D107
hypocenterC107#2817
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #20292569C2semidirect
4. #20292565C2semidirect
7. #20292548C2semidirect
8. #20292548C2semidirect
9. #20292547C2semidirect
11. #400×C107C2semidirect
126. C107#2817semidirect
131. #400D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×#168120872
#202925692
#202925652
#202925562
#202925542
#202925482
#202925472
#202924892
#400×C1072
#2817107