Finite Group Databasebeta

← #28339798

Subgroups — #28339798

Order 13696.

864 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133818C211yesmaximal, characteristic
2C2×#16812093C212yesmaximal, complemented
3C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
4#20292548C212yesmaximal, complemented
5#20292545C212yesmaximal, complemented
6#20292504C212yesmaximal, complemented
7#20292501C214yesmaximal, complemented
8#381×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—24no—
11C22×Dic214C2211yescharacteristic
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C23×Dic107C2211yescharacteristic
15C23×Dic107C2211yescharacteristic
16C22×C4×D107C2212yes—
17#16812093—28no—
18C2×C4⋊Dic107C2214yes—
19C2×Dic107⋊C4—24no—
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2214yes—
23C2×(C2×D107)⋊C4C2214yes—
24C107⋊#95C2214yes—
25C23×C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2212yes—
27#95×C107C2212yes—
28#95×C107C2212yes—
29C23×D107C2312yes—
30#13133818—232no—
31C2×Dic214—22no—
32C2×Dic214—24no—
33C2×D428—24no—
34C2×D428—22no—
35C2×C22⋊D107—24no—
36C2×C22⋊D107—24no—
37C2×C22⋊D107D418yes—
38C22×Dic107—22no—
39C22×Dic107—24no—
40C22×Dic107—22no—
41C22×Dic107—24no—
42C22×Dic107C2312yes—
43C22×Dic107C2312yes—
44C22×Dic107C2312yes—
45C2×C4×D107—24no—
46C2×C4×D107—24no—
47C2×C4×D107—24no—
48C4⋊Dic107—28no—
49Dic107⋊C4—24no—
50Dic107⋊C4—24no—
51(C2×D107)⋊C4—28no—
52(C2×D107)⋊C4—28no—
53C24×C107C2311yescharacteristic
54C22×C4×C107—24no—
55C22×C4×C107D414yes—
56C22×C4×C107C2314yes—
57C22×C4×C107C2311yescharacteristic
58C22×C4×C107C2311yescharacteristic
59C22⋊C4×C107—24no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107C2×D414yes—
63C23×C107C2×D412yes—
64C23×C107C2411yescharacteristic
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—24no—
68C22×D107—24no—
69C22×D107C2×D414yes—
70C2×C4×C107—28no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—24no—
74C2×C4×C107—24no—
75C2×C4×C107—24no—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2×D412yes—
78C22⋊D107—216no—
79C2×Dic107—28no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—24no—
85C2×Dic107—22no—
86C2×Dic107—44no—
87C2×Dic107C2×D414yes—
88C2×Dic107#4114yescomplemented
89D428—28no—
90C4×D107—216no—
91Dic214—28no—
92C22×C107—22no—
93C22×C107—22no—
94C22×C107—28no—
95C22×C107#12014yes—
96C22×C107C22×D412yes—
97C22×C107C22×D411yescharacteristic
98C22×C107C2×#4112yes—
99C22×C1072−1+412yes—
100C2×D107—24no—
101C2×D107—28no—
102C2×D107—42no—
103C2×D107—42no—
104C2×D107—44no—
105C4×C107—22no—
106C4×C107—22no—
107C4×C107—44no—
108Dic107—24no—
109Dic107—24no—
110Dic107—42no—
111C2×C107—24no—
112C2×C107C2×#12011yescharacteristic
113C2×C107#52511yescharacteristic
114C2×C107#53611yescharacteristic
115C2×C107#54714yes—
116D107—44no—
117#2406—1071nomaximal, Sylow
118C107#240611yescharacteristic, complemented, Sylow
119C22×#41—1071no—
120C2×#123—1072no—
121C22×C4⋊C4—1071no—
122#395—1072no—
123#394—1072no—
124#386—1072no—
125#385—1074no—
126#381D10711yescharacteristic, complemented
127C2×#41—2148no—
128C2×#41—2144no—
129C22×Q8—1071no—
130C22×D4—1072no—
131C22×D4—1071no—
132C23×C4—1071no—
133C23×C4—1071no—
134C23×C4—1072no—
135C23×C4C2×D10711yescharacteristic
136#123—2148no—
137C2×C4⋊C4—1072no—
138C2×C4⋊C4—1072no—
139C2×C4⋊C4—1074no—
140C2×C4⋊C4—2144no—
141C2×C22⋊C4—1074no—
142C2×C22⋊C4—1074no—
143C2×C22⋊C4C2×D10712yes—
144#95—1074no—
145#95C2×D10712yes—
146#95C2×D10712yes—
147C24—1072no—
148C24C22×D10711yescharacteristic
149#41—21432no—
150C2×Q8—2144no—
151C2×Q8—2142no—
152C2×D4—1078no—
153C2×D4—2144no—
154C2×D4—2144no—
155C2×D4—2144no—
156C2×D4—2142no—
157C22×C4—24no—
158C22×C4—1072no—
159C22×C4—1072no—
160C22×C4—1072no—
161C22×C4—2144no—
162C22×C4—2144no—
163C22×C4—2144no—
164C22×C4—2144no—
165C22×C4—2142no—
166C22×C4—2142no—
167C22×C4—2144no—
168C22×C4D42814yes—
169C22×C4C22×D10714yes—
170C22×C4C22×D10711yescharacteristic
171C22×C4C22×D10711yescharacteristic
172C4⋊C4—2148no—
173C4⋊C4—2144no—
174C4⋊C4—2144no—
175C22⋊C4—24no—
176C22⋊C4—2148no—
177C22⋊C4—2148no—
178C23—22no—
179C23—22no—
180C23—1074no—
181C23—2142no—
182C23—2142no—
183C23—2144no—
184C23—2144no—
185C23D4×D10714yes—
186C23C2×D42812yes—
187C23C23×D10711yescharacteristic
188Q8—2148no—
189D4—2148no—
190D4—21416no—
191C2×C4—28no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—1074no—
198C2×C4—1074no—
199C2×C4—4284no—
200C2×C4—21416no—
201C2×C4—2144no—
202C2×C4—2142no—
203C2×C4—2148no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4C2×D42812yes—
209C2×C4C2×D42812yes—
210C22—22no—
211C22—22no—
212C22—28no—
213C22—4282no—
214C22—4282no—
215C22—4284no—
216C22—2148no—
217C22—2144no—
218C22#1681207914yes—
219C22C2×D4×D10712yes—
220C22C22×D42811yescharacteristic
221C22#41×D10712yes—
222C22#1681215612yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—4282no—
227C4—2144no—
228C4—2144no—
229C2—24no—
230C2—4284no—
231C2C2×#1681207911yescharacteristic
232C2#2029313911yescharacteristic
233C2#2029320311yescharacteristic
234C2#2029327514yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#381×C107C2
prime radicalC22×#13133818C2
hypercenter#381D107
hypocenterC107#2406
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#16812093C2semidirect
3. C22×Dic107⋊C4C2semidirect
4. #20292548C2semidirect
5. #20292545C2semidirect
6. #20292504C2semidirect
7. #20292501C2semidirect
8. #381×C107C2semidirect
88. C2×Dic107#41semidirect
118. C107#2406semidirect
126. #381D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168120932
C22×Dic107⋊C42
#202925482
#202925452
#202925042
#202925012
#381×C1072
#2406107