Finite Group Databasebeta

← #28335378

Subgroups — #28335378

Order 13696.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293189C211yesmaximal, characteristic, complemented
2#20292761C212yesmaximal, complemented
3#20292360C212yesmaximal, complemented
4#20292351C211yesmaximal, characteristic, complemented
5#392×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107—21no—
7C2×#13133803—22no—
8#16812093—22no—
9#16812083—21no—
10#16812083C2212yes—
11#16812079—22no—
12#16812030—22no—
13C22⋊(C107⋊C8)C2212yescomplemented
14C22×D4×C107—21no—
15C22×D4×C107C2211yescharacteristic, complemented
16C2×C22⋊C4×C107—21no—
17C2×C22⋊C4×C107C2211yescharacteristic
18#95×C107C2211yescharacteristic, complemented
19C23×D107—21no—
20C2×D428—21no—
21C2×D428D412yescomplemented
22C2×C22⋊D107—22no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—42no—
25C2×C22⋊D107—42no—
26C22×Dic107—21no—
27#13133803—44no—
28C2×C107⋊C8—22no—
29C4⋊Dic107—21no—
30C4⋊Dic107D412yescomplemented
31(C2×D107)⋊C4—22no—
32(C2×D107)⋊C4—22no—
33C24×C107—21no—
34C24×C107D411yescharacteristic
35C2×D4×C107—22no—
36C2×D4×C107—42no—
37C2×D4×C107—42no—
38C2×D4×C107—42no—
39C2×D4×C107—42no—
40C22×C4×C107—21no—
41C22×C4×C107D411yescharacteristic
42C22×C4×C107C2311yescharacteristic
43C22⋊C4×C107—22no—
44C22⋊C4×C107—42no—
45C23×C107—21no—
46C23×C107—21no—
47C23×C107—22no—
48C23×C107—41no—
49C23×C107—41no—
50C23×C107—41no—
51C23×C107—41no—
52C23×C107—41no—
53C23×C107—41no—
54C23×C107—41no—
55C23×C107—41no—
56C23×C107D812yescomplemented
57C23×C107C2×D411yescharacteristic
58C22×D107—22no—
59C22×D107—42no—
60C22×D107—81no—
61C22×D107—81no—
62D4×C107—24no—
63D4×C107—42no—
64D4×C107—48no—
65C2×C4×C107—22no—
66C2×C4×C107—21no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107C2×D412yes—
72C22⋊D107—44no—
73C22⋊D107—42no—
74C22⋊D107—42no—
75C2×Dic107—22no—
76C2×Dic107—42no—
77D428—42no—
78D428—81no—
79C107⋊C8—42no—
80C22×C107—21no—
81C22×C107—21no—
82C22×C107—22no—
83C22×C107—42no—
84C22×C107—42no—
85C22×C107—42no—
86C22×C107—41no—
87C22×C107—41no—
88C22×C107—42no—
89C22×C107—81no—
90C22×C107—81no—
91C22×C107—81no—
92C22×C107—81no—
93C22×C107—81no—
94C22×C107#12011yescharacteristic
95C22×C107C2×D811yescharacteristic
96C22×C107AGL(1,ℤ/8)11yescharacteristic
97C2×D107—42no—
98C2×D107—42no—
99C2×D107—42no—
100C2×D107—84no—
101C4×C107—22no—
102C4×C107—42no—
103C4×C107—41no—
104Dic107—42no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107—42no—
108C2×C107—82no—
109C2×C107#44711yescharacteristic
110C2×C107#45311yescharacteristic
111C2×C107B(4,𝔽2)11yescharacteristic
112D107—82no—
113#1595—1071nomaximal, Sylow
114C107#159511yescharacteristic, complemented, Sylow
115#534—1071no—
116#447—1072no—
117#392D10711yescharacteristic, complemented
118#327—1072no—
119#323—1071no—
120C22×D4—21no—
121C22×D4—2141no—
122C22×D4C2×D10711yescharacteristic, complemented
123C2×D8—2142no—
124#123—2142no—
125#121—1072no—
126#121—2141no—
127#120—2142no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C2×D10711yescharacteristic
130#102—2142no—
131C22⋊C8—1072no—
132#95C2×D10711yescharacteristic, complemented
133C24—21no—
134C24—2141no—
135C24C22⋊D10711yescharacteristic
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—1072no—
142C2×D4—4282no—
143C2×D4—4282no—
144C2×D4—2141no—
145C2×D4—2142no—
146C2×D4—2142no—
147C22×C4—21no—
148C22×C4—2141no—
149C22×C4C22⋊D10711yescharacteristic
150C22×C4C22×D10711yescharacteristic
151D8—4284no—
152C2×C8—2142no—
153C4⋊C4—1072no—
154C4⋊C4—2141no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C22⋊C4—2142no—
158C22⋊C4—2142no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—8561no—
171C23—8561no—
172C23—4282no—
173C23—2142no—
174C23#1313380312yescomplemented
175C23C2×C22⋊D10711yescharacteristic
176D4—24no—
177D4—42no—
178D4—48no—
179D4—8561no—
180D4—4282no—
181D4—4284no—
182D4—4282no—
183D4—4282no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—4282no—
191C2×C4—2142no—
192C2×C4D4×D10712yes—
193C8—4282no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—8564no—
209C22—4282no—
210C22—4282no—
211C22—4282no—
212C22#1681207711yescharacteristic
213C22C2×#1313380311yescharacteristic
214C22#1681213411yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—4282no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—8562no—
224C2#2029276011yescharacteristic
225C2#2029279411yescharacteristic
226C2#2029281311yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#392×C107C2
socleC22×C107#120
hypercenter#392D107
hypocenterC107#1595
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #20293189C2semidirect
2. #20292761C2semidirect
3. #20292360C2semidirect
4. #20292351C2semidirect
5. #392×C107C2semidirect
13. C22⋊(C107⋊C8)C22semidirect
15. C22×D4×C107C22semidirect
18. #95×C107C22semidirect
21. C2×D428D4semidirect
30. C4⋊Dic107D4semidirect
56. C23×C107D8semidirect
114. C107#1595semidirect
117. #392D107semidirect
122. C22×D4C2×D107semidirect
132. #95C2×D107semidirect
174. C23#13133803semidirect

Maximal subgroups

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

subgroupindex
#202931892
#202927612
#202923602
#202923512
#392×C1072
#1595107