Finite Group Databasebeta

← #28337709

Subgroups — #28337709

Order 13696.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16812024C212yesmaximal, complemented
2#20292507C212yesmaximal, complemented
3#20292499C211yesmaximal, characteristic, complemented
4#20292394C211yesmaximal, characteristic, complemented
5#534×C107C211yesmaximal, characteristic
6C23×Dic107C2211yescharacteristic
7C2×Dic107⋊C4—21no—
8C2×C22⋊Dic107—22no—
9C2×C22⋊Dic107C2212yes—
10C2×C22⋊Dic107C2211yescharacteristic
11#16812024—24no—
12C107⋊#95—21no—
13C107⋊#95—22no—
14C22×D4×C107C2212yes—
15#123×C107—22no—
16#121×C107—21no—
17#121×C107—21no—
18#120×C107—22no—
19C2×C22⋊C4×C107C2211yescharacteristic
20C22×Dic107—22no—
21C22×Dic107—21no—
22C22×Dic107—21no—
23C22×Dic107—22no—
24C22×Dic107—42no—
25Dic107⋊C4—42no—
26C22⋊Dic107—21no—
27C22⋊Dic107—22no—
28C22⋊Dic107—22no—
29C22⋊Dic107—44no—
30C22⋊Dic107D412yescomplemented
31C24×C107D412yes—
32C24×C107C2311yescharacteristic
33C2×D4×C107—22no—
34C2×D4×C107—22no—
35C2×D4×C107—22no—
36C2×D4×C107—22no—
37C2×D4×C107—22no—
38C2×D4×C107—22no—
39C2×D4×C107—21no—
40C2×D4×C107—21no—
41C22×C4×C107D412yes—
42C22×C4×C107D411yescharacteristic
43C22×C4×C107D411yescharacteristic
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C22⋊C4×C107—22no—
47C22⋊C4×C107—22no—
48C22⋊C4×C107—22no—
49C23×C107—22no—
50C23×C107—22no—
51C23×C107—22no—
52C23×C107—41no—
53C23×C107—41no—
54C23×C107—42no—
55C23×C107—42no—
56C23×C107C2×D411yescharacteristic
57C23×C107C2×D412yes—
58C23×C107C2×D411yescharacteristic
59C23×C107#4111yescharacteristic
60C23×C107#4112yes—
61D4×C107—42no—
62D4×C107—42no—
63D4×C107—42no—
64D4×C107—42no—
65D4×C107—81no—
66D4×C107—81no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×Dic107—24no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—41no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—42no—
81C2×Dic107—41no—
82C2×Dic107—42no—
83C2×Dic107—42no—
84C2×Dic107—42no—
85C22×C107—21no—
86C22×C107—22no—
87C22×C107—21no—
88C22×C107—22no—
89C22×C107—41no—
90C22×C107—41no—
91C22×C107—42no—
92C22×C107—44no—
93C22×C107—42no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107—82no—
97C22×C107#12011yescharacteristic
98C22×C107#12111yescharacteristic
99C22×C107#12112yes—
100C22×C107#12312yes—
101C22×C107#12411yescharacteristic
102C4×C107—42no—
103C4×C107—41no—
104C4×C107—41no—
105Dic107—42no—
106Dic107—42no—
107Dic107—82no—
108C2×C107—21no—
109C2×C107—22no—
110C2×C107—22no—
111C2×C107—21no—
112C2×C107—82no—
113C2×C107#39411yescharacteristic
114C2×C107B(4,𝔽2)11yescharacteristic
115C2×C107#45811yescharacteristic
116#2044—1071nomaximal, Sylow
117C107#204411yescharacteristic, complemented, Sylow
118#534D10711yescharacteristic
119C2×#99—1072no—
120#386—1072no—
121#385—1071no—
122#342—1071no—
123C22×D4C2×D10712yes—
124C23×C4—1071no—
125#123—22no—
126#121—21no—
127#121—21no—
128#120—22no—
129C2×C4⋊C4—2141no—
130C2×C22⋊C4—1071no—
131C2×C22⋊C4—1072no—
132C2×C22⋊C4—2142no—
133C2×C22⋊C4C2×D10711yescharacteristic
134#99—2144no—
135#95—2141no—
136#95—2142no—
137C24C22⋊D10712yes—
138C24C22×D10711yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C22×C4—4282no—
148C22×C4—2142no—
149C22×C4—2141no—
150C22×C4—2142no—
151C22×C4—2141no—
152C22×C4C22⋊D10712yes—
153C22×C4C22⋊D10711yescharacteristic
154C22×C4C22⋊D10711yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—4282no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—1072no—
162C22⋊C4—4284no—
163C22⋊C4—2142no—
164C22⋊C4—2142no—
165C22⋊C4—2141no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23D4×D10711yescharacteristic
174C23C2×C22⋊D10711yescharacteristic
175C23C2×C22⋊D10712yes—
176C23#1313381711yescharacteristic
177C23#1313381712yes—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—4282no—
192C2×C4—4282no—
193C2×C4—4282no—
194C2×C4—4281no—
195C2×C4—4281no—
196C2×C4—4282no—
197C2×C4—4281no—
198C2×C4—4281no—
199C2×C4—2144no—
200C2×C4—2142no—
201C2×C4—2142no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#1681207811yescharacteristic
215C22#1681208011yescharacteristic
216C22#1681208112yes—
217C22#1681209012yes—
218C22#1681209611yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—8562no—
223C4—4282no—
224C4—4282no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#2029254111yescharacteristic
231C2#2029281411yescharacteristic
232C2#2029281711yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC24×C107C23
FrattiniC24C22×D107
Fitting, prime radical#534×C107C2
socleC22×C107#120
hypercenter#534D107
hypocenterC107#2044
LCSG ▷ C24×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#16812024C2semidirect
2. #20292507C2semidirect
3. #20292499C2semidirect
4. #20292394C2semidirect
30. C22⋊Dic107D4semidirect
117. C107#2044semidirect

Maximal subgroups

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

subgroupindex
C2×#168120242
#202925072
#202924992
#202923942
#534×C1072
#2044107