Finite Group Databasebeta

← #28344162

Subgroups — #28344162

Order 13696.

472 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 137 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#20292566C211yesmaximal, characteristic, complemented
2#20292550C212yesmaximal, complemented
3#20292547C211yesmaximal, characteristic, complemented
4#20292539C212yesmaximal, complemented
5#20292511C212yesmaximal, complemented
6#20292507C212yesmaximal
7#20292499C212yesmaximal, complemented
8#20292485C212yesmaximal, complemented
9#390×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C23×Dic107C2212yes—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×Dic107⋊C4C2212yes—
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2212yes—
19C107⋊#95C2211yescharacteristic
20C107⋊#95C2212yes—
21C107⋊#95C2211yescharacteristic
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×D4×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2212yes—
29C2×C22⋊C4×C107C2212yes—
30C2×C42×C107C2211yescharacteristic
31C2×Dic214—41no—
32C2×Dic214—42no—
33C22×Dic107—24no—
34C22×Dic107—42no—
35C22×Dic107—42no—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C4⋊Dic107—41no—
41Dic107⋊C4—24no—
42Dic107⋊C4—42no—
43Dic107⋊C4—42no—
44C22⋊Dic107—24no—
45C22⋊Dic107—42no—
46C22⋊Dic107—42no—
47C24×C107C2312yes—
48C2×D4×C107—41no—
49C2×D4×C107—41no—
50C2×D4×C107Q814yescomplemented
51C22×C4×C107C2311yescharacteristic
52C22×C4×C107C2312yes—
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—41no—
56C22⋊C4×C107—42no—
57C22⋊C4×C107—42no—
58C42×C107—41no—
59C23×C107—22no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—22no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107C2×Q814yes—
66C23×C107C2411yescharacteristic
67D4×C107—44no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—21no—
76C2×C4×C107—21no—
77C2×C4×C107C2×Q812yes—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—48no—
90C2×Dic107#4114yes—
91Dic214—44no—
92C22×C107—24no—
93C22×C107—24no—
94C22×C107—24no—
95C22×C107—42no—
96C22×C107—42no—
97C22×C107—42no—
98C22×C107—42no—
99C22×C107C22×Q811yescharacteristic
100C22×C107C2×#4112yes—
101C22×C107H(5,𝔽2)11yescharacteristic
102C22×C107H(5,𝔽2)12yes—
103C22×C107H(5,𝔽2)11yescharacteristic
104C4×C107—22no—
105C4×C107—42no—
106C4×C107—41no—
107C4×C107—41no—
108Dic107—24no—
109Dic107—42no—
110Dic107—42no—
111Dic107—42no—
112C2×C107—44no—
113C2×C107#53812yes—
114C2×C107#54311yescharacteristic
115C2×C107#55412yes—
116C2×C107#55511yescharacteristic
117C2×C107#56011yescharacteristic
118#2784—1071nomaximal, Sylow
119C107#278411yescharacteristic, complemented, Sylow
120#400—1071no—
121#396—1072no—
122#395—1071no—
123#393—1072no—
124#390D10711yescharacteristic
125#387—1072no—
126#386—1072no—
127#385—1072no—
128#380—1072no—
129C22×Q8—1071no—
130C22×D4C2×D10711yescharacteristic
131C23×C4—1072no—
132C2×C4⋊C4—1072no—
133C2×C4⋊C4—1072no—
134C2×C4⋊C4—1071no—
135C2×C4⋊C4—1072no—
136C2×C4⋊C4C2×D10711yescharacteristic
137C2×C22⋊C4—1072no—
138C2×C22⋊C4—1072no—
139C2×C22⋊C4—1072no—
140C2×C22⋊C4C2×D10712yes—
141C2×C22⋊C4C2×D10712yes—
142C2×C42C2×D10711yescharacteristic
143#95—1071no—
144#95—1072no—
145#95—1071no—
146#95—1072no—
147#95—1072no—
148#95—1072no—
149#95—1072no—
150C24C22×D10712yes—
151C2×Q8—4282no—
152C2×Q8—4281no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic21414yescomplemented
156C22×C4—1072no—
157C22×C4—1072no—
158C22×C4—1072no—
159C22×C4—1072no—
160C22×C4—4282no—
161C22×C4—4282no—
162C22×C4—2144no—
163C22×C4C22×D10711yescharacteristic
164C22×C4C22×D10712yes—
165C22×C4C22×D10711yescharacteristic
166C22×C4C22×D10711yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—4282no—
169C4⋊C4—4282no—
170C4⋊C4—4281no—
171C4⋊C4—2144no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—4282no—
175C22⋊C4—4282no—
176C22⋊C4—2144no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic21414yes—
185C23C23×D10711yescharacteristic
186Q8—4284no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—1074no—
198C2×C4—4288no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4—2142no—
209C2×C4—2142no—
210C2×C4C2×Dic21412yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic21411yescharacteristic
219C22#41×D10712yes—
220C22#1681215411yescharacteristic
221C22#1681215412yes—
222C22#1681215411yescharacteristic
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—4282no—
228C4—4282no—
229C4—4282no—
230C4—2144no—
231C2—44no—
232C2#2029320912yes—
233C2#2029325311yescharacteristic
234C2#2029333612yes—
235C2#2029333911yescharacteristic
236C2#2029336411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
prime radicalC22×D4×C107C22
hypercenter#390D107
hypocenterC107#2784
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20292566C2semidirect
2. #20292550C2semidirect
3. #20292547C2semidirect
4. #20292539C2semidirect
5. #20292511C2semidirect
7. #20292499C2semidirect
8. #20292485C2semidirect
50. C2×D4×C107Q8semidirect
119. C107#2784semidirect
155. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202925662
#202925502
#202925472
#202925392
#202925112
#202925072
#202924992
#202924852
#390×C1072
#2784107