Finite Group Databasebeta

← #28337489

Subgroups — #28337489

Order 13696.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16812139C211yesmaximal, characteristic, complemented
2#20293191C211yesmaximal, characteristic
3C2×#16812097C211yesmaximal, characteristic, complemented
4#20292637C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7C2×#104×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10#16812139—22no—
11#16812139—22no—
12C2×#13133808—21no—
13C2×#13133806—21no—
14C2×#13133802C2211yescharacteristic
15#16812105—21no—
16#16812098—21no—
17#16812097—22no—
18#16812097—22no—
19#16812083—22no—
20#16812077—22no—
21#16812068C2211yescharacteristic
22C2×(C2×D107)⋊C4—22no—
23#16812034—22no—
24#16812025—22no—
25C2×#41×C107C2211yescharacteristic
26C2×M16×C107C2211yescharacteristic
27C2×C42×C107C2211yescharacteristic
28#104×C107—22no—
29C23×D107D412yes—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214—21no—
34C2×Dic214—21no—
35C2×Dic214D412yescomplemented
36C2×D428—21no—
37C2×D428—21no—
38C2×D428—21no—
39C2×D428—21no—
40C2×D428—21no—
41C2×D428—21no—
42C2×C22⋊D107—24no—
43C22×Dic107—21no—
44#13133808—24no—
45#13133806—24no—
46#13133802—21no—
47#13133802D412yescomplemented
48C2×C107⋊C8—21no—
49C4⋊Dic107D412yes—
50C22⋊Dic107—21no—
51(C2×D107)⋊C4—28no—
52(C2×D107)⋊C4—24no—
53C4×Dic107—21no—
54#41×C107—21no—
55#41×C107—21no—
56#41×C107—21no—
57#41×C107—21no—
58C2×Q8×C107D411yescharacteristic
59C2×D4×C107—21no—
60C2×D4×C107D411yescharacteristic
61C22×C4×C107—21no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64M16×C107—21no—
65M16×C107—21no—
66C2×C8×C107—21no—
67C42×C107—21no—
68C42×C107D412yescomplemented
69C23×C107—21no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—22no—
74C22×D107—44no—
75Q8×C107—21no—
76Q8×C107—21no—
77D4×C107—21no—
78D4×C107—21no—
79D4×C107—41no—
80D4×C107—41no—
81C2×C4×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—42no—
84C2×C4×C107—41no—
85C2×C4×C107—41no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107#4111yescharacteristic
92C22⋊D107—84no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—41no—
96C2×Dic107—41no—
97D428—41no—
98D428—41no—
99D428—41no—
100D428—41no—
101Dic214—22no—
102Dic214—22no—
103Dic214—41no—
104Dic214—41no—
105C8×C107—41no—
106C107⋊C8—22no—
107C22×C107—22no—
108C22×C107—41no—
109C22×C107—42no—
110C22×C107#12011yescharacteristic
111C22×C107#12011yescharacteristic
112C22×C107#12111yescharacteristic
113C2×D107—44no—
114C2×D107—42no—
115C2×D107—44no—
116C2×D107—82no—
117C4×C107—42no—
118C4×C107—41no—
119C4×C107#12011yescharacteristic
120C4×C107#12111yescharacteristic
121C4×C107#12111yescharacteristic
122C4×C107#12711yescharacteristic
123Dic107—42no—
124Dic107—42no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—81no—
128C2×C107#39211yescharacteristic
129C2×C107#45512yes—
130D107—82no—
131#2004—1071nomaximal, Sylow
132C107#200411yescharacteristic, complemented, Sylow
133C2×#137—1071no—
134#535—1071no—
135C2×#124—1071no—
136C2×#104D10711yescharacteristic, complemented
137#419—1071no—
138C2×#100—1071no—
139#339—1071no—
140C2×#41C2×D10711yescharacteristic
141C22×Q8—1071no—
142C22×D4—1071no—
143#137—2142no—
144#137—2142no—
145C2×Q16—2141no—
146C2×SD16—2141no—
147C2×M16—1071no—
148C2×M16C2×D10711yescharacteristic
149#127—2141no—
150#124—2142no—
151#124—2141no—
152#124—2142no—
153#121—2142no—
154#120—2142no—
155#117—1071no—
156C2×C22⋊C4—2142no—
157C2×C42C2×D10711yescharacteristic
158#104—22no—
159#103—2142no—
160#100—2142no—
161C24—1072no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—1072no—
167C2×Q8—2141no—
168C2×Q8—2141no—
169C2×Q8—2141no—
170C2×Q8—2141no—
171C2×Q8—2141no—
172C2×Q8D42811yescharacteristic
173C2×D4—21no—
174C2×D4—2144no—
175C2×D4—2141no—
176C2×D4—2141no—
177C2×D4—2141no—
178C2×D4—2141no—
179C2×D4—2141no—
180C2×D4—2141no—
181C2×D4D42811yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2141no—
185C22×C4C22×D10711yescharacteristic
186Q16—2144no—
187SD16—2144no—
188M16—21no—
189M16—21no—
190M16—1072no—
191M16—2141no—
192C2×C8—21no—
193C2×C8—2141no—
194C4⋊C4—1072no—
195C22⋊C4—2148no—
196C22⋊C4—2144no—
197C22⋊C4—2141no—
198C42—21no—
199C42—2141no—
200C42C22⋊D10712yescomplemented
201C23—21no—
202C23—4284no—
203C23—2142no—
204C23—2142no—
205C23—2142no—
206C23D4×D10711yescharacteristic
207Q8—21no—
208Q8—21no—
209Q8—4281no—
210Q8—4281no—
211Q8—2142no—
212Q8—2142no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—8564no—
218D4—4281no—
219D4—4281no—
220D4—4281no—
221D4—4281no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—4281no—
228C2×C4—4281no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4D4×D10711yescharacteristic
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×D42811yescharacteristic
236C2×C4#1313381811yescharacteristic
237C8—41no—
238C8—2142no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8562no—
243C22—4284no—
244C22—4282no—
245C22—4284no—
246C22#1681207711yescharacteristic
247C22#1681207911yescharacteristic
248C22#1681208211yescharacteristic
249C4—42no—
250C4—41no—
251C4—4282no—
252C4—4282no—
253C4#1681207911yescharacteristic
254C4#1681208211yescharacteristic
255C4#1681208311yescharacteristic
256C4#1681210511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—8562no—
261C2#2029253611yescharacteristic
262C2#2029280612yes—
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#104×C107C2
socleC22×C107#120
prime radical#20293191C2
hypercenterC2×#104D107
hypocenterC107#2004
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
3. C2×#16812097C2semidirect
4. #20292637C2semidirect
5. C2×#16812025C2semidirect
6. #20292387C2semidirect
7. C2×#104×C107C2semidirect
35. C2×Dic214D4semidirect
47. #13133802D4semidirect
68. C42×C107D4semidirect
132. C107#2004semidirect
136. C2×#104D107semidirect
200. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
#202931912
C2×#168120972
#202926372
C2×#168120252
#202923872
C2×#104×C1072
#2004107