Finite Group Databasebeta

← #28337487

Subgroups — #28337487

Order 13696.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#20293191C211yesmaximal, characteristic
2C2×#16812098C211yesmaximal, characteristic, complemented
3C2×#16812036C211yesmaximal, characteristic, complemented
4#20292636C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6#20292388C211yesmaximal, characteristic, complemented
7C2×#137×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10#16812105—21no—
11#16812098—22no—
12#16812098—21no—
13#16812098—22no—
14#16812083—22no—
15#16812077—22no—
16#16812068C2211yescharacteristic
17C2×(C2×D107)⋊C4—22no—
18C2×C4×Dic107C2211yescharacteristic
19#16812036—22no—
20#16812033—22no—
21#16812025—22no—
22C2×#41×C107C2211yescharacteristic
23C22×Q8×C107C2211yescharacteristic
24#137×C107—22no—
25#137×C107—22no—
26C2×Q16×C107—21no—
27C2×SD16×C107—21no—
28C2×M16×C107C2211yescharacteristic
29C23×D107D412yes—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—21no—
36C2×C22⋊D107—24no—
37C22×Dic107—21no—
38#13133802—21no—
39#13133802—21no—
40C2×C107⋊C8—21no—
41C4⋊Dic107D412yes—
42C22⋊Dic107—21no—
43(C2×D107)⋊C4—28no—
44(C2×D107)⋊C4—24no—
45C4×Dic107—21no—
46C4×Dic107—21no—
47C4×Dic107D412yescomplemented
48#41×C107—21no—
49#41×C107—21no—
50#41×C107—21no—
51#41×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—21no—
55C2×Q8×C107—21no—
56C2×Q8×C107—21no—
57C2×Q8×C107D411yescharacteristic
58C2×Q8×C107D412yescomplemented
59C2×D4×C107—21no—
60C2×D4×C107D411yescharacteristic
61C22×C4×C107—21no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64Q16×C107—24no—
65SD16×C107—24no—
66M16×C107—21no—
67M16×C107D412yescomplemented
68C2×C8×C107—21no—
69C23×C107—21no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—22no—
73C22×D107—22no—
74C22×D107—44no—
75Q8×C107—22no—
76Q8×C107—21no—
77Q8×C107—22no—
78Q8×C107—21no—
79Q8×C107—41no—
80Q8×C107—41no—
81D4×C107—21no—
82D4×C107—21no—
83D4×C107—41no—
84D4×C107—41no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107—41no—
88C2×C4×C107—41no—
89C2×C4×C107—41no—
90C2×C4×C107—41no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107#4111yescharacteristic
97C22⋊D107—84no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—42no—
101D428—41no—
102D428—41no—
103D428—41no—
104D428—41no—
105C8×C107—22no—
106C107⋊C8—41no—
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—41no—
118C4×C107—42no—
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×#137D10711yescharacteristic, complemented
134#535—1071no—
135C2×#124—1071no—
136C2×#104—1071no—
137#419—1071no—
138C2×#100—1071no—
139#339—1071no—
140C2×#41C2×D10711yescharacteristic
141C22×Q8C2×D10711yescharacteristic
142C22×D4—1071no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
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×C42—1071no—
158#104—2142no—
159#103—2142no—
160#100—2142no—
161C24—1072no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D10711yescharacteristic
172C2×Q8C22⋊D10712yescomplemented
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×D4C22⋊D10711yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2141no—
185C22×C4C22×D10711yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—2141no—
190M16—2141no—
191M16C22⋊D10712yescomplemented
192C2×C8—21no—
193C2×C8—2141no—
194C4⋊C4—1072no—
195C22⋊C4—2148no—
196C22⋊C4—2144no—
197C22⋊C4—2141no—
198C42—1072no—
199C42—2141no—
200C42—2141no—
201C23—21no—
202C23—4284no—
203C23—2142no—
204C23—2142no—
205C23—2142no—
206C23D4×D10711yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
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—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—4282no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4C2×C22⋊D10711yescharacteristic
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×C22⋊D10711yescharacteristic
236C2×C4#1313381711yescharacteristic
237C8—22no—
238C8—4281no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—8562no—
243C22—4284no—
244C22—4282no—
245C22—4284no—
246C22#1681207711yescharacteristic
247C22#1681207711yescharacteristic
248C22#1681208011yescharacteristic
249C4—41no—
250C4—42no—
251C4—4282no—
252C4—4282no—
253C4#1681207811yescharacteristic
254C4#1681208111yescharacteristic
255C4#1681208111yescharacteristic
256C4#1681210511yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—8562no—
261C2#2029253511yescharacteristic
262C2#2029280312yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#137×C107C2
socleC22×C107#120
prime radical#20293191C2
hypercenterC2×#137D107
hypocenterC107#2004
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16812098C2semidirect
3. C2×#16812036C2semidirect
4. #20292636C2semidirect
5. C2×#16812025C2semidirect
6. #20292388C2semidirect
7. C2×#137×C107C2semidirect
47. C4×Dic107D4semidirect
58. C2×Q8×C107D4semidirect
67. M16×C107D4semidirect
132. C107#2004semidirect
133. C2×#137D107semidirect
172. C2×Q8C22⋊D107semidirect
191. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202931912
C2×#168120982
C2×#168120362
#202926362
C2×#168120252
#202923882
C2×#137×C1072
#2004107