Finite Group Databasebeta

← #28337490

Subgroups — #28337490

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
1C2×#16812140C211yesmaximal, characteristic, complemented
2#20293194C211yesmaximal, characteristic
3C2×#16812037C211yesmaximal, characteristic, complemented
4#20292636C211yesmaximal, characteristic, complemented
5C2×#16812026C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7C2×#124×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—22no—
10#16812140—22no—
11C2×#13133808—21no—
12C2×#13133805—21no—
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812105—21no—
16#16812096—21no—
17#16812080—22no—
18#16812078—22no—
19#16812068C2211yescharacteristic
20#16812037—22no—
21#16812033—22no—
22#16812026—22no—
23C22×Q8×C107C2211yescharacteristic
24C22×D4×C107C2211yescharacteristic
25#124×C107—22no—
26#124×C107—22no—
27C2×C22⋊C4×C107—22no—
28C2×C42×C107C2211yescharacteristic
29#13133818—21no—
30#13133818—21no—
31#13133818—21no—
32#13133818—21no—
33C2×Dic214D411yescharacteristic
34C2×D428D411yescharacteristic
35C2×C22⋊D107—21no—
36C2×C22⋊D107—24no—
37C2×C4×D107—21no—
38#13133808—24no—
39#13133805—24no—
40#13133802—21no—
41#13133802—21no—
42#13133802—21no—
43#13133802D412yescomplemented
44C2×C107⋊C8—21no—
45C2×C107⋊C8—21no—
46C4⋊Dic107D412yes—
47C22⋊Dic107—24no—
48C22⋊Dic107—21no—
49C4×Dic107—21no—
50C24×C107D412yes—
51C2×Q8×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—21no—
55C2×Q8×C107—21no—
56C2×Q8×C107D412yescomplemented
57C2×D4×C107—21no—
58C2×D4×C107—21no—
59C2×D4×C107—21no—
60C2×D4×C107—21no—
61C2×D4×C107—21no—
62C2×D4×C107—21no—
63C22×C4×C107—21no—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66C22⋊C4×C107—28no—
67C42×C107—21no—
68C42×C107D412yescomplemented
69C23×C107—22no—
70C23×C107—22no—
71C23×C107—22no—
72C23×C107—44no—
73C23×C107C2×D411yescharacteristic
74C22×D107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77Q8×C107—41no—
78Q8×C107—41no—
79D4×C107—41no—
80D4×C107—41no—
81D4×C107—41no—
82D4×C107—41no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107—41no—
87C2×C4×C107—41no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107#4111yescharacteristic
94C22⋊D107—41no—
95C22⋊D107—41no—
96C22⋊D107—84no—
97C2×Dic107—22no—
98C2×Dic107—21no—
99D428—21no—
100D428—21no—
101C4×D107—41no—
102C4×D107—41no—
103Dic214—21no—
104Dic214—21no—
105C107⋊C8—22no—
106C107⋊C8—41no—
107C22×C107—22no—
108C22×C107—44no—
109C22×C107—42no—
110C22×C107—44no—
111C22×C107—82no—
112C22×C107#12011yescharacteristic
113C22×C107#12011yescharacteristic
114C22×C107#12111yescharacteristic
115C2×D107—41no—
116C2×D107—42no—
117C4×C107—42no—
118C4×C107—42no—
119C4×C107#12011yescharacteristic
120C4×C107#12111yescharacteristic
121C4×C107#12111yescharacteristic
122C4×C107#12711yescharacteristic
123Dic107—41no—
124Dic107—42no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—82no—
128C2×C107#39211yescharacteristic
129C2×C107#45512yes—
130D107—81no—
131#2004—1071nomaximal, Sylow
132C107#200411yescharacteristic, complemented, Sylow
133C2×#137—1071no—
134#535—1071no—
135C2×#124D10711yescharacteristic, complemented
136C2×#104—1071no—
137#419—1071no—
138C2×#100—1071no—
139#339—1071no—
140C2×#41—1071no—
141C22×Q8C2×D10711yescharacteristic
142C22×D4C2×D10711yescharacteristic
143#137—2142no—
144#137—2142no—
145C2×Q16—2141no—
146C2×SD16—2141no—
147C2×M16—1071no—
148C2×M16—1071no—
149#127—2141no—
150#124—22no—
151#124—22no—
152#124—2141no—
153#121—2142no—
154#120—2142no—
155#117—1071no—
156C2×C22⋊C4—22no—
157C2×C42C2×D10711yescharacteristic
158#104—2142no—
159#103—2142no—
160#100—2142no—
161C24C22⋊D10712yes—
162#41—2141no—
163#41—2141no—
164#41—2141no—
165#41—2141no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1071no—
172C2×Q8C22⋊D10712yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—1071no—
180C2×D4—2144no—
181C2×D4—2141no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2141no—
185C22×C4C22×D10711yescharacteristic
186Q16—2144no—
187SD16—2144no—
188M16—1072no—
189M16—2141no—
190M16—2141no—
191M16—2141no—
192C2×C8—2141no—
193C2×C8—2141no—
194C4⋊C4—1072no—
195C22⋊C4—28no—
196C22⋊C4—2144no—
197C22⋊C4—2141no—
198C42—21no—
199C42—2141no—
200C42C22⋊D10712yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—2141no—
206C23C2×C22⋊D10711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—2141no—
212Q8—2141no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—8564no—
218D4—4281no—
219D4—4281no—
220D4—2141no—
221D4—2141no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—4281no—
228C2×C4—4281no—
229C2×C4—2142no—
230C2×C4—2141no—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4D4×D10711yescharacteristic
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×C22⋊D10711yescharacteristic
236C2×C4#1313381711yescharacteristic
237C8—4281no—
238C8—2142no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—4281no—
245C22—4282no—
246C22#1681207711yescharacteristic
247C22#1681207811yescharacteristic
248C22#1681208111yescharacteristic
249C4—42no—
250C4—42no—
251C4—4281no—
252C4—4282no—
253C4#1681207711yescharacteristic
254C4#1681208011yescharacteristic
255C4#1681208111yescharacteristic
256C4#1681210511yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—8561no—
261C2#2029253511yescharacteristic
262C2#2029280412yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#124×C107C2
socleC22×C107#120
prime radical#20293194C2
hypercenterC2×#124D107
hypocenterC107#2004
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
3. C2×#16812037C2semidirect
4. #20292636C2semidirect
5. C2×#16812026C2semidirect
6. #20292387C2semidirect
7. C2×#124×C107C2semidirect
43. #13133802D4semidirect
56. C2×Q8×C107D4semidirect
68. C42×C107D4semidirect
132. C107#2004semidirect
135. C2×#124D107semidirect
172. C2×Q8C22⋊D107semidirect
200. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
#202931942
C2×#168120372
#202926362
C2×#168120262
#202923872
C2×#124×C1072
#2004107