Finite Group Databasebeta

← #28337521

Subgroups — #28337521

Order 13696.

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

normal subgroups only

This group has 348 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
1#20293425C211yesmaximal, characteristic
2C2×#16812140C211yesmaximal, characteristic, complemented
3#20292644C211yesmaximal, characteristic, complemented
4#20292636C211yesmaximal, characteristic, complemented
5#20292600C211yesmaximal, characteristic, complemented
6#20292397C211yesmaximal, characteristic, complemented
7#536×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—22no—
10#16812140—22no—
11#16812140—21no—
12#16812139—21no—
13#16812135—21no—
14#16812134—21no—
15#16812129—22no—
16C2×#13133808—21no—
17C2×#13133805—21no—
18C2×#13133802C2211yescharacteristic
19C2×#13133802C2211yescharacteristic
20#16812068C2211yescharacteristic
21#16812036—22no—
22#16812033—22no—
23#16812023—22no—
24C2×#41×C107C2211yescharacteristic
25C22×Q8×C107C2211yescharacteristic
26C4⋊Q8×C107—21no—
27#124×C107—21no—
28#123×C107—22no—
29C22⋊Q8×C107—22no—
30#117×C107C2211yescharacteristic
31#13133818—21no—
32#13133818—21no—
33#13133818—21no—
34#13133818—21no—
35C2×Dic214D411yescharacteristic
36C2×D428D411yescharacteristic
37C2×C22⋊D107—21no—
38C2×C4×D107—21no—
39#13133808—24no—
40#13133808—22no—
41#13133806—22no—
42#13133805—24no—
43#13133805—22no—
44#13133803—22no—
45#13133802—21no—
46#13133802—21no—
47#13133802—21no—
48#13133802D412yes—
49C2×C107⋊C8—21no—
50C2×C107⋊C8D412yes—
51C4⋊Dic107—21no—
52C22⋊Dic107—21no—
53C4×Dic107—21no—
54#41×C107—21no—
55#41×C107—21no—
56#41×C107—21no—
57#41×C107—21no—
58C2×Q8×C107—21no—
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107—21no—
62C2×Q8×C107—21no—
63C2×Q8×C107D412yes—
64C2×Q8×C107D411yescharacteristic
65C2×D4×C107—21no—
66C2×D4×C107D411yescharacteristic
67C22×C4×C107—21no—
68C22×C4×C107—21no—
69C22×C4×C107C2311yescharacteristic
70C4⋊C4×C107—24no—
71C4⋊C4×C107—21no—
72C22⋊C4×C107—24no—
73C22⋊C4×C107D412yes—
74C42×C107—21no—
75C23×C107—21no—
76C23×C107C2×D411yescharacteristic
77C22×D107—21no—
78Q8×C107—22no—
79Q8×C107—21no—
80Q8×C107—22no—
81Q8×C107—21no—
82Q8×C107—41no—
83Q8×C107—41no—
84D4×C107—21no—
85D4×C107—21no—
86D4×C107—41no—
87D4×C107—41no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—41no—
92C2×C4×C107—41no—
93C2×C4×C107—41no—
94C2×C4×C107—41no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107#4111yescharacteristic
101C22⋊D107—41no—
102C22⋊D107—41no—
103C2×Dic107—21no—
104C2×Dic107—41no—
105D428—21no—
106D428—21no—
107C4×D107—41no—
108C4×D107—41no—
109Dic214—21no—
110Dic214—21no—
111C107⋊C8—22no—
112C107⋊C8—22no—
113C22×C107—41no—
114C22×C107—42no—
115C22×C107—41no—
116C22×C107#12011yescharacteristic
117C22×C107#12111yescharacteristic
118C22×C107#12711yescharacteristic
119C2×D107—41no—
120C2×D107—81no—
121C4×C107—42no—
122C4×C107—41no—
123C4×C107—42no—
124C4×C107#12011yescharacteristic
125C4×C107#12011yescharacteristic
126C4×C107#12111yescharacteristic
127C4×C107#12111yescharacteristic
128Dic107—41no—
129Dic107—81no—
130C2×C107—21no—
131C2×C107—21no—
132C2×C107—21no—
133C2×C107—81no—
134C2×C107#39211yescharacteristic
135D107—81no—
136#2009—1071nomaximal, Sylow
137C107#200911yescharacteristic, complemented, Sylow
138#575—1071no—
139C2×#137—1071no—
140#536D10711yescharacteristic, complemented
141#421—1071no—
142#419—1071no—
143#410—1071no—
144#344—1071no—
145C2×#41—1071no—
146C2×#41C2×D10711yescharacteristic
147C22×Q8C2×D10711yescharacteristic
148#137—2142no—
149#137—2141no—
150#137—2142no—
151#137—2141no—
152AGL(1,ℤ/8)—2141no—
153AGL(1,ℤ/8)—2141no—
154#135—2142no—
155C2×Q16—2141no—
156C2×SD16—2141no—
157C2×M16—1071no—
158C2×M16—1071no—
159C4⋊Q8—21no—
160#124—21no—
161#123—22no—
162C22⋊Q8—22no—
163#117—1071no—
164#117C2×D10711yescharacteristic
165#104—2142no—
166#103—2142no—
167#99—2142no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—2141no—
173#41—2141no—
174#41—2141no—
175#41—2141no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1071no—
182C2×Q8C22⋊D10712yes—
183C2×Q8C22⋊D10711yescharacteristic
184C2×D4—21no—
185C2×D4—1071no—
186C2×D4—2141no—
187C2×D4C22⋊D10711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—2141no—
191C22×C4C22×D10711yescharacteristic
192Q16—2144no—
193Q16—2142no—
194SD16—2142no—
195SD16—2144no—
196SD16—2142no—
197D8—2142no—
198M16—1072no—
199M16—2141no—
200M16—2141no—
201M16—2141no—
202C2×C8—1072no—
203C2×C8—2141no—
204C4⋊C4—24no—
205C4⋊C4—21no—
206C4⋊C4—2141no—
207C22⋊C4—24no—
208C22⋊C4—2141no—
209C22⋊C4C22⋊D10712yes—
210C42—21no—
211C42—2141no—
212C23—21no—
213C23—2141no—
214C23C2×C22⋊D10711yescharacteristic
215Q8—22no—
216Q8—21no—
217Q8—22no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—2141no—
222Q8—2141no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—4281no—
228D4—4281no—
229D4—2141no—
230D4—2141no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—4281no—
239C2×C4—4281no—
240C2×C4—4281no—
241C2×C4—2141no—
242C2×C4D4×D10711yescharacteristic
243C2×C4D4×D10711yescharacteristic
244C2×C4D4×D10711yescharacteristic
245C2×C4C2×C22⋊D10711yescharacteristic
246C2×C4C2×C22⋊D10711yescharacteristic
247C2×C4#1313381711yescharacteristic
248C8—2142no—
249C8—2142no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—8561no—
254C22—4281no—
255C22#1681207811yescharacteristic
256C22#1681208011yescharacteristic
257C22#1681210511yescharacteristic
258C4—42no—
259C4—41no—
260C4—42no—
261C4—8561no—
262C4—4281no—
263C4#1681207711yescharacteristic
264C4#1681207711yescharacteristic
265C4#1681208111yescharacteristic
266C4#1681208111yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—8561no—
272C2#2029253511yescharacteristic
subgroupG/H
centerC2#20292535
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#536×C107C2
socleC2×C107#392
prime radical#20293425C2
hypercenter#536D107
hypocenterC107#2009
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20292644C2semidirect
4. #20292636C2semidirect
5. #20292600C2semidirect
6. #20292397C2semidirect
7. #536×C107C2semidirect
137. C107#2009semidirect
140. #536D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
C2×#168121402
#202926442
#202926362
#202926002
#202923972
#536×C1072
#2009107