Finite Group Databasebeta

← #28345969

Subgroups — #28345969

Order 13696.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812134C211yesmaximal, characteristic
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293101C211yesmaximal, characteristic, complemented
4#20292839C214yesmaximal, complemented
5#20292829C214yesmaximal, complemented
6#20292654C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292623C211yesmaximal, characteristic, complemented
9#536×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11#16812134—24no—
12C2×#13133805—24no—
13C2×#13133805C2214yes—
14C2×#13133805C2212yes—
15C2×#13133803C2212yes—
16C2×#13133802C2211yescharacteristic
17C22×C107⋊C8C2211yescharacteristic
18C4×D428C2214yes—
19#16812071—22no—
20#16812068C2211yescharacteristic
21C22⋊C4×D107—22no—
22C4⋊(C107⋊C8)C2214yescomplemented
23#16812033C2212yescomplemented
24#16812033C2214yescomplemented
25#16812030C2212yescomplemented
26C2×#41×C107C2211yescharacteristic
27C22×Q8×C107C2211yescharacteristic
28C4⋊Q8×C107C2212yescomplemented
29#124×C107C2212yescomplemented
30#123×C107—22no—
31C22⋊Q8×C107—22no—
32#117×C107C2211yescharacteristic
33C23×D107—21no—
34C2×D428—21no—
35C2×D428—22no—
36C2×D428D414yes—
37C2×D428C2312yes—
38C2×C4×D107—24no—
39#13133805—28no—
40#13133805—24no—
41#13133803—24no—
42#13133802—22no—
43C2×C107⋊C8—22no—
44C2×C107⋊C8C2312yes—
45C2×C107⋊C8C2312yes—
46C4⋊Dic107C2312yes—
47C22⋊Dic107—21no—
48(C2×D107)⋊C4—24no—
49C4×Dic107—21no—
50#41×C107—22no—
51#41×C107—22no—
52C2×Q8×C107—21no—
53C2×Q8×C107—22no—
54C2×Q8×C107—22no—
55C2×Q8×C107C2312yes—
56C2×Q8×C107C2311yescharacteristic
57C2×D4×C107—21no—
58C2×D4×C107C2311yescharacteristic
59C22×C4×C107—21no—
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62C4⋊C4×C107—24no—
63C4⋊C4×C107D412yes—
64C22⋊C4×C107—24no—
65C22⋊C4×C107D412yes—
66C42×C107C2312yes—
67C23×C107—21no—
68C23×C107C2×D411yescharacteristic
69C22×D107—22no—
70C22×D107—24no—
71C22×D107—82no—
72Q8×C107—22no—
73Q8×C107—21no—
74Q8×C107—22no—
75Q8×C107—21no—
76Q8×C107—42no—
77D4×C107—21no—
78D4×C107—21no—
79D4×C107—42no—
80C2×C4×C107—22no—
81C2×C4×C107—21no—
82C2×C4×C107—42no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D414yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—22no—
90D428—24no—
91D428—22no—
92D428C2×D414yes—
93C4×D107—84no—
94C107⋊C8—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107#12112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—24no—
102C2×D107—44no—
103C2×D107—84no—
104C4×C107—24no—
105C4×C107—41no—
106C4×C107—42no—
107C4×C107#12112yes—
108C4×C107C22×D411yescharacteristic
109C4×C107C2×#4111yescharacteristic
110Dic107—42no—
111C2×C107—22no—
112C2×C107—81no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—44no—
116#3042—1071nomaximal, Sylow
117C107#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1071no—
119C22×SD16—1071no—
120#536D10711yescharacteristic, complemented
121#518—1071no—
122#463—1074no—
123#461—1074no—
124#424—1071no—
125#419—1071no—
126#416—1071no—
127C2×#41C2×D10711yescharacteristic
128C22×Q8C2×D10711yescharacteristic
129C22×D4—1071no—
130AGL(1,ℤ/8)—2144no—
131C2×SD16—1074no—
132C2×SD16—1072no—
133C2×SD16—2144no—
134C2×D8—1072no—
135C2×M16—1071no—
136C22×C8—1071no—
137C4⋊Q8C2×D10712yescomplemented
138#124C2×D10712yescomplemented
139#123—22no—
140C22⋊Q8—22no—
141C4×D4—1074no—
142C4×D4—2142no—
143#117—1071no—
144#117C2×D10711yescharacteristic
145C2×C22⋊C4—2142no—
146C4⋊C8—1074no—
147#103—1072no—
148#103—1074no—
149#102—1072no—
150C24—2141no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8C22×D10712yes—
157C2×Q8C22×D10711yescharacteristic
158C2×D4—21no—
159C2×D4—1074no—
160C2×D4—1072no—
161C2×D4—2141no—
162C2×D4—2142no—
163C2×D4C22×D10711yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2144no—
167C22×C4C22×D10711yescharacteristic
168SD16—2148no—
169SD16—2144no—
170D8—2144no—
171M16—2142no—
172C2×C8—1072no—
173C2×C8—1072no—
174C2×C8—2142no—
175C4⋊C4—24no—
176C4⋊C4—1072no—
177C4⋊C4C22⋊D10712yes—
178C22⋊C4—24no—
179C22⋊C4—2141no—
180C22⋊C4—2144no—
181C22⋊C4C22⋊D10712yes—
182C42—2141no—
183C42C22×D10712yes—
184C23—21no—
185C23—8562no—
186C23—2144no—
187C23—2142no—
188C23C2×C22⋊D10711yescharacteristic
189Q8—22no—
190Q8—21no—
191Q8—22no—
192Q8—21no—
193Q8—42no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—1074no—
198D4—2144no—
199D4—2142no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—8564no—
205C2×C4—2142no—
206C2×C4D4×D10712yes—
207C2×C4C2×C22⋊D10714yes—
208C2×C4C2×C22⋊D10711yescharacteristic
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—2142no—
212C8—2142no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—8564no—
217C22—4284no—
218C22—2144no—
219C22#1681208012yes—
220C22C22×C22⋊D10711yescharacteristic
221C4—24no—
222C4—41no—
223C4—42no—
224C4—4282no—
225C4#1681208012yes—
226C4C2×D4×D10711yescharacteristic
227C4C2×#1313381711yescharacteristic
228C2—22no—
229C2—81no—
230C2—4284no—
231C2C2×#1681208011yescharacteristic
232C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812134C2
hypercenter#536D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#13133805C2semidirect
3. #20293101C2semidirect
4. #20292839C2semidirect
5. #20292829C2semidirect
6. #20292654C2semidirect
7. #20292636C2semidirect
8. #20292623C2semidirect
9. #536×C107C2semidirect
22. C4⋊(C107⋊C8)C22semidirect
23. #16812033C22semidirect
24. #16812033C22semidirect
25. #16812030C22semidirect
28. C4⋊Q8×C107C22semidirect
29. #124×C107C22semidirect
117. C107#3042semidirect
120. #536D107semidirect
137. C4⋊Q8C2×D107semidirect
138. #124C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C22×#131338052
#202931012
#202928392
#202928292
#202926542
#202926362
#202926232
#536×C1072
#3042107