Finite Group Databasebeta

← #28345970

Subgroups — #28345970

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×#16812135C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic, complemented
3#20293102C211yesmaximal, characteristic, complemented
4#20292844C214yesmaximal, complemented
5#20292830C214yesmaximal, complemented
6#20292653C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292624C211yesmaximal, characteristic, complemented
9C22×SD16×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812135—24no—
12C2×#13133806C2212yes—
13C2×#13133803C2212yes—
14C2×#13133802C2211yescharacteristic
15#16812107C2212yescomplemented
16#16812098C2212yescomplemented
17#16812092—22no—
18#16812085—22no—
19D4×Dic107—22no—
20D4×Dic107C2214yes—
21#16812068C2211yescharacteristic
22#16812068C2211yescharacteristic
23C2×C22⋊Dic107—22no—
24Dic107⋊C8C2214yescomplemented
25#16812034C2212yescomplemented
26#16812033C2214yescomplemented
27#16812031C2212yescomplemented
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30C2×SD16×C107—24no—
31C2×SD16×C107C2214yes—
32C22×C8×C107C2211yescharacteristic
33#13133818—22no—
34#13133818—22no—
35C2×Dic214C2311yescharacteristic
36C2×D428C2311yescharacteristic
37C2×C22⋊D107—21no—
38C22×Dic107—24no—
39C2×C4×D107—21no—
40#13133806—24no—
41#13133803—24no—
42#13133802—22no—
43C2×C107⋊C8C2312yes—
44C4⋊Dic107D412yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—24no—
47C22⋊Dic107—21no—
48C22⋊Dic107—24no—
49C22⋊Dic107D412yes—
50(C2×D107)⋊C4—24no—
51C4×Dic107—21no—
52C4×Dic107C2312yes—
53C24×C107—21no—
54C2×Q8×C107—21no—
55C2×Q8×C107—22no—
56C2×Q8×C107—22no—
57C2×Q8×C107C2312yes—
58C2×D4×C107—21no—
59C2×D4×C107—22no—
60C2×D4×C107D414yes—
61C2×D4×C107C2312yes—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64SD16×C107—28no—
65C2×C8×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—82no—
70C23×C107C2×D411yescharacteristic
71C22×D107—21no—
72Q8×C107—22no—
73Q8×C107—22no—
74Q8×C107—42no—
75D4×C107—24no—
76D4×C107—22no—
77D4×C107C2×D414yes—
78C2×C4×C107—22no—
79C2×C4×C107—42no—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D412yes—
82C2×C4×C107#4112yes—
83C2×C4×C107C2411yescharacteristic
84C22⋊D107—42no—
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107—84no—
88C2×Dic107C2×D414yes—
89D428—21no—
90D428—21no—
91C4×D107—42no—
92Dic214—21no—
93Dic214—21no—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—24no—
98C22×C107—44no—
99C22×C107—84no—
100C22×C107#12112yes—
101C22×C107C22×D411yescharacteristic
102C2×D107—41no—
103C2×D107—42no—
104C4×C107—42no—
105C4×C107#12112yes—
106C4×C107C22×D411yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—24no—
109Dic107—41no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—44no—
113C2×C107C2×#12111yescharacteristic
114C2×C107#57712yes—
115D107—81no—
116#3042—1071nomaximal, Sylow
117C107#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1071no—
119C22×SD16D10711yescharacteristic, complemented
120#536—1071no—
121#518—1071no—
122#463—1074no—
123#461—1074no—
124#424—1071no—
125#419—1071no—
126#416—1071no—
127C2×#41—1071no—
128C22×Q8C2×D10711yescharacteristic
129C22×D4C2×D10711yescharacteristic
130AGL(1,ℤ/8)—2144no—
131C2×SD16—24no—
132C2×SD16—1072no—
133C2×SD16C2×D10714yes—
134C2×D8—1072no—
135C2×M16—1071no—
136C22×C8C2×D10711yescharacteristic
137C4⋊Q8—1072no—
138#124—1072no—
139#123—2142no—
140C22⋊Q8—2142no—
141C4×D4—1074no—
142C4×D4—2142no—
143#117—1071no—
144#117—1071no—
145C2×C22⋊C4—2142no—
146C4⋊C8—1074no—
147#103—1074no—
148#103—1072no—
149#102—1072no—
150C24—21no—
151#41—2142no—
152#41—2142no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1071no—
157C2×Q8C22×D10712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1071no—
161C2×D4—2141no—
162C2×D4C22⋊D10714yes—
163C2×D4C22×D10712yes—
164C22×C4—21no—
165C22×C4—2144no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168SD16—28no—
169SD16—2144no—
170D8—2144no—
171M16—2142no—
172C2×C8—22no—
173C2×C8—1072no—
174C2×C8C22×D10712yes—
175C4⋊C4—1072no—
176C4⋊C4—1072no—
177C4⋊C4—2144no—
178C22⋊C4—1072no—
179C22⋊C4—2144no—
180C22⋊C4—2144no—
181C22⋊C4—2141no—
182C42—1072no—
183C42—2141no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2141no—
188C23D4×D10711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2141no—
193Q8—2141no—
194D4—24no—
195D4—22no—
196D4—4282no—
197D4—2141no—
198D4—2141no—
199D4C2×C22⋊D10714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—8564no—
203C2×C4—1074no—
204C2×C4—4282no—
205C2×C4—2142no—
206C2×C4—2141no—
207C2×C4D4×D10711yescharacteristic
208C2×C4C2×C22⋊D10712yes—
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—4282no—
218C22—4281no—
219C22#1681208112yes—
220C22C2×D4×D10711yescharacteristic
221C4—42no—
222C4—4282no—
223C4—4281no—
224C4—2144no—
225C4#1681208112yes—
226C4C22×C22⋊D10711yescharacteristic
227C4C2×#1313381711yescharacteristic
228C2—22no—
229C2—44no—
230C2—8561no—
231C2C2×#1681208111yescharacteristic
232C2#2029343312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812135C2
hypercenterC22×SD16D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #20293201C2semidirect
3. #20293102C2semidirect
4. #20292844C2semidirect
5. #20292830C2semidirect
6. #20292653C2semidirect
7. #20292636C2semidirect
8. #20292624C2semidirect
9. C22×SD16×C107C2semidirect
15. #16812107C22semidirect
16. #16812098C22semidirect
24. Dic107⋊C8C22semidirect
25. #16812034C22semidirect
26. #16812033C22semidirect
27. #16812031C22semidirect
117. C107#3042semidirect
119. C22×SD16D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
#202932012
#202931022
#202928442
#202928302
#202926532
#202926362
#202926242
C22×SD16×C1072
#3042107