Finite Group Databasebeta

← #28345975

Subgroups — #28345975

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
1C22×#13133806C211yesmaximal, characteristic, complemented
2#20293198C211yesmaximal, characteristic, complemented
3#20293102C211yesmaximal, characteristic, complemented
4#20292840C214yesmaximal, complemented
5#20292830C214yesmaximal, complemented
6#20292655C211yesmaximal, characteristic, complemented
7#20292637C211yesmaximal, characteristic, complemented
8#20292623C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×#13133806—24no—
12C2×#13133806C2214yes—
13C22×C107⋊C8C2211yescharacteristic
14#16812107C2212yescomplemented
15#16812096C2212yescomplemented
16#16812090—22no—
17#16812088—22no—
18D4×Dic107—22no—
19D4×Dic107C2214yes—
20#16812068C2211yescharacteristic
21#16812068C2211yescharacteristic
22C2×C22⋊Dic107—22no—
23Dic107⋊C8C2214yescomplemented
24#16812034C2214yescomplemented
25#16812033C2212yescomplemented
26#16812030C2212yescomplemented
27C2×#41×C107C2211yescharacteristic
28C22×D4×C107C2211yescharacteristic
29AGL(1,ℤ/8)×C107—24no—
30C2×SD16×C107C2212yes—
31C2×D8×C107C2212yes—
32C2×M16×C107C2211yescharacteristic
33C2×Dic214—22no—
34C2×Dic214—22no—
35C2×Dic214—21no—
36C2×Dic214C2312yes—
37C22×Dic107—21no—
38C22×Dic107—24no—
39#13133806—28no—
40C2×C107⋊C8—22no—
41C2×C107⋊C8C2312yes—
42C4⋊Dic107D412yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—24no—
45C22⋊Dic107—21no—
46C22⋊Dic107—24no—
47C22⋊Dic107—24no—
48C22⋊Dic107D412yes—
49C4×Dic107—21no—
50C4×Dic107C2312yes—
51C24×C107—21no—
52#41×C107—22no—
53#41×C107—22no—
54C2×Q8×C107C2311yescharacteristic
55C2×D4×C107—21no—
56C2×D4×C107—21no—
57C2×D4×C107—22no—
58C2×D4×C107D414yes—
59C2×D4×C107C2311yescharacteristic
60C2×D4×C107C2312yes—
61C22×C4×C107—21no—
62C22×C4×C107C2311yescharacteristic
63SD16×C107—24no—
64D8×C107—24no—
65M16×C107—22no—
66C2×C8×C107C2312yes—
67C23×C107—22no—
68C23×C107—24no—
69C23×C107—21no—
70C23×C107—82no—
71C23×C107C2×D411yescharacteristic
72Q8×C107—21no—
73Q8×C107—21no—
74D4×C107—24no—
75D4×C107—21no—
76D4×C107—22no—
77D4×C107—21no—
78D4×C107—42no—
79D4×C107C2×D414yes—
80C2×C4×C107—21no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107#4112yes—
85C2×C4×C107C2411yescharacteristic
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—42no—
89C2×Dic107—84no—
90C2×Dic107C2×D414yes—
91Dic214—22no—
92Dic214—22no—
93Dic214—42no—
94C8×C107—22no—
95C107⋊C8—22no—
96C22×C107—22no—
97C22×C107—24no—
98C22×C107—44no—
99C22×C107—41no—
100C22×C107—42no—
101C22×C107—84no—
102C22×C107#12112yes—
103C22×C107C22×D411yescharacteristic
104C4×C107—41no—
105C4×C107#12112yes—
106C4×C107C22×D411yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—24no—
109Dic107—42no—
110Dic107—42no—
111C2×C107—22no—
112C2×C107—44no—
113C2×C107—81no—
114C2×C107C2×#12111yescharacteristic
115C2×C107#57712yes—
116#3042—1071nomaximal, Sylow
117C107#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)D10711yescharacteristic
119C22×SD16—1071no—
120#536—1071no—
121#518—1071no—
122#463—1074no—
123#461—1074no—
124#424—1071no—
125#419—1071no—
126#416—1071no—
127C2×#41C2×D10711yescharacteristic
128C22×Q8—1071no—
129C22×D4C2×D10711yescharacteristic
130AGL(1,ℤ/8)—24no—
131C2×SD16—1074no—
132C2×SD16—2144no—
133C2×SD16C2×D10712yes—
134C2×D8C2×D10712yes—
135C2×M16C2×D10711yescharacteristic
136C22×C8—1071no—
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—1072no—
148#103—1074no—
149#102—1072no—
150C24—21no—
151#41—22no—
152#41—22no—
153C2×Q8—1072no—
154C2×Q8—2142no—
155C2×Q8—2142no—
156C2×Q8—2141no—
157C2×Q8C22×D10711yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4C22⋊D10714yes—
162C2×D4C22×D10711yescharacteristic
163C2×D4C22×D10712yes—
164C22×C4—21no—
165C22×C4—2144no—
166C22×C4—2141no—
167C22×C4C22×D10711yescharacteristic
168SD16—24no—
169SD16—2148no—
170D8—24no—
171M16—22no—
172C2×C8—1072no—
173C2×C8—2142no—
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—21no—
187C23—82no—
188C23D4×D10711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—4282no—
192Q8—2142no—
193Q8—2142no—
194D4—24no—
195D4—21no—
196D4—22no—
197D4—21no—
198D4—42no—
199D4C2×C22⋊D10714yes—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—8564no—
203C2×C4—1074no—
204C2×C4—4282no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4D4×D10711yescharacteristic
208C2×C4C2×C22⋊D10712yes—
209C2×C4#1313381712yes—
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2142no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—42no—
217C22—41no—
218C22—84no—
219C22#1681208112yes—
220C22C2×D4×D10711yescharacteristic
221C4—41no—
222C4—4282no—
223C4—4282no—
224C4—2144no—
225C4#1681208112yes—
226C4C22×C22⋊D10711yescharacteristic
227C4C2×#1313381711yescharacteristic
228C2—22no—
229C2—44no—
230C2—81no—
231C2C2×#1681208111yescharacteristic
232C2#2029343712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3042
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#13133806C2semidirect
2. #20293198C2semidirect
3. #20293102C2semidirect
4. #20292840C2semidirect
5. #20292830C2semidirect
6. #20292655C2semidirect
7. #20292637C2semidirect
8. #20292623C2semidirect
14. #16812107C22semidirect
15. #16812096C22semidirect
23. Dic107⋊C8C22semidirect
24. #16812034C22semidirect
25. #16812033C22semidirect
26. #16812030C22semidirect
117. C107#3042semidirect

Maximal subgroups

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

subgroupindex
C22×#131338062
#202931982
#202931022
#202928402
#202928302
#202926552
#202926372
#202926232
C2×AGL(1,ℤ/8)×C1072
#3042107