Finite Group Databasebeta

← #28349098

Subgroups — #28349098

Order 13696.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293352C211yesmaximal, characteristic, complemented
2D4×Dic214C211yesmaximal, characteristic
3#20292977C211yesmaximal, characteristic, complemented
4#20292976C212yesmaximal, complemented
5#20292913C211yesmaximal, characteristic, complemented
6#20292910C212yesmaximal, complemented
7#20292830C212yesmaximal
8#20292751C211yesmaximal, characteristic, complemented
9SD16×Dic107C212yesmaximal
10C4×#13133806C211yesmaximal, characteristic
11#461×C107C211yesmaximal, characteristic
12C22×Dic214—21no—
13C2×#13133806C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812108C2212yes—
16#16812107C2212yes—
17#16812100—21no—
18#16812099—22no—
19#16812088—21no—
20#16812086—22no—
21Q8×Dic107C2212yes—
22C4×Dic214C2211yescharacteristic
23C4×Dic214C2211yescharacteristic
24#16812074—21no—
25D4×Dic107C2212yes—
26#16812041C2212yes—
27#16812040C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812034C2212yes—
30#16812033C2212yes—
31#16812033C2212yes—
32#16812032C2211yescharacteristic
33#16812030C2211yescharacteristic
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×SD16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214—44no—
44C2×Dic214D411yescharacteristic
45C2×Dic214C2311yescharacteristic
46C22×Dic107—22no—
47#13133806—22no—
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2311yescharacteristic
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2312yes—
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—21no—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59Dic107⋊C4D412yes—
60C22⋊Dic107—22no—
61C4×Dic107—21no—
62C4×Dic107—22no—
63C4×Dic107C2312yes—
64C2×Q8×C107C2312yes—
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67SD16×C107—24no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2311yescharacteristic
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77D4×C107—21no—
78D4×C107#4112yes—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—22no—
88C2×Dic107—44no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91Dic214—21no—
92Dic214—21no—
93Dic214—42no—
94Dic214—44no—
95Dic214SD1614yescomplemented
96C8×C107—22no—
97C107⋊C8—21no—
98C107⋊C8—21no—
99C22×C107—22no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107C22×D411yescharacteristic
103C4×C107—21no—
104C4×C107—42no—
105C4×C107—41no—
106C4×C107C2×SD1612yes—
107C4×C107C2×#4111yescharacteristic
108C4×C1072−1+411yescharacteristic
109Dic107—22no—
110Dic107—41no—
111Dic107—42no—
112Dic107—41no—
113Dic107C2×SD1614yes—
114C2×C107—42no—
115C2×C107#54811yescharacteristic
116C2×C107C22×SD1611yescharacteristic
117C2×C107#57811yescharacteristic
118#3359—1071nomaximal, Sylow
119C107#335911yescharacteristic, complemented, Sylow
120#557—1071no—
121D4×Q8—1071no—
122#487—1071no—
123#487—1072no—
124#475—1071no—
125#475—1072no—
126#461—1072no—
127#461D10711yescharacteristic
128C8×Q8—1071no—
129C4×SD16—1072no—
130C4×SD16—1071no—
131C22×Q8—2141no—
132C2×SD16—1071no—
133C2×SD16C2×D10712yes—
134C4⋊Q8—1072no—
135C4⋊Q8—1072no—
136C4⋊Q8—1071no—
137C4⋊Q8C2×D10711yescharacteristic
138#125—2142no—
139#125—2141no—
140C22⋊Q8—2142no—
141C22⋊Q8—2141no—
142C4×Q8—1071no—
143C4×Q8—1072no—
144C4×Q8—1071no—
145C4×Q8—2141no—
146C4×D4—1072no—
147C4×D4C2×D10711yescharacteristic
148#106—1072no—
149#106—1071no—
150C4⋊C8—1072no—
151C4⋊C8C2×D10711yescharacteristic
152#103—1072no—
153#103—1072no—
154#103—1072no—
155#103—1071no—
156#103C2×D10712yes—
157#102—1071no—
158C4×C8—1072no—
159C4×C8—1071no—
160C2×Q8—1071no—
161C2×Q8—1071no—
162C2×Q8—4284no—
163C2×Q8—2142no—
164C2×Q8—2141no—
165C2×Q8C22×D10712yes—
166C2×D4C22×D10711yescharacteristic
167C22×C4—21no—
168C22×C4—2142no—
169SD16—24no—
170SD16—2142no—
171C2×C8—1071no—
172C2×C8—1071no—
173C2×C8C22×D10712yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—1072no—
177C4⋊C4—1071no—
178C4⋊C4—1071no—
179C4⋊C4—1072no—
180C4⋊C4—2142no—
181C4⋊C4—2142no—
182C4⋊C4—2142no—
183C4⋊C4—2142no—
184C4⋊C4—2141no—
185C4⋊C4—2141no—
186C4⋊C4C22×D10711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—2142no—
189C42—1072no—
190C42—2142no—
191C42—2141no—
192C42C22×D10711yescharacteristic
193C23—21no—
194Q8—22no—
195Q8—22no—
196Q8—1074no—
197Q8—4284no—
198Q8—4282no—
199Q8—2141no—
200Q8—2141no—
201D4—21no—
202D4#1313381712yes—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—1072no—
207C2×C4—1072no—
208C2×C4—4284no—
209C2×C4—2141no—
210C2×C4—2141no—
211C2×C4—2142no—
212C2×C4D4×D10711yescharacteristic
213C2×C4D4×D10711yescharacteristic
214C2×C4C23×D10711yescharacteristic
215C8—22no—
216C8—2141no—
217C8—2141no—
218C22—22no—
219C22—41no—
220C22—41no—
221C22C2×D4×D10711yescharacteristic
222C4—21no—
223C4—41no—
224C4—42no—
225C4—1074no—
226C4—4282no—
227C4—4281no—
228C4—4281no—
229C4—2142no—
230C4SD16×D10712yes—
231C4C2×#1313381711yescharacteristic
232C4#1681215611yescharacteristic
233C2—42no—
234C2#2029328311yescharacteristic
235C2C2×SD16×D10711yescharacteristic
236C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×C107C23
hypercenter#461D107
hypocenterC107#3359
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20293352C2semidirect
3. #20292977C2semidirect
4. #20292976C2semidirect
5. #20292913C2semidirect
6. #20292910C2semidirect
8. #20292751C2semidirect
95. Dic214SD16semidirect
119. C107#3359semidirect

Maximal subgroups

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

subgroupindex
#202933522
D4×Dic2142
#202929772
#202929762
#202929132
#202929102
#202928302
#202927512
SD16×Dic1072
C4×#131338062
#461×C1072
#3359107