Finite Group Databasebeta

← #28349110

Subgroups — #28349110

Order 13696.

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

normal subgroups only

This group has 350 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#20293350C211yesmaximal, characteristic, complemented
2#20292984C212yesmaximal, complemented
3#20292979C211yesmaximal, characteristic, complemented
4#20292931C212yesmaximal, complemented
5#20292914C211yesmaximal, characteristic, complemented
6#20292842C212yesmaximal
7#20292832C211yesmaximal, characteristic, complemented
8Q8×C107⋊C8C211yesmaximal, characteristic, complemented
9C4×#13133806C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic
11#554×C107C211yesmaximal, characteristic, complemented
12C2×#13133806C2212yes—
13C2×#13133803C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812104—22no—
16#16812097C2212yes—
17#16812085—21no—
18C4×Dic214C2212yes—
19C4×D428C2211yescharacteristic
20#16812067—21no—
21#16812042C2211yescharacteristic
22#16812040C2212yes—
23C4⋊(C107⋊C8)C2211yescharacteristic
24C4⋊(C107⋊C8)C2212yes—
25#16812032C2212yes—
26#16812032C2212yes—
27#16812032C2212yes—
28#16812030C2211yescharacteristic
29#16812029C2212yes—
30#16812029C2211yescharacteristic
31C4×C107⋊C8C2211yescharacteristic
32C4×C107⋊C8C2212yes—
33C4⋊Q8×C107C2211yescharacteristic
34#125×C107C2212yes—
35C22⋊Q8×C107—21no—
36C22⋊Q8×C107—22no—
37C4×Q8×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4×D4×C107C2212yes—
40C2×C4⋊C4×C107—21no—
41C2×Dic214C2312yes—
42C2×D428C2311yescharacteristic
43C2×C4×D107—21no—
44#13133806—24no—
45#13133803—22no—
46C2×C107⋊C8C2311yescharacteristic
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2312yes—
49C4⋊Dic107—21no—
50C4⋊Dic107—21no—
51C4⋊Dic107C2311yescharacteristic
52Dic107⋊C4—22no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—21no—
55(C2×D107)⋊C4—22no—
56C4×Dic107—22no—
57C2×Q8×C107—21no—
58C2×Q8×C107D411yescharacteristic
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C22×C4×C107—22no—
62C4⋊C4×C107—22no—
63C4⋊C4×C107—21no—
64C4⋊C4×C107—22no—
65C4⋊C4×C107D411yescharacteristic
66C4⋊C4×C107D412yes—
67C4⋊C4×C107C2312yes—
68C4⋊C4×C107C2311yescharacteristic
69C22⋊C4×C107—21no—
70C22⋊C4×C107—22no—
71C42×C107C2311yescharacteristic
72C42×C107C2312yes—
73C23×C107—21no—
74C22×D107—21no—
75Q8×C107—22no—
76Q8×C107—81no—
77D4×C107—21no—
78D4×C107#4112yes—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—42no—
82C2×C4×C107—82no—
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C2×Dic107—21no—
89C2×Dic107—22no—
90D428—21no—
91D428—21no—
92C4×D107—42no—
93Dic214—22no—
94Dic214—22no—
95C107⋊C8—21no—
96C107⋊C8—22no—
97C107⋊C8—21no—
98C22×C107—22no—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107C22×D411yescharacteristic
102C2×D107—41no—
103C2×D107—41no—
104C2×D107—41no—
105C4×C107—22no—
106C4×C107—22no—
107C4×C107—21no—
108C4×C107—41no—
109C4×C107—42no—
110C4×C107#13512yes—
111C4×C107C2×#4111yescharacteristic
112C4×C1072−1+411yescharacteristic
113Dic107—42no—
114Dic107—41no—
115C2×C107—42no—
116C2×C107#54811yescharacteristic
117C2×C107C2×#13511yescharacteristic
118C2×C107#57711yescharacteristic
119D107—81no—
120#3360—1071nomaximal, Sylow
121C107#336011yescharacteristic, complemented, Sylow
122#556—1071no—
123#554D10711yescharacteristic, complemented
124#488—1072no—
125#487—1071no—
126#479—1072no—
127#476—1071no—
128#463—1072no—
129#461—1071no—
130C8×Q8—1071no—
131C4×SD16—1072no—
132C4×D8—1071no—
133C2×SD16—1072no—
134C2×D8—1071no—
135C4⋊Q8—1071no—
136C4⋊Q8C2×D10711yescharacteristic
137#126—2142no—
138#125C2×D10712yes—
139#124—1072no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—2141no—
143C4×Q8—1072no—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1071no—
146C4×D4C2×D10711yescharacteristic
147C4×D4C2×D10712yes—
148#117—2141no—
149C2×C4⋊C4—21no—
150#107—1071no—
151#106—1072no—
152C4⋊C8—1071no—
153C4⋊C8—1072no—
154#103—1072no—
155#103—1072no—
156#103—1072no—
157#102—1071no—
158#102—1071no—
159#102—1072no—
160C4×C8—1071no—
161C4×C8—1072no—
162C2×Q8—21no—
163C2×Q8—1072no—
164C2×Q8C22⋊D10711yescharacteristic
165C2×D4—1071no—
166C2×D4C22×D10711yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—2141no—
170SD16—2144no—
171D8—2142no—
172C2×C8—1071no—
173C2×C8—1072no—
174C2×C8—1071no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—1071no—
179C4⋊C4—2141no—
180C4⋊C4—2142no—
181C4⋊C4—2141no—
182C4⋊C4C22⋊D10711yescharacteristic
183C4⋊C4C22⋊D10712yes—
184C4⋊C4C22×D10712yes—
185C4⋊C4C22×D10711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—2142no—
189C22⋊C4—2141no—
190C22⋊C4—2142no—
191C42—2142no—
192C42C22×D10711yescharacteristic
193C42C22×D10712yes—
194C23—21no—
195C23—2141no—
196Q8—22no—
197Q8—81no—
198Q8—2142no—
199Q8—2142no—
200D4—21no—
201D4—2141no—
202D4—2141no—
203D4#1313381912yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—4282no—
209C2×C4—2141no—
210C2×C4—2142no—
211C2×C4C2×C22⋊D10712yes—
212C2×C4C2×C22⋊D10712yes—
213C2×C4C2×C22⋊D10711yescharacteristic
214C2×C4C2×C22⋊D10711yescharacteristic
215C2×C4C23×D10711yescharacteristic
216C8—2141no—
217C8—2142no—
218C8—2141no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—4281no—
223C22—4281no—
224C22—4281no—
225C22C22×C22⋊D10711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—4281no—
232C4—4282no—
233C4#1681212912yes—
234C4C2×#1313381911yescharacteristic
235C4#1681215711yescharacteristic
236C2—42no—
237C2—8561no—
238C2#2029328811yescharacteristic
239C2C2×#1681212911yescharacteristic
240C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#554×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#554D107
hypocenterC107#3360
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
2. #20292984C2semidirect
3. #20292979C2semidirect
4. #20292931C2semidirect
5. #20292914C2semidirect
7. #20292832C2semidirect
8. Q8×C107⋊C8C2semidirect
9. C4×#13133806C2semidirect
11. #554×C107C2semidirect
121. C107#3360semidirect
123. #554D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202929842
#202929792
#202929312
#202929142
#202928422
#202928322
Q8×C107⋊C82
C4×#131338062
C4×#131338032
#554×C1072
#3360107