Finite Group Databasebeta

← #28349106

Subgroups — #28349106

Order 13696.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293337C211yesmaximal, characteristic, complemented
2#20292981C212yesmaximal, complemented
3#20292977C211yesmaximal, characteristic, complemented
4#20292934C212yesmaximal, complemented
5#20292917C211yesmaximal, characteristic, complemented
6#20292839C212yesmaximal
7#20292829C211yesmaximal, characteristic, complemented
8Q8×C107⋊C8C211yesmaximal, characteristic, complemented
9C4×#13133805C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic
11#556×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2212yes—
13C2×#13133803C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812101C2212yes—
16#16812087—22no—
17#16812085—21no—
18C4×D428C2211yescharacteristic
19C4×D428C2212yes—
20C4⋊C4×D107—21no—
21#16812042C2211yescharacteristic
22#16812040C2212yes—
23C4⋊(C107⋊C8)C2211yescharacteristic
24C4⋊(C107⋊C8)C2212yes—
25#16812033C2212yes—
26#16812033C2212yes—
27#16812033C2212yes—
28#16812030C2212yes—
29#16812030C2211yescharacteristic
30#16812029C2211yescharacteristic
31C4×C107⋊C8C2211yescharacteristic
32C4×C107⋊C8C2212yes—
33C4⋊Q8×C107C2211yescharacteristic
34#126×C107—22no—
35#124×C107C2212yes—
36C22⋊Q8×C107—21no—
37C4×Q8×C107C2212yes—
38C4×Q8×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40#117×C107—21no—
41C2×Dic214—21no—
42C2×D428C2311yescharacteristic
43C2×C4×D107—21no—
44C2×C4×D107—22no—
45#13133805—24no—
46#13133803—22no—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—21no—
51C4⋊Dic107C2312yes—
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55(C2×D107)⋊C4—22no—
56(C2×D107)⋊C4—21no—
57C2×Q8×C107D411yescharacteristic
58C2×Q8×C107C2312yes—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107—21no—
61C4⋊C4×C107—21no—
62C4⋊C4×C107—22no—
63C4⋊C4×C107—21no—
64C4⋊C4×C107D411yescharacteristic
65C4⋊C4×C107D412yes—
66C4⋊C4×C107C2311yescharacteristic
67C22⋊C4×C107—22no—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—22no—
70C42×C107—22no—
71C42×C107C2312yes—
72C42×C107C2311yescharacteristic
73C23×C107—21no—
74C22×D107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77Q8×C107—22no—
78D4×C107—21no—
79D4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—42no—
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#4112yes—
92C4×D107—42no—
93C4×D107—82no—
94Dic214—81no—
95C107⋊C8—21no—
96C107⋊C8—22no—
97C107⋊C8—21no—
98C22×C107—41no—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107C22×D411yescharacteristic
102C2×D107—22no—
103C2×D107—41no—
104C2×D107—41no—
105C4×C107—22no—
106C4×C107—21no—
107C4×C107—22no—
108C4×C107—42no—
109C4×C107—41no—
110C4×C107#13512yes—
111C4×C107C2×#4111yescharacteristic
112C4×C1072−1+411yescharacteristic
113Dic107—41no—
114Dic107—42no—
115C2×C107—81no—
116C2×C107#54811yescharacteristic
117C2×C107C2×#13511yescharacteristic
118C2×C107#57711yescharacteristic
119D107—42no—
120#3360—1071nomaximal, Sylow
121C107#336011yescharacteristic, complemented, Sylow
122#556D10711yescharacteristic, complemented
123#554—1071no—
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—22no—
138#125—1072no—
139#124C2×D10712yes—
140C22⋊Q8—21no—
141C22⋊Q8—2141no—
142C22⋊Q8—2142no—
143C4×Q8C2×D10712yes—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1071no—
146C4×D4—1072no—
147C4×D4C2×D10711yescharacteristic
148#117—21no—
149C2×C4⋊C4—2141no—
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—1072no—
159#102—1071no—
160C4×C8—1071no—
161C4×C8—1072no—
162C2×Q8—2141no—
163C2×Q8C22⋊D10711yescharacteristic
164C2×Q8C22×D10712yes—
165C2×D4—1071no—
166C2×D4C22×D10711yescharacteristic
167C22×C4—21no—
168C22×C4—2142no—
169C22×C4—2141no—
170SD16—2144no—
171D8—2142no—
172C2×C8—1071no—
173C2×C8—1072no—
174C2×C8—1071no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—1072no—
179C4⋊C4—1071no—
180C4⋊C4—2142no—
181C4⋊C4—2141no—
182C4⋊C4—2142no—
183C4⋊C4C22⋊D10711yescharacteristic
184C4⋊C4C22⋊D10712yes—
185C4⋊C4C22×D10711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—2141no—
190C22⋊C4—2142no—
191C42—22no—
192C42C22×D10711yescharacteristic
193C42C22×D10712yes—
194C23—21no—
195C23—2141no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—8561no—
200D4—21no—
201D4—21no—
202D4—1072no—
203D4—2141no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—8562no—
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—41no—
220C22—41no—
221C22—41no—
222C22—4281no—
223C22—4281no—
224C22—2142no—
225C22C22×C22⋊D10711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—4281no—
232C4—4282no—
233C4#1681212912yes—
234C4#41×D10711yescharacteristic
235C4#1681215611yescharacteristic
236C2—81no—
237C2—4282no—
238C2#2029328411yescharacteristic
239C2C2×#1681212911yescharacteristic
240C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#556×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#556D107
hypocenterC107#3360
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20293337C2semidirect
2. #20292981C2semidirect
3. #20292977C2semidirect
4. #20292934C2semidirect
5. #20292917C2semidirect
7. #20292829C2semidirect
8. Q8×C107⋊C8C2semidirect
9. C4×#13133805C2semidirect
11. #556×C107C2semidirect
121. C107#3360semidirect
122. #556D107semidirect

Maximal subgroups

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

subgroupindex
#202933372
#202929812
#202929772
#202929342
#202929172
#202928392
#202928292
Q8×C107⋊C82
C4×#131338052
C4×#131338032
#556×C1072
#3360107