Finite Group Databasebeta

← #28349112

Subgroups — #28349112

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#20293350C211yesmaximal, characteristic, complemented
2#20293337C211yesmaximal, characteristic, complemented
3#20292982C212yesmaximal, complemented
4#20292978C211yesmaximal, characteristic, complemented
5#20292935C212yesmaximal, complemented
6#20292918C211yesmaximal, characteristic, complemented
7#20292843C212yesmaximal
8#20292833C211yesmaximal, characteristic, complemented
9C4×#13133807C212yesmaximal, complemented
10C4×D856C211yesmaximal, characteristic
11C8×Q8×C107C211yesmaximal, characteristic, complemented
12C2×#13133807C2212yes—
13C2×D856C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812104—22no—
17#16812101C2212yes—
18#16812097C2212yes—
19#16812087—22no—
20#16812085—21no—
21#16812085—21no—
22C4×Dic214C2212yes—
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25C4×D428C2212yes—
26#16812067—21no—
27C4⋊C4×D107—21no—
28#16812043C2211yescharacteristic
29#16812041C2212yes—
30#16812034C2212yes—
31#16812034C2212yes—
32#16812034C2212yes—
33#16812031C2212yes—
34#16812031C2211yescharacteristic
35#16812031C2211yescharacteristic
36C4×Q8×C107C2211yescharacteristic
37C4⋊C8×C107C2211yescharacteristic
38C4⋊C8×C107C2212yes—
39C4×C8×C107C2211yescharacteristic
40C4×C8×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214C2312yes—
43C2×D428C2311yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C4×D107—21no—
46C2×C4×D107—21no—
47C2×C4×D107—22no—
48#13133807—24no—
49D856—22no—
50C4⋊Dic107—21no—
51C4⋊Dic107—21no—
52C4⋊Dic107—21no—
53C4⋊Dic107C2312yes—
54C4⋊Dic107C2311yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—21no—
63(C2×D107)⋊C4—22no—
64C4×Dic107—22no—
65C2×Q8×C107D411yescharacteristic
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C2×C8×C107C2312yes—
69C4⋊C4×C107D411yescharacteristic
70C4⋊C4×C107D412yes—
71C42×C107C2311yescharacteristic
72C42×C107C2312yes—
73C22×D107—21no—
74C22×D107—21no—
75Q8×C107—22no—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2411yescharacteristic
81C2×Dic107—21no—
82C2×Dic107—22no—
83C2×Dic107—21no—
84C2×Dic107—22no—
85D428—21no—
86D428—21no—
87D428—21no—
88D428#4112yes—
89C4×D107—42no—
90C4×D107—42no—
91C4×D107—82no—
92Dic214—22no—
93Dic214—22no—
94Dic214—81no—
95C8×C107—21no—
96C8×C107—22no—
97C8×C107—21no—
98C22×C107C22×D411yescharacteristic
99C2×D107—22no—
100C2×D107—41no—
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C2×D107—41no—
105C4×C107—22no—
106C4×C107—21no—
107C4×C107—22no—
108C4×C107#13512yes—
109C4×C107C2×#4111yescharacteristic
110C4×C1072−1+411yescharacteristic
111Dic107—42no—
112Dic107—41no—
113Dic107—41no—
114Dic107—42no—
115C2×C107#54811yescharacteristic
116C2×C107C2×#13511yescharacteristic
117C2×C107#57711yescharacteristic
118D107—42no—
119D107—81no—
120#3360—1071nomaximal, Sylow
121C107#336011yescharacteristic, complemented, Sylow
122#556—1071no—
123#554—1071no—
124#488—1072no—
125#487—1071no—
126#479—1072no—
127#476—1071no—
128#463—1072no—
129#461—1071no—
130C8×Q8D10711yescharacteristic, complemented
131C4×SD16—1072no—
132C4×D8—1071no—
133C2×SD16—1072no—
134C2×D8—1071no—
135C4⋊Q8—1071no—
136C4⋊Q8—1071no—
137#126—2142no—
138#125—1072no—
139#124—1072no—
140C22⋊Q8—2141no—
141C22⋊Q8—2141no—
142C22⋊Q8—2142no—
143C4×Q8—1072no—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1071no—
146C4×D4—1072no—
147C4×D4—1071no—
148#117—2141no—
149C2×C4⋊C4—2141no—
150#107—1071no—
151#106—1072no—
152C4⋊C8C2×D10711yescharacteristic
153C4⋊C8C2×D10712yes—
154#103—1072no—
155#103—1072no—
156#103—1072no—
157#102—1071no—
158#102—1071no—
159#102—1072no—
160C4×C8C2×D10711yescharacteristic
161C4×C8C2×D10712yes—
162C2×Q8—1072no—
163C2×Q8—2141no—
164C2×Q8D42811yescharacteristic
165C2×D4—1071no—
166C2×D4—1071no—
167C22×C4—2142no—
168C22×C4—2141no—
169C22×C4—2141no—
170SD16—2144no—
171D8—2142no—
172C2×C8C22×D10711yescharacteristic
173C2×C8C22×D10712yes—
174C2×C8C22×D10711yescharacteristic
175C4⋊C4—1072no—
176C4⋊C4—1071no—
177C4⋊C4—1071no—
178C4⋊C4—2142no—
179C4⋊C4—2141no—
180C4⋊C4—2141no—
181C4⋊C4—2142no—
182C4⋊C4—2142no—
183C4⋊C4—2141no—
184C4⋊C4D42811yescharacteristic
185C4⋊C4D42812yes—
186C22⋊C4—2142no—
187C22⋊C4—2141no—
188C22⋊C4—2142no—
189C22⋊C4—2141no—
190C22⋊C4—2142no—
191C42—2142no—
192C42C22×D10711yescharacteristic
193C42C22×D10712yes—
194C23—2141no—
195C23—2141no—
196Q8—22no—
197Q8—8561no—
198Q8—2142no—
199Q8—2142no—
200D4—1072no—
201D4—2141no—
202D4—2141no—
203D4—2141no—
204C2×C4—8562no—
205C2×C4—4282no—
206C2×C4—4282no—
207C2×C4—2141no—
208C2×C4—2142no—
209C2×C4—2141no—
210C2×C4—2142no—
211C2×C4C2×D42812yes—
212C2×C4C2×D42812yes—
213C2×C4C2×D42811yescharacteristic
214C2×C4C2×D42811yescharacteristic
215C2×C4C23×D10711yescharacteristic
216C8—21no—
217C8—22no—
218C8—21no—
219C22—4281no—
220C22—4281no—
221C22—4281no—
222C22—4281no—
223C22—4281no—
224C22—2142no—
225C22C22×D42811yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—4281no—
230C4—4281no—
231C4—4282no—
232C4—4282no—
233C4#1681213112yes—
234C4#41×D10711yescharacteristic
235C4#1681215711yescharacteristic
236C2—8561no—
237C2—4282no—
238C2#2029328911yescharacteristic
239C2C2×#1681213111yescharacteristic
240C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×Q8×C107C2
socleC22×C107C22×D4
prime radicalC2×D856C22
hypercenterC8×Q8D107
hypocenterC107#3360
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
2. #20293337C2semidirect
3. #20292982C2semidirect
4. #20292978C2semidirect
5. #20292935C2semidirect
6. #20292918C2semidirect
8. #20292833C2semidirect
9. C4×#13133807C2semidirect
11. C8×Q8×C107C2semidirect
121. C107#3360semidirect
130. C8×Q8D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202933372
#202929822
#202929782
#202929352
#202929182
#202928432
#202928332
C4×#131338072
C4×D8562
C8×Q8×C1072
#3360107