Finite Group Databasebeta

← #28347923

Subgroups — #28347923

Order 13696.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#20293250C212yesmaximal, complemented
2#20292920C212yesmaximal, complemented
3#20292906C212yesmaximal, complemented
4#20292875C212yesmaximal, complemented
5#20292863C211yesmaximal, characteristic
6#20292851C212yesmaximal, complemented
7#20292845C212yesmaximal, complemented
8#20292653C211yesmaximal, characteristic, complemented
9#467×C107C211yesmaximal, characteristic, complemented
10C2×#13133805C2212yes—
11C2×D856C2211yescharacteristic
12C2×#13133802C2211yescharacteristic
13#16812108C2212yes—
14#16812104—22no—
15#16812098C2212yes—
16#16812095—22no—
17#16812091—22no—
18#16812089—22no—
19#16812083C2212yes—
20Q8×Dic107C2212yes—
21#16812071C2212yes—
22#16812068C2211yescharacteristic
23#16812043C2211yescharacteristic
24#16812042C2212yes—
25Dic107⋊C8C2212yes—
26Dic107⋊C8C2212yes—
27#16812033C2212yes—
28#16812031C2212yes—
29#16812029C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C22×C8×C107C2211yescharacteristic
32C22⋊Q8×C107C2212yes—
33#107×C107C2211yescharacteristic
34#103×C107C2212yes—
35C2×Dic214—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133805—42no—
40D856—41no—
41#13133802—41no—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107—22no—
44C4⋊Dic107D411yescharacteristic
45C4⋊Dic107D411yescharacteristic
46C4⋊Dic107C2312yes—
47Dic107⋊C4—22no—
48C22⋊Dic107D412yes—
49(C2×D107)⋊C4—22no—
50(C2×D107)⋊C4—22no—
51(C2×D107)⋊C4—22no—
52(C2×D107)⋊C4—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C22×C4×C107C2311yescharacteristic
57Q16×C107—41no—
58C2×C8×C107—41no—
59C2×C8×C107C2311yescharacteristic
60C2×C8×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107C2312yes—
63C22⋊C4×C107—22no—
64C23×C107C2×D411yescharacteristic
65C22×D107—22no—
66Q8×C107—22no—
67Q8×C107—22no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2411yescharacteristic
74C22⋊D107—82no—
75C2×Dic107—22no—
76C2×Dic107C2×D412yes—
77C2×Dic107C2×D412yes—
78D428—22no—
79D428—22no—
80C4×D107—82no—
81Dic214—82no—
82C8×C107—21no—
83C8×C107—21no—
84C107⋊C8—22no—
85C22×C107—21no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107C22×D411yescharacteristic
89C2×D107—42no—
90C2×D107—42no—
91C2×D107—42no—
92C4×C107—21no—
93C4×C107—42no—
94C4×C107—42no—
95C4×C107H(5,𝔽2)11yescharacteristic
96C4×C1072−1+411yescharacteristic
97Dic107—22no—
98Dic107—22no—
99Dic107—42no—
100C2×C107—41no—
101C2×C107#53711yescharacteristic
102C2×C107#57611yescharacteristic
103C2×C107#57811yescharacteristic
104D107—82no—
105#3262—1071nomaximal, Sylow
106C107#326211yescharacteristic, complemented, Sylow
107#542—1072no—
108#477—1072no—
109#474—1072no—
110#468—1072no—
111#467D10711yescharacteristic, complemented
112#466—1071no—
113#464—1072no—
114#463—1072no—
115#424—1071no—
116C2×Q16C2×D10711yescharacteristic
117C2×SD16—1072no—
118C2×D8—1071no—
119C2×M16—1071no—
120C22×C8C2×D10711yescharacteristic
121C4⋊Q8—1072no—
122#126—2142no—
123#124—1072no—
124#124—2142no—
125#123—2142no—
126C22⋊Q8—2142no—
127C22⋊Q8C2×D10712yes—
128#121—1072no—
129C4×Q8—1072no—
130C4×D4—1072no—
131#117—1071no—
132#107—1071no—
133#107—1072no—
134#107C2×D10711yescharacteristic
135C4⋊C8—1072no—
136C4⋊C8—1072no—
137#103—1072no—
138#103C2×D10712yes—
139#102—1072no—
140#102—1072no—
141C2×Q8—2142no—
142C2×Q8C22×D10712yes—
143C2×D4—1072no—
144C2×D4—2142no—
145C22×C4—2142no—
146C22×C4C22×D10711yescharacteristic
147Q16—41no—
148SD16—4282no—
149D8—4281no—
150M16—4281no—
151C2×C8—41no—
152C2×C8—1072no—
153C2×C8C22×D10711yescharacteristic
154C2×C8C22×D10711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—1071no—
157C4⋊C4—1071no—
158C4⋊C4—1072no—
159C4⋊C4—2142no—
160C4⋊C4—2142no—
161C4⋊C4C22×D10712yes—
162C22⋊C4—22no—
163C22⋊C4—1072no—
164C22⋊C4—2142no—
165C22⋊C4—2142no—
166C22⋊C4—2142no—
167C22⋊C4—2142no—
168C42—1072no—
169C42—2142no—
170C23—2142no—
171C23D4×D10711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—8562no—
175D4—8562no—
176D4—2142no—
177D4—2142no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—8562no—
183C2×C4—1072no—
184C2×C4—1072no—
185C2×C4—2142no—
186C2×C4D4×D10711yescharacteristic
187C2×C4C23×D10711yescharacteristic
188C8—21no—
189C8—21no—
190C8—2142no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—4282no—
195C22—4282no—
196C22—4282no—
197C22C2×D4×D10711yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—4282no—
202C4—2142no—
203C4—2142no—
204C4#1681215411yescharacteristic
205C4#1681215611yescharacteristic
206C2—41no—
207C2—8562no—
208C2#2029320411yescharacteristic
209C2#2029342911yescharacteristic
210C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radical#20292863C2
hypercenter#467D107
hypocenterC107#3262
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20293250C2semidirect
2. #20292920C2semidirect
3. #20292906C2semidirect
4. #20292875C2semidirect
6. #20292851C2semidirect
7. #20292845C2semidirect
8. #20292653C2semidirect
9. #467×C107C2semidirect
106. C107#3262semidirect
111. #467D107semidirect

Maximal subgroups

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

subgroupindex
#202932502
#202929202
#202929062
#202928752
#202928632
#202928512
#202928452
#202926532
#467×C1072
#3262107