Finite Group Databasebeta

← #28347897

Subgroups — #28347897

Order 13696.

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

normal subgroups only

This group has 370 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#20293230C212yesmaximal, complemented
2#20292929C212yesmaximal, complemented
3#20292926C212yesmaximal, complemented
4#20292880C211yesmaximal, characteristic, complemented
5#20292861C212yesmaximal, complemented
6#20292821C212yesmaximal, complemented
7#20292820C212yesmaximal, complemented
8#20292655C211yesmaximal, characteristic, complemented
9#469×C107C211yesmaximal, characteristic, complemented
10C2×D856C2211yescharacteristic
11C2×#13133803C2212yes—
12C22×C107⋊C8C2211yescharacteristic
13#16812105C2212yes—
14#16812100C2212yes—
15#16812093—22no—
16#16812091—22no—
17#16812083C2212yes—
18#16812081—22no—
19#16812080—22no—
20D4×Dic107C2212yes—
21#16812071C2212yes—
22#16812068C2211yescharacteristic
23#16812042C2212yes—
24#16812041C2211yescharacteristic
25Dic107⋊C8C2212yes—
26Dic107⋊C8C2212yes—
27#16812031C2212yes—
28#16812030C2212yes—
29#16812029C2212yes—
30C2×D8×C107C2211yescharacteristic
31C2×M16×C107C2211yescharacteristic
32#121×C107C2212yes—
33#106×C107C2211yescharacteristic
34#102×C107C2212yes—
35C2×D428C2312yes—
36C2×C22⋊D107—22no—
37C2×C22⋊D107—22no—
38C2×C22⋊D107—22no—
39C22×Dic107—22no—
40C2×C4×D107—22no—
41D856—41no—
42#13133803—42no—
43C2×C107⋊C8—41no—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—22no—
46C4⋊Dic107D412yes—
47C4⋊Dic107C2312yes—
48Dic107⋊C4—22no—
49C22⋊Dic107—22no—
50C22⋊Dic107D411yescharacteristic
51C22⋊Dic107D411yescharacteristic
52(C2×D107)⋊C4—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107C2312yes—
55C2×D4×C107—22no—
56C2×D4×C107C2312yes—
57C22×C4×C107C2311yescharacteristic
58D8×C107—41no—
59M16×C107—41no—
60C2×C8×C107C2311yescharacteristic
61C2×C8×C107C2311yescharacteristic
62C4⋊C4×C107C2312yes—
63C22⋊C4×C107—22no—
64C23×C107—22no—
65C23×C107C2×D411yescharacteristic
66C22×D107—22no—
67D4×C107—22no—
68D4×C107—22no—
69D4×C107—82no—
70C2×C4×C107—21no—
71C2×C4×C107—21no—
72C2×C4×C107—22no—
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2411yescharacteristic
75C22⋊D107—82no—
76C22⋊D107—82no—
77C22⋊D107—82no—
78C2×Dic107—22no—
79C2×Dic107—82no—
80C2×Dic107C2×D412yes—
81C2×Dic107C2×D412yes—
82D428—22no—
83D428—22no—
84C4×D107—82no—
85C8×C107—21no—
86C8×C107—21no—
87C107⋊C8—22no—
88C22×C107—21no—
89C22×C107—22no—
90C22×C107—42no—
91C22×C107—42no—
92C22×C107—42no—
93C22×C107C22×D411yescharacteristic
94C2×D107—42no—
95C2×D107—42no—
96C2×D107—42no—
97C4×C107—21no—
98C4×C107—42no—
99C4×C107H(5,𝔽2)11yescharacteristic
100C4×C1072−1+411yescharacteristic
101Dic107—22no—
102Dic107—22no—
103Dic107—42no—
104C2×C107—41no—
105C2×C107—82no—
106C2×C107#53711yescharacteristic
107C2×C107#57612yes—
108D107—82no—
109#3259—1071nomaximal, Sylow
110C107#325911yescharacteristic, complemented, Sylow
111#540—1072no—
112#478—1072no—
113#478—1072no—
114#469—1071no—
115#469D10711yescharacteristic, complemented
116#466—1072no—
117#459—1072no—
118#459—1072no—
119#424—1071no—
120C2×D8—1072no—
121C2×D8—1071no—
122C2×D8C2×D10711yescharacteristic
123C2×M16C2×D10711yescharacteristic
124C22×C8—1071no—
125#127—1072no—
126#125—1072no—
127#123—2142no—
128#123—2142no—
129#121—1072no—
130#121—2142no—
131#121—2142no—
132#121C2×D10712yes—
133C4×D4—1072no—
134C4×D4—1072no—
135#117—1071no—
136#107—1072no—
137#106—1071no—
138#106C2×D10711yescharacteristic
139C4⋊C8—1072no—
140C4⋊C8—1072no—
141#102—1072no—
142#102—1072no—
143#102—1072no—
144#102C2×D10712yes—
145C2×D4—22no—
146C2×D4—1072no—
147C2×D4—2142no—
148C2×D4—2142no—
149C2×D4—2142no—
150C2×D4C22×D10712yes—
151C22×C4—2142no—
152C22×C4—2142no—
153C22×C4C22×D10711yescharacteristic
154D8—41no—
155D8—4281no—
156D8—4282no—
157M16—41no—
158C2×C8—1072no—
159C2×C8—4281no—
160C2×C8C22×D10711yescharacteristic
161C2×C8C22×D10711yescharacteristic
162C4⋊C4—1072no—
163C4⋊C4—1072no—
164C4⋊C4—2142no—
165C4⋊C4—2142no—
166C4⋊C4C22×D10712yes—
167C22⋊C4—22no—
168C22⋊C4—1071no—
169C22⋊C4—1071no—
170C22⋊C4—2142no—
171C22⋊C4—2142no—
172C22⋊C4—2142no—
173C42—1072no—
174C23—22no—
175C23—2142no—
176C23D4×D10711yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—8562no—
181D4—8562no—
182D4—8562no—
183D4—2142no—
184D4—2142no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—8562no—
189C2×C4—8562no—
190C2×C4—1072no—
191C2×C4—1072no—
192C2×C4—2142no—
193C2×C4D4×D10711yescharacteristic
194C2×C4C23×D10711yescharacteristic
195C8—21no—
196C8—21no—
197C8—2142no—
198C22—22no—
199C22—21no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—4282no—
204C22—4282no—
205C22—4282no—
206C22C2×D4×D10711yescharacteristic
207C4—21no—
208C4—42no—
209C4—4282no—
210C4—2142no—
211C4—2142no—
212C4#1681215411yescharacteristic
213C4#1681215611yescharacteristic
214C2—41no—
215C2—82no—
216C2—8562no—
217C2#2029320411yescharacteristic
218C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107#3259
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #20293230C2semidirect
2. #20292929C2semidirect
3. #20292926C2semidirect
4. #20292880C2semidirect
5. #20292861C2semidirect
6. #20292821C2semidirect
7. #20292820C2semidirect
8. #20292655C2semidirect
9. #469×C107C2semidirect
110. C107#3259semidirect
115. #469D107semidirect

Maximal subgroups

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

subgroupindex
#202932302
#202929292
#202929262
#202928802
#202928612
#202928212
#202928202
#202926552
#469×C1072
#3259107