Finite Group Databasebeta

← #28347896

Subgroups — #28347896

Order 13696.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293232C211yesmaximal, characteristic, complemented
2#20292927C212yesmaximal, complemented
3#20292924C212yesmaximal, complemented
4#20292878C212yesmaximal, complemented
5#20292861C212yesmaximal, complemented
6#20292822C212yesmaximal, complemented
7#20292819C212yesmaximal, complemented
8#20292654C211yesmaximal, characteristic, complemented
9#540×C107C211yesmaximal, characteristic, complemented
10C2×#13133803C2212yes—
11C2×#13133803C2212yes—
12C2×#13133802C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812106C2211yescharacteristic
15#16812101C2211yescharacteristic
16#16812094—22no—
17#16812083C2212yes—
18#16812082—22no—
19C4×D428C2212yes—
20#16812042C2212yes—
21#16812040C2212yes—
22C4⋊(C107⋊C8)C2212yes—
23C4⋊(C107⋊C8)C2212yes—
24#16812030C2212yes—
25#16812030C2212yes—
26#16812029C2212yes—
27#16812029C2212yes—
28#127×C107C2211yescharacteristic
29#125×C107C2211yescharacteristic
30#123×C107—22no—
31#121×C107—22no—
32#121×C107C2212yes—
33C4×D4×C107C2212yes—
34#117×C107C2211yescharacteristic
35C2×D428—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133803—42no—
40#13133803—42no—
41#13133802—41no—
42C2×C107⋊C8—41no—
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107C2312yes—
46Dic107⋊C4—22no—
47(C2×D107)⋊C4—22no—
48(C2×D107)⋊C4—22no—
49C2×D4×C107—22no—
50C2×D4×C107—22no—
51C2×D4×C107C2312yes—
52C22×C4×C107—22no—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—22no—
55C4⋊C4×C107D412yes—
56C4⋊C4×C107C2312yes—
57C22⋊C4×C107—22no—
58C22⋊C4×C107—22no—
59C22⋊C4×C107D412yes—
60C42×C107C2311yescharacteristic
61C42×C107C2311yescharacteristic
62C23×C107—22no—
63C23×C107C2×D411yescharacteristic
64C22×D107—22no—
65D4×C107—22no—
66D4×C107—22no—
67D4×C107—82no—
68D4×C107—82no—
69C2×C4×C107—21no—
70C2×C4×C107—21no—
71C2×C4×C107—22no—
72C2×C4×C107—82no—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2411yescharacteristic
77C22⋊D107—82no—
78C2×Dic107—22no—
79D428—22no—
80D428—22no—
81D428—82no—
82C4×D107—82no—
83C107⋊C8—22no—
84C107⋊C8—22no—
85C22×C107—21no—
86C22×C107—22no—
87C22×C107—42no—
88C22×C107—42no—
89C22×C107—42no—
90C22×C107C22×D411yescharacteristic
91C2×D107—42no—
92C2×D107—42no—
93C2×D107—42no—
94C4×C107—21no—
95C4×C107—22no—
96C4×C107—22no—
97C4×C107—42no—
98C4×C107H(5,𝔽2)11yescharacteristic
99C4×C1072−1+411yescharacteristic
100Dic107—42no—
101C2×C107—41no—
102C2×C107—82no—
103C2×C107#53711yescharacteristic
104C2×C107#57612yes—
105D107—82no—
106#3259—1071nomaximal, Sylow
107C107#325911yescharacteristic, complemented, Sylow
108#540—1071no—
109#540D10711yescharacteristic, complemented
110#478—1072no—
111#478—1072no—
112#469—1072no—
113#466—1072no—
114#459—1072no—
115#459—1072no—
116#424—1071no—
117C2×D8—1072no—
118C2×D8—1072no—
119C2×M16—1071no—
120C22×C8—1071no—
121#127—1071no—
122#127C2×D10711yescharacteristic
123#125—1071no—
124#125C2×D10711yescharacteristic
125#123—22no—
126#123—2142no—
127#121—22no—
128#121—1072no—
129#121—2142no—
130#121C2×D10712yes—
131C4×D4—1072no—
132C4×D4C2×D10712yes—
133#117C2×D10711yescharacteristic
134#107—1072no—
135#106—1072no—
136C4⋊C8—1072no—
137C4⋊C8—1072no—
138#102—1072no—
139#102—1072no—
140#102—1072no—
141#102—1072no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—1072no—
145C2×D4—2142no—
146C2×D4—2142no—
147C2×D4C22×D10712yes—
148C22×C4—22no—
149C22×C4—2142no—
150C22×C4C22×D10711yescharacteristic
151D8—4282no—
152D8—4282no—
153M16—4281no—
154C2×C8—1072no—
155C2×C8—1072no—
156C2×C8—4281no—
157C4⋊C4—22no—
158C4⋊C4—1072no—
159C4⋊C4—2142no—
160C4⋊C4C22⋊D10712yes—
161C4⋊C4C22×D10712yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—2142no—
165C22⋊C4—2142no—
166C22⋊C4C22⋊D10712yes—
167C42C22×D10711yescharacteristic
168C42C22×D10711yescharacteristic
169C23—22no—
170C23—2142no—
171C23C2×C22⋊D10711yescharacteristic
172D4—22no—
173D4—22no—
174D4—82no—
175D4—82no—
176D4—8562no—
177D4—8562no—
178D4—2142no—
179D4—2142no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—8562no—
185C2×C4—2142no—
186C2×C4C2×C22⋊D10712yes—
187C2×C4C2×C22⋊D10712yes—
188C2×C4C2×C22⋊D10711yescharacteristic
189C2×C4C23×D10711yescharacteristic
190C8—2142no—
191C8—2142no—
192C22—22no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—4282no—
198C22—4282no—
199C22—4282no—
200C22C22×C22⋊D10711yescharacteristic
201C4—22no—
202C4—22no—
203C4—21no—
204C4—42no—
205C4—4282no—
206C4#1681215511yescharacteristic
207C4#1681215611yescharacteristic
208C2—41no—
209C2—82no—
210C2—8562no—
211C2#2029320511yescharacteristic
212C2#2029343012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#540×C107C2
socleC22×C107C22×D4
hypercenter#540D107
hypocenterC107#3259
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20293232C2semidirect
2. #20292927C2semidirect
3. #20292924C2semidirect
4. #20292878C2semidirect
5. #20292861C2semidirect
6. #20292822C2semidirect
7. #20292819C2semidirect
8. #20292654C2semidirect
9. #540×C107C2semidirect
107. C107#3259semidirect
109. #540D107semidirect

Maximal subgroups

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

subgroupindex
#202932322
#202929272
#202929242
#202928782
#202928612
#202928222
#202928192
#202926542
#540×C1072
#3259107