Finite Group Databasebeta

← #28335483

Subgroups — #28335483

Order 13696.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293189C211yesmaximal, characteristic
2#20292946C212yesmaximal, complemented
3#20292563C211yesmaximal, characteristic, complemented
4#20292362C212yesmaximal, complemented
5#323×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107—21no—
7#16812093—22no—
8#16812083—21no—
9#16812083C2212yes—
10#16812079—22no—
11C2×C4⋊Dic107C2211yescharacteristic, complemented
12C2×Dic107⋊C4—21no—
13C2×C22⋊Dic107—21no—
14#16812041—22no—
15#16812031—22no—
16C107⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yescomplemented
19C23×D107—21no—
20C2×D428—21no—
21C2×D428D412yes—
22C2×C22⋊D107—22no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—42no—
25C2×C22⋊D107—42no—
26C22×Dic107—21no—
27C22×Dic107—21no—
28C22×Dic107—21no—
29C4⋊Dic107—21no—
30C4⋊Dic107—22no—
31C4⋊Dic107D412yes—
32Dic107⋊C4—42no—
33C22⋊Dic107—42no—
34(C2×D107)⋊C4—22no—
35(C2×D107)⋊C4—22no—
36C24×C107D411yescharacteristic
37C22×C4×C107D411yescharacteristic
38C22×C4×C107C2311yescharacteristic
39C2×C8×C107—22no—
40C22⋊C4×C107—22no—
41C23×C107—21no—
42C23×C107—21no—
43C23×C107—41no—
44C23×C107—41no—
45C23×C107SD1612yes—
46C23×C107C2×D411yescharacteristic
47C22×D107—22no—
48C22×D107—42no—
49C22×D107—81no—
50C22×D107—81no—
51C2×C4×C107—21no—
52C2×C4×C107—41no—
53C2×C4×C107—41no—
54C2×C4×C107C2×D412yes—
55C22⋊D107—44no—
56C22⋊D107—42no—
57C22⋊D107—42no—
58C2×Dic107—22no—
59C2×Dic107—22no—
60C2×Dic107—41no—
61C2×Dic107—42no—
62C2×Dic107—41no—
63C2×Dic107—41no—
64C2×Dic107—41no—
65C2×Dic107—41no—
66D428—42no—
67D428—81no—
68C8×C107—42no—
69C22×C107—21no—
70C22×C107—21no—
71C22×C107—22no—
72C22×C107—42no—
73C22×C107—41no—
74C22×C107—41no—
75C22×C107—81no—
76C22×C107#12011yescharacteristic
77C22×C107C2×SD1611yescharacteristic
78C22×C107AGL(1,ℤ/8)11yescharacteristic
79C2×D107—42no—
80C2×D107—42no—
81C2×D107—42no—
82C2×D107—84no—
83C4×C107—22no—
84C4×C107—41no—
85Dic107—42no—
86Dic107—42no—
87Dic107—81no—
88C2×C107—21no—
89C2×C107—21no—
90C2×C107—42no—
91C2×C107#45011yescharacteristic
92C2×C107#45311yescharacteristic
93C2×C107#45811yescharacteristic
94D107—82no—
95#1613—1071nomaximal, Sylow
96C107#161311yescharacteristic, complemented, Sylow
97#534—1071no—
98#481—1072no—
99#399—1071no—
100#327—1072no—
101#323D10711yescharacteristic, complemented
102C22×D4—2141no—
103#123—2142no—
104#121—1072no—
105#121—2141no—
106#120—2142no—
107C2×C4⋊C4—1071no—
108C2×C4⋊C4—2141no—
109C2×C22⋊C4—2141no—
110C2×C22⋊C4C2×D10711yescharacteristic
111#106—2142no—
112#102—2142no—
113C22⋊C8C2×D10712yescomplemented
114#95—1071no—
115C24—2141no—
116C24D42811yescharacteristic
117C2×D4—1072no—
118C2×D4—4282no—
119C2×D4—4282no—
120C2×D4—2141no—
121C2×D4—2142no—
122C2×D4—2142no—
123C22×C4—2141no—
124C22×C4—2141no—
125C22×C4—2141no—
126C22×C4D42811yescharacteristic
127C22×C4C22×D10711yescharacteristic
128C2×C8—22no—
129C4⋊C4—1072no—
130C4⋊C4—4282no—
131C4⋊C4—2141no—
132C4⋊C4—2142no—
133C22⋊C4—22no—
134C22⋊C4—4282no—
135C22⋊C4—2142no—
136C22⋊C4—2142no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—8561no—
142C23—8561no—
143C23—4282no—
144C23—2142no—
145C23#1313380712yes—
146C23C2×D42811yescharacteristic
147D4—8561no—
148D4—4282no—
149D4—4284no—
150D4—4282no—
151D4—4282no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—4281no—
156C2×C4—4281no—
157C2×C4—4281no—
158C2×C4—4282no—
159C2×C4—4281no—
160C2×C4—4281no—
161C2×C4—2142no—
162C2×C4—2142no—
163C2×C4D4×D10712yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—8564no—
173C22—4282no—
174C22—4282no—
175C22—4282no—
176C22#1681207911yescharacteristic
177C22C2×#1313380711yescharacteristic
178C22#1681213711yescharacteristic
179C4—22no—
180C4—41no—
181C4—8561no—
182C4—4282no—
183C4—4282no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—8562no—
188C2#2029278011yescharacteristic
189C2#2029279711yescharacteristic
190C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107#120
prime radical#20293189C2
hypercenter#323D107
hypocenterC107#1613
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292946C2semidirect
3. #20292563C2semidirect
4. #20292362C2semidirect
5. #323×C107C2semidirect
11. C2×C4⋊Dic107C22semidirect
16. C107⋊#95C22semidirect
18. C22⋊C8×C107C22semidirect
96. C107#1613semidirect
101. #323D107semidirect
113. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931892
#202929462
#202925632
#202923622
#323×C1072
#1613107