Finite Group Databasebeta

← #28335481

Subgroups — #28335481

Order 13696.

288 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 57 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#20292943C212yesmaximal, complemented
2#20292563C211yesmaximal, characteristic, complemented
3#20292361C212yesmaximal, complemented
4#20292351C211yesmaximal, characteristic, complemented
5#534×C107C211yesmaximal, characteristic
6C2×C4⋊Dic107C2211yescharacteristic, complemented
7C2×Dic107⋊C4—21no—
8C2×C22⋊Dic107—21no—
9#16812040—22no—
10#16812030—22no—
11C22⋊(C107⋊C8)C2212yescomplemented
12C107⋊#95C2211yescharacteristic, complemented
13C22×D4×C107—21no—
14#123×C107—22no—
15#121×C107—21no—
16#121×C107C2212yes—
17#120×C107—22no—
18C2×C22⋊C4×C107C2211yescharacteristic
19C22×Dic107—21no—
20C22×Dic107—21no—
21C2×C107⋊C8—22no—
22C4⋊Dic107—22no—
23Dic107⋊C4—42no—
24C22⋊Dic107—42no—
25C24×C107—21no—
26C24×C107D411yescharacteristic
27C2×D4×C107—22no—
28C2×D4×C107—22no—
29C2×D4×C107—21no—
30C2×D4×C107—42no—
31C2×D4×C107—42no—
32C2×D4×C107D412yes—
33C22×C4×C107—21no—
34C22×C4×C107D411yescharacteristic
35C22×C4×C107C2311yescharacteristic
36C4⋊C4×C107—21no—
37C4⋊C4×C107D412yes—
38C22⋊C4×C107—22no—
39C22⋊C4×C107—22no—
40C22⋊C4×C107—22no—
41C23×C107—21no—
42C23×C107—21no—
43C23×C107—22no—
44C23×C107—41no—
45C23×C107—41no—
46C23×C107—42no—
47C23×C107—81no—
48C23×C107—81no—
49C23×C107SD1612yes—
50C23×C107C2×D411yescharacteristic
51D4×C107—42no—
52D4×C107—44no—
53D4×C107—42no—
54D4×C107—42no—
55D4×C107—81no—
56C2×C4×C107—21no—
57C2×C4×C107—22no—
58C2×C4×C107—41no—
59C2×C4×C107—42no—
60C2×C4×C107—41no—
61C2×C4×C107C2×D412yes—
62C2×Dic107—22no—
63C2×Dic107—41no—
64C2×Dic107—41no—
65C2×Dic107—41no—
66C2×Dic107—41no—
67C2×Dic107—41no—
68C107⋊C8—42no—
69C22×C107—21no—
70C22×C107—21no—
71C22×C107—22no—
72C22×C107—42no—
73C22×C107—41no—
74C22×C107—41no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107—42no—
78C22×C107—81no—
79C22×C107—84no—
80C22×C107#12011yescharacteristic
81C22×C107C2×SD1611yescharacteristic
82C22×C107AGL(1,ℤ/8)11yescharacteristic
83C4×C107—22no—
84C4×C107—42no—
85C4×C107—41no—
86Dic107—42no—
87Dic107—81no—
88C2×C107—21no—
89C2×C107—21no—
90C2×C107—42no—
91C2×C107—82no—
92C2×C107#45011yescharacteristic
93C2×C107#45311yescharacteristic
94C2×C107#45811yescharacteristic
95#1613—1071nomaximal, Sylow
96C107#161311yescharacteristic, complemented, Sylow
97#534D10711yescharacteristic
98#481—1072no—
99#399—1071no—
100#327—1072no—
101#323—1071no—
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C2×D10712yes—
106#120—22no—
107C2×C4⋊C4—1071no—
108C2×C4⋊C4—2141no—
109C2×C22⋊C4—2141no—
110C2×C22⋊C4C2×D10711yescharacteristic
111#106—2142no—
112#102—2142no—
113C22⋊C8—1072no—
114#95—1071no—
115C24—21no—
116C24C22⋊D10711yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C22⋊D10712yes—
123C22×C4—21no—
124C22×C4—2141no—
125C22×C4—2141no—
126C22×C4C22⋊D10711yescharacteristic
127C22×C4C22×D10711yescharacteristic
128C2×C8—2142no—
129C4⋊C4—21no—
130C4⋊C4—4282no—
131C4⋊C4—2142no—
132C4⋊C4C22⋊D10712yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—4282no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#1313380612yes—
146C23C2×C22⋊D10711yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—4281no—
158C2×C4—4281no—
159C2×C4—4281no—
160C2×C4—4281no—
161C2×C4—4281no—
162C2×C4—2142no—
163C2×C4C2×C22⋊D10712yes—
164C8—4282no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#1681207811yescharacteristic
177C22C2×#1313380611yescharacteristic
178C22#1681213511yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—8561no—
183C4—4282no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#2029277811yescharacteristic
189C2#2029279511yescharacteristic
190C2#2029281711yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting, prime radical#534×C107C2
socleC22×C107#120
hypercenter#534D107
hypocenterC107#1613
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20292943C2semidirect
2. #20292563C2semidirect
3. #20292361C2semidirect
4. #20292351C2semidirect
6. C2×C4⋊Dic107C22semidirect
11. C22⋊(C107⋊C8)C22semidirect
12. C107⋊#95C22semidirect
96. C107#1613semidirect

Maximal subgroups

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

subgroupindex
#202929432
#202925632
#202923612
#202923512
#534×C1072
#1613107