Finite Group Databasebeta

← #28344456

Subgroups — #28344456

Order 13696.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16812078C211yesmaximal, characteristic, complemented
2#20292555C213yesmaximal, complemented
3#20292542C213yesmaximal, complemented
4#20292541C213yesmaximal, complemented
5#20292535C213yesmaximal, complemented
6#20292483C211yesmaximal, characteristic, complemented
7C2×#120×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107C2213yes—
9C22×C22⋊D107C2211yescharacteristic, complemented
10#16812078—24no—
11C2×Dic107⋊C4C2213yescomplemented
12C2×C22⋊Dic107C2213yes—
13C2×C22⋊Dic107C2213yes—
14C2×C22⋊Dic107C2213yes—
15C2×C22⋊Dic107C2213yescomplemented
16C2×(C2×D107)⋊C4C2213yescomplemented
17C107⋊#95C2213yescomplemented
18C107⋊#95C2213yescomplemented
19C25×C107C2211yescharacteristic
20C22×D4×C107C2213yes—
21#120×C107—24no—
22C2×C22⋊C4×C107C2213yes—
23C23×D107C2311yescharacteristic
24C2×C22⋊D107—26no—
25C2×C22⋊D107—43no—
26C2×C22⋊D107—43no—
27C2×C22⋊D107—43no—
28C22×Dic107C2313yes—
29C22×Dic107C2311yescharacteristic
30C22×Dic107C2313yes—
31Dic107⋊C4—43no—
32C22⋊Dic107—26no—
33C22⋊Dic107—43no—
34C22⋊Dic107—43no—
35C22⋊Dic107—43no—
36(C2×D107)⋊C4—43no—
37C24×C107—43no—
38C24×C107—43no—
39C24×C107D414yescomplemented
40C24×C107C2311yescharacteristic
41C24×C107C2313yes—
42C2×D4×C107—26no—
43C2×D4×C107—43no—
44C2×D4×C107—43no—
45C22×C4×C107C2313yes—
46C22⋊C4×C107—26no—
47C23×C107—23no—
48C23×C107—23no—
49C23×C107—23no—
50C23×C107—23no—
51C23×C107—23no—
52C23×C107—23no—
53C23×C107—21no—
54C23×C107—23no—
55C23×C107—23no—
56C23×C107—412no—
57C23×C107—412no—
58C23×C107—44no—
59C23×C107C2×D416yes—
60C23×C107C2411yescharacteristic
61C22×D107—21no—
62C22×D107—23no—
63C22×D107—23no—
64D4×C107—412no—
65C2×C4×C107—23no—
66C2×C4×C107—23no—
67C2×C4×C107—23no—
68C22⋊D107—412no—
69C22⋊D107—44no—
70C2×Dic107—23no—
71C2×Dic107—23no—
72C2×Dic107—23no—
73C2×Dic107—23no—
74C2×Dic107—23no—
75C2×Dic107—23no—
76C2×Dic107—23no—
77C22×C107—26no—
78C22×C107—26no—
79C22×C107—26no—
80C22×C107—416no—
81C22×C107—43no—
82C22×C107—43no—
83C22×C107—43no—
84C22×C107—43no—
85C22×C107—43no—
86C22×C107—43no—
87C22×C107—41no—
88C22×C107C22×D411yescharacteristic
89C22×C107H(5,𝔽2)13yes—
90C22×C107H(5,𝔽2)13yes—
91C2×D107—43no—
92C2×D107—43no—
93C2×D107—41no—
94C4×C107—43no—
95Dic107—43no—
96Dic107—43no—
97Dic107—41no—
98C2×C107—46no—
99C2×C107—81no—
100C2×C107#53413yes—
101C2×C107#56013yes—
102C2×C107H(3,𝔽4)11yescharacteristic
103D107—81no—
104#2812—1071nomaximal, Sylow
105C107#281211yescharacteristic, complemented, Sylow
106C2×#120—1071no—
107C2×#120D10711yescharacteristic, complemented
108#397—1073no—
109#394—1073no—
110#394—1073no—
111#392—1073no—
112#379—1071no—
113C25C2×D10711yescharacteristic
114C22×D4—1071no—
115C22×D4—1073no—
116C22×D4C2×D10713yes—
117#120—24no—
118#120—2144no—
119C2×C4⋊C4—1073no—
120C2×C22⋊C4—1073no—
121C2×C22⋊C4—1073no—
122C2×C22⋊C4—1073no—
123C2×C22⋊C4—1073no—
124C2×C22⋊C4—1073no—
125C2×C22⋊C4C2×D10713yes—
126#95—1073no—
127#95—1073no—
128C24—43no—
129C24—43no—
130C24—1071no—
131C24C22⋊D10714yescomplemented
132C24C22×D10711yescharacteristic
133C24C22×D10713yes—
134C2×D4—26no—
135C2×D4—43no—
136C2×D4—43no—
137C2×D4—4283no—
138C2×D4—4283no—
139C2×D4—4283no—
140C2×D4—2146no—
141C22×C4—1071no—
142C22×C4—1073no—
143C22×C4—1073no—
144C22×C4C22×D10713yes—
145C4⋊C4—4283no—
146C22⋊C4—26no—
147C22⋊C4—4283no—
148C22⋊C4—4283no—
149C22⋊C4—4283no—
150C22⋊C4—4283no—
151C22⋊C4—2146no—
152C23—23no—
153C23—23no—
154C23—23no—
155C23—23no—
156C23—23no—
157C23—23no—
158C23—21no—
159C23—23no—
160C23—23no—
161C23—412no—
162C23—412no—
163C23—44no—
164C23—2141no—
165C23—2143no—
166C23—2143no—
167C23C2×C22⋊D10716yes—
168C23C23×D10711yescharacteristic
169D4—412no—
170D4—42812no—
171D4—4284no—
172C2×C4—23no—
173C2×C4—23no—
174C2×C4—23no—
175C2×C4—2143no—
176C2×C4—2143no—
177C2×C4—2143no—
178C2×C4—2143no—
179C2×C4—2143no—
180C2×C4—2143no—
181C2×C4—2143no—
182C22—26no—
183C22—26no—
184C22—26no—
185C22—416no—
186C22—43no—
187C22—43no—
188C22—43no—
189C22—43no—
190C22—43no—
191C22—43no—
192C22—41no—
193C22—4283no—
194C22—4283no—
195C22—4281no—
196C22C22×C22⋊D10711yescharacteristic
197C22#1681215413yes—
198C22#1681215413yes—
199C4—43no—
200C4—4283no—
201C4—4283no—
202C4—4281no—
203C2—46no—
204C2—81no—
205C2—8561no—
206C2#2029318613yes—
207C2#2029336613yes—
208C2#2029337011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#120×C107C2
hypercenterC2×#120D107
hypocenterC107#2812
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16812078C2semidirect
2. #20292555C2semidirect
3. #20292542C2semidirect
4. #20292541C2semidirect
5. #20292535C2semidirect
6. #20292483C2semidirect
7. C2×#120×C107C2semidirect
9. C22×C22⋊D107C22semidirect
11. C2×Dic107⋊C4C22semidirect
15. C2×C22⋊Dic107C22semidirect
16. C2×(C2×D107)⋊C4C22semidirect
17. C107⋊#95C22semidirect
18. C107⋊#95C22semidirect
39. C24×C107D4semidirect
105. C107#2812semidirect
107. C2×#120D107semidirect
131. C24C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120782
#202925552
#202925422
#202925412
#202925352
#202924832
C2×#120×C1072
#2812107