Finite Group Databasebeta

← C2×#20292948 #28346342

Subgroups — C2×#20292948 #28346342

Order 13696.

468 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812068C211yesmaximal, characteristic, complemented
2#20292948C218yesmaximal, direct product factor
3C2×#16812042C211yesmaximal, characteristic, complemented
4C2×#16812040C211yesmaximal, characteristic, complemented
5C2×#16812030C211yesmaximal, characteristic, complemented
6C2×#16812030C211yesmaximal, characteristic, complemented
7C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic
9C22×C107⋊C8C2211yescharacteristic
10C22×C107⋊C8C2211yescharacteristic
11#16812068—22no—
12#16812068C2214yescomplemented
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×C22⋊Dic107—21no—
16C2×C4×Dic107—21no—
17#16812042C2214yescomplemented
18#16812040C2214yescomplemented
19#16812030C2214yescomplemented
20#16812030C2214yescomplemented
21C22⋊(C107⋊C8)C2214yescomplemented
22C22×D4×C107—21no—
23C22×D4×C107C2211yescharacteristic
24C23×C4×C107C2211yescharacteristic
25#121×C107—22no—
26#121×C107C2214yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107—21no—
29C22×Dic107—21no—
30C22×Dic107—21no—
31C2×C107⋊C8—22no—
32C2×C107⋊C8—22no—
33C2×C107⋊C8C2312yes—
34C2×C107⋊C8C2312yes—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107C2312yes—
38C4⋊Dic107C2312yes—
39C22⋊Dic107—24no—
40C4×Dic107—24no—
41C24×C107—21no—
42C24×C107D411yescharacteristic
43C2×D4×C107—24no—
44C2×D4×C107—21no—
45C2×D4×C107—22no—
46C2×D4×C107—22no—
47C2×D4×C107—44no—
48C2×D4×C107C2312yes—
49C22×C4×C107—22no—
50C22×C4×C107—22no—
51C22×C4×C107—21no—
52C22×C4×C107D411yescharacteristic
53C22×C4×C107D412yes—
54C22×C4×C107C2312yes—
55C22×C4×C107C2311yescharacteristic
56C4⋊C4×C107—21no—
57C4⋊C4×C107C2312yes—
58C22⋊C4×C107—24no—
59C23×C107—22no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—22no—
63C23×C107—42no—
64C23×C107—42no—
65C23×C107—42no—
66C23×C107C2×D412yes—
67C23×C107C2×D411yescharacteristic
68D4×C107—22no—
69D4×C107—22no—
70D4×C107—48no—
71D4×C107—42no—
72C2×C4×C107—22no—
73C2×C4×C107—24no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—42no—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107#4112yes—
81C2×C4×C107#4112yes—
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—42no—
86C2×Dic107—42no—
87C107⋊C8—22no—
88C107⋊C8—22no—
89C22×C107—22no—
90C22×C107—22no—
91C22×C107—22no—
92C22×C107—44no—
93C22×C107—42no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107—84no—
97C22×C107#12312yes—
98C22×C107#13512yes—
99C22×C107AGL(1,ℤ/8)12yes—
100C22×C107C22×D411yescharacteristic
101C4×C107—22no—
102C4×C107—42no—
103C4×C107#12312yes—
104C4×C107C2×#4111yescharacteristic
105C4×C107C2×#4111yescharacteristic
106Dic107—42no—
107Dic107—42no—
108C2×C107—42no—
109C2×C107—82no—
110C2×C107#48214yescomplemented
111C2×C107C2×#12311yescharacteristic
112C2×C107C2×#13511yescharacteristic
113C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
114C2×#482—1071nomaximal, Sylow
115C107C2×#48211yescharacteristic, complemented, Sylow
116C2×#121D10711yescharacteristic
117C2×#117—1071no—
118#482—1078no—
119C2×#107—1071no—
120C2×#106—1071no—
121C2×#102—1071no—
122C2×#102—1071no—
123C2×C22⋊C8—1071no—
124C22×D4—21no—
125C22×D4C2×D10711yescharacteristic
126C23×C4C2×D10711yescharacteristic
127C22×C8—1071no—
128C22×C8—1071no—
129#121—22no—
130#121C2×D10714yes—
131#117—1074no—
132#117—2142no—
133C2×C4⋊C4—1071no—
134C2×C4⋊C4—1071no—
135C2×C4⋊C4C2×D10711yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—2141no—
138C2×C42—2141no—
139#107—1074no—
140#106—1074no—
141#102—1074no—
142#102—1074no—
143C22⋊C8—1074no—
144C24—21no—
145C24C22⋊D10711yescharacteristic
146C2×D4—24no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—44no—
151C2×D4C22×D10712yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—2141no—
156C22×C4—2141no—
157C22×C4C22⋊D10712yes—
158C22×C4C22⋊D10711yescharacteristic
159C22×C4C22×D10712yes—
160C22×C4C22×D10711yescharacteristic
161C2×C8—1072no—
162C2×C8—1072no—
163C2×C8—2142no—
164C2×C8—2142no—
165C4⋊C4—21no—
166C4⋊C4—1072no—
167C4⋊C4—1072no—
168C4⋊C4—2141no—
169C4⋊C4—2141no—
170C4⋊C4C22×D10712yes—
171C22⋊C4—24no—
172C22⋊C4—2144no—
173C42—2144no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D10712yes—
182C23C2×C22⋊D10711yescharacteristic
183D4—22no—
184D4—22no—
185D4—48no—
186D4—42no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—4282no—
194C2×C4—4282no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4C2×C22⋊D10712yes—
198C2×C4C2×C22⋊D10711yescharacteristic
199C2×C4#1313381712yes—
200C2×C4#1313381712yes—
201C2×C4C23×D10711yescharacteristic
202C8—2142no—
203C8—2142no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—84no—
212C22#1681209012yes—
213C22#1681212912yes—
214C22#1681213512yes—
215C22C22×C22⋊D10711yescharacteristic
216C4—22no—
217C4—42no—
218C4—4282no—
219C4—4282no—
220C4#1681209012yes—
221C4C2×#1313381711yescharacteristic
222C4C2×#1313381711yescharacteristic
223C2—42no—
224C2—82no—
225C2#2029294814yesdirect product factor
226C2C2×#1681209011yescharacteristic
227C2C2×#1681212911yescharacteristic
228C2C2×#1681213511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×#121×C107C2
socleC23×C107C2×D4
hypercenterC2×#121D107
hypocenterC107C2×#482
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16812068C2semidirect
2. #20292948C2direct (N × H)
3. C2×#16812042C2semidirect
4. C2×#16812040C2semidirect
5. C2×#16812030C2semidirect
6. C2×#16812030C2semidirect
7. C2×C22⋊(C107⋊C8)C2semidirect
12. #16812068C22semidirect
17. #16812042C22semidirect
18. #16812040C22semidirect
19. #16812030C22semidirect
20. #16812030C22semidirect
21. C22⋊(C107⋊C8)C22semidirect
110. C2×C107#482semidirect
115. C107C2×#482semidirect
225. C2#20292948direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168120682
#202929482
C2×#168120422
C2×#168120402
C2×#168120302
C2×C22⋊(C107⋊C8)2
C2×#121×C1072
C2×#482107