Finite Group Databasebeta

← C4×#16812023 #28336250

Subgroups — C4×#16812023 #28336250

Order 13696.

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

normal subgroups only

This group has 412 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
1C2×C4×C22⋊D107C211yesmaximal, characteristic
2C2×#16812023C212yesmaximal
3C4×C22⋊Dic107C211yesmaximal, characteristic, complemented
4#20292393C212yesmaximal, complemented
5C4×C22⋊C4×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107C2211yescharacteristic
7C22×C4×D107C411yescharacteristic, complemented
8C4×C22⋊D107—21no—
9C4×C22⋊D107—21no—
10C4×C22⋊D107—21no—
11C4×C22⋊D107—21no—
12C2×Dic107⋊C4C411yescharacteristic, complemented
13C2×C22⋊Dic107C2211yescharacteristic
14C2×C22⋊Dic107C2212yes—
15C2×(C2×D107)⋊C4C411yescharacteristic, complemented
16C2×C4×Dic107—21no—
17C2×C4×Dic107C411yescharacteristic, complemented
18#16812023C418yesdirect product factor
19C107⋊#95—21no—
20C23×C4×C107C2211yescharacteristic
21C2×C22⋊C4×C107C2212yes—
22C2×C42×C107—21no—
23#95×C107—21no—
24C23×D107C2×C411yescharacteristic
25C2×C22⋊D107—21no—
26C2×C22⋊D107—21no—
27C2×C22⋊D107—22no—
28C2×C22⋊D107C2×C414yes—
29C22×Dic107—22no—
30C22×Dic107C2×C411yescharacteristic
31C22×Dic107C2×C411yescharacteristic
32C22×Dic107C2×C411yescharacteristic
33C2×C4×D107—21no—
34C2×C4×D107—21no—
35C2×C4×D107—42no—
36Dic107⋊C4—21no—
37Dic107⋊C4—21no—
38C22⋊Dic107—21no—
39C22⋊Dic107—21no—
40C22⋊Dic107—22no—
41C22⋊Dic107C2×C414yescomplemented
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44C4×Dic107—21no—
45C4×Dic107—21no—
46C4×Dic107—42no—
47C24×C107C2311yescharacteristic
48C22×C4×C107—22no—
49C22×C4×C107—42no—
50C22×C4×C107C2×C411yescharacteristic
51C22×C4×C107C2×C411yescharacteristic
52C22×C4×C107D411yescharacteristic
53C22×C4×C107D411yescharacteristic
54C22×C4×C107D411yescharacteristic
55C22×C4×C107D411yescharacteristic
56C22⋊C4×C107—22no—
57C22⋊C4×C107C2×C414yescomplemented
58C42×C107—42no—
59C23×C107—41no—
60C23×C107—41no—
61C23×C107C22×C411yescharacteristic
62C23×C107C22×C412yes—
63C23×C107C2×D411yescharacteristic
64C23×C107C2×D411yescharacteristic
65C23×C107#4112yes—
66C22×D107—21no—
67C22×D107—21no—
68C22×D107—41no—
69C22×D107—41no—
70C22×D107C4212yescomplemented
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—24no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107—41no—
78C2×C4×C107—42no—
79C2×C4×C107—42no—
80C2×C4×C107C22⋊C411yescharacteristic
81C2×C4×C107C22⋊C411yescharacteristic
82C2×C4×C107C22⋊C411yescharacteristic
83C2×C4×C107C22⋊C411yescharacteristic
84C22⋊D107—24no—
85C22⋊D107—24no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—24no—
91C2×Dic107—42no—
92C2×Dic107—42no—
93C2×Dic107—41no—
94C2×Dic107—41no—
95C2×Dic107C4212yescomplemented
96C4×D107—42no—
97C4×D107—42no—
98C22×C107—21no—
99C22×C107—21no—
100C22×C107—21no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—21no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107C2×C4211yescharacteristic
107C22×C107C2×C22⋊C411yescharacteristic
108C22×C107#11711yescharacteristic
109C22×C107C4×D412yes—
110C22×C107C4×D412yes—
111C2×D107—22no—
112C2×D107—42no—
113C2×D107—41no—
114C2×D107—41no—
115C2×D107—42no—
116C4×C107—21no—
117C4×C107—21no—
118C4×C107—21no—
119C4×C107—44no—
120C4×C107#9912yescomplemented
121Dic107—22no—
122Dic107—44no—
123Dic107—41no—
124Dic107—41no—
125C2×C107—22no—
126C2×C107—22no—
127C2×C107—21no—
128C2×C107—21no—
129C2×C107C4×C22⋊C411yescharacteristic
130C2×C107C2×#9911yescharacteristic
131C2×C107#41011yescharacteristic
132D107—42no—
133C4×#99—1071nomaximal, Sylow
134C107C4×#9911yescharacteristic, complemented, Sylow
135C2×C4×D4—1071no—
136C2×#99—1072no—
137C4×C22⋊C4—1071no—
138C4×C22⋊C4D10711yescharacteristic, complemented
139#342—1072no—
140C22×D4—1071no—
141C23×C4—1071no—
142C23×C4C2×D10711yescharacteristic
143C4×D4—2141no—
144C4×D4—2141no—
145C4×D4—2141no—
146C4×D4—2141no—
147C2×C4⋊C4—1071no—
148C2×C22⋊C4—1072no—
149C2×C22⋊C4—1071no—
150C2×C22⋊C4—1071no—
151C2×C22⋊C4C2×D10712yes—
152C2×C42—21no—
153C2×C42—1071no—
154C2×C42—2141no—
155#99—1078no—
156#95—21no—
157#95—2141no—
158C24—1071no—
159C24C22×D10711yescharacteristic
160C2×D4—1074no—
161C2×D4—2141no—
162C2×D4—2141no—
163C2×D4—2142no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—1071no—
167C22×C4—1071no—
168C22×C4—1071no—
169C22×C4—4282no—
170C22×C4—2142no—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4C4×D10711yescharacteristic
174C22×C4C4×D10711yescharacteristic
175C22×C4D42811yescharacteristic
176C22×C4D42811yescharacteristic
177C22×C4C22⋊D10711yescharacteristic
178C22×C4C22⋊D10711yescharacteristic
179C4⋊C4—2141no—
180C4⋊C4—2141no—
181C22⋊C4—22no—
182C22⋊C4—1074no—
183C22⋊C4—2141no—
184C22⋊C4—2141no—
185C22⋊C4—2142no—
186C22⋊C4—2141no—
187C22⋊C4—2141no—
188C22⋊C4C4×D10714yescomplemented
189C42—42no—
190C42—4282no—
191C42—2141no—
192C42—2141no—
193C23—41no—
194C23—41no—
195C23—1072no—
196C23—4281no—
197C23—4281no—
198C23—2141no—
199C23—2141no—
200C23C2×C4×D10711yescharacteristic
201C23C2×C4×D10712yes—
202C23C2×C22⋊D10711yescharacteristic
203C23C2×D42811yescharacteristic
204C23#1313381812yes—
205D4—2144no—
206D4—2144no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—1072no—
217C2×C4—4282no—
218C2×C4—4282no—
219C2×C4—4282no—
220C2×C4—4282no—
221C2×C4—4281no—
222C2×C4—4281no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2144no—
226C2×C4—2141no—
227C2×C4—2141no—
228C2×C4(C2×D107)⋊C411yescharacteristic
229C2×C4(C2×D107)⋊C411yescharacteristic
230C2×C4(C2×D107)⋊C411yescharacteristic
231C2×C4(C2×D107)⋊C411yescharacteristic
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—4282no—
241C22—4282no—
242C22—4281no—
243C22—4281no—
244C22—2142no—
245C22C42×D10711yescharacteristic
246C22C2×(C2×D107)⋊C411yescharacteristic
247C22#1681206611yescharacteristic
248C22C4×C22⋊D10712yes—
249C22C4×D42812yes—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—44no—
254C4—4284no—
255C4—4281no—
256C4—4281no—
257C4—2142no—
258C4#1681202312yesdirect product factor
259C2—22no—
260C2—22no—
261C2—21no—
262C2—21no—
263C2—4282no—
264C2C4×(C2×D107)⋊C411yescharacteristic
265C2C2×#1681202311yescharacteristic
266C2#2029260011yescharacteristic
subgroupG/H
centerC2×C4(C2×D107)⋊C4
commutatorC22×C107C2×C42
FrattiniC24C22×D107
FittingC4×C22⋊C4×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC22×C22⋊D107C22
cosocleC24×C107C23
hypercenterC4×C22⋊C4D107
hypocenterC107C4×#99
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C4×C22⋊Dic107C2semidirect
4. #20292393C2semidirect
5. C4×C22⋊C4×C107C2semidirect
7. C22×C4×D107C4semidirect
12. C2×Dic107⋊C4C4semidirect
15. C2×(C2×D107)⋊C4C4semidirect
17. C2×C4×Dic107C4semidirect
18. #16812023C4direct (N × H)
41. C22⋊Dic107C2×C4semidirect
57. C22⋊C4×C107C2×C4semidirect
70. C22×D107C42semidirect
95. C2×Dic107C42semidirect
120. C4×C107#99semidirect
134. C107C4×#99semidirect
138. C4×C22⋊C4D107semidirect
188. C22⋊C4C4×D107semidirect
258. C4#16812023direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1072
C2×#168120232
C4×C22⋊Dic1072
#202923932
C4×C22⋊C4×C1072
C4×#99107