Finite Group Databasebeta

← C2×#20292998 #28346866

Subgroups — C2×#20292998 #28346866

Order 13696.

656 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×#13133807C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3C22×#13133805C211yesmaximal, characteristic, complemented
4C2×#16812107C211yesmaximal, characteristic, complemented
5C2×#16812105C211yesmaximal, characteristic
6#20292998C218yesmaximal, direct product factor
7C2×C8×Dic107C211yesmaximal, characteristic, complemented
8C22×SD16×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11C22×C22⋊D107—22no—
12C2×#13133807—24no—
13C2×#13133807C2214yescomplemented
14C2×#13133806—24no—
15C2×#13133806C2214yescomplemented
16C2×#13133805—24no—
17C2×#13133805C2214yescomplemented
18C22×C107⋊C8C2211yescharacteristic
19#16812107—22no—
20#16812107C2214yescomplemented
21#16812105—22no—
22#16812105C2214yes—
23C2×Dic107⋊C4—22no—
24C2×C4×Dic107C2211yescharacteristic
25C8×Dic107C2214yescomplemented
26C22×Q8×C107C2211yescharacteristic
27C22×D4×C107C2211yescharacteristic
28C2×SD16×C107—24no—
29C2×SD16×C107C2214yescomplemented
30C22×C8×C107C2211yescharacteristic
31C23×D107—21no—
32C2×Dic214—22no—
33C2×Dic214—22no—
34C2×Dic214—21no—
35C2×Dic214C2312yes—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—28no—
41C2×C22⋊D107—48no—
42C22×Dic107—21no—
43C22×Dic107D412yes—
44#13133807—28no—
45#13133806—28no—
46#13133805—28no—
47C2×C107⋊C8D414yescomplemented
48C2×C107⋊C8C2312yes—
49Dic107⋊C4—28no—
50C4×Dic107D412yes—
51C4×Dic107C2312yes—
52C24×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—22no—
55C2×Q8×C107—22no—
56C2×Q8×C107C2312yes—
57C2×D4×C107—22no—
58C2×D4×C107—22no—
59C2×D4×C107—21no—
60C2×D4×C107C2312yes—
61C22×C4×C107—21no—
62C22×C4×C107C2311yescharacteristic
63SD16×C107—28no—
64C2×C8×C107D414yescomplemented
65C2×C8×C107C2312yes—
66C23×C107—22no—
67C23×C107—42no—
68C23×C107—44no—
69C23×C107C2×D411yescharacteristic
70C22×D107—22no—
71C22×D107—42no—
72C22×D107—44no—
73Q8×C107—22no—
74Q8×C107—22no—
75Q8×C107—42no—
76D4×C107—22no—
77D4×C107—22no—
78D4×C107—42no—
79C2×C4×C107—22no—
80C2×C4×C107—42no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—416no—
86C2×Dic107—22no—
87C2×Dic107—42no—
88C2×Dic107SD1618yescomplemented
89C2×Dic107C2×D414yes—
90D428—22no—
91D428—22no—
92D428—42no—
93Dic214—22no—
94Dic214—22no—
95Dic214—42no—
96C8×C107C2×D414yescomplemented
97C107⋊C8C2×D414yescomplemented
98C22×C107—42no—
99C22×C107—44no—
100C22×C107—84no—
101C22×C107#12712yes—
102C22×C107C2×SD1614yes—
103C22×C107C22×D411yescharacteristic
104C2×D107—42no—
105C2×D107—44no—
106C2×D107—84no—
107C4×C107—42no—
108C4×C107#12712yes—
109C4×C107C22×D411yescharacteristic
110C4×C107C22×D411yescharacteristic
111Dic107—42no—
112Dic107C2×SD1618yescomplemented
113C2×C107—82no—
114C2×C107#49214yescomplemented
115C2×C107C2×#12711yescharacteristic
116C2×C107C22×SD1612yes—
117D107—82no—
118C2×#492—1071nomaximal, Sylow
119C107C2×#49211yescharacteristic, complemented, Sylow
120C22×SD16—1071no—
121C22×SD16—1071no—
122C22×SD16—1071no—
123C22×SD16D10711yescharacteristic, complemented
124C2×C4⋊Q8—1071no—
125C2×#127—1071no—
126#492—1078no—
127C2×C4×C8—1071no—
128C22×Q8—1071no—
129C22×Q8C2×D10711yescharacteristic
130C22×D4—1071no—
131C22×D4—2142no—
132C22×D4C2×D10711yescharacteristic
133C2×SD16—24no—
134C2×SD16—1074no—
135C2×SD16—1074no—
136C2×SD16—1074no—
137C2×SD16—2144no—
138C2×SD16—2144no—
139C2×SD16—2144no—
140C2×SD16C2×D10714yescomplemented
141C22×C8—1071no—
142C22×C8C2×D10711yescharacteristic
143C4⋊Q8—1074no—
144C4⋊Q8—2142no—
145#127—1074no—
146#127—2142no—
147C2×C4⋊C4—2142no—
148C2×C42—1071no—
149C4×C8—1074no—
150C24—21no—
151C24—2141no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1072no—
156C2×Q8—2142no—
157C2×Q8—2142no—
158C2×Q8—2141no—
159C2×Q8C22×D10712yes—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—1072no—
164C2×D4—4288no—
165C2×D4—2148no—
166C2×D4—2141no—
167C2×D4—2142no—
168C2×D4—2142no—
169C2×D4C22×D10712yes—
170C22×C4—21no—
171C22×C4—1072no—
172C22×C4—2141no—
173C22×C4C22×D10711yescharacteristic
174SD16—28no—
175SD16—2148no—
176SD16—2148no—
177SD16—2148no—
178C2×C8—1074no—
179C2×C8—1072no—
180C2×C8C22⋊D10714yescomplemented
181C2×C8C22×D10712yes—
182C4⋊C4—2148no—
183C42—1072no—
184C42—1072no—
185C23—22no—
186C23—42no—
187C23—44no—
188C23—4282no—
189C23—4284no—
190C23—2142no—
191C23D4×D10711yescharacteristic
192Q8—22no—
193Q8—22no—
194Q8—42no—
195Q8—4282no—
196Q8—2142no—
197Q8—2142no—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—42816no—
202D4—4282no—
203D4—2142no—
204D4—2142no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1078no—
208C2×C4—1074no—
209C2×C4—4282no—
210C2×C4—2142no—
211C2×C4D4×D10711yescharacteristic
212C2×C4D4×D10712yes—
213C2×C4C2×C22⋊D10712yes—
214C2×C4C23×D10711yescharacteristic
215C8—1074no—
216C8C2×C22⋊D10714yescomplemented
217C22—44no—
218C22—42no—
219C22—84no—
220C22—8564no—
221C22—4284no—
222C22—4282no—
223C22#1681210512yes—
224C22SD16×D10714yes—
225C22C2×D4×D10711yescharacteristic
226C4—42no—
227C4—1078no—
228C4—4282no—
229C4#1681210512yes—
230C4C2×D4×D10711yescharacteristic
231C4C22×C22⋊D10711yescharacteristic
232C2—82no—
233C2—8562no—
234C2#2029299814yesdirect product factor
235C2C2×#1681210511yescharacteristic
236C2C2×SD16×D10712yes—
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812105C2
hypercenterC22×SD16D107
hypocenterC107C2×#492
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
2. C22×#13133806C2semidirect
3. C22×#13133805C2semidirect
4. C2×#16812107C2semidirect
6. #20292998C2direct (N × H)
7. C2×C8×Dic107C2semidirect
8. C22×SD16×C107C2semidirect
13. C2×#13133807C22semidirect
15. C2×#13133806C22semidirect
17. C2×#13133805C22semidirect
20. #16812107C22semidirect
25. C8×Dic107C22semidirect
29. C2×SD16×C107C22semidirect
47. C2×C107⋊C8D4semidirect
64. C2×C8×C107D4semidirect
88. C2×Dic107SD16semidirect
96. C8×C107C2×D4semidirect
97. C107⋊C8C2×D4semidirect
112. Dic107C2×SD16semidirect
114. C2×C107#492semidirect
119. C107C2×#492semidirect
123. C22×SD16D107semidirect
140. C2×SD16C2×D107semidirect
180. C2×C8C22⋊D107semidirect
216. C8C2×C22⋊D107semidirect
234. C2#20292998direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338072
C22×#131338062
C22×#131338052
C2×#168121072
C2×#168121052
#202929982
C2×C8×Dic1072
C22×SD16×C1072
C2×#492107