Finite Group Databasebeta

← C2×#20295768 #28377152

Subgroups — C2×#20295768 #28377152

Order 13952.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#20295768C2116yesmaximal, direct product factor
2C2×#16812688C212yesmaximal, complemented
3C2×#16812687C212yesmaximal, complemented
4C2×#16812685C214yesmaximal, complemented
5C2×#16812669C214yesmaximal, complemented
6C2×#16812667C212yesmaximal, complemented
7C2×#124×C109C211yesmaximal, characteristic, complemented
8C24×D109C2212yescomplemented
9C22×Dic218C2212yescomplemented
10C22×D436C2212yes—
11C22×D436C2212yes—
12C22×C22⋊D109C2214yes—
13#16812688C22116yescomplemented
14#16812687C22116yescomplemented
15#16812685C22132yescomplemented
16#16812669C22132yescomplemented
17#16812667C22116yescomplemented
18C2×C22⋊Dic109C2212yescomplemented
19C2×#13182006C2214yescomplemented
20C2×#13182006C2214yescomplemented
21C2×#13182006C2214yescomplemented
22C2×C4×Dic109C2212yescomplemented
23C22×Q8×C109C2211yescharacteristic, complemented
24C22×D4×C109C2211yescharacteristic
25#124×C109C2218yescomplemented
26C2×C22⋊C4×C109C2214yescomplemented
27C2×C42×C109C2211yescharacteristic, complemented
28C23×D109—48no—
29C23×D109—44no—
30C23×D109C2314yes—
31C23×D109C2318yescomplemented
32C2×Dic218—44no—
33C2×Dic218C2318yescomplemented
34C2×D436—44no—
35C2×D436—44no—
36C2×D436C2318yes—
37C2×D436C2318yes—
38C2×C22⋊D109—48no—
39C2×C22⋊D109C23116yes—
40C22×Dic109C2314yes—
41C22⋊Dic109C2318yescomplemented
42#13182006C23116yescomplemented
43#13182006C23116yescomplemented
44#13182006C23116yescomplemented
45C4×Dic109C2318yescomplemented
46C24×C109C2312yes—
47C2×Q8×C109—42no—
48C2×Q8×C109C2314yescomplemented
49C2×D4×C109—42no—
50C2×D4×C109C2314yes—
51C22×C4×C109C2312yes—
52C22×C4×C109C2311yescharacteristic
53C22×C4×C109C2312yes—
54C22⋊C4×C109C23116yescomplemented
55C42×C109C2314yescomplemented
56C23×C109—24no—
57C23×C109—28no—
58C23×C109C2414yes—
59C23×C109C2411yescharacteristic
60C22×D109—28no—
61C22×D109—28no—
62C22×D109—28no—
63C22×D109—48no—
64C22×D109—416no—
65C22×D109—432no—
66C22×D109C2418yes—
67Q8×C109—44no—
68D4×C109—44no—
69C2×C4×C109—22no—
70C2×C4×C109—24no—
71C2×C4×C109—24no—
72C2×C4×C109C2414yes—
73C2×C4×C109C2412yes—
74C2×C4×C109C2414yes—
75C22⋊D109—416no—
76C2×Dic109—28no—
77C2×Dic109C2418yes—
78D436—48no—
79D436—48no—
80Dic218—48no—
81C22×C109—28no—
82C22×C109—24no—
83C22×C109—48no—
84C22×C109H(5,𝔽2)12yes—
85C22×C109H(5,𝔽2)14yes—
86C22×C109C2511yescharacteristic
87C2×D109—28no—
88C2×D109—28no—
89C2×D109—28no—
90C2×D109—432no—
91C2×D109—416no—
92C4×C109—24no—
93C4×C109—24no—
94C4×C109—22no—
95Dic109—28no—
96C2×C109—44no—
97C2×C109H(3,𝔽4)14yescomplemented
98C2×C109C2×H(5,𝔽2)11yescharacteristic
99C2×C109C2×H(5,𝔽2)12yes—
100D109—48no—
101C2×H(3,𝔽4)—1091nomaximal, Sylow
102C109C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
103H(3,𝔽4)—10916no—
104C2×#124—1092no—
105C2×#124—1094no—
106C2×#124—1092no—
107C2×#124D10911yescharacteristic, complemented
108C2×#120—1092no—
109C2×#120—1094no—
110C25—1092no—
111C22×Q8—1092no—
112C22×Q8C2×D10911yescharacteristic, complemented
113C22×D4—1094no—
114C22×D4—1092no—
115C22×D4—1092no—
116C22×D4C2×D10911yescharacteristic
117#124—10916no—
118#124—10932no—
119#124—10916no—
120#124C2×D10918yescomplemented
121#120—10916no—
122#120—10932no—
123C2×C22⋊C4—1092no—
124C2×C22⋊C4—1094no—
125C2×C22⋊C4—1094no—
126C2×C22⋊C4—1094no—
127C2×C22⋊C4C2×D10914yescomplemented
128C2×C42—1092no—
129C2×C42C2×D10911yescharacteristic, complemented
130C24—1098no—
131C24—1094no—
132C24—4364no—
133C24—4368no—
134C24C22×D10912yes—
135C2×Q8—42no—
136C2×Q8—1098no—
137C2×Q8—4364no—
138C2×Q8C22×D10914yescomplemented
139C2×D4—42no—
140C2×D4—10916no—
141C2×D4—1098no—
142C2×D4—1098no—
143C2×D4—4364no—
144C2×D4—4364no—
145C2×D4—4368no—
146C2×D4C22×D10914yes—
147C22×C4—1094no—
148C22×C4C22×D10912yes—
149C22×C4C22×D10911yescharacteristic
150C22×C4C22×D10912yes—
151C22⋊C4—1098no—
152C22⋊C4—10916no—
153C22⋊C4—10916no—
154C22⋊C4—10916no—
155C22⋊C4C22×D109116yescomplemented
156C42—1098no—
157C42C22×D10914yescomplemented
158C23—24no—
159C23—28no—
160C23—1098no—
161C23—4368no—
162C23—43616no—
163C23—43632no—
164C23—2188no—
165C23—2188no—
166C23—2188no—
167C23C23×D10914yes—
168C23C23×D10911yescharacteristic
169Q8—44no—
170Q8—4368no—
171D4—44no—
172D4—4368no—
173D4—4368no—
174D4—43616no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—1098no—
179C2×C4—2188no—
180C2×C4C23×D10914yes—
181C2×C4C23×D10912yes—
182C2×C4C23×D10914yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—43632no—
187C22—43616no—
188C22—2188no—
189C22—2188no—
190C22—2188no—
191C22#1681274412yes—
192C22#1681274514yes—
193C22C24×D10911yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—2188no—
198C2—44no—
199C2—4368no—
200C2#2029576814yesdirect product factor
201C2C2×#1681274411yescharacteristic
202C2C2×#1681274512yes—
subgroupG/H
centerC23C23×D109
commutator, cosocleC22×C109C25
FrattiniC22C24×D109
FittingC2×#124×C109C2
socleC23×C109C24
hypercenterC2×#124D109
hypocenterC109C2×H(3,𝔽4)
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20295768C2direct (N × H)
2. C2×#16812688C2semidirect
3. C2×#16812687C2semidirect
4. C2×#16812685C2semidirect
5. C2×#16812669C2semidirect
6. C2×#16812667C2semidirect
7. C2×#124×C109C2semidirect
8. C24×D109C22semidirect
9. C22×Dic218C22semidirect
13. #16812688C22semidirect
14. #16812687C22semidirect
15. #16812685C22semidirect
16. #16812669C22semidirect
17. #16812667C22semidirect
18. C2×C22⋊Dic109C22semidirect
19. C2×#13182006C22semidirect
20. C2×#13182006C22semidirect
21. C2×#13182006C22semidirect
22. C2×C4×Dic109C22semidirect
23. C22×Q8×C109C22semidirect
25. #124×C109C22semidirect
26. C2×C22⋊C4×C109C22semidirect
27. C2×C42×C109C22semidirect
31. C23×D109C23semidirect
33. C2×Dic218C23semidirect
41. C22⋊Dic109C23semidirect
42. #13182006C23semidirect
43. #13182006C23semidirect
44. #13182006C23semidirect
45. C4×Dic109C23semidirect
48. C2×Q8×C109C23semidirect
54. C22⋊C4×C109C23semidirect
55. C42×C109C23semidirect
97. C2×C109H(3,𝔽4)semidirect
102. C109C2×H(3,𝔽4)semidirect
107. C2×#124D109semidirect
112. C22×Q8C2×D109semidirect
120. #124C2×D109semidirect
127. C2×C22⋊C4C2×D109semidirect
129. C2×C42C2×D109semidirect
138. C2×Q8C22×D109semidirect
155. C22⋊C4C22×D109semidirect
157. C42C22×D109semidirect
200. C2#20295768direct (N × H)

Maximal subgroups

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

subgroupindex
#202957682
C2×#168126882
C2×#168126872
C2×#168126852
C2×#168126692
C2×#168126672
C2×#124×C1092
C2×H(3,𝔽4)109