Finite Group Databasebeta

← C22×#16812105 #28349732

Subgroups — C22×#16812105 #28349732

Order 13696.

3000 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 725 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
1C23×D428C211yesmaximal, characteristic, complemented
2C23×C22⋊D107C214yesmaximal, complemented
3C2×#16812105C2124yesmaximal, direct product factor
4C22×C4×Dic107C211yesmaximal, characteristic, complemented
5C23×D4×C107C211yesmaximal, characteristic, complemented
6C24×D107C2212yes—
7C22×D428—28no—
8C22×D428C22112yescomplemented
9C22×C22⋊D107—232no—
10C22×C22⋊D107C22148yescomplemented
11C23×Dic107C2212yes—
12#16812105C22164yesdirect product factor
13C2×C4×Dic107C22112yescomplemented
14C25×C107C2212yes—
15C22×D4×C107—28no—
16C22×D4×C107C22112yescomplemented
17C23×C4×C107C2211yescharacteristic
18C23×D107—216no—
19C23×D107—28no—
20C23×D107C23112yes—
21C2×D428—248no—
22C2×D428C23116yescomplemented
23C2×C22⋊D107—2192no—
24C2×C22⋊D107C23164yescomplemented
25C22×Dic107D4116yescomplemented
26C22×Dic107C23112yes—
27C4×Dic107C23116yescomplemented
28C24×C107—216no—
29C24×C107—28no—
30C24×C107C23112yes—
31C24×C107C2311yescharacteristic
32C2×D4×C107—248no—
33C2×D4×C107C23116yescomplemented
34C22×C4×C107D418yescomplemented
35C22×C4×C107C2316yes—
36C23×C107—224no—
37C23×C107—248no—
38C23×C107—432no—
39C23×C107C2×D418yes—
40C23×C107C2×D414yes—
41C23×C107C2418yes—
42C23×C107C2413yes—
43C22×D107—248no—
44C22×D107—224no—
45C22×D107—432no—
46C22×D107C2418yes—
47D4×C107—232no—
48C2×C4×C107C2×D4124yescomplemented
49C2×C4×C107C2414yes—
50C22⋊D107—2128no—
51C2×Dic107C2×D4148yescomplemented
52C2×Dic107C2418yes—
53D428—232no—
54C22×C107—216no—
55C22×C107—28no—
56C22×C107—448no—
57C22×C107#127116yescomplemented
58C22×C107C22×D4112yes—
59C22×C107C22×D416yes—
60C22×C107C2511yescharacteristic
61C2×D107—28no—
62C2×D107—216no—
63C2×D107—448no—
64C4×C107C22×D418yescomplemented
65Dic107C22×D4116yescomplemented
66C2×C107—48no—
67C2×C107C2×#127112yescomplemented
68C2×C107C23×D412yes—
69C2×C107C23×D411yescharacteristic
70D107—48no—
71C22×#127—1071nomaximal, Sylow
72C107C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—1074no—
74C23×D4—1071no—
75C23×D4D10711yescharacteristic, complemented
76C2×#127—10724no—
77C22×C42—1071no—
78C25—1072no—
79C25C2×D10712yes—
80C22×D4—28no—
81C22×D4—10748no—
82C22×D4—10712no—
83C22×D4—21432no—
84C22×D4—2148no—
85C22×D4C2×D107112yescomplemented
86C23×C4—1072no—
87C23×C4C2×D10711yescharacteristic
88#127—10764no—
89C2×C42—10712no—
90C24—216no—
91C24—28no—
92C24—10712no—
93C24—21416no—
94C24—2148no—
95C24C22×D107112yes—
96C24C22×D10711yescharacteristic
97C2×D4—248no—
98C2×D4—10764no—
99C2×D4—10716no—
100C2×D4—214192no—
101C2×D4—21448no—
102C2×D4C22×D107116yescomplemented
103C22×C4—10716no—
104C22×C4—10712no—
105C22×C4C22⋊D10718yescomplemented
106C22×C4C22×D10716yes—
107C42—10716no—
108C23—248no—
109C23—224no—
110C23—432no—
111C23—1078no—
112C23—42832no—
113C23—21448no—
114C23—21424no—
115C23D4×D10718yes—
116C23C2×C22⋊D10714yes—
117C23C23×D10718yes—
118C23C23×D10713yes—
119D4—232no—
120D4—214128no—
121D4—21432no—
122C2×C4—10748no—
123C2×C4—1078no—
124C2×C4C2×C22⋊D107124yescomplemented
125C2×C4C23×D10714yes—
126C22—216no—
127C22—28no—
128C22—448no—
129C22—42848no—
130C22—21416no—
131C22—2148no—
132C22#16812105116yesdirect product factor
133C22C2×D4×D107112yes—
134C22C22×C22⋊D10716yes—
135C22C24×D10711yescharacteristic
136C4—10716no—
137C4C22×C22⋊D10718yescomplemented
138C2—48no—
139C2—4288no—
140C2C2×#16812105112yesdirect product factor
141C2C22×D4×D10712yes—
142C2C23×C22⋊D10711yescharacteristic
subgroupG/H
centerC24C22×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC23×D4×C107C2
socleC24×C107C23
hypercenterC23×D4D107
hypocenterC107C22×#127
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×D428C2semidirect
2. C23×C22⋊D107C2semidirect
3. C2×#16812105C2direct (N × H)
4. C22×C4×Dic107C2semidirect
5. C23×D4×C107C2semidirect
8. C22×D428C22semidirect
10. C22×C22⋊D107C22semidirect
12. #16812105C22direct (N × H)
13. C2×C4×Dic107C22semidirect
16. C22×D4×C107C22semidirect
22. C2×D428C23semidirect
24. C2×C22⋊D107C23semidirect
25. C22×Dic107D4semidirect
27. C4×Dic107C23semidirect
33. C2×D4×C107C23semidirect
34. C22×C4×C107D4semidirect
48. C2×C4×C107C2×D4semidirect
51. C2×Dic107C2×D4semidirect
57. C22×C107#127semidirect
64. C4×C107C22×D4semidirect
65. Dic107C22×D4semidirect
67. C2×C107C2×#127semidirect
72. C107C22×#127semidirect
75. C23×D4D107semidirect
85. C22×D4C2×D107semidirect
102. C2×D4C22×D107semidirect
105. C22×C4C22⋊D107semidirect
124. C2×C4C2×C22⋊D107semidirect
132. C22#16812105direct (N × H)
137. C4C22×C22⋊D107semidirect
140. C2C2×#16812105direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4282
C23×C22⋊D1072
C2×#168121052
C22×C4×Dic1072
C23×D4×C1072
C22×#127107