Finite Group Databasebeta

← C22×#16812080 #28349680

Subgroups — C22×#16812080 #28349680

Order 13696.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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×C22⋊D107C212yesmaximal, complemented
2C23×C4×D107C211yesmaximal, characteristic, complemented
3C2×#16812080C2124yesmaximal, direct product factor
4C22×C4⋊Dic107C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic107C212yesmaximal, complemented
6C23×D4×C107C211yesmaximal, characteristic, complemented
7C24×D107C2211yescharacteristic
8C22×C22⋊D107—216no—
9C22×C22⋊D107C22124yescomplemented
10C23×Dic107C2211yescharacteristic
11C23×Dic107C2212yes—
12C22×C4×D107—28no—
13C22×C4×D107C22112yescomplemented
14#16812080C22164yesdirect product factor
15C2×C4⋊Dic107C22112yescomplemented
16C2×C22⋊Dic107C22124yescomplemented
17C25×C107C2212yes—
18C22×D4×C107—28no—
19C22×D4×C107C22112yescomplemented
20C23×C4×C107C2211yescharacteristic
21C23×D107—24no—
22C23×D107—24no—
23C23×D107D418yescomplemented
24C23×D107C2316yes—
25C2×C22⋊D107—296no—
26C2×C22⋊D107C23132yescomplemented
27C22×Dic107—24no—
28C22×Dic107—28no—
29C22×Dic107C2316yes—
30C22×Dic107C23112yes—
31C2×C4×D107—248no—
32C2×C4×D107C23116yescomplemented
33C4⋊Dic107C23116yescomplemented
34C22⋊Dic107C23132yescomplemented
35C24×C107—28no—
36C24×C107—28no—
37C24×C107—28no—
38C24×C107C23112yes—
39C24×C107C2311yescharacteristic
40C2×D4×C107—248no—
41C2×D4×C107C23116yescomplemented
42C22×C4×C107D418yescomplemented
43C22×C4×C107C2316yes—
44C23×C107—224no—
45C23×C107—224no—
46C23×C107—224no—
47C23×C107—432no—
48C23×C107C2×D414yes—
49C23×C107C2×D414yes—
50C23×C107#4114yes—
51C23×C107C2418yes—
52C23×C107C2413yes—
53C22×D107—232no—
54C22×D107—212no—
55C22×D107—212no—
56C22×D107C2×D4124yescomplemented
57C22×D107C2414yes—
58D4×C107—232no—
59C2×C4×C107C2×D4124yescomplemented
60C2×C4×C107C2414yes—
61C22⋊D107—264no—
62C2×Dic107—212no—
63C2×Dic107—224no—
64C2×Dic107C2414yes—
65C2×Dic107C2418yes—
66C4×D107—232no—
67C22×C107—28no—
68C22×C107—28no—
69C22×C107—28no—
70C22×C107—448no—
71C22×C107#121116yescomplemented
72C22×C107C22×D416yes—
73C22×C107C22×D416yes—
74C22×C107C2×#4116yes—
75C22×C107C2511yescharacteristic
76C2×D107—248no—
77C2×D107—24no—
78C2×D107—24no—
79C2×D107C22×D418yescomplemented
80C4×C107C22×D418yescomplemented
81Dic107—24no—
82Dic107—28no—
83C2×C107—48no—
84C2×C107C2×#121112yescomplemented
85C2×C107C23×D411yescharacteristic
86C2×C107C23×D411yescharacteristic
87C2×C107C22×#4111yescharacteristic
88D107—28no—
89C22×#121—1071nomaximal, Sylow
90C107C22×#12111yescharacteristic, complemented, Sylow
91C23×D4—1072no—
92C23×D4D10711yescharacteristic, complemented
93C24×C4—1071no—
94C2×#121—10724no—
95C22×C4⋊C4—1071no—
96C22×C22⋊C4—1072no—
97C25—1071no—
98C25C2×D10712yes—
99C22×D4—28no—
100C22×D4—10724no—
101C22×D4—21416no—
102C22×D4C2×D107112yescomplemented
103C23×C4—1071no—
104C23×C4—1072no—
105C23×C4—10712no—
106C23×C4—2148no—
107C23×C4C2×D10711yescharacteristic
108#121—10764no—
109C2×C4⋊C4—10712no—
110C2×C22⋊C4—10724no—
111C24—28no—
112C24—28no—
113C24—28no—
114C24—1078no—
115C24—1076no—
116C24—2144no—
117C24—2144no—
118C24C22×D107112yes—
119C24C22×D10711yescharacteristic
120C2×D4—248no—
121C2×D4—10732no—
122C2×D4—21496no—
123C2×D4C22×D107116yescomplemented
124C22×C4—1076no—
125C22×C4—10712no—
126C22×C4—10716no—
127C22×C4—2148no—
128C22×C4—2144no—
129C22×C4—21448no—
130C22×C4C22⋊D10718yescomplemented
131C22×C4C22×D10716yes—
132C4⋊C4—10716no—
133C22⋊C4—10732no—
134C23—224no—
135C23—224no—
136C23—224no—
137C23—432no—
138C23—10724no—
139C23—1074no—
140C23—21432no—
141C23—21412no—
142C23—21412no—
143C23D4×D10714yes—
144C23C2×C22⋊D10714yes—
145C23#1313381714yes—
146C23C23×D10718yes—
147C23C23×D10713yes—
148D4—232no—
149D4—21464no—
150C2×C4—1074no—
151C2×C4—1078no—
152C2×C4—21424no—
153C2×C4—21412no—
154C2×C4—21432no—
155C2×C4C2×C22⋊D107124yescomplemented
156C2×C4C23×D10714yes—
157C22—28no—
158C22—28no—
159C22—28no—
160C22—448no—
161C22—1078no—
162C22—2144no—
163C22—2144no—
164C22—21448no—
165C22#16812080116yesdirect product factor
166C22C2×D4×D10716yes—
167C22C22×C22⋊D10716yes—
168C22C2×#1313381716yes—
169C22C24×D10711yescharacteristic
170C4—2148no—
171C4—2144no—
172C4C22×C22⋊D10718yescomplemented
173C2—48no—
174C2—2148no—
175C2C2×#16812080112yesdirect product factor
176C2C22×D4×D10711yescharacteristic
177C2C23×C22⋊D10711yescharacteristic
178C2C22×#1313381711yescharacteristic
subgroupG/H
centerC24C22×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC23×D4×C107C2
socleC24×C107C23
hypercenterC23×D4D107
hypocenterC107C22×#121
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D107C2semidirect
2. C23×C4×D107C2semidirect
3. C2×#16812080C2direct (N × H)
4. C22×C4⋊Dic107C2semidirect
5. C22×C22⋊Dic107C2semidirect
6. C23×D4×C107C2semidirect
9. C22×C22⋊D107C22semidirect
13. C22×C4×D107C22semidirect
14. #16812080C22direct (N × H)
15. C2×C4⋊Dic107C22semidirect
16. C2×C22⋊Dic107C22semidirect
19. C22×D4×C107C22semidirect
23. C23×D107D4semidirect
26. C2×C22⋊D107C23semidirect
32. C2×C4×D107C23semidirect
33. C4⋊Dic107C23semidirect
34. C22⋊Dic107C23semidirect
41. C2×D4×C107C23semidirect
42. C22×C4×C107D4semidirect
56. C22×D107C2×D4semidirect
59. C2×C4×C107C2×D4semidirect
71. C22×C107#121semidirect
79. C2×D107C22×D4semidirect
80. C4×C107C22×D4semidirect
84. C2×C107C2×#121semidirect
90. C107C22×#121semidirect
92. C23×D4D107semidirect
102. C22×D4C2×D107semidirect
123. C2×D4C22×D107semidirect
130. C22×C4C22⋊D107semidirect
155. C2×C4C2×C22⋊D107semidirect
165. C22#16812080direct (N × H)
172. C4C22×C22⋊D107semidirect
175. C2C2×#16812080direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1072
C23×C4×D1072
C2×#168120802
C22×C4⋊Dic1072
C22×C22⋊Dic1072
C23×D4×C1072
C22×#121107