Finite Group Databasebeta

← C23×#14579180 #35202053

Subgroups — C23×#14579180 #35202053

Order 20224.

11422 conjugacy classes of subgroups; only the 3659 normal subgroups are shown.

normal subgroups only

This group has 11422 conjugacy classes of subgroups — too many to draw. Only its 3659 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×#11929623C212yesmaximal, complemented
2C22×#14579180C2156yesmaximal, direct product factor
3C23×Dic316C211yesmaximal, characteristic, complemented
4C23×#11929612C212yesmaximal, complemented
5C23×C8×D79C211yesmaximal, characteristic, complemented
6C23×D8×C79C211yesmaximal, characteristic, complemented
7C22×#11929623—228no—
8C22×#11929623C22156yescomplemented
9C23×Dic158C2212yes—
10C23×C22⋊D79—22no—
11C24×Dic79—22no—
12C23×C4×D79C2211yescharacteristic
13C2×#14579180C221448yesdirect product factor
14C22×Dic316C22128yescomplemented
15C22×#11929612C22156yescomplemented
16C23×C79⋊C8C2211yescharacteristic
17C22×C8×D79C22128yescomplemented
18C23×D4×C79C2212yes—
19C22×D8×C79C22128yescomplemented
20C23×C8×C79C2211yescharacteristic
21C24×D79D411yescharacteristic
22C2×#11929623—2336no—
23C2×#11929623C231224yescomplemented
24C22×Dic158—214no—
25C22×Dic158C23128yes—
26C22×C22⋊D79—256no—
27C23×Dic79—22no—
28C23×Dic79—256no—
29C23×Dic79D411yescharacteristic
30C22×C4×D79D4114yes—
31C22×C4×D79C23114yes—
32#14579180C231512yesdirect product factor
33C2×Dic316C231112yescomplemented
34C2×#11929612C231224yescomplemented
35C22×C79⋊C8C23114yes—
36C2×C8×D79C231112yescomplemented
37C25×C79—22no—
38C22×D4×C79—214no—
39C22×D4×C79C23128yes—
40C23×C4×C79C2311yescharacteristic
41C2×D8×C79C231112yescomplemented
42C22×C8×C79C23114yes—
43C23×D79—28no—
44C23×D79C2×D4114yes—
45#11929623—2448no—
46#11929623C241128yescomplemented
47C2×Dic158—284no—
48C2×Dic158C24156yes—
49C2×C22⋊D79—2224no—
50C22×Dic79—2224no—
51C22×Dic79—228no—
52C22×Dic79C2×D4114yes—
53C2×C4×D79C2×D4184yes—
54C2×C4×D79C24128yes—
55Dic316C24164yescomplemented
56#11929612C241128yescomplemented
57C2×C79⋊C8C24128yes—
58C8×D79C24164yescomplemented
59C24×C79—228no—
60C24×C79—416no—
61C24×C79C2×D411yescharacteristic
62C2×D4×C79—284no—
63C2×D4×C79C24156yes—
64C22×C4×C79C2×D417yes—
65C22×C4×C79C2417yes—
66D8×C79C24164yescomplemented
67C2×C8×C79C24128yes—
68C23×C79—256no—
69C23×C79—4112no—
70C23×C79#13518yescomplemented
71C23×C79C22×D417yes—
72C22×D79—256no—
73C22×D79C22×D4128yes—
74D4×C79—256no—
75D4×C79C25116yes—
76C2×C4×C79C22×D4121yes—
77C2×C4×C79C2517yes—
78C22⋊D79—2128no—
79C2×Dic79—256no—
80C2×Dic79—2128no—
81C2×Dic79C22×D4128yes—
82C4×D79C22×D4156yes—
83C4×D79C2518yes—
84Dic158—256no—
85Dic158C25116yes—
86C8×C79C2518yes—
87C79⋊C8C2518yes—
88C22×C79—216no—
89C22×C79—4112no—
90C22×C79C2×#135128yescomplemented
91C22×C79C23×D417yes—
92C2×D79—256no—
93C2×D79C23×D418yes—
94C4×C79C23×D417yes—
95C4×C79C2611yescharacteristic
96Dic79—216no—
97Dic79C23×D418yes—
98C23×#135—791nomaximal, Sylow
99C2×C79—416no—
100C2×C79C22×#135114yescomplemented
101C2×C79C24×D411yescharacteristic
102D79—28no—
103C23×#41—792no—
104C22×#135—7956no—
105C23×Q16—791no—
106C23×SD16—792no—
107C23×D8D7911yescharacteristic, complemented
108C24×C8—791no—
109C79C23×#13511yescharacteristic, complemented, Sylow
110C22×#41—7956no—
111C22×#41—15828no—
112C23×Q8—792no—
113C23×D4—1582no—
114C23×D4C2×D7912yes—
115C24×C4—791no—
116C24×C4—1582no—
117C2×#135—79448no—
118C22×Q16—7928no—
119C22×SD16—7956no—
120C22×D8C2×D79128yescomplemented
121C23×C8—791no—
122C23×C8—7928no—
123C23×C8C2×D7911yescharacteristic
124C25—22no—
125C25—791no—
126C2×#41—79224no—
127C2×#41—158336no—
128C22×Q8—7928no—
129C22×Q8—15814no—
130C22×D4—214no—
131C22×D4—15856no—
132C22×D4C22×D79128yes—
133C23×C4—791no—
134C23×C4—7914no—
135C23×C4—7914no—
136C23×C4—15856no—
137C23×C4—1582no—
138C23×C4C22×D7911yescharacteristic
139#135—79512no—
140C2×Q16—79112no—
141C2×SD16—79224no—
142C2×D8C22×D791112yescomplemented
143C22×C8—7914no—
144C22×C8—79112no—
145C22×C8C22×D79114yes—
146C24—228no—
147C24—416no—
148C24—7914no—
149C24—1588no—
150C24D4×D7911yescharacteristic
151#41—79128no—
152#41—158448no—
153C2×Q8—7956no—
154C2×Q8—15884no—
155C2×D4—284no—
156C2×D4—158224no—
157C2×D4C23×D79156yes—
158C22×C4—7914no—
159C22×C4—7928no—
160C22×C4—7984no—
161C22×C4—158224no—
162C22×C4—15828no—
163C22×C4D4×D7917yes—
164C22×C4C23×D7917yes—
165Q16—7964no—
166SD16—79128no—
167D8C23×D79164yescomplemented
168C2×C8—7928no—
169C2×C8—7964no—
170C2×C8C23×D79128yes—
171C23—256no—
172C23—4112no—
173C23—7928no—
174C23—15856no—
175C23#1457918018yesdirect product factor
176C23C2×D4×D7917yes—
177Q8—7916no—
178Q8—15856no—
179D4—256no—
180D4—158128no—
181D4C24×D79116yes—
182C2×C4—7928no—
183C2×C4—798no—
184C2×C4—7956no—
185C2×C4—158128no—
186C2×C4—15856no—
187C2×C4C2×D4×D79121yes—
188C2×C4C24×D7917yes—
189C8—798no—
190C8C24×D7918yes—
191C22—216no—
192C22—4112no—
193C22—798no—
194C22—15856no—
195C22C2×#14579180128yesdirect product factor
196C22C22×D4×D7917yes—
197C4—798no—
198C4—15816no—
199C4C22×D4×D7917yes—
200C4C25×D7911yescharacteristic
201C2—416no—
202C2—1588no—
203C2C22×#14579180114yesdirect product factor
204C2C23×D4×D7911yescharacteristic
subgroupG/H
centerC24D4×D79
commutator, cosocleC4×C79C26
FrattiniC4C25×D79
FittingC23×D8×C79C2
socleC24×C79C2×D4
hypercenterC23×D8D79
hypocenterC79C23×#135
LCSG ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C23×#11929623C2semidirect
2. C22×#14579180C2direct (N × H)
3. C23×Dic316C2semidirect
4. C23×#11929612C2semidirect
5. C23×C8×D79C2semidirect
6. C23×D8×C79C2semidirect
8. C22×#11929623C22semidirect
13. C2×#14579180C22direct (N × H)
14. C22×Dic316C22semidirect
15. C22×#11929612C22semidirect
17. C22×C8×D79C22semidirect
19. C22×D8×C79C22semidirect
23. C2×#11929623C23semidirect
32. #14579180C23direct (N × H)
33. C2×Dic316C23semidirect
34. C2×#11929612C23semidirect
36. C2×C8×D79C23semidirect
41. C2×D8×C79C23semidirect
46. #11929623C24semidirect
55. Dic316C24semidirect
56. #11929612C24semidirect
58. C8×D79C24semidirect
66. D8×C79C24semidirect
70. C23×C79#135semidirect
90. C22×C79C2×#135semidirect
100. C2×C79C22×#135semidirect
107. C23×D8D79semidirect
109. C79C23×#135semidirect
120. C22×D8C2×D79semidirect
142. C2×D8C22×D79semidirect
167. D8C23×D79semidirect
175. C23#14579180direct (N × H)
195. C22C2×#14579180direct (N × H)
203. C2C22×#14579180direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119296232
C22×#145791802
C23×Dic3162
C23×#119296122
C23×C8×D792
C23×D8×C792
C23×#13579