Finite Group Databasebeta

← #36378297

Subgroups — #36378297

Order 22592.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#19150736C211yesmaximal, characteristic, complemented
2C22×C4×D353C211yesmaximal, characteristic, complemented
3#26014974C212yesmaximal, complemented
4#26014972C211yesmaximal, characteristic, complemented
5#26014971C211yesmaximal, characteristic, complemented
6#26014963C211yesmaximal, characteristic, complemented
7#26014960C212yesmaximal, complemented
8D4×Dic353C212yesmaximal, complemented
9#26014952C212yesmaximal, complemented
10#26014950C211yesmaximal, characteristic, complemented
11#121×C353C211yesmaximal, characteristic, complemented
12C23×D353C2211yescharacteristic
13#19150736—24no—
14C2×Dic706C2211yescharacteristic, complemented
15C2×C22⋊D353C2212yes—
16C2×C22⋊D353C2212yes—
17C22×Dic353C2211yescharacteristic
18C22×Dic353C2212yescomplemented
19C2×C4×D353—24no—
20C2×C4×D353C2211yescharacteristic
21C2×C4×D353C2211yescharacteristic
22C2×C4×D353C2211yescharacteristic
23C2×C4×D353C2211yescharacteristic
24C4⋊Dic353C2211yescharacteristic, complemented
25C4⋊Dic353C2212yescomplemented
26#19150716C2212yescomplemented
27C22⋊Dic353C2212yes—
28C22⋊Dic353C2212yescomplemented
29#19150714C2212yes—
30#19150714C2212yes—
31C4×Dic353C2212yescomplemented
32C2×D4×C353C2212yes—
33C2×D4×C353C2211yescharacteristic, complemented
34C22×C4×C353C2211yescharacteristic
35C4⋊C4×C353C2211yescharacteristic, complemented
36C22⋊C4×C353C2212yescomplemented
37C22×D353—22no—
38C22×D353—22no—
39C22×D353—22no—
40C22×D353C2311yescharacteristic
41C22×D353C2311yescharacteristic
42Dic706—22no—
43C22⋊D353—24no—
44C22⋊D353—24no—
45C2×Dic353—22no—
46C2×Dic353—24no—
47C2×Dic353C2312yes—
48C2×Dic353C2312yes—
49C2×Dic353C2311yescharacteristic
50C2×Dic353C2311yescharacteristic
51C4×D353—22no—
52C4×D353—22no—
53C4×D353—22no—
54C4×D353D414yescomplemented
55C23×C353C2312yes—
56C23×C353C2311yescharacteristic
57D4×C353—24no—
58D4×C353—22no—
59C2×C4×C353—22no—
60C2×C4×C353C2311yescharacteristic
61C2×C4×C353C2312yes—
62C2×C4×C353C2311yescharacteristic
63C22×C353—21no—
64C22×C353—21no—
65C22×C353—22no—
66C22×C353—22no—
67C22×C353—22no—
68C22×C353#4112yescomplemented
69C22×C353C2411yescharacteristic
70C2×D353—28no—
71C2×D353—21no—
72C2×D353—21no—
73C2×D353—21no—
74C2×D353—21no—
75C2×D353C2×D412yes—
76C2×D353#4112yescomplemented
77C4×C353—21no—
78C4×C353—22no—
79C4×C353C2×D412yes—
80Dic353—21no—
81Dic353—22no—
82Dic353—22no—
83Dic353C2×D412yes—
84C2×C353—22no—
85C2×C353—42no—
86C2×C353C22×D411yescharacteristic
87C2×C353C2×#4111yescharacteristic
88C2×C353C2×#4111yescharacteristic
89D353—22no—
90D353—22no—
91C353#52511yescharacteristic, complemented, Sylow
92#525—3531nomaximal, Sylow
93C2×#41—3531no—
94C23×C4—3531no—
95#123—3532no—
96C22⋊Q8—3531no—
97C22⋊Q8—3531no—
98#121—3531no—
99#121D35311yescharacteristic, complemented
100#120—3532no—
101C4×D4—3532no—
102C4×D4—3532no—
103#117—3531no—
104C24—3531no—
105#41—7064no—
106C2×Q8—3531no—
107C2×D4—3532no—
108C2×D4—3532no—
109C2×D4C2×D35312yes—
110C2×D4C2×D35311yescharacteristic, complemented
111C22×C4—3531no—
112C22×C4—3531no—
113C22×C4—3532no—
114C22×C4—3531no—
115C22×C4—3531no—
116C22×C4—3531no—
117C22×C4—7064no—
118C22×C4C2×D35311yescharacteristic
119C4⋊C4—3531no—
120C4⋊C4—3532no—
121C4⋊C4—3532no—
122C4⋊C4C2×D35311yescharacteristic, complemented
123C22⋊C4—3532no—
124C22⋊C4—3532no—
125C22⋊C4—3532no—
126C22⋊C4—3532no—
127C22⋊C4C2×D35312yescomplemented
128C42—3532no—
129C23—3531no—
130C23—3531no—
131C23—7062no—
132C23—7062no—
133C23—7062no—
134C23C22×D35312yes—
135C23C22×D35311yescharacteristic
136Q8—7062no—
137D4—24no—
138D4—22no—
139D4—7064no—
140D4—7064no—
141C2×C4—22no—
142C2×C4—3531no—
143C2×C4—3534no—
144C2×C4—3532no—
145C2×C4—3531no—
146C2×C4—3532no—
147C2×C4—7062no—
148C2×C4—7062no—
149C2×C4—7062no—
150C2×C4—7064no—
151C2×C4—7062no—
152C2×C4C22×D35312yes—
153C2×C4C22×D35311yescharacteristic
154C2×C4C22×D35311yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3532no—
161C22—3532no—
162C22—7068no—
163C22—7061no—
164C22—7061no—
165C22—7061no—
166C22—7061no—
167C22#1915073612yescomplemented
168C22C23×D35311yescharacteristic
169C4—21no—
170C4—22no—
171C4—3532no—
172C4—7061no—
173C4—7062no—
174C4—7062no—
175C4D4×D35312yes—
176C2—22no—
177C2—42no—
178C2—7062no—
179C2—7062no—
180C2C2×D4×D35311yescharacteristic
181C2#41×D35311yescharacteristic
182C2C2×#1915073611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#121×C353C2
hypercenter#121D353
hypocenterC353#525
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19150736C2semidirect
2. C22×C4×D353C2semidirect
3. #26014974C2semidirect
4. #26014972C2semidirect
5. #26014971C2semidirect
6. #26014963C2semidirect
7. #26014960C2semidirect
8. D4×Dic353C2semidirect
9. #26014952C2semidirect
10. #26014950C2semidirect
11. #121×C353C2semidirect
14. C2×Dic706C22semidirect
18. C22×Dic353C22semidirect
24. C4⋊Dic353C22semidirect
25. C4⋊Dic353C22semidirect
26. #19150716C22semidirect
28. C22⋊Dic353C22semidirect
31. C4×Dic353C22semidirect
33. C2×D4×C353C22semidirect
35. C4⋊C4×C353C22semidirect
36. C22⋊C4×C353C22semidirect
54. C4×D353D4semidirect
68. C22×C353#41semidirect
76. C2×D353#41semidirect
91. C353#525semidirect
99. #121D353semidirect
110. C2×D4C2×D353semidirect
122. C4⋊C4C2×D353semidirect
127. C22⋊C4C2×D353semidirect
167. C22#19150736semidirect

Maximal subgroups

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

subgroupindex
C2×#191507362
C22×C4×D3532
#260149742
#260149722
#260149712
#260149632
#260149602
D4×Dic3532
#260149522
#260149502
#121×C3532
#525353