Finite Group Databasebeta

← #35203733

Subgroups — #35203733

Order 20288.

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×#19099573C211yesmaximal, characteristic, complemented
2C22×C4×D317C211yesmaximal, characteristic, complemented
3#23778762C212yesmaximal, complemented
4#23778760C211yesmaximal, characteristic, complemented
5#23778759C211yesmaximal, characteristic, complemented
6#23778751C211yesmaximal, characteristic, complemented
7#23778748C212yesmaximal, complemented
8D4×Dic317C212yesmaximal, complemented
9#23778740C212yesmaximal, complemented
10#23778738C211yesmaximal, characteristic, complemented
11#121×C317C211yesmaximal, characteristic, complemented
12C23×D317C2211yescharacteristic
13#19099573—24no—
14C2×Dic634C2211yescharacteristic, complemented
15C2×C22⋊D317C2212yes—
16C2×C22⋊D317C2212yes—
17C22×Dic317C2211yescharacteristic
18C22×Dic317C2212yescomplemented
19C2×C4×D317—24no—
20C2×C4×D317C2211yescharacteristic
21C2×C4×D317C2211yescharacteristic
22C2×C4×D317C2211yescharacteristic
23C2×C4×D317C2211yescharacteristic
24C4⋊Dic317C2211yescharacteristic, complemented
25C4⋊Dic317C2212yescomplemented
26#19099553C2212yescomplemented
27C22⋊Dic317C2212yes—
28C22⋊Dic317C2212yescomplemented
29#19099551C2212yes—
30#19099551C2212yes—
31C4×Dic317C2212yescomplemented
32C2×D4×C317C2212yes—
33C2×D4×C317C2211yescharacteristic, complemented
34C22×C4×C317C2211yescharacteristic
35C4⋊C4×C317C2211yescharacteristic, complemented
36C22⋊C4×C317C2212yescomplemented
37C22×D317—22no—
38C22×D317—22no—
39C22×D317—22no—
40C22×D317C2311yescharacteristic
41C22×D317C2311yescharacteristic
42Dic634—22no—
43C22⋊D317—24no—
44C22⋊D317—24no—
45C2×Dic317—22no—
46C2×Dic317—24no—
47C2×Dic317C2312yes—
48C2×Dic317C2312yes—
49C2×Dic317C2311yescharacteristic
50C2×Dic317C2311yescharacteristic
51C4×D317—22no—
52C4×D317—22no—
53C4×D317—22no—
54C4×D317D414yescomplemented
55C23×C317C2312yes—
56C23×C317C2311yescharacteristic
57D4×C317—24no—
58D4×C317—22no—
59C2×C4×C317—22no—
60C2×C4×C317C2311yescharacteristic
61C2×C4×C317C2312yes—
62C2×C4×C317C2311yescharacteristic
63C22×C317—21no—
64C22×C317—21no—
65C22×C317—22no—
66C22×C317—22no—
67C22×C317—22no—
68C22×C317#4112yescomplemented
69C22×C317C2411yescharacteristic
70C2×D317—28no—
71C2×D317—21no—
72C2×D317—21no—
73C2×D317—21no—
74C2×D317—21no—
75C2×D317C2×D412yes—
76C2×D317#4112yescomplemented
77C4×C317—21no—
78C4×C317—22no—
79C4×C317C2×D412yes—
80Dic317—21no—
81Dic317—22no—
82Dic317—22no—
83Dic317C2×D412yes—
84C2×C317—22no—
85C2×C317—42no—
86C2×C317C22×D411yescharacteristic
87C2×C317C2×#4111yescharacteristic
88C2×C317C2×#4111yescharacteristic
89D317—22no—
90D317—22no—
91C317#52511yescharacteristic, complemented, Sylow
92#525—3171nomaximal, Sylow
93C2×#41—3171no—
94C23×C4—3171no—
95#123—3172no—
96C22⋊Q8—3171no—
97C22⋊Q8—3171no—
98#121—3171no—
99#121D31711yescharacteristic, complemented
100#120—3172no—
101C4×D4—3172no—
102C4×D4—3172no—
103#117—3171no—
104C24—3171no—
105#41—6344no—
106C2×Q8—3171no—
107C2×D4—3172no—
108C2×D4—3172no—
109C2×D4C2×D31712yes—
110C2×D4C2×D31711yescharacteristic, complemented
111C22×C4—3171no—
112C22×C4—3171no—
113C22×C4—3172no—
114C22×C4—3171no—
115C22×C4—3171no—
116C22×C4—3171no—
117C22×C4—6344no—
118C22×C4C2×D31711yescharacteristic
119C4⋊C4—3171no—
120C4⋊C4—3172no—
121C4⋊C4—3172no—
122C4⋊C4C2×D31711yescharacteristic, complemented
123C22⋊C4—3172no—
124C22⋊C4—3172no—
125C22⋊C4—3172no—
126C22⋊C4—3172no—
127C22⋊C4C2×D31712yescomplemented
128C42—3172no—
129C23—3171no—
130C23—3171no—
131C23—6342no—
132C23—6342no—
133C23—6342no—
134C23C22×D31712yes—
135C23C22×D31711yescharacteristic
136Q8—6342no—
137D4—24no—
138D4—22no—
139D4—6344no—
140D4—6344no—
141C2×C4—22no—
142C2×C4—3171no—
143C2×C4—3174no—
144C2×C4—3172no—
145C2×C4—3171no—
146C2×C4—3172no—
147C2×C4—6342no—
148C2×C4—6342no—
149C2×C4—6342no—
150C2×C4—6344no—
151C2×C4—6342no—
152C2×C4C22×D31712yes—
153C2×C4C22×D31711yescharacteristic
154C2×C4C22×D31711yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3172no—
161C22—3172no—
162C22—6348no—
163C22—6341no—
164C22—6341no—
165C22—6341no—
166C22—6341no—
167C22#1909957312yescomplemented
168C22C23×D31711yescharacteristic
169C4—21no—
170C4—22no—
171C4—3172no—
172C4—6341no—
173C4—6342no—
174C4—6342no—
175C4D4×D31712yes—
176C2—22no—
177C2—42no—
178C2—6342no—
179C2—6342no—
180C2C2×D4×D31711yescharacteristic
181C2#41×D31711yescharacteristic
182C2C2×#1909957311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D317
commutator, socle, cosocleC22×C317C24
Fitting#121×C317C2
hypercenter#121D317
hypocenterC317#525
LCSG ▷ C22×C317 ▷ C317
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#19099573C2semidirect
2. C22×C4×D317C2semidirect
3. #23778762C2semidirect
4. #23778760C2semidirect
5. #23778759C2semidirect
6. #23778751C2semidirect
7. #23778748C2semidirect
8. D4×Dic317C2semidirect
9. #23778740C2semidirect
10. #23778738C2semidirect
11. #121×C317C2semidirect
14. C2×Dic634C22semidirect
18. C22×Dic317C22semidirect
24. C4⋊Dic317C22semidirect
25. C4⋊Dic317C22semidirect
26. #19099553C22semidirect
28. C22⋊Dic317C22semidirect
31. C4×Dic317C22semidirect
33. C2×D4×C317C22semidirect
35. C4⋊C4×C317C22semidirect
36. C22⋊C4×C317C22semidirect
54. C4×D317D4semidirect
68. C22×C317#41semidirect
76. C2×D317#41semidirect
91. C317#525semidirect
99. #121D317semidirect
110. C2×D4C2×D317semidirect
122. C4⋊C4C2×D317semidirect
127. C22⋊C4C2×D317semidirect
167. C22#19099573semidirect

Maximal subgroups

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

subgroupindex
C2×#190995732
C22×C4×D3172
#237787622
#237787602
#237787592
#237787512
#237787482
D4×Dic3172
#237787402
#237787382
#121×C3172
#525317