Finite Group Databasebeta

← #42122317

Subgroups — #42122317

Order 27712.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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×#20294096C211yesmaximal, characteristic, complemented
2C23×Dic433C211yesmaximal, characteristic, complemented
3#28357204C211yesmaximal, characteristic, complemented
4#28357201C211yesmaximal, characteristic, complemented
5#28357197C212yesmaximal, complemented
6#28357192C212yesmaximal, complemented
7#28357189C211yesmaximal, characteristic, complemented
8D4×Dic433C212yesmaximal, complemented
9#28357180C212yesmaximal, complemented
10#28357176C211yesmaximal, characteristic, complemented
11#120×C433C211yesmaximal, characteristic, complemented
12#20294096—24no—
13C2×Dic866C2211yescharacteristic, complemented
14C2×C22⋊D433C2212yes—
15C2×C22⋊D433C2211yescharacteristic
16C2×C22⋊D433C2211yescharacteristic, complemented
17C22×Dic433—24no—
18C22×Dic433C2212yes—
19C22×Dic433C2212yes—
20C22×Dic433C2211yescharacteristic
21C22×Dic433C2211yescharacteristic
22C22×Dic433C2211yescharacteristic, complemented
23C2×C4×D433C2211yescharacteristic, complemented
24C4⋊Dic433C2212yescomplemented
25#20294076C2212yescomplemented
26#20294076C2212yescomplemented
27C22⋊Dic433C2211yescharacteristic
28C22⋊Dic433C2212yes—
29C22⋊Dic433C2212yescomplemented
30#20294074C2212yescomplemented
31C4×Dic433C2212yescomplemented
32C24×C433C2211yescharacteristic
33C2×D4×C433C2212yes—
34C2×D4×C433C2211yescharacteristic
35C22⋊C4×C433C2211yescharacteristic
36C22⋊C4×C433C2212yes—
37C22×D433C2311yescharacteristic
38Dic866—22no—
39C22⋊D433—22no—
40C22⋊D433—24no—
41C22⋊D433—22no—
42C2×Dic433—22no—
43C2×Dic433—24no—
44C2×Dic433—24no—
45C2×Dic433—22no—
46C2×Dic433D414yescomplemented
47C2×Dic433C2312yes—
48C2×Dic433C2312yes—
49C2×Dic433C2311yescharacteristic
50C2×Dic433C2311yescharacteristic
51C2×Dic433C2311yescharacteristic
52C4×D433—22no—
53C23×C433—22no—
54C23×C433—24no—
55C23×C433C2311yescharacteristic
56C23×C433C2311yescharacteristic
57C23×C433C2312yes—
58D4×C433—22no—
59D4×C433—24no—
60C2×C4×C433C2312yes—
61C2×C4×C433C2311yescharacteristic
62C22×C433—28no—
63C22×C433—22no—
64C22×C433—22no—
65C22×C433—22no—
66C22×C433—21no—
67C22×C433—22no—
68C22×C433C2×D412yes—
69C22×C433#4114yescomplemented
70C22×C433C2411yescharacteristic
71C2×D433—21no—
72C2×D433—22no—
73C4×C433—21no—
74C4×C433—22no—
75Dic433—22no—
76Dic433—21no—
77Dic433—22no—
78Dic433C2×D412yes—
79Dic433C2×D412yes—
80C2×C433—22no—
81C2×C433—24no—
82C2×C433—41no—
83C2×C433C22×D411yescharacteristic
84C2×C433C2×#4112yes—
85D433—41no—
86C433#52511yescharacteristic, complemented, Sylow
87#525—4331nomaximal, Sylow
88C2×#41—4331no—
89C23×C4—4331no—
90#123—4331no—
91#123—4331no—
92C22⋊Q8—4332no—
93#121—4332no—
94#120—4331no—
95#120D43311yescharacteristic, complemented
96C4×D4—4332no—
97C4×D4—4332no—
98#117—4331no—
99C24C2×D43311yescharacteristic
100#41—8664no—
101C2×Q8—4331no—
102C2×D4—4332no—
103C2×D4—4331no—
104C2×D4—4331no—
105C2×D4C2×D43312yes—
106C2×D4C2×D43311yescharacteristic
107C22×C4—8664no—
108C22×C4—4332no—
109C22×C4—4331no—
110C22×C4—4332no—
111C22×C4—4331no—
112C22×C4—4331no—
113C22×C4—4331no—
114C4⋊C4—4332no—
115C4⋊C4—4332no—
116C4⋊C4—4332no—
117C22⋊C4—4331no—
118C22⋊C4—4332no—
119C22⋊C4—4332no—
120C22⋊C4—4332no—
121C22⋊C4C2×D43311yescharacteristic
122C22⋊C4C2×D43312yes—
123C42—4332no—
124C23—22no—
125C23—24no—
126C23—4331no—
127C23C22×D43311yescharacteristic
128C23C22×D43311yescharacteristic
129C23C22×D43312yes—
130Q8—8662no—
131D4—22no—
132D4—24no—
133D4—8662no—
134D4—8664no—
135D4—8662no—
136C2×C4—8662no—
137C2×C4—8662no—
138C2×C4—8664no—
139C2×C4—8664no—
140C2×C4—8662no—
141C2×C4—4332no—
142C2×C4—4332no—
143C2×C4—4331no—
144C2×C4—4334no—
145C2×C4—4331no—
146C2×C4—4331no—
147C2×C4C22×D43312yes—
148C2×C4C22×D43311yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—8661no—
156C22—8662no—
157C22D4×D43312yes—
158C22#2029409614yescomplemented
159C22C23×D43311yescharacteristic
160C4—21no—
161C4—22no—
162C4—8662no—
163C4—8661no—
164C4—8662no—
165C4—4332no—
166C4—4332no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—17321no—
171C2C2×D4×D43311yescharacteristic
172C2C2×#2029409612yes—
subgroupG/H
center, FrattiniC22C23×D433
commutator, socle, cosocleC22×C433C24
Fitting#120×C433C2
hypercenter#120D433
hypocenterC433#525
LCSG ▷ C22×C433 ▷ C433
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20294096C2semidirect
2. C23×Dic433C2semidirect
3. #28357204C2semidirect
4. #28357201C2semidirect
5. #28357197C2semidirect
6. #28357192C2semidirect
7. #28357189C2semidirect
8. D4×Dic433C2semidirect
9. #28357180C2semidirect
10. #28357176C2semidirect
11. #120×C433C2semidirect
13. C2×Dic866C22semidirect
16. C2×C22⋊D433C22semidirect
22. C22×Dic433C22semidirect
23. C2×C4×D433C22semidirect
24. C4⋊Dic433C22semidirect
25. #20294076C22semidirect
26. #20294076C22semidirect
29. C22⋊Dic433C22semidirect
30. #20294074C22semidirect
31. C4×Dic433C22semidirect
46. C2×Dic433D4semidirect
69. C22×C433#41semidirect
86. C433#525semidirect
95. #120D433semidirect
158. C22#20294096semidirect

Maximal subgroups

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

subgroupindex
C2×#202940962
C23×Dic4332
#283572042
#283572012
#283571972
#283571922
#283571892
D4×Dic4332
#283571802
#283571762
#120×C4332
#525433