Finite Group Databasebeta

← #37554694

Subgroups — #37554694

Order 23488.

330 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 107 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×#19154576C211yesmaximal, characteristic, complemented
2C22×C4×D367C211yesmaximal, characteristic, complemented
3#26044244C211yesmaximal, characteristic, complemented
4#26044239C212yesmaximal, complemented
5#26044234C212yesmaximal, complemented
6#26044231C211yesmaximal, characteristic, complemented
7#26044229C211yesmaximal, characteristic, complemented
8#26044223C212yesmaximal, complemented
9(C22⋊D367)⋊C4C212yesmaximal, complemented
10#26044217C211yesmaximal, characteristic, complemented
11#123×C367C211yesmaximal, characteristic, complemented
12C23×D367C2211yescharacteristic
13#19154576—24no—
14C2×Dic734C2211yescharacteristic, complemented
15C2×D1468C2212yes—
16C2×C22⋊D367C2212yes—
17C2×C22⋊D367C2211yescharacteristic
18C2×C22⋊D367C2211yescharacteristic, complemented
19C22×Dic367C2211yescharacteristic
20C22×Dic367C2211yescharacteristic, complemented
21C2×C4×D367—24no—
22C2×C4×D367C2212yes—
23C2×C4×D367C2212yes—
24C2×C4×D367C2211yescharacteristic, complemented
25Dic367⋊C4C2212yescomplemented
26Dic367⋊C4C2212yescomplemented
27C22⋊Dic367C2211yescharacteristic
28(C2×D367)⋊C4C2212yes—
29(C2×D367)⋊C4C2212yes—
30(C2×D367)⋊C4C2212yescomplemented
31C4×Dic367C2212yescomplemented
32C2×D4×C367C2211yescharacteristic
33C22×C4×C367C2211yescharacteristic
34C4⋊C4×C367C2212yescomplemented
35C22⋊C4×C367C2211yescharacteristic
36C22⋊C4×C367C2212yescomplemented
37C22×D367—22no—
38C22×D367—24no—
39C22×D367C2311yescharacteristic
40C22×D367C2312yes—
41Dic734—22no—
42D1468—24no—
43C22⋊D367—22no—
44C22⋊D367—24no—
45C22⋊D367—22no—
46C2×Dic367—22no—
47C2×Dic367D414yescomplemented
48C2×Dic367C2312yes—
49C2×Dic367C2311yescharacteristic
50C2×Dic367C2311yescharacteristic
51C2×Dic367C2311yescharacteristic
52C4×D367—24no—
53C4×D367—24no—
54C4×D367—22no—
55C23×C367C2311yescharacteristic
56C23×C367C2311yescharacteristic
57D4×C367—22no—
58C2×C4×C367—22no—
59C2×C4×C367C2312yes—
60C2×C4×C367C2312yes—
61C2×C4×C367C2311yescharacteristic
62C22×C367—22no—
63C22×C367—21no—
64C22×C367—22no—
65C22×C367C2×D412yes—
66C22×C367C2411yescharacteristic
67C2×D367—28no—
68C2×D367—22no—
69C2×D367—22no—
70C2×D367—21no—
71C2×D367—22no—
72C2×D367#4114yescomplemented
73C4×C367—22no—
74C4×C367—21no—
75C4×C367—22no—
76Dic367—21no—
77Dic367—22no—
78Dic367C2×D412yes—
79Dic367C2×D412yes—
80C2×C367—22no—
81C2×C367—41no—
82C2×C367C22×D411yescharacteristic
83C2×C367C2×#4112yes—
84D367—24no—
85D367—41no—
86C367#52511yescharacteristic, complemented, Sylow
87#525—3671nomaximal, Sylow
88C2×#41—3671no—
89C23×C4—3671no—
90#123—3671no—
91#123D36711yescharacteristic, complemented
92C22⋊Q8—3672no—
93#121—3672no—
94#120—3671no—
95#120—3671no—
96C4×D4—3672no—
97C4×D4—3672no—
98#117—3671no—
99C24—3671no—
100#41—7344no—
101C2×Q8—3671no—
102C2×D4—3672no—
103C2×D4—3671no—
104C2×D4—3671no—
105C2×D4—3672no—
106C2×D4C2×D36711yescharacteristic
107C22×C4—3671no—
108C22×C4—3671no—
109C22×C4—3672no—
110C22×C4—3671no—
111C22×C4—3672no—
112C22×C4—7344no—
113C22×C4C2×D36711yescharacteristic
114C4⋊C4—3672no—
115C4⋊C4—3672no—
116C4⋊C4C2×D36712yescomplemented
117C22⋊C4—3671no—
118C22⋊C4—3672no—
119C22⋊C4—3672no—
120C22⋊C4—3672no—
121C22⋊C4C2×D36711yescharacteristic
122C22⋊C4C2×D36712yescomplemented
123C42—3672no—
124C23—3672no—
125C23—3671no—
126C23—7344no—
127C23—7342no—
128C23C22×D36711yescharacteristic
129C23C22×D36711yescharacteristic
130Q8—7342no—
131D4—22no—
132D4—7344no—
133D4—7342no—
134D4—7344no—
135D4—7342no—
136C2×C4—22no—
137C2×C4—3674no—
138C2×C4—3671no—
139C2×C4—3671no—
140C2×C4—3672no—
141C2×C4—3671no—
142C2×C4—7344no—
143C2×C4—7344no—
144C2×C4—7342no—
145C2×C4—7342no—
146C2×C4C22×D36712yes—
147C2×C4C22×D36711yescharacteristic
148C2×C4C22×D36712yes—
149C22—22no—
150C22—21no—
151C22—22no—
152C22—3674no—
153C22—7348no—
154C22—7341no—
155C22—7342no—
156C22—7342no—
157C22—7342no—
158C22D4×D36712yes—
159C22C23×D36711yescharacteristic
160C4—22no—
161C4—21no—
162C4—22no—
163C4—3672no—
164C4—3672no—
165C4—7341no—
166C4—7342no—
167C2—22no—
168C2—41no—
169C2—14681no—
170C2—7344no—
171C2C2×D4×D36711yescharacteristic
172C2#41×D36712yes—
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
Fitting#123×C367C2
hypercenter#123D367
hypocenterC367#525
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#19154576C2semidirect
2. C22×C4×D367C2semidirect
3. #26044244C2semidirect
4. #26044239C2semidirect
5. #26044234C2semidirect
6. #26044231C2semidirect
7. #26044229C2semidirect
8. #26044223C2semidirect
9. (C22⋊D367)⋊C4C2semidirect
10. #26044217C2semidirect
11. #123×C367C2semidirect
14. C2×Dic734C22semidirect
18. C2×C22⋊D367C22semidirect
20. C22×Dic367C22semidirect
24. C2×C4×D367C22semidirect
25. Dic367⋊C4C22semidirect
26. Dic367⋊C4C22semidirect
30. (C2×D367)⋊C4C22semidirect
31. C4×Dic367C22semidirect
34. C4⋊C4×C367C22semidirect
36. C22⋊C4×C367C22semidirect
47. C2×Dic367D4semidirect
72. C2×D367#41semidirect
86. C367#525semidirect
91. #123D367semidirect
116. C4⋊C4C2×D367semidirect
122. C22⋊C4C2×D367semidirect

Maximal subgroups

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

subgroupindex
C2×#191545762
C22×C4×D3672
#260442442
#260442392
#260442342
#260442312
#260442292
#260442232
(C22⋊D367)⋊C42
#260442172
#123×C3672
#525367