Finite Group Databasebeta

← #36334381

Subgroups — #36334381

Order 21376.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#24923874C211yesmaximal, characteristic, complemented
2#24923593C212yesmaximal, complemented
3#24923569C212yesmaximal, complemented
4#24923464C212yesmaximal, complemented
5#24923403C212yesmaximal, complemented
6#24923379C212yesmaximal, complemented
7C4×#14581935C212yesmaximal, complemented
8#24923330C211yesmaximal, characteristic, complemented
9#518×C167C211yesmaximal, characteristic, complemented
10C2×#14581935C2212yes—
11C2×#14581935C2212yes—
12#19128829C2211yescharacteristic, complemented
13#19128824C2211yescharacteristic, complemented
14#19128817—22no—
15#19128806C2212yes—
16#19128805—22no—
17C4×D668C2212yescomplemented
18#19128765C2212yescomplemented
19#19128763C2212yescomplemented
20C4⋊(C167⋊C8)C2212yescomplemented
21#19128753C2212yes—
22#19128753C2212yes—
23#19128752C2212yescomplemented
24#19128752C2212yescomplemented
25C22⋊(C167⋊C8)C2212yes—
26#19128743C2211yescharacteristic, complemented
27C4×C167⋊C8C2211yescharacteristic, complemented
28C22×D4×C167C2211yescharacteristic
29C4×D4×C167—22no—
30C4×D4×C167C2212yes—
31C4×D4×C167C2212yes—
32#117×C167C2211yescharacteristic
33#117×C167C2211yescharacteristic
34C2×C22⋊C4×C167—22no—
35C2×D668—22no—
36C2×D668C2312yes—
37C2×C22⋊D167—22no—
38C2×C4×D167—22no—
39#14581935—24no—
40#14581935—24no—
41C2×C167⋊C8C2312yes—
42C2×C167⋊C8C2312yes—
43C4⋊Dic167C2312yes—
44Dic167⋊C4—22no—
45(C2×D167)⋊C4—22no—
46(C2×D167)⋊C4—22no—
47C24×C167—21no—
48C2×D4×C167—24no—
49C2×D4×C167—21no—
50C2×D4×C167C2312yes—
51C22×C4×C167—22no—
52C22×C4×C167—22no—
53C22×C4×C167C2311yescharacteristic
54C4⋊C4×C167D412yes—
55C4⋊C4×C167C2312yes—
56C22⋊C4×C167—21no—
57C22⋊C4×C167—22no—
58C22⋊C4×C167—22no—
59C22⋊C4×C167D412yes—
60C42×C167—21no—
61C42×C167C2311yescharacteristic
62C42×C167C2311yescharacteristic
63C22×D167—22no—
64D668—22no—
65D668—22no—
66D668—82no—
67C22⋊D167—82no—
68C2×Dic167—22no—
69C4×D167—82no—
70C167⋊C8—22no—
71C167⋊C8—22no—
72C23×C167—24no—
73C23×C167—22no—
74C23×C167—82no—
75C23×C167C2×D411yescharacteristic
76D4×C167—24no—
77D4×C167—22no—
78D4×C167#4114yes—
79C2×C4×C167—22no—
80C2×C4×C167—22no—
81C2×C4×C167—82no—
82C2×C4×C167—82no—
83C2×C4×C167C2×D411yescharacteristic
84C2×C4×C167C2×D412yes—
85C2×C4×C167C2×D412yes—
86C2×C4×C167C2411yescharacteristic
87C22×C167—21no—
88C22×C167—24no—
89C22×C167—22no—
90C22×C167—42no—
91C22×C167—42no—
92C22×C167—84no—
93C22×C167C22×D411yescharacteristic
94C2×D167—42no—
95C2×D167—42no—
96C2×D167—42no—
97C4×C167—21no—
98C4×C167—22no—
99C4×C167—22no—
100C4×C167—42no—
101C4×C167C2×#4112yes—
102Dic167—42no—
103C2×C167—41no—
104C2×C167—44no—
105C2×C167#52511yescharacteristic
106C2×C167#57612yes—
107D167—82no—
108C167#312211yescharacteristic, complemented, Sylow
109#3122—1671nomaximal, Sylow
110#540—1671no—
111#518D16711yescharacteristic, complemented
112#482—1672no—
113#478—1672no—
114#459—1672no—
115#447—1672no—
116#442—1672no—
117C4×D8—1672no—
118#433—1671no—
119C22×D4C2×D16711yescharacteristic
120C2×D8—1672no—
121C2×D8—1672no—
122#127—1671no—
123#125—1671no—
124#123—3342no—
125#121—3342no—
126#121—1672no—
127C4×D4—22no—
128C4×D4—1672no—
129C4×D4C2×D16712yes—
130C4×D4C2×D16712yes—
131#117C2×D16711yescharacteristic
132#117C2×D16711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1672no—
135#106—1672no—
136C4⋊C8—1672no—
137#102—1672no—
138#102—1672no—
139#102—1672no—
140#102—1672no—
141C22⋊C8—1672no—
142#97—1671no—
143C4×C8—1671no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3342no—
148C2×D4—3342no—
149C2×D4—1672no—
150C2×D4C22×D16712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—3342no—
154C22×C4C22×D16711yescharacteristic
155D8—3344no—
156D8—3344no—
157C2×C8—1672no—
158C2×C8—1672no—
159C4⋊C4—3342no—
160C4⋊C4—1672no—
161C4⋊C4C22⋊D16712yes—
162C4⋊C4C22×D16712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—3342no—
167C22⋊C4—3342no—
168C22⋊C4C22⋊D16712yes—
169C42—21no—
170C42C22×D16711yescharacteristic
171C42C22×D16711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—3342no—
176C23C2×C22⋊D16711yescharacteristic
177D4—24no—
178D4—22no—
179D4—13362no—
180D4—13362no—
181D4—3342no—
182D4—3342no—
183D4#1458195014yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—13362no—
189C2×C4—3342no—
190C2×C4C2×C22⋊D16711yescharacteristic
191C2×C4C2×C22⋊D16712yes—
192C2×C4C2×C22⋊D16712yes—
193C2×C4C23×D16711yescharacteristic
194C8—3342no—
195C8—3342no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—6682no—
203C22—6682no—
204C22—6682no—
205C22C22×C22⋊D16711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—6682no—
211C4C2×#1458195012yes—
212C2—41no—
213C2—44no—
214C2—13362no—
215C2#2492378011yescharacteristic
216C2#2492407212yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#518×C167C2
socleC22×C167C22×D4
hypercenter#518D167
hypocenterC167#3122
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #24923874C2semidirect
2. #24923593C2semidirect
3. #24923569C2semidirect
4. #24923464C2semidirect
5. #24923403C2semidirect
6. #24923379C2semidirect
7. C4×#14581935C2semidirect
8. #24923330C2semidirect
9. #518×C167C2semidirect
12. #19128829C22semidirect
13. #19128824C22semidirect
17. C4×D668C22semidirect
18. #19128765C22semidirect
19. #19128763C22semidirect
20. C4⋊(C167⋊C8)C22semidirect
23. #19128752C22semidirect
24. #19128752C22semidirect
26. #19128743C22semidirect
27. C4×C167⋊C8C22semidirect
108. C167#3122semidirect
111. #518D167semidirect

Maximal subgroups

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

subgroupindex
#249238742
#249235932
#249235692
#249234642
#249234032
#249233792
C4×#145819352
#249233302
#518×C1672
#3122167