Finite Group Databasebeta

← #36337550

Subgroups — #36337550

Order 21376.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#24923868C216yesmaximal, complemented
2#24923859C216yesmaximal, complemented
3#24923831C213yesmaximal, complemented
4#24923805C216yesmaximal, complemented
5#24923782C216yesmaximal, complemented
6C2×(C22⋊D167)⋊C4C213yesmaximal, complemented
7#543×C167C211yesmaximal, characteristic, complemented
8C2×#14581951C2212yescomplemented
9C22×C22⋊D167C2213yes—
10C23×Dic167C2211yescharacteristic
11C22×C4×D167C2213yescomplemented
12#19128827C22112yescomplemented
13#19128823C2216yescomplemented
14#19128821C2216yescomplemented
15#19128817C22112yescomplemented
16#19128816C22112yescomplemented
17#19128808C2216yescomplemented
18#19128807C2216yescomplemented
19#19128806C2216yes—
20#19128806C2216yes—
21#19128802C22112yescomplemented
22Q8×Dic167C2212yescomplemented
23#19128794C2216yescomplemented
24(C22⋊D167)⋊C4C22112yescomplemented
25(C22⋊D167)⋊C4C22112yescomplemented
26#19128790C2216yescomplemented
27C2×Dic167⋊C4C2213yescomplemented
28C2×(C2×D167)⋊C4C2213yescomplemented
29C2×C4×Dic167C2213yescomplemented
30C22⋊Q8×C167C2216yescomplemented
31C22⋊Q8×C167C2216yescomplemented
32C2×C22⋊C4×C167C2213yes—
33C23×D167C2313yes—
34#14581951—44no—
35C2×D668C2316yes—
36C2×D668C2316yes—
37C2×C22⋊D167—212no—
38C2×C22⋊D167C23112yes—
39C22×Dic167—42no—
40C22×Dic167C2313yes—
41C22×Dic167C2313yes—
42C22×Dic167C2313yes—
43C2×C4×D167—46no—
44C2×C4×D167C2316yes—
45C2×C4×D167C2316yes—
46C4⋊Dic167C2316yes—
47C4⋊Dic167C2316yes—
48Dic167⋊C4C23112yes—
49(C2×D167)⋊C4C23112yes—
50(C2×D167)⋊C4C23112yes—
51(C2×D167)⋊C4C23112yes—
52C4×Dic167C2316yes—
53C4×Dic167C2316yes—
54C24×C167C2311yescharacteristic
55C2×Q8×C167C2312yes—
56C2×Q8×C167C2312yes—
57C22×C4×C167C2313yes—
58C22×C4×C167C2313yes—
59C4⋊C4×C167C2316yes—
60C4⋊C4×C167C2316yes—
61C22⋊C4×C167C23112yes—
62C22×D167—26no—
63C22×D167—43no—
64C22×D167—43no—
65C22×D167C2416yes—
66D668—46no—
67D668—46no—
68C22⋊D167—224no—
69C2×Dic167—26no—
70C2×Dic167—26no—
71C2×Dic167—43no—
72C2×Dic167C2413yes—
73C2×Dic167C2416yes—
74C2×Dic167C2411yescharacteristic
75C4×D167—212no—
76C4×D167—46no—
77C23×C167—41no—
78C23×C167—41no—
79C23×C167#4114yescomplemented
80C23×C167C2413yes—
81Q8×C167—42no—
82Q8×C167—42no—
83C2×C4×C167—43no—
84C2×C4×C167—43no—
85C2×C4×C167C2416yes—
86C2×C4×C167C2416yes—
87C22×C167—23no—
88C22×C167—23no—
89C22×C167—44no—
90C22×C167C2×#4116yes—
91C22×C167C2511yescharacteristic
92C2×D167—26no—
93C2×D167—26no—
94C2×D167—26no—
95C2×D167—412no—
96C4×C167—26no—
97C4×C167—26no—
98Dic167—26no—
99Dic167—23no—
100Dic167H(5,𝔽2)12yes—
101C2×C167—26no—
102C2×C167C22×#4111yescharacteristic
103C2×C167C2×H(5,𝔽2)11yescharacteristic
104C2×C167#58511yescharacteristic
105D167—46no—
106C167#345911yescharacteristic, complemented, Sylow
107#3459—1671nomaximal, Sylow
108#543D16711yescharacteristic, complemented
109#539—1676no—
110#538—1676no—
111#534—1673no—
112#529—1676no—
113#525—1676no—
114C2×C4×D4—1673no—
115C2×#41—1672no—
116C22×D4—1673no—
117C23×C4—1671no—
118C23×C4—1673no—
119#126—16712no—
120#125—1676no—
121#124—1676no—
122#123—16712no—
123#123—16712no—
124C22⋊Q8—1676no—
125C22⋊Q8C2×D16716yescomplemented
126C22⋊Q8C2×D16716yescomplemented
127#121—1676no—
128#121—1676no—
129#121—1676no—
130#120—16712no—
131C4×Q8—1672no—
132C4×D4—1676no—
133C4×D4—16712no—
134C4×D4—16712no—
135#117—1676no—
136C2×C4⋊C4—1673no—
137C2×C22⋊C4—1673no—
138C2×C22⋊C4C2×D16713yes—
139C2×C42—1673no—
140C24—1673no—
141C24C22×D16711yescharacteristic
142#41—6684no—
143C2×Q8C22×D16712yes—
144C2×Q8C22×D16712yes—
145C2×D4—33412no—
146C2×D4—16712no—
147C2×D4—1676no—
148C2×D4—1676no—
149C22×C4—6686no—
150C22×C4—6682no—
151C22×C4—1673no—
152C22×C4—1673no—
153C22×C4—1676no—
154C22×C4—1673no—
155C22×C4—1676no—
156C22×C4C22×D16713yes—
157C22×C4C22×D16713yes—
158C4⋊C4—16712no—
159C4⋊C4—1676no—
160C4⋊C4—1676no—
161C4⋊C4C22×D16716yes—
162C4⋊C4C22×D16716yes—
163C22⋊C4—16712no—
164C22⋊C4—16712no—
165C22⋊C4—16712no—
166C22⋊C4C22×D167112yes—
167C42—1676no—
168C42—1676no—
169C23—41no—
170C23—41no—
171C23—3346no—
172C23—6683no—
173C23—6683no—
174C23—1676no—
175C23#1458195114yescomplemented
176C23C23×D16713yes—
177Q8—42no—
178Q8—42no—
179D4—33424no—
180D4—6686no—
181D4—6686no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—33412no—
185C2×C4—3346no—
186C2×C4—3346no—
187C2×C4—6686no—
188C2×C4—6683no—
189C2×C4—1676no—
190C2×C4—1673no—
191C2×C4—1671no—
192C2×C4C23×D16716yes—
193C2×C4C23×D16716yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—3346no—
198C22—3346no—
199C22—3346no—
200C22—66812no—
201C22C2×#1458195116yes—
202C22C24×D16711yescharacteristic
203C4—26no—
204C4—26no—
205C4—3346no—
206C4—3343no—
207C4—1672no—
208C2—26no—
209C2—6686no—
210C2C22×#1458195111yescharacteristic
211C2H(5,𝔽2)×D16711yescharacteristic
212C2#2492410411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D167
commutator, socle, cosocleC22×C167C25
Fitting#543×C167C2
hypercenter#543D167
hypocenterC167#3459
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #24923868C2semidirect
2. #24923859C2semidirect
3. #24923831C2semidirect
4. #24923805C2semidirect
5. #24923782C2semidirect
6. C2×(C22⋊D167)⋊C4C2semidirect
7. #543×C167C2semidirect
8. C2×#14581951C22semidirect
11. C22×C4×D167C22semidirect
12. #19128827C22semidirect
13. #19128823C22semidirect
14. #19128821C22semidirect
15. #19128817C22semidirect
16. #19128816C22semidirect
17. #19128808C22semidirect
18. #19128807C22semidirect
21. #19128802C22semidirect
22. Q8×Dic167C22semidirect
23. #19128794C22semidirect
24. (C22⋊D167)⋊C4C22semidirect
25. (C22⋊D167)⋊C4C22semidirect
26. #19128790C22semidirect
27. C2×Dic167⋊C4C22semidirect
28. C2×(C2×D167)⋊C4C22semidirect
29. C2×C4×Dic167C22semidirect
30. C22⋊Q8×C167C22semidirect
31. C22⋊Q8×C167C22semidirect
79. C23×C167#41semidirect
106. C167#3459semidirect
108. #543D167semidirect
125. C22⋊Q8C2×D167semidirect
126. C22⋊Q8C2×D167semidirect
175. C23#14581951semidirect

Maximal subgroups

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

subgroupindex
#249238682
#249238592
#249238312
#249238052
#249237822
C2×(C22⋊D167)⋊C42
#543×C1672
#3459167