Finite Group Databasebeta

← #36334382

Subgroups — #36334382

Order 21376.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#24923743C211yesmaximal, characteristic, complemented
3#24923594C212yesmaximal, complemented
4#24923570C212yesmaximal, complemented
5#24923465C212yesmaximal, complemented
6#24923405C212yesmaximal, complemented
7#24923382C212yesmaximal, complemented
8C4×D1336C212yesmaximal, complemented
9#433×C167C211yesmaximal, characteristic, complemented
10C22×D668C2211yescharacteristic
11C2×D1336C2212yes—
12C2×D1336C2212yes—
13#19128829C2211yescharacteristic, complemented
14#19128824C2211yescharacteristic, complemented
15#19128817—22no—
16#19128806C2212yes—
17#19128805—22no—
18C4×D668C2212yes—
19C4×D668C2212yes—
20C4×D668C2212yescomplemented
21#19128794—22no—
22#19128791C2211yescharacteristic
23C22⋊C4×D167—22no—
24#19128766C2212yescomplemented
25#19128764C2212yescomplemented
26#19128754C2212yes—
27#19128754C2212yes—
28#19128754C2212yescomplemented
29#19128754C2212yescomplemented
30#117×C167C2211yescharacteristic
31C4⋊C8×C167C2212yescomplemented
32C22⋊C8×C167C2212yes—
33#97×C167C2211yescharacteristic, complemented
34C4×C8×C167C2211yescharacteristic, complemented
35C23×D167—21no—
36C2×D668—24no—
37C2×D668—21no—
38C2×D668—22no—
39C2×D668C2312yes—
40C2×D668C2312yes—
41C2×C22⋊D167—22no—
42C2×C4×D167—22no—
43C2×C4×D167—22no—
44C2×C4×D167—22no—
45D1336—24no—
46D1336—24no—
47C4⋊Dic167C2312yes—
48C4⋊Dic167C2312yes—
49Dic167⋊C4—22no—
50C22⋊Dic167—21no—
51(C2×D167)⋊C4—22no—
52(C2×D167)⋊C4—22no—
53(C2×D167)⋊C4—22no—
54(C2×D167)⋊C4—22no—
55C4×Dic167—21no—
56C22×C4×C167C2311yescharacteristic
57C2×C8×C167C2312yes—
58C2×C8×C167C2312yes—
59C4⋊C4×C167D412yes—
60C22⋊C4×C167D412yes—
61C42×C167C2311yescharacteristic
62C42×C167C2311yescharacteristic
63C22×D167—22no—
64C22×D167—24no—
65C22×D167—22no—
66C22×D167—82no—
67D668—24no—
68D668—22no—
69D668—22no—
70D668—22no—
71D668—82no—
72D668#4114yes—
73C22⋊D167—82no—
74C2×Dic167—22no—
75C2×Dic167—22no—
76C4×D167—82no—
77C4×D167—82no—
78C4×D167—82no—
79C23×C167C2×D411yescharacteristic
80C2×C4×C167—22no—
81C2×C4×C167C2×D412yes—
82C2×C4×C167C2×D412yes—
83C2×C4×C167C2×D411yescharacteristic
84C2×C4×C167C2411yescharacteristic
85C8×C167—22no—
86C8×C167—22no—
87C22×C167—21no—
88C22×C167—22no—
89C22×C167C22×D411yescharacteristic
90C2×D167—24no—
91C2×D167—42no—
92C2×D167—42no—
93C2×D167—42no—
94C2×D167—42no—
95C2×D167—42no—
96C2×D167—84no—
97C4×C167—21no—
98C4×C167—22no—
99C4×C167—22no—
100C4×C167C2×#4112yes—
101Dic167—42no—
102Dic167—42no—
103C2×C167—41no—
104C2×C167#52511yescharacteristic
105C2×C167#57612yes—
106D167—44no—
107D167—82no—
108C167#312211yescharacteristic, complemented, Sylow
109#3122—1671nomaximal, Sylow
110#540—1671no—
111#518—1671no—
112#482—1672no—
113#478—1672no—
114#459—1672no—
115#447—1672no—
116#442—1672no—
117C4×D8—1672no—
118#433D16711yescharacteristic, complemented
119C22×D4—1671no—
120C2×D8—1672no—
121C2×D8—1672no—
122#127—1671no—
123#125—1671no—
124#123—3342no—
125#121—3342no—
126#121—1672no—
127C4×D4—3342no—
128C4×D4—1672no—
129C4×D4—1672no—
130C4×D4—1672no—
131#117—1671no—
132#117C2×D16711yescharacteristic
133C2×C22⋊C4—3342no—
134#107—1672no—
135#106—1672no—
136C4⋊C8C2×D16712yescomplemented
137#102—1672no—
138#102—1672no—
139#102—1672no—
140#102—1672no—
141C22⋊C8C2×D16712yes—
142#97C2×D16711yescharacteristic, complemented
143C4×C8C2×D16711yescharacteristic, complemented
144C24—3341no—
145C2×D4—3344no—
146C2×D4—3342no—
147C2×D4—3342no—
148C2×D4—3341no—
149C2×D4—1672no—
150C2×D4—1672no—
151C22×C4—3342no—
152C22×C4—3342no—
153C22×C4—3342no—
154C22×C4C22×D16711yescharacteristic
155D8—3344no—
156D8—3344no—
157C2×C8C22×D16712yes—
158C2×C8C22×D16712yes—
159C4⋊C4—3342no—
160C4⋊C4—1672no—
161C4⋊C4—1672no—
162C4⋊C4D66812yes—
163C22⋊C4—3341no—
164C22⋊C4—3342no—
165C22⋊C4—3342no—
166C22⋊C4—3342no—
167C22⋊C4—3342no—
168C22⋊C4D66812yes—
169C42—3341no—
170C42C22×D16711yescharacteristic
171C42C22×D16711yescharacteristic
172C23—13362no—
173C23—3344no—
174C23—3342no—
175C23—3342no—
176C23C2×D66811yescharacteristic
177D4—13362no—
178D4—13362no—
179D4—3344no—
180D4—3342no—
181D4—3342no—
182D4—3342no—
183D4—1674no—
184C2×C4—22no—
185C2×C4—13362no—
186C2×C4—13362no—
187C2×C4—13362no—
188C2×C4—3342no—
189C2×C4—3342no—
190C2×C4C2×D66811yescharacteristic
191C2×C4C2×D66812yes—
192C2×C4C2×D66812yes—
193C2×C4C23×D16711yescharacteristic
194C8—22no—
195C8—22no—
196C22—21no—
197C22—22no—
198C22—13364no—
199C22—3344no—
200C22—6682no—
201C22—6682no—
202C22—6682no—
203C22—6682no—
204C22—6682no—
205C22C22×D66811yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—6682no—
210C4—6682no—
211C4#41×D16712yes—
212C2—41no—
213C2—13362no—
214C2—6684no—
215C2#2492378111yescharacteristic
216C2#2492407412yes—
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#433×C167C2
socleC22×C167C22×D4
hypercenter#433D167
hypocenterC167#3122
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #24923874C2semidirect
2. #24923743C2semidirect
3. #24923594C2semidirect
4. #24923570C2semidirect
5. #24923465C2semidirect
6. #24923405C2semidirect
7. #24923382C2semidirect
8. C4×D1336C2semidirect
9. #433×C167C2semidirect
13. #19128829C22semidirect
14. #19128824C22semidirect
20. C4×D668C22semidirect
24. #19128766C22semidirect
25. #19128764C22semidirect
28. #19128754C22semidirect
29. #19128754C22semidirect
31. C4⋊C8×C167C22semidirect
33. #97×C167C22semidirect
34. C4×C8×C167C22semidirect
108. C167#3122semidirect
118. #433D167semidirect
136. C4⋊C8C2×D167semidirect
142. #97C2×D167semidirect
143. C4×C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249238742
#249237432
#249235942
#249235702
#249234652
#249234052
#249233822
C4×D13362
#433×C1672
#3122167