Finite Group Databasebeta

← #36334069

Subgroups — #36334069

Order 21376.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#24923780C211yesmaximal, characteristic
2#24923610C214yesmaximal, complemented
3#24923593C214yesmaximal, complemented
4#24923309C212yesmaximal, complemented
5#24923268C212yesmaximal, complemented
6#24923233C211yesmaximal, characteristic, complemented
7C2×#117×C167C211yesmaximal, characteristic, complemented
8C2×#14581950C2211yescharacteristic
9C2×#14581934C2212yes—
10#19128815—21no—
11#19128811C2212yes—
12#19128806C2212yes—
13#19128801—21no—
14C4×C22⋊D167—22no—
15#19128791C2211yescharacteristic
16#19128765C2214yescomplemented
17#19128763C2214yescomplemented
18#19128755C2214yescomplemented
19#19128752C2214yescomplemented
20C22⋊(C167⋊C8)C2214yes—
21C23×C4×C167C2211yescharacteristic
22#117×C167—22no—
23#117×C167C2214yes—
24C2×C4⋊C4×C167C2212yes—
25C2×C22⋊C4×C167—21no—
26C2×C42×C167—21no—
27#14581950—24no—
28C2×Dic334C2311yescharacteristic
29C2×D668C2311yescharacteristic
30C2×C22⋊D167—21no—
31C2×C22⋊D167—22no—
32C2×C4×D167—21no—
33#14581934—24no—
34C2×C167⋊C8C2314yes—
35C4⋊Dic167C2312yes—
36Dic167⋊C4—22no—
37C22⋊Dic167—21no—
38C22⋊Dic167—22no—
39(C2×D167)⋊C4—22no—
40C4×Dic167—21no—
41C24×C167D411yescharacteristic
42C22×C4×C167—24no—
43C22×C4×C167—22no—
44C22×C4×C167D411yescharacteristic
45C22×C4×C167D412yes—
46C22×C4×C167C2312yes—
47C22×C4×C167C2311yescharacteristic
48C4⋊C4×C167—22no—
49C4⋊C4×C167C2314yes—
50C22⋊C4×C167—24no—
51C42×C167—24no—
52C22×D167—21no—
53Dic334—22no—
54D668—22no—
55C22⋊D167—44no—
56C22⋊D167—42no—
57C2×Dic167—22no—
58C2×Dic167—21no—
59C4×D167—42no—
60C167⋊C8—24no—
61C23×C167—22no—
62C23×C167—42no—
63C23×C167C2×D412yes—
64C23×C167C2×D411yescharacteristic
65C2×C4×C167—22no—
66C2×C4×C167—24no—
67C2×C4×C167—24no—
68C2×C4×C167—24no—
69C2×C4×C167—44no—
70C2×C4×C167C2×D412yes—
71C2×C4×C167C2×D411yescharacteristic
72C2×C4×C167#4114yes—
73C2×C4×C167C2411yescharacteristic
74C22×C167—22no—
75C22×C167—22no—
76C22×C167—22no—
77C22×C167—22no—
78C22×C167—44no—
79C22×C167#12312yes—
80C22×C167C22×D411yescharacteristic
81C2×D167—41no—
82C2×D167—42no—
83C4×C167—22no—
84C4×C167—44no—
85C4×C167#12312yes—
86C4×C167C2×#4112yes—
87Dic167—41no—
88Dic167—42no—
89C2×C167—22no—
90C2×C167—42no—
91C2×C167C2×#12311yescharacteristic
92C2×C167#57512yes—
93D167—81no—
94C167#309311yescharacteristic, complemented, Sylow
95#3093—1671nomaximal, Sylow
96#525—1671no—
97C2×#117D16711yescharacteristic, complemented
98#485—1674no—
99#482—1674no—
100#428—1672no—
101#417—1672no—
102#407—1671no—
103C2×#41—1671no—
104C23×C4C2×D16711yescharacteristic
105C2×M16—1672no—
106#123—3341no—
107C22⋊Q8—1672no—
108#121—1672no—
109#120—3341no—
110C4×D4—3342no—
111#117—22no—
112#117—1671no—
113#117C2×D16714yes—
114C2×C4⋊C4C2×D16712yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—1674no—
118#106—1674no—
119#103—1674no—
120#102—1674no—
121C22⋊C8—1674no—
122C24C22⋊D16711yescharacteristic
123#41—3344no—
124C2×Q8—1671no—
125C2×D4—3342no—
126C2×D4—3341no—
127C2×D4—1671no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—3341no—
131C22×C4C22⋊D16711yescharacteristic
132C22×C4C22⋊D16712yes—
133C22×C4C22×D16712yes—
134C22×C4C22×D16711yescharacteristic
135M16—3344no—
136C2×C8—1674no—
137C4⋊C4—22no—
138C4⋊C4—3342no—
139C4⋊C4—1672no—
140C4⋊C4C22×D16714yes—
141C22⋊C4—24no—
142C22⋊C4—3341no—
143C22⋊C4—3342no—
144C22⋊C4—3342no—
145C42—24no—
146C42—3341no—
147C23—22no—
148C23—42no—
149C23—3341no—
150C23C2×C22⋊D16712yes—
151C23C2×C22⋊D16711yescharacteristic
152Q8—3342no—
153D4—3342no—
154D4—6684no—
155D4—6682no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—3342no—
162C2×C4—3341no—
163C2×C4—6682no—
164C2×C4C2×C22⋊D16712yes—
165C2×C4C2×C22⋊D16711yescharacteristic
166C2×C4#1458195014yes—
167C2×C4C23×D16711yescharacteristic
168C8—3344no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—6681no—
175C22—6682no—
176C22#1912881512yes—
177C22C22×C22⋊D16711yescharacteristic
178C4—22no—
179C4—44no—
180C4—6681no—
181C4—6682no—
182C4#1912881512yes—
183C4C2×#1458195012yes—
184C2—22no—
185C2—42no—
186C2—13361no—
187C2C2×#1912881511yescharacteristic
188C2#2492406712yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#117×C167C2
socleC22×C167C22×D4
prime radical#24923780C2
hypercenterC2×#117D167
hypocenterC167#3093
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #24923610C2semidirect
3. #24923593C2semidirect
4. #24923309C2semidirect
5. #24923268C2semidirect
6. #24923233C2semidirect
7. C2×#117×C167C2semidirect
16. #19128765C22semidirect
17. #19128763C22semidirect
18. #19128755C22semidirect
19. #19128752C22semidirect
94. C167#3093semidirect
97. C2×#117D167semidirect

Maximal subgroups

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

subgroupindex
#249237802
#249236102
#249235932
#249233092
#249232682
#249232332
C2×#117×C1672
#3093167