Finite Group Databasebeta

← #36334067

Subgroups — #36334067

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
1C2×#19128791C211yesmaximal, characteristic, complemented
2#24923607C214yesmaximal, complemented
3#24923590C214yesmaximal, complemented
4#24923312C212yesmaximal, complemented
5#24923270C212yesmaximal, complemented
6#24923233C211yesmaximal, characteristic, complemented
7#525×C167C211yesmaximal, characteristic
8C2×#14581934C2212yes—
9#19128791—22no—
10#19128791C2214yes—
11C2×C4⋊Dic167C2212yes—
12C2×C22⋊Dic167—21no—
13C2×C4×Dic167—21no—
14#19128765C2214yescomplemented
15#19128763C2214yescomplemented
16#19128756C2214yescomplemented
17#19128753C2214yescomplemented
18C22⋊(C167⋊C8)C2214yes—
19C2×#41×C167C2211yescharacteristic
20C23×C4×C167C2211yescharacteristic
21#123×C167—21no—
22C22⋊Q8×C167C2212yes—
23#121×C167C2212yes—
24#120×C167—21no—
25C4×D4×C167—22no—
26#117×C167C2211yescharacteristic
27C22×Dic167—22no—
28#14581934—24no—
29C2×C167⋊C8C2314yes—
30C4⋊Dic167—22no—
31C4⋊Dic167C2314yes—
32C22⋊Dic167—24no—
33C4×Dic167—24no—
34C24×C167D411yescharacteristic
35#41×C167—24no—
36C2×Q8×C167C2311yescharacteristic
37C2×D4×C167—22no—
38C2×D4×C167—21no—
39C2×D4×C167C2311yescharacteristic
40C22×C4×C167—24no—
41C22×C4×C167—21no—
42C22×C4×C167D412yes—
43C22×C4×C167D411yescharacteristic
44C22×C4×C167C2312yes—
45C22×C4×C167C2311yescharacteristic
46C4⋊C4×C167—22no—
47C4⋊C4×C167C2312yes—
48C22⋊C4×C167—21no—
49C22⋊C4×C167—22no—
50C22⋊C4×C167—22no—
51C42×C167—21no—
52C2×Dic167—24no—
53C2×Dic167—44no—
54C167⋊C8—24no—
55C23×C167—22no—
56C23×C167—21no—
57C23×C167—42no—
58C23×C167C2×D412yes—
59C23×C167C2×D411yescharacteristic
60Q8×C167—22no—
61D4×C167—22no—
62D4×C167—44no—
63D4×C167—42no—
64C2×C4×C167—22no—
65C2×C4×C167—24no—
66C2×C4×C167—24no—
67C2×C4×C167—22no—
68C2×C4×C167—21no—
69C2×C4×C167—42no—
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—41no—
80C22×C167—42no—
81C22×C167#12312yes—
82C22×C167C22×D411yescharacteristic
83C4×C167—22no—
84C4×C167—41no—
85C4×C167—42no—
86C4×C167#12312yes—
87C4×C167C2×#4112yes—
88Dic167—44no—
89C2×C167—22no—
90C2×C167—42no—
91C2×C167—81no—
92C2×C167C2×#12311yescharacteristic
93C2×C167#57512yes—
94C167#309311yescharacteristic, complemented, Sylow
95#3093—1671nomaximal, Sylow
96#525D16711yescharacteristic
97C2×#117—1671no—
98#485—1674no—
99#482—1674no—
100#428—1672no—
101#417—1672no—
102#407—1671no—
103C2×#41C2×D16711yescharacteristic
104C23×C4C2×D16711yescharacteristic
105C2×M16—1672no—
106#123—21no—
107C22⋊Q8C2×D16712yes—
108#121C2×D16712yes—
109#120—21no—
110C4×D4—22no—
111#117—3342no—
112#117—1674no—
113#117C2×D16711yescharacteristic
114C2×C4⋊C4—1672no—
115C2×C22⋊C4—3341no—
116C2×C42—3341no—
117#107—1674no—
118#106—1674no—
119#103—1674no—
120#102—1674no—
121C22⋊C8—1674no—
122C24C22⋊D16711yescharacteristic
123#41—24no—
124C2×Q8C22×D16711yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D16711yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—3342no—
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—1674no—
140C4⋊C4C22×D16712yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3344no—
145C42—21no—
146C42—3344no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D16712yes—
151C23C2×C22⋊D16711yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—3344no—
163C2×C4—6684no—
164C2×C4C2×C22⋊D16712yes—
165C2×C4C2×C22⋊D16711yescharacteristic
166C2×C4#1458194914yes—
167C2×C4C23×D16711yescharacteristic
168C8—3344no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1912881312yes—
177C22C22×C22⋊D16711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—6684no—
182C4#1912881312yes—
183C4C2×#1458194912yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1912881311yescharacteristic
188C2#2492406712yes—
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radical#525×C167C2
socleC22×C167C22×D4
hypercenter#525D167
hypocenterC167#3093
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19128791C2semidirect
2. #24923607C2semidirect
3. #24923590C2semidirect
4. #24923312C2semidirect
5. #24923270C2semidirect
6. #24923233C2semidirect
14. #19128765C22semidirect
15. #19128763C22semidirect
16. #19128756C22semidirect
17. #19128753C22semidirect
94. C167#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#191287912
#249236072
#249235902
#249233122
#249232702
#249232332
#525×C1672
#3093167