Finite Group Databasebeta

← #36323463

Subgroups — #36323463

Order 21376.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#24924013C211yesmaximal, characteristic
2#24923638C211yesmaximal, characteristic, complemented
3#24923629C211yesmaximal, characteristic, complemented
4#24923628C211yesmaximal, characteristic, complemented
5#24923010C211yesmaximal, characteristic, complemented
6#24923008C211yesmaximal, characteristic, complemented
7#329×C167C211yesmaximal, characteristic, complemented
8#19128832C2211yescharacteristic
9#19128824C2211yescharacteristic
10#19128820C2211yescharacteristic
11#19128816—21no—
12#19128814—21no—
13#19128809—21no—
14#19128808—21no—
15#19128805—21no—
16#19128804—21no—
17#19128757—21no—
18#19128757—21no—
19#19128757—21no—
20#19128756—21no—
21#19128754—21no—
22#19128753—21no—
23#19128743C2211yescharacteristic
24#124×C167C2211yescharacteristic
25#97×C167C2211yescharacteristic
26#97×C167C2211yescharacteristic
27C2×Dic334—21no—
28C2×Dic334D411yescharacteristic
29C2×D668D411yescharacteristic
30C2×C22⋊D167—21no—
31C2×C22⋊D167—21no—
32C22×Dic167—21no—
33C2×C4×D167—21no—
34C2×C167⋊C8—21no—
35C4⋊Dic167—21no—
36C4⋊Dic167—21no—
37C4⋊Dic167—21no—
38C4⋊Dic167D411yescharacteristic
39C4⋊Dic167D411yescharacteristic
40Dic167⋊C4—21no—
41Dic167⋊C4—21no—
42Dic167⋊C4—21no—
43C22⋊Dic167—21no—
44(C2×D167)⋊C4—21no—
45(C2×D167)⋊C4—21no—
46(C2×D167)⋊C4—21no—
47C2×Q8×C167D411yescharacteristic
48C2×D4×C167D411yescharacteristic
49C2×C8×C167—21no—
50C2×C8×C167—21no—
51C22⋊C4×C167—21no—
52C22⋊C4×C167—21no—
53C42×C167C2311yescharacteristic
54C22×D167—21no—
55Dic334—41no—
56Dic334—42no—
57D668—41no—
58C22⋊D167—81no—
59C22⋊D167—81no—
60C2×Dic167—21no—
61C2×Dic167—21no—
62C2×Dic167—21no—
63C2×Dic167—41no—
64C2×Dic167—81no—
65C4×D167—81no—
66C167⋊C8—41no—
67C23×C167—21no—
68Q8×C167—41no—
69D4×C167—41no—
70C2×C4×C167—21no—
71C2×C4×C167C2×D411yescharacteristic
72C2×C4×C167C2×D411yescharacteristic
73C2×C4×C167C2×D411yescharacteristic
74C8×C167—41no—
75C8×C167—41no—
76C22×C167—41no—
77C22×C167—41no—
78C22×C167—41no—
79C22×C167#12011yescharacteristic
80C2×D167—41no—
81C2×D167—41no—
82C2×D167—41no—
83C4×C167—21no—
84C4×C167—21no—
85C4×C167—21no—
86C4×C167—41no—
87Dic167—41no—
88Dic167—41no—
89Dic167—41no—
90Dic167—81no—
91C2×C167—81no—
92C2×C167#45411yescharacteristic
93C2×C167#45511yescharacteristic
94C2×C167#45511yescharacteristic
95D167—81no—
96C167#166211yescharacteristic, complemented, Sylow
97#1662—1671nomaximal, Sylow
98#562—1671no—
99#491—1671no—
100#489—1671no—
101#489—1671no—
102#330—1671no—
103#329—1671no—
104#329D16711yescharacteristic, complemented
105C4⋊Q8—1671no—
106#125—1671no—
107#124—1671no—
108#124C2×D16711yescharacteristic
109#123—3341no—
110#123—3341no—
111C22⋊Q8—3341no—
112C22⋊Q8—3341no—
113#121—3341no—
114#121—3341no—
115#103—3341no—
116#103—3341no—
117#103—3341no—
118#103—3341no—
119#102—3341no—
120#102—3341no—
121#97—1671no—
122#97C2×D16711yescharacteristic
123#97C2×D16711yescharacteristic
124C2×Q8—3341no—
125C2×Q8—1671no—
126C2×Q8D66811yescharacteristic
127C2×D4—3341no—
128C2×D4—3341no—
129C2×D4—1671no—
130C2×D4D66811yescharacteristic
131C22×C4—3341no—
132C22×C4—3341no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3341no—
136C4⋊C4—3341no—
137C4⋊C4—3341no—
138C4⋊C4—3341no—
139C4⋊C4—3341no—
140C4⋊C4—3341no—
141C4⋊C4—3341no—
142C4⋊C4—1671no—
143C4⋊C4—1671no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—3341no—
147C22⋊C4—3341no—
148C22⋊C4—3341no—
149C22⋊C4—3341no—
150C42C22×D16711yescharacteristic
151C23—21no—
152C23—3341no—
153Q8—41no—
154Q8—6682no—
155Q8—6681no—
156D4—41no—
157D4—13361no—
158D4—13361no—
159D4—6681no—
160C2×C4—21no—
161C2×C4—13361no—
162C2×C4—13361no—
163C2×C4—3341no—
164C2×C4—3341no—
165C2×C4—3341no—
166C2×C4—6681no—
167C2×C4D4×D16711yescharacteristic
168C2×C4D4×D16711yescharacteristic
169C2×C4C2×D66811yescharacteristic
170C8—41no—
171C8—41no—
172C8—6681no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—6681no—
177C22—6681no—
178C22—6681no—
179C22#1912880211yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—13361no—
185C4—6681no—
186C4—6681no—
187C4—6681no—
188C2—81no—
189C2—13361no—
190C2#2492344311yescharacteristic
191C2#2492344811yescharacteristic
192C2#2492344911yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#329×C167C2
socleC22×C167#120
prime radical#24924013C2
hypercenter#329D167
hypocenterC167#1662
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #24923638C2semidirect
3. #24923629C2semidirect
4. #24923628C2semidirect
5. #24923010C2semidirect
6. #24923008C2semidirect
7. #329×C167C2semidirect
96. C167#1662semidirect
104. #329D167semidirect

Maximal subgroups

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

subgroupindex
#249240132
#249236382
#249236292
#249236282
#249230102
#249230082
#329×C1672
#1662167