Finite Group Databasebeta

← #36323287

Subgroups — #36323287

Order 21376.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#24923923C211yesmaximal, characteristic
2#24923619C211yesmaximal, characteristic, complemented
3#24923556C211yesmaximal, characteristic, complemented
4#24923016C211yesmaximal, characteristic, complemented
5D4⋊(C167⋊C8)C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic, complemented
7#476×C167C211yesmaximal, characteristic, complemented
8C2×#14581949—21no—
9#19128832C2211yescharacteristic
10#19128816—22no—
11#19128812—21no—
12#19128806—21no—
13C4×D668C2211yescharacteristic
14C2×C4⋊Dic167—21no—
15#19128763—21no—
16C4⋊(C167⋊C8)C2211yescharacteristic
17#19128756—22no—
18#19128752—21no—
19C4×C167⋊C8C2211yescharacteristic
20C4⋊Q8×C167C2211yescharacteristic
21C4×D4×C167C2211yescharacteristic
22#106×C167—21no—
23C4⋊C8×C167C2211yescharacteristic
24#102×C167—21no—
25#14581949—42no—
26#14581949—42no—
27C2×Dic334—21no—
28C2×D668D411yescharacteristic
29C2×C22⋊D167—22no—
30C22×Dic167—22no—
31C2×C4×D167—21no—
32C2×C167⋊C8—21no—
33C2×C167⋊C8—21no—
34C4⋊Dic167—22no—
35C4⋊Dic167—21no—
36C4⋊Dic167D412yes—
37C4⋊Dic167D411yescharacteristic
38(C2×D167)⋊C4—22no—
39(C2×D167)⋊C4—21no—
40C2×Q8×C167—21no—
41C2×D4×C167D411yescharacteristic
42C22×C4×C167—21no—
43C2×C8×C167—21no—
44C4⋊C4×C167—21no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167D411yescharacteristic
47C22⋊C4×C167—21no—
48C42×C167C2311yescharacteristic
49C22×D167—21no—
50Dic334—41no—
51Dic334—41no—
52D668—21no—
53D668—21no—
54C22⋊D167—44no—
55C2×Dic167—21no—
56C2×Dic167—22no—
57C2×Dic167—44no—
58C4×D167—41no—
59C4×D167—41no—
60C167⋊C8—41no—
61C167⋊C8—41no—
62C23×C167—21no—
63Q8×C167—42no—
64D4×C167—21no—
65D4×C167SD1612yes—
66C2×C4×C167—21no—
67C2×C4×C167—41no—
68C2×C4×C167—41no—
69C2×C4×C167—41no—
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167C2×D411yescharacteristic
72C2×C4×C167C2×D411yescharacteristic
73C8×C167—41no—
74C22×C167—22no—
75C22×C167—41no—
76C22×C167—41no—
77C22×C167#12011yescharacteristic
78C2×D167—41no—
79C2×D167—41no—
80C2×D167—41no—
81C4×C167—21no—
82C4×C167—41no—
83C4×C167—81no—
84C4×C167C2×SD1611yescharacteristic
85C4×C167#13511yescharacteristic
86C4×C167AGL(1,ℤ/8)11yescharacteristic
87C4×C167#13711yescharacteristic
88Dic167—41no—
89Dic167—42no—
90C2×C167—42no—
91C2×C167#45011yescharacteristic
92C2×C167#45211yescharacteristic
93C2×C167#45411yescharacteristic
94D167—81no—
95C167#163511yescharacteristic, complemented, Sylow
96#1635—1671nomaximal, Sylow
97#547—1671no—
98#487—1671no—
99#476—1671no—
100#476D16711yescharacteristic, complemented
101#332—1671no—
102D4⋊C8—1671no—
103D4⋊C8—1671no—
104C2×#41—3341no—
105C4⋊Q8—1671no—
106C4⋊Q8C2×D16711yescharacteristic
107#123—3342no—
108C22⋊Q8—3341no—
109#121—3341no—
110C4×D4—1671no—
111C4×D4C2×D16711yescharacteristic
112C2×C4⋊C4—3341no—
113#106—21no—
114#106—3341no—
115C4⋊C8—1671no—
116C4⋊C8C2×D16711yescharacteristic
117#103—3342no—
118#102—21no—
119#102—3341no—
120C4×C8—1671no—
121#41—6682no—
122#41—6682no—
123C2×Q8—21no—
124C2×Q8—3341no—
125C2×D4—3342no—
126C2×D4—1671no—
127C2×D4C22⋊D16711yescharacteristic
128C22×C4—21no—
129C22×C4—3342no—
130C22×C4—3341no—
131C2×C8—21no—
132C2×C8—3341no—
133C2×C8—3341no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—3342no—
137C4⋊C4—3341no—
138C4⋊C4—1672no—
139C4⋊C4—1671no—
140C4⋊C4C22⋊D16711yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—3342no—
143C22⋊C4—3341no—
144C42C22×D16711yescharacteristic
145C23—21no—
146C23—3341no—
147Q8—42no—
148Q8—6681no—
149Q8—6681no—
150D4—21no—
151D4—3341no—
152D4—3341no—
153D4—6684no—
154D4#1458193712yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3341no—
160C2×C4—3342no—
161C2×C4—6684no—
162C2×C4—6681no—
163C2×C4—6681no—
164C2×C4D4×D16711yescharacteristic
165C2×C4D4×D16711yescharacteristic
166C2×C4C2×C22⋊D16711yescharacteristic
167C8—41no—
168C8—6681no—
169C8—6681no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—6681no—
174C22—6681no—
175C22—6681no—
176C22#1912880011yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—6681no—
181C4—6682no—
182C4C2×#1458193711yescharacteristic
183C4#1912885311yescharacteristic
184C4#1912885711yescharacteristic
185C4#1912886111yescharacteristic
186C2—42no—
187C2—13361no—
188C2#2492341911yescharacteristic
189C2#2492342811yescharacteristic
190C2#2492344011yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#476×C167C2
socleC22×C167#120
prime radical#24923923C2
hypercenter#476D167
hypocenterC167#1635
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #24923619C2semidirect
3. #24923556C2semidirect
4. #24923016C2semidirect
5. D4⋊(C167⋊C8)C2semidirect
6. #24922996C2semidirect
7. #476×C167C2semidirect
95. C167#1635semidirect
100. #476D167semidirect

Maximal subgroups

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

subgroupindex
#249239232
#249236192
#249235562
#249230162
D4⋊(C167⋊C8)2
#249229962
#476×C1672
#1635167