Finite Group Databasebeta

← #36323312

Subgroups — #36323312

Order 21376.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#24923989C211yesmaximal, characteristic
2#24923620C211yesmaximal, characteristic, complemented
3#24923565C211yesmaximal, characteristic, complemented
4#24923559C211yesmaximal, characteristic
5#24923014C211yesmaximal, characteristic, complemented
6#24923001C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic
8#19128832C2211yescharacteristic
9#19128832C2211yescharacteristic
10#19128825—22no—
11#19128819—21no—
12#19128811—21no—
13C4×Dic334C2211yescharacteristic
14#19128797—21no—
15#19128791—21no—
16#19128766—21no—
17#19128763—21no—
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128757—21no—
20#19128757—22no—
21#19128753—21no—
22C4×D4×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24C4×C8×C167C2211yescharacteristic
25C2×Dic334—21no—
26C2×Dic334—21no—
27C2×Dic334D411yescharacteristic
28C2×C167⋊C8—21no—
29C4⋊Dic167—21no—
30C4⋊Dic167—21no—
31C4⋊Dic167—21no—
32C4⋊Dic167D412yes—
33C4⋊Dic167D411yescharacteristic
34Dic167⋊C4—22no—
35Dic167⋊C4—21no—
36C22⋊Dic167—22no—
37C22⋊Dic167—22no—
38C4×Dic167—22no—
39C4×Dic167—21no—
40C2×D4×C167D411yescharacteristic
41C22×C4×C167—21no—
42C2×C8×C167—21no—
43C2×C8×C167—21no—
44C4⋊C4×C167D411yescharacteristic
45C22⋊C4×C167—21no—
46C42×C167C2311yescharacteristic
47Dic334—21no—
48Dic334—21no—
49Dic334—41no—
50Dic334—41no—
51Dic334—42no—
52C2×Dic167—21no—
53C2×Dic167—21no—
54C2×Dic167—22no—
55C2×Dic167—41no—
56C167⋊C8—41no—
57C23×C167—21no—
58D4×C167—21no—
59D4×C167—21no—
60C2×C4×C167—21no—
61C2×C4×C167—41no—
62C2×C4×C167—41no—
63C2×C4×C167C2×D411yescharacteristic
64C2×C4×C167C2×D411yescharacteristic
65C2×C4×C167C2×D411yescharacteristic
66C8×C167—41no—
67C8×C167—41no—
68C22×C167—41no—
69C22×C167—41no—
70C22×C167—41no—
71C22×C167#12011yescharacteristic
72C4×C167—21no—
73C4×C167—41no—
74C4×C167#13511yescharacteristic
75C4×C167#13511yescharacteristic
76C4×C167#13711yescharacteristic
77C4×C167#13711yescharacteristic
78Dic167—41no—
79Dic167—42no—
80Dic167—41no—
81Dic167—81no—
82C2×C167—81no—
83C2×C167#45211yescharacteristic
84C2×C167#45211yescharacteristic
85C2×C167#45411yescharacteristic
86C167#163811yescharacteristic, complemented, Sylow
87#1638—1671nomaximal, Sylow
88#556—1671no—
89#487—1671no—
90#477—1671no—
91#476—1671no—
92#332—1671no—
93Q8⋊C8—1671no—
94D4⋊C8D16711yescharacteristic
95C4⋊Q8—1671no—
96C4⋊Q8—1671no—
97#126—3342no—
98#124—3341no—
99C22⋊Q8—3341no—
100C4×Q8—3341no—
101C4×Q8—1671no—
102C4×D4C2×D16711yescharacteristic
103#117—3341no—
104#107—3341no—
105#106—3341no—
106C4⋊C8—1671no—
107C4⋊C8C2×D16711yescharacteristic
108#103—3341no—
109#103—3342no—
110#102—3341no—
111C4×C8C2×D16711yescharacteristic
112C2×Q8—3341no—
113C2×Q8—3341no—
114C2×Q8—1671no—
115C2×D4D66811yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3341no—
120C4⋊C4—3342no—
121C4⋊C4—3341no—
122C4⋊C4—3341no—
123C4⋊C4—3341no—
124C4⋊C4—3341no—
125C4⋊C4—1672no—
126C4⋊C4—1671no—
127C4⋊C4D66811yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3342no—
130C22⋊C4—3342no—
131C42—3342no—
132C42—3341no—
133C42C22×D16711yescharacteristic
134C23—21no—
135Q8—3341no—
136Q8—3341no—
137Q8—6682no—
138Q8—6681no—
139Q8—6681no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3341no—
146C2×C4—3341no—
147C2×C4—3342no—
148C2×C4—6681no—
149C2×C4D4×D16711yescharacteristic
150C2×C4D4×D16711yescharacteristic
151C2×C4C2×D66811yescharacteristic
152C8—41no—
153C8—41no—
154C8—6681no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1912880211yescharacteristic
159C4—21no—
160C4—41no—
161C4—13361no—
162C4—6681no—
163C4—6682no—
164C4—6681no—
165C4#1912885111yescharacteristic
166C4#1912885411yescharacteristic
167C4#1912886111yescharacteristic
168C4#1912886511yescharacteristic
169C2—81no—
170C2#2492343211yescharacteristic
171C2#2492343311yescharacteristic
172C2#2492344311yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
FittingD4⋊C8×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenterD4⋊C8D167
hypocenterC167#1638
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #24923620C2semidirect
3. #24923565C2semidirect
5. #24923014C2semidirect
6. #24923001C2semidirect
86. C167#1638semidirect

Maximal subgroups

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

subgroupindex
#249239892
#249236202
#249235652
#249235592
#249230142
#249230012
D4⋊C8×C1672
#1638167