Finite Group Databasebeta

← #36324686

Subgroups — #36324686

Order 21376.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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#24924036C211yesmaximal, characteristic, complemented
2#24923253C211yesmaximal, characteristic, complemented
3C2×#19128749C212yesmaximal, complemented
4#24923039C212yesmaximal, complemented
5#535×C167C211yesmaximal, characteristic
6#19128839—24no—
7C2×#14581934—22no—
8C2×#14581934C2212yes—
9C2×#14581934C2212yes—
10#19128751—21no—
11#19128749—24no—
12#19128749—21no—
13C2×#41×C167C2211yescharacteristic
14C22×D4×C167C2211yescharacteristic
15#127×C167—21no—
16#124×C167—21no—
17#121×C167—22no—
18#120×C167C414yes—
19#117×C167C2211yescharacteristic
20#14581934—24no—
21#14581934—22no—
22#14581934—22no—
23#14581934—22no—
24#14581934D414yescomplemented
25C2×C167⋊C8—24no—
26C2×C167⋊C8—22no—
27C2×C167⋊C8—22no—
28C24×C167C2×C412yes—
29#41×C167—22no—
30#41×C167—22no—
31C2×Q8×C167D411yescharacteristic
32C2×D4×C167—24no—
33C2×D4×C167—21no—
34C2×D4×C167—21no—
35C2×D4×C167—22no—
36C2×D4×C167—22no—
37C2×D4×C167C2×C412yes—
38C2×D4×C167D411yescharacteristic
39C22×C4×C167D412yes—
40C22×C4×C167C2311yescharacteristic
41C4⋊C4×C167—21no—
42C22⋊C4×C167—24no—
43C22⋊C4×C167C2×C412yes—
44C42×C167—21no—
45C167⋊C8—24no—
46C167⋊C8—42no—
47C23×C167—22no—
48C23×C167—24no—
49C23×C167—82no—
50C23×C167C22⋊C412yes—
51C23×C167C22×C411yescharacteristic
52Q8×C167—22no—
53D4×C167—24no—
54D4×C167—22no—
55D4×C167—41no—
56D4×C167—42no—
57D4×C167—41no—
58D4×C167—84no—
59C2×C4×C167—22no—
60C2×C4×C167—22no—
61C2×C4×C167—22no—
62C2×C4×C167C22⋊C412yes—
63C2×C4×C167C2×D412yes—
64C2×C4×C167C2×D411yescharacteristic
65C2×C4×C167C2×D411yescharacteristic
66C2×C4×C167#4112yes—
67C22×C167—22no—
68C22×C167—41no—
69C22×C167—42no—
70C22×C167—44no—
71C22×C167—42no—
72C22×C167—84no—
73C22×C167C2×C22⋊C411yescharacteristic
74C22×C167C4×D412yes—
75C4×C167—22no—
76C4×C167—42no—
77C4×C167#12011yescharacteristic
78C4×C167#12112yes—
79C4×C167#12311yescharacteristic
80C2×C167—21no—
81C2×C167—22no—
82C2×C167—42no—
83C2×C167—82no—
84C2×C167#38611yescharacteristic
85C167#191411yescharacteristic, complemented, Sylow
86#1914—1671nomaximal, Sylow
87#568—1671no—
88#535D16711yescharacteristic
89#413—1671no—
90C2×#100—1672no—
91#344—1672no—
92C2×#41C2×D16711yescharacteristic
93C22×D4C2×D16711yescharacteristic
94#131—3344no—
95C2×M16—3342no—
96C2×M16—1672no—
97C2×M16—1672no—
98#127—21no—
99#124—21no—
100#121—22no—
101#120Dic16714yes—
102#117C2×D16711yescharacteristic
103#101—3341no—
104#100—3341no—
105#100—3344no—
106C24C2×Dic16712yes—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D16711yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×Dic16712yes—
116C2×D4C22⋊D16711yescharacteristic
117C22×C4C22⋊D16712yes—
118C22×C4C22×D16711yescharacteristic
119M16—3344no—
120M16—3342no—
121M16—3342no—
122M16—3342no—
123M16—1674no—
124C2×C8—3344no—
125C2×C8—3342no—
126C2×C8—3342no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×Dic16712yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊Dic16712yes—
135C23C22×Dic16711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊Dic16712yes—
147C2×C4D4×D16712yes—
148C2×C4C2×C22⋊D16711yescharacteristic
149C2×C4C2×C22⋊D16711yescharacteristic
150C2×C4#1458194912yes—
151C8—3344no—
152C8—6682no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊Dic16711yescharacteristic
160C22D4×Dic16712yes—
161C4—22no—
162C4—42no—
163C4#1912880011yescharacteristic
164C4#1912880412yes—
165C4#1912881311yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2492314911yescharacteristic
subgroupG/H
centerC2#24923149
commutatorC23×C167C22×C4
FrattiniC22×C4C22×D167
Fitting, prime radical#535×C167C2
socleC2×C167#386
cosocleC22×C4×C167C23
hypercenter#535D167
hypocenterC167#1914
LCSG ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #24924036C2semidirect
2. #24923253C2semidirect
3. C2×#19128749C2semidirect
4. #24923039C2semidirect
24. #14581934D4semidirect
85. C167#1914semidirect

Maximal subgroups

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

subgroupindex
#249240362
#249232532
C2×#191287492
#249230392
#535×C1672
#1914167