Finite Group Databasebeta

← #36323143

Subgroups — #36323143

Order 21376.

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

normal subgroups only

This group has 254 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#24923850C211yesmaximal, characteristic
2#24923583C212yesmaximal, complemented
3#24923199C211yesmaximal, characteristic, complemented
4#24923004C212yesmaximal, complemented
5#324×C167C211yesmaximal, characteristic, complemented
6C2×#14581950—21no—
7#19128812—22no—
8#19128807—22no—
9#19128806—21no—
10#19128806C2212yes—
11C2×C4⋊Dic167C2211yescharacteristic
12C2×Dic167⋊C4—21no—
13#19128766—22no—
14#19128754—22no—
15C167⋊#95—21no—
16C167⋊#95C2211yescharacteristic
17C2×C4⋊C4×C167C2211yescharacteristic
18C22⋊C8×C167C2212yes—
19#14581950—42no—
20#14581950—42no—
21C2×Dic334—21no—
22C2×D668—21no—
23C2×D668—21no—
24C2×D668D412yes—
25C2×C22⋊D167—22no—
26C22×Dic167—21no—
27C22×Dic167—21no—
28C2×C4×D167—22no—
29C4⋊Dic167—21no—
30C4⋊Dic167—22no—
31C4⋊Dic167D412yes—
32Dic167⋊C4—42no—
33(C2×D167)⋊C4—22no—
34(C2×D167)⋊C4—22no—
35C22×C4×C167D411yescharacteristic
36C22×C4×C167D411yescharacteristic
37C22×C4×C167C2311yescharacteristic
38C2×C8×C167—22no—
39C4⋊C4×C167—22no—
40C22×D167—22no—
41Dic334—42no—
42D668—42no—
43D668—42no—
44D668—81no—
45C22⋊D167—42no—
46C22⋊D167—42no—
47C2×Dic167—22no—
48C2×Dic167—22no—
49C2×Dic167—41no—
50C2×Dic167—41no—
51C2×Dic167—41no—
52C2×Dic167—41no—
53C2×Dic167—41no—
54C4×D167—44no—
55C23×C167C2×D411yescharacteristic
56C2×C4×C167—21no—
57C2×C4×C167—21no—
58C2×C4×C167—21no—
59C2×C4×C167—41no—
60C2×C4×C167—41no—
61C2×C4×C167D812yes—
62C2×C4×C167C2×D412yes—
63C8×C167—42no—
64C22×C167—21no—
65C22×C167—21no—
66C22×C167#12011yescharacteristic
67C22×C167C2×D811yescharacteristic
68C22×C167AGL(1,ℤ/8)11yescharacteristic
69C2×D167—42no—
70C2×D167—42no—
71C2×D167—42no—
72C4×C167—22no—
73C4×C167—22no—
74C4×C167—41no—
75Dic167—42no—
76Dic167—42no—
77Dic167—81no—
78C2×C167—21no—
79C2×C167—21no—
80C2×C167#44711yescharacteristic
81C2×C167#45611yescharacteristic
82C2×C167#45811yescharacteristic
83D167—82no—
84C167#161011yescharacteristic, complemented, Sylow
85#1610—1671nomaximal, Sylow
86#537—1671no—
87#480—1672no—
88#398—1671no—
89#327—1672no—
90#324D16711yescharacteristic, complemented
91C2×#41—3341no—
92C22⋊Q8—3342no—
93#121—3342no—
94#121—3341no—
95#121—1672no—
96C2×C4⋊C4—3341no—
97C2×C4⋊C4—1671no—
98C2×C4⋊C4C2×D16711yescharacteristic
99#107—3342no—
100#102—3342no—
101C22⋊C8C2×D16712yes—
102#95—3341no—
103#95—1671no—
104#41—6682no—
105#41—6682no—
106C2×Q8—3341no—
107C2×D4—3341no—
108C2×D4—3341no—
109C2×D4—3342no—
110C2×D4—1672no—
111C22×C4—3342no—
112C22×C4—3341no—
113C22×C4—3341no—
114C22×C4D66811yescharacteristic
115C22×C4D66811yescharacteristic
116C22×C4C22×D16711yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—3341no—
120C4⋊C4—3342no—
121C4⋊C4—6682no—
122C4⋊C4—1672no—
123C22⋊C4—3342no—
124C22⋊C4—3342no—
125C23—3342no—
126C23C2×D66811yescharacteristic
127Q8—6682no—
128D4—13361no—
129D4—6682no—
130D4—6682no—
131D4—6682no—
132D4—6682no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—3342no—
139C2×C4—3342no—
140C2×C4—6684no—
141C2×C4—6681no—
142C2×C4—6681no—
143C2×C4—6681no—
144C2×C4—6681no—
145C2×C4—6681no—
146C2×C4D133612yes—
147C2×C4D4×D16712yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—6682no—
152C22—6682no—
153C22—6682no—
154C22#1912880211yescharacteristic
155C22C2×D133611yescharacteristic
156C22#1912886011yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—13361no—
161C4—6682no—
162C4—6682no—
163C2—21no—
164C2—21no—
165C2—13362no—
166C2#2492340511yescharacteristic
167C2#2492345411yescharacteristic
168C2#2492346011yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#324×C167C2
socleC22×C167#120
prime radical#24923850C2
hypercenter#324D167
hypocenterC167#1610
LCSG ▷ C22×C4×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #24923583C2semidirect
3. #24923199C2semidirect
4. #24923004C2semidirect
5. #324×C167C2semidirect
84. C167#1610semidirect
90. #324D167semidirect

Maximal subgroups

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

subgroupindex
#249238502
#249235832
#249231992
#249230042
#324×C1672
#1610167