Finite Group Databasebeta

← #36323182

Subgroups — #36323182

Order 21376.

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

normal subgroups only

This group has 202 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#24923900C211yesmaximal, characteristic
2#24923605C212yesmaximal
3#24923208C211yesmaximal, characteristic
4#24923007C212yesmaximal
5#324×C167C211yesmaximal, characteristic
6#19128831—22no—
7#19128823—22no—
8#19128811—21no—
9#19128811C2212yes—
10#19128791—21no—
11C2×C4⋊Dic167C2211yescharacteristic
12#19128766—22no—
13#19128757—22no—
14C167⋊#95—21no—
15C167⋊#95—21no—
16C167⋊#95C2211yescharacteristic
17C2×C4⋊C4×C167C2211yescharacteristic
18C22⋊C8×C167C2212yes—
19C2×Dic334—21no—
20C2×Dic334D412yes—
21C22×Dic167—21no—
22C22×Dic167—21no—
23C4⋊Dic167—21no—
24C4⋊Dic167—21no—
25C4⋊Dic167—21no—
26C4⋊Dic167—22no—
27C4⋊Dic167D412yes—
28Dic167⋊C4—22no—
29Dic167⋊C4—22no—
30C22⋊Dic167—22no—
31C4×Dic167—22no—
32C22×C4×C167D411yescharacteristic
33C22×C4×C167D411yescharacteristic
34C22×C4×C167C2311yescharacteristic
35C2×C8×C167—22no—
36C4⋊C4×C167—22no—
37Dic334—42no—
38Dic334—81no—
39C2×Dic167—22no—
40C2×Dic167—22no—
41C2×Dic167—41no—
42C2×Dic167—41no—
43C2×Dic167—42no—
44C2×Dic167—41no—
45C2×Dic167—41no—
46C23×C167C2×D411yescharacteristic
47C2×C4×C167—21no—
48C2×C4×C167—21no—
49C2×C4×C167—21no—
50C2×C4×C167—41no—
51C2×C4×C167—41no—
52C2×C4×C167Q1612yes—
53C2×C4×C167C2×D412yes—
54C8×C167—42no—
55C22×C167—21no—
56C22×C167—21no—
57C22×C167#12011yescharacteristic
58C22×C167C2×Q1611yescharacteristic
59C22×C167#13711yescharacteristic
60C4×C167—22no—
61C4×C167—22no—
62C4×C167—41no—
63Dic167—42no—
64Dic167—42no—
65Dic167—81no—
66Dic167—81no—
67C2×C167—21no—
68C2×C167—21no—
69C2×C167#45111yescharacteristic
70C2×C167#45411yescharacteristic
71C2×C167#45811yescharacteristic
72C167#161711yescharacteristic, complemented, Sylow
73#1617—1671nomaximal, Sylow
74#544—1671no—
75#484—1672no—
76#400—1671no—
77#328—1672no—
78#324D16711yescharacteristic
79C4⋊Q8—3342no—
80#125—3342no—
81C22⋊Q8—3341no—
82C22⋊Q8—1672no—
83#117—3341no—
84C2×C4⋊C4—1671no—
85C2×C4⋊C4C2×D16711yescharacteristic
86#107—3342no—
87#103—3342no—
88C22⋊C8C2×D16712yes—
89#95—3341no—
90#95—3341no—
91#95—1671no—
92C2×Q8—3341no—
93C2×Q8—1672no—
94C22×C4—3341no—
95C22×C4—3341no—
96C22×C4D66811yescharacteristic
97C22×C4D66811yescharacteristic
98C22×C4C22×D16711yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—3341no—
102C4⋊C4—3341no—
103C4⋊C4—3341no—
104C4⋊C4—3342no—
105C4⋊C4—3342no—
106C4⋊C4—3342no—
107C4⋊C4—1672no—
108C22⋊C4—3342no—
109C42—3342no—
110C23C2×D66811yescharacteristic
111Q8—13361no—
112Q8—6682no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—3342no—
119C2×C4—3342no—
120C2×C4—6681no—
121C2×C4—6681no—
122C2×C4—6682no—
123C2×C4—6681no—
124C2×C4—6681no—
125C2×C4Dic66812yes—
126C2×C4D4×D16712yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#1912880211yescharacteristic
131C22C2×Dic66811yescharacteristic
132C22#1912886511yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—13361no—
137C4—13361no—
138C4—6682no—
139C4—6682no—
140C2—21no—
141C2—21no—
142C2#2492342711yescharacteristic
143C2#2492344411yescharacteristic
144C2#2492346011yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
Fitting#324×C167C2
socleC22×C167#120
prime radicalC23×C167C2×D4
hypercenter#324D167
hypocenterC167#1617
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
72. C167#1617semidirect

Maximal subgroups

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

subgroupindex
#249239002
#249236052
#249232082
#249230072
#324×C1672
#1617167