Finite Group Databasebeta

← #36323472

Subgroups — #36323472

Order 21376.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#24923679C211yesmaximal, characteristic, complemented
2#24923639C211yesmaximal, characteristic, complemented
3#24923628C211yesmaximal, characteristic
4#24923011C211yesmaximal, characteristic, complemented
5#24923011C211yesmaximal, characteristic, complemented
6#24923009C211yesmaximal, characteristic
7#563×C167C211yesmaximal, characteristic
8#19128832C2211yescharacteristic
9#19128765—21no—
10#19128763—21no—
11#19128756—21no—
12#19128755—21no—
13#19128755—21no—
14#19128753—21no—
15#19128743C2211yescharacteristic
16#19128743C2211yescharacteristic
17#19128743C2211yescharacteristic
18C4⋊Q8×C167—21no—
19#126×C167—21no—
20#126×C167—21no—
21#125×C167C2211yescharacteristic
22#125×C167C2211yescharacteristic
23#124×C167C2211yescharacteristic
24#123×C167—21no—
25C22⋊Q8×C167—21no—
26C22⋊Q8×C167—21no—
27C2×Dic334—21no—
28C2×C167⋊C8—21no—
29C2×C167⋊C8—21no—
30C2×C167⋊C8—21no—
31C4⋊Dic167—21no—
32C4⋊Dic167—21no—
33C2×Q8×C167—21no—
34C2×Q8×C167D411yescharacteristic
35C2×D4×C167D411yescharacteristic
36C22×C4×C167—21no—
37C4⋊C4×C167—21no—
38C4⋊C4×C167—21no—
39C4⋊C4×C167—21no—
40C4⋊C4×C167—21no—
41C4⋊C4×C167—21no—
42C4⋊C4×C167—21no—
43C4⋊C4×C167D411yescharacteristic
44C4⋊C4×C167D411yescharacteristic
45C4⋊C4×C167D411yescharacteristic
46C4⋊C4×C167D411yescharacteristic
47C22⋊C4×C167—21no—
48C22⋊C4×C167—21no—
49C22⋊C4×C167—21no—
50C22⋊C4×C167—21no—
51C22⋊C4×C167—21no—
52C42×C167—21no—
53C42×C167C2311yescharacteristic
54Dic334—42no—
55C2×Dic167—41no—
56C167⋊C8—22no—
57C167⋊C8—41no—
58C167⋊C8—41no—
59C23×C167—21no—
60Q8×C167—41no—
61Q8×C167—81no—
62D4×C167—41no—
63C2×C4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—21no—
66C2×C4×C167—21no—
67C2×C4×C167—21no—
68C2×C4×C167—81no—
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167C2×D411yescharacteristic
72C22×C167—41no—
73C22×C167—41no—
74C22×C167—41no—
75C22×C167#12011yescharacteristic
76C4×C167—21no—
77C4×C167—21no—
78C4×C167—21no—
79C4×C167—41no—
80C4×C167—41no—
81C4×C167—41no—
82C4×C167—41no—
83C4×C167—41no—
84Dic167—81no—
85C2×C167—81no—
86C2×C167#45411yescharacteristic
87C2×C167#45511yescharacteristic
88C2×C167#45611yescharacteristic
89C167#166411yescharacteristic, complemented, Sylow
90#1664—1671nomaximal, Sylow
91#563D16711yescharacteristic
92#501—1671no—
93#491—1671no—
94#489—1671no—
95#330—1671no—
96#330—1671no—
97#329—1671no—
98C4⋊Q8—21no—
99C4⋊Q8—1671no—
100#126—21no—
101#126—21no—
102#125C2×D16711yescharacteristic
103#125C2×D16711yescharacteristic
104#124C2×D16711yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—3341no—
109#106—3341no—
110#103—3341no—
111#103—3341no—
112#103—3341no—
113#102—3341no—
114#97—1671no—
115#97—1671no—
116#97—1671no—
117C2×Q8—21no—
118C2×Q8—3341no—
119C2×Q8C22⋊D16711yescharacteristic
120C2×D4C22⋊D16711yescharacteristic
121C22×C4—21no—
122C2×C8—3341no—
123C2×C8—3341no—
124C2×C8—3341no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3341no—
132C4⋊C4—3341no—
133C4⋊C4C22⋊D16711yescharacteristic
134C4⋊C4C22⋊D16711yescharacteristic
135C4⋊C4C22⋊D16711yescharacteristic
136C4⋊C4C22⋊D16711yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D16711yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—6682no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—6681no—
156C2×C4C2×C22⋊D16711yescharacteristic
157C2×C4C2×C22⋊D16711yescharacteristic
158C2×C4C2×C22⋊D16711yescharacteristic
159C8—3342no—
160C8—6681no—
161C8—6681no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1912880111yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—13361no—
175C2—81no—
176C2#2492344211yescharacteristic
177C2#2492344711yescharacteristic
178C2#2492345211yescharacteristic
subgroupG/H
centerC22#19128801
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#563×C167C2
socleC22×C167#120
prime radicalC2×D4×C167D4
hypercenter#563D167
hypocenterC167#1664
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #24923679C2semidirect
2. #24923639C2semidirect
4. #24923011C2semidirect
5. #24923011C2semidirect
89. C167#1664semidirect

Maximal subgroups

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

subgroupindex
#249236792
#249236392
#249236282
#249230112
#249230092
#563×C1672
#1664167