Finite Group Databasebeta

← #36352534

Subgroups — #36352534

Order 22144.

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

normal subgroups only

This group has 282 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#26012072C211yesmaximal, characteristic
2#26011632C212yesmaximal, complemented
3#26011411C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7C22×Dic346C2211yescharacteristic
8C2×#14583815—22no—
9#19149659—22no—
10#19149654—22no—
11#19149653—21no—
12#19149653C2212yes—
13C2×#14583805—21no—
14#19149604—22no—
15#19149586—21no—
16#19149586C2211yescharacteristic
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19#14583826—42no—
20#14583826—42no—
21C2×Dic346—21no—
22C2×Dic346—22no—
23C2×Dic346—42no—
24C2×D692—21no—
25C2×D692—21no—
26C2×D692D412yes—
27C2×C22⋊D173—22no—
28C22×Dic173—21no—
29C22×Dic173—21no—
30C2×C4×D173—22no—
31#14583815—44no—
32C4⋊Dic173—21no—
33C4⋊Dic173D412yes—
34#14583805—42no—
35#14583803—22no—
36#14583803—22no—
37C22×C4×C173D411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C2×C8×C173—22no—
41C4⋊C4×C173—22no—
42C22×D173—22no—
43Dic346—24no—
44Dic346—42no—
45Dic346—42no—
46D692—42no—
47D692—42no—
48D692—81no—
49C22⋊D173—42no—
50C22⋊D173—42no—
51C2×Dic173—22no—
52C2×Dic173—22no—
53C2×Dic173—22no—
54C2×Dic173—41no—
55C2×Dic173—41no—
56C2×Dic173—41no—
57C2×Dic173—41no—
58C4×D173—44no—
59C23×C173C2×D411yescharacteristic
60C2×C4×C173—21no—
61C2×C4×C173—21no—
62C2×C4×C173—21no—
63C2×C4×C173—41no—
64C2×C4×C173—41no—
65C2×C4×C173SD1612yescomplemented
66C2×C4×C173C2×D412yes—
67C8×C173—42no—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173#12011yescharacteristic
71C22×C173C2×SD1611yescharacteristic
72C22×C173AGL(1,ℤ/8)11yescharacteristic
73C2×D173—42no—
74C2×D173—42no—
75C2×D173—42no—
76C4×C173—22no—
77C4×C173—22no—
78C4×C173—41no—
79Dic173—42no—
80Dic173—42no—
81Dic173—42no—
82C2×C173—21no—
83C2×C173—21no—
84C2×C173#45011yescharacteristic
85C2×C173#45611yescharacteristic
86C2×C173B(4,𝔽2)11yescharacteristic
87D173—82no—
88C173#160411yescharacteristic, complemented, Sylow
89#1604—1731nomaximal, Sylow
90#537—1731no—
91#448—1732no—
92#395—1731no—
93#327—1732no—
94#324D17311yescharacteristic, complemented
95C2×#41—3461no—
96C22×Q8—1731no—
97C2×SD16—3462no—
98C22⋊Q8—3462no—
99#121—3462no—
100#121—3461no—
101#121—1732no—
102C2×C4⋊C4—3461no—
103C2×C4⋊C4C2×D17311yescharacteristic
104#103—3462no—
105C22⋊C8C2×D17312yes—
106#95—3461no—
107#95—1731no—
108#41—6922no—
109#41—6922no—
110C2×Q8—3461no—
111C2×Q8—3462no—
112C2×Q8—6922no—
113C2×D4—3461no—
114C2×D4—3461no—
115C2×D4—3462no—
116C2×D4—1732no—
117C22×C4—3462no—
118C22×C4—3461no—
119C22×C4—3461no—
120C22×C4D69211yescharacteristic
121C22×C4D69211yescharacteristic
122C22×C4C22×D17311yescharacteristic
123SD16—6924no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—3461no—
127C4⋊C4—1732no—
128C4⋊C4—6922no—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C23—3462no—
132C23C2×D69211yescharacteristic
133Q8—3464no—
134Q8—6922no—
135Q8—6922no—
136D4—13841no—
137D4—6922no—
138D4—6922no—
139D4—6922no—
140D4—6922no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3462no—
147C2×C4—3462no—
148C2×C4—3462no—
149C2×C4—6924no—
150C2×C4—6921no—
151C2×C4—6921no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4#1458381512yescomplemented
155C2×C4D4×D17312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—6922no—
160C22—6922no—
161C22—6922no—
162C22#1914964911yescharacteristic
163C22C2×#1458381511yescharacteristic
164C22#1914970711yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—6922no—
169C4—6922no—
170C4—6922no—
171C2—21no—
172C2—21no—
173C2—13842no—
174C2#2601164411yescharacteristic
175C2#2601167611yescharacteristic
176C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173#120
prime radical#26012072C2
hypercenter#324D173
hypocenterC173#1604
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #26011632C2semidirect
3. #26011411C2semidirect
4. #26011226C2semidirect
5. #324×C173C2semidirect
65. C2×C4×C173SD16semidirect
88. C173#1604semidirect
94. #324D173semidirect
154. C2×C4#14583815semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260116322
#260114112
#260112262
#324×C1732
#1604173