Finite Group Databasebeta

← #36352522

Subgroups — #36352522

Order 22144.

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

normal subgroups only

This group has 256 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#26012117C211yesmaximal, characteristic
2#26011649C212yesmaximal, complemented
3#26011403C211yesmaximal, characteristic
4#26011229C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic
6C22×Dic346C2211yescharacteristic
7C2×Dic692—22no—
8#19149658—21no—
9#19149658C2212yes—
10#19149656—22no—
11#19149656—22no—
12C2×C22⋊Dic173—21no—
13C2×C22⋊Dic173—21no—
14#19149604—22no—
15#19149586—21no—
16#19149586C2211yescharacteristic
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C2×Dic346—21no—
20C2×Dic346—22no—
21C2×Dic346—42no—
22C2×Dic346D412yes—
23C22×Dic173—21no—
24C22×Dic173—21no—
25C22×Dic173—21no—
26C22×Dic173—21no—
27Dic692—44no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583805—22no—
31#14583805—22no—
32C22⋊Dic173—22no—
33C22⋊Dic173—22no—
34C22⋊Dic173—42no—
35C24×C173D411yescharacteristic
36C22×C4×C173D411yescharacteristic
37C22×C4×C173C2311yescharacteristic
38C2×C8×C173—22no—
39C22⋊C4×C173—22no—
40Dic346—24no—
41Dic346—42no—
42Dic346—42no—
43Dic346—81no—
44C2×Dic173—22no—
45C2×Dic173—22no—
46C2×Dic173—22no—
47C2×Dic173—41no—
48C2×Dic173—41no—
49C2×Dic173—41no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C2×Dic173—81no—
53C2×Dic173—81no—
54C23×C173—21no—
55C23×C173—21no—
56C23×C173—41no—
57C23×C173—41no—
58C23×C173Q1612yescomplemented
59C23×C173C2×D411yescharacteristic
60C2×C4×C173—21no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173C2×D412yes—
64C8×C173—42no—
65C22×C173—21no—
66C22×C173—21no—
67C22×C173—22no—
68C22×C173—42no—
69C22×C173—41no—
70C22×C173—41no—
71C22×C173—81no—
72C22×C173#12011yescharacteristic
73C22×C173C2×Q1611yescharacteristic
74C22×C173#13711yescharacteristic
75C4×C173—22no—
76C4×C173—41no—
77Dic173—42no—
78Dic173—42no—
79Dic173—42no—
80Dic173—81no—
81C2×C173—21no—
82C2×C173—21no—
83C2×C173—42no—
84C2×C173#45111yescharacteristic
85C2×C173#45511yescharacteristic
86C2×C173B(4,𝔽2)11yescharacteristic
87C173#160211yescharacteristic, complemented, Sylow
88#1602—1731nomaximal, Sylow
89#543—1731no—
90#451—1732no—
91#393—1731no—
92#328—1732no—
93#323D17311yescharacteristic
94C22×Q8—1731no—
95C2×Q16—3462no—
96C22⋊Q8—3462no—
97C22⋊Q8—3462no—
98C22⋊Q8—3461no—
99C22⋊Q8—1732no—
100C2×C22⋊C4—3461no—
101C2×C22⋊C4—3461no—
102C2×C22⋊C4C2×D17311yescharacteristic
103#103—3462no—
104C22⋊C8C2×D17312yes—
105#95—3461no—
106#95—1731no—
107C24D69211yescharacteristic
108C2×Q8—3461no—
109C2×Q8—3462no—
110C2×Q8—1732no—
111C2×Q8—6922no—
112C22×C4—3461no—
113C22×C4—3461no—
114C22×C4—3461no—
115C22×C4—3461no—
116C22×C4D69211yescharacteristic
117C22×C4C22×D17311yescharacteristic
118Q16—6924no—
119C2×C8—22no—
120C4⋊C4—3462no—
121C4⋊C4—3461no—
122C4⋊C4—3462no—
123C4⋊C4—1732no—
124C22⋊C4—22no—
125C22⋊C4—3462no—
126C22⋊C4—3462no—
127C22⋊C4—6922no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic69212yescomplemented
133C23C2×D69211yescharacteristic
134Q8—3464no—
135Q8—13841no—
136Q8—6922no—
137Q8—6922no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3462no—
142C2×C4—3462no—
143C2×C4—3462no—
144C2×C4—13841no—
145C2×C4—13841no—
146C2×C4—6921no—
147C2×C4—6921no—
148C2×C4—6921no—
149C2×C4—6921no—
150C2×C4—6921no—
151C2×C4D4×D17312yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1914964911yescharacteristic
161C22C2×Dic69211yescharacteristic
162C22#1914971211yescharacteristic
163C4—22no—
164C4—41no—
165C4—13841no—
166C4—6922no—
167C4—6922no—
168C4—6922no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2601164911yescharacteristic
173C2#2601167111yescharacteristic
174C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radicalC24×C173D4
hypercenter#323D173
hypocenterC173#1602
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26011649C2semidirect
4. #26011229C2semidirect
58. C23×C173Q16semidirect
87. C173#1602semidirect
132. C23Dic692semidirect

Maximal subgroups

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

subgroupindex
#260121172
#260116492
#260114032
#260112292
#323×C1732
#1602173