Finite Group Databasebeta

← #36369399

Subgroups — #36369399

Order 22144.

584 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#19149740C212yesmaximal, complemented
2C2×#19149739C212yesmaximal, complemented
3#26012463C212yesmaximal, complemented
4C22×#14583826C211yesmaximal, characteristic
5C2×#14583835C2214yes—
6#19149740—24no—
7#19149739—24no—
8#19149736—24no—
9C2×#14583826—24no—
10C2×#14583826—24no—
11C2×#14583826—24no—
12C22×Dic346C2211yescharacteristic
13C22×D692C411yescharacteristic
14C22×C22⋊D173C411yescharacteristic
15C22×C22⋊D173C2211yescharacteristic
16C22×C4×D173C411yescharacteristic
17C22×C4×D173C2211yescharacteristic
18C23×C4×C173C411yescharacteristic
19#14583835—24no—
20#14583835—24no—
21C2×#13063443—24no—
22C23×D173C2×C411yescharacteristic
23C23×D173C2×C411yescharacteristic
24#14583826—232no—
25C2×Dic346—22no—
26C2×Dic346—22no—
27C2×Dic346D414yes—
28C2×D692—22no—
29C2×D692—22no—
30C2×D692—22no—
31C2×C22⋊D173—22no—
32C2×C22⋊D173—22no—
33C2×C22⋊D173—22no—
34C2×C22⋊D173—22no—
35C2×C22⋊D173—22no—
36C2×C22⋊D173D414yes—
37C22×Dic173C2×C411yescharacteristic
38C22×Dic173C2311yescharacteristic
39C2×C4×D173—22no—
40C2×C4×D173—22no—
41C2×C4×D173—22no—
42C2×C4×D173—22no—
43C2×C4×D173—22no—
44C2×C4×D173D414yes—
45C24×C173C2×C411yescharacteristic
46C22×C4×C173—22no—
47C22×C4×C173—22no—
48C22×C4×C173—22no—
49C22×C4×C173C2×C411yescharacteristic
50C22×C4×C173C2×C411yescharacteristic
51#13063443—44no—
52C22×D173—21no—
53C22×D173—21no—
54C22×D173—21no—
55C22×D173—21no—
56C22×D173—42no—
57C22×D173—42no—
58C22×D173C22⋊C412yes—
59C22×D173C22⋊C412yes—
60Dic346—28no—
61D692—28no—
62C22⋊D173—28no—
63C22⋊D173—28no—
64C2×Dic173—21no—
65C2×Dic173—21no—
66C2×Dic173C22⋊C412yes—
67C2×Dic173C2×D412yes—
68C2×Dic173C2×D412yes—
69C2×Dic173C2×D412yes—
70C4×D173—28no—
71C4×D173—28no—
72C23×C173—21no—
73C23×C173—21no—
74C23×C173—42no—
75C23×C173C22⋊C412yes—
76C23×C173C22×C411yescharacteristic
77C2×C4×C173—28no—
78C2×C4×C173—21no—
79C2×C4×C173—21no—
80C2×C4×C173—21no—
81C2×C4×C173—21no—
82C2×C4×C173C22⋊C412yes—
83C2×C4×C173C22⋊C412yes—
84C22×C173—22no—
85C22×C173—22no—
86C22×C173—42no—
87C22×C173—42no—
88C22×C173—42no—
89C22×C173C2×C22⋊C411yescharacteristic
90C22×C173C2×C22⋊C411yescharacteristic
91C22×C173C2×C22⋊C411yescharacteristic
92C2×D173—22no—
93C2×D173—22no—
94C2×D173—24no—
95C2×D173—42no—
96C2×D173—42no—
97C2×D173—42no—
98C2×D173—42no—
99C2×D173—42no—
100C4×C173—22no—
101C4×C173—22no—
102Dic173—22no—
103Dic173#12014yes—
104C2×C173—21no—
105C2×C173—21no—
106C2×C173—42no—
107C2×C173#37911yescharacteristic
108C2×C173#41312yes—
109D173—42no—
110D173—42no—
111C173#178511yescharacteristic, complemented, Sylow
112#1785—1731nomaximal, Sylow
113C22×#41—1731no—
114C2×#101—1732no—
115C2×#100—1732no—
116#407—1732no—
117C2×#41—3464no—
118C2×#41—3464no—
119C2×#41—3464no—
120C22×Q8—1731no—
121C22×D4—1731no—
122C22×D4—1731no—
123C22×D4—1731no—
124C23×C4—1731no—
125C23×C4—1731no—
126C23×C4#1062192411yescharacteristic
127C2×M16—1734no—
128#101—3464no—
129#100—3464no—
130C22⋊C8—3464no—
131C24—1731no—
132C24—1731no—
133C24C2×#1062192411yescharacteristic
134#41—34632no—
135C2×Q8—3462no—
136C2×Q8—3462no—
137C2×Q8—1734no—
138C2×D4—3462no—
139C2×D4—3462no—
140C2×D4—3462no—
141C2×D4—3462no—
142C2×D4—3462no—
143C2×D4—3462no—
144C2×D4—3462no—
145C2×D4—3462no—
146C2×D4—1734no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—3462no—
151C22×C4—3462no—
152C22×C4—3462no—
153C22×C4—3462no—
154C22×C4—3462no—
155C22×C4—1731no—
156C22×C4—1734no—
157C22×C4—1731no—
158C22×C4C2×#1062192411yescharacteristic
159C22×C4C2×#1062192411yescharacteristic
160M16—3464no—
161M16—3464no—
162C2×C8—3464no—
163C23—21no—
164C23—21no—
165C23—42no—
166C23—3461no—
167C23—3461no—
168C23—3461no—
169C23—3461no—
170C23—1732no—
171C23—1732no—
172C23—6922no—
173C23—6922no—
174C23C22⋊#1062192412yes—
175C23C22×#1062192411yescharacteristic
176Q8—3468no—
177D4—3468no—
178D4—3468no—
179D4—3468no—
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—3468no—
186C2×C4—3468no—
187C2×C4—3461no—
188C2×C4—3461no—
189C2×C4—1732no—
190C2×C4—1732no—
191C2×C4—1732no—
192C2×C4—1732no—
193C2×C4C22⋊#1062192412yes—
194C2×C4C22⋊#1062192412yes—
195C8—6924no—
196C22—22no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—3462no—
202C22—3462no—
203C22—3464no—
204C22—6922no—
205C22—6922no—
206C22—6922no—
207C22—6922no—
208C22—6922no—
209C22C2×C22⋊#1062192411yescharacteristic
210C22C2×C22⋊#1062192411yescharacteristic
211C22C2×C22⋊#1062192411yescharacteristic
212C4—22no—
213C4—22no—
214C4—3462no—
215C4—1734no—
216C2—21no—
217C2—21no—
218C2—42no—
219C2—6922no—
220C2—6922no—
221C2#2601243211yescharacteristic
222C2#2601247712yes—
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC23×C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×#14583826C2
cosocleC22×Dic173C23
hypercenterC23×C4#10621924
hypocenterC173#1785
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#19149740C2semidirect
2. C2×#19149739C2semidirect
3. #26012463C2semidirect
111. C173#1785semidirect

Maximal subgroups

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

subgroupindex
C2×#191497402
C2×#191497392
#260124632
C22×#145838262
#1785173