Finite Group Databasebeta

← #36354695

Subgroups — #36354695

Order 22144.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#26012053C211yesmaximal, characteristic
2C2×#19149663C212yesmaximal, complemented
3C2×#19149593C212yesmaximal, complemented
4#26011352C211yesmaximal, characteristic, complemented
5#342×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173C2212yes—
7C23×Dic173C2211yescharacteristic
8#19149663—24no—
9#19149663—22no—
10#19149663—24no—
11#19149653—22no—
12#19149649—22no—
13C2×C4⋊Dic173—22no—
14C2×C22⋊Dic173C2211yescharacteristic
15C2×#14583803—22no—
16#19149593—24no—
17#19149586—22no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C2×C22⋊C4×C173C2212yes—
20C23×D173D412yes—
21C2×D692—22no—
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—22no—
25C2×C22⋊D173—22no—
26C2×C22⋊D173—22no—
27C2×C22⋊D173—22no—
28C22×Dic173—21no—
29C22×Dic173—22no—
30C22×Dic173—22no—
31C22×Dic173—21no—
32C22×Dic173—21no—
33C22×Dic173D412yes—
34C22×Dic173D412yescomplemented
35C4⋊Dic173—28no—
36C4⋊Dic173—22no—
37C22⋊Dic173—22no—
38#14583803—28no—
39#14583803—24no—
40C24×C173C2311yescharacteristic
41C22×C4×C173—22no—
42C22×C4×C173D412yes—
43C22⋊C4×C173—22no—
44C22⋊C4×C173—22no—
45C22⋊C4×C173D414yescomplemented
46C22×D173—22no—
47C22×D173—22no—
48C22×D173—22no—
49C22×D173—44no—
50D692—82no—
51C22⋊D173—42no—
52C22⋊D173—42no—
53C22⋊D173—42no—
54C22⋊D173—42no—
55C2×Dic173—24no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—41no—
61C2×Dic173—42no—
62C2×Dic173—41no—
63C2×Dic173—42no—
64C2×Dic173—41no—
65C23×C173—42no—
66C23×C173C2×D412yes—
67C23×C173C2×D411yescharacteristic
68C23×C173C2×D411yescharacteristic
69C23×C173C2×D412yes—
70C23×C173#4111yescharacteristic
71C2×C4×C173—22no—
72C2×C4×C173—24no—
73C2×C4×C173—44no—
74C2×C4×C173—42no—
75C22×C173—22no—
76C22×C173—24no—
77C22×C173—42no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173#12012yes—
81C22×C173#12011yescharacteristic
82C22×C173#12111yescharacteristic
83C22×C173#12112yes—
84C22×C173#12711yescharacteristic
85C2×D173—44no—
86C2×D173—42no—
87C2×D173—44no—
88C2×D173—82no—
89C4×C173—44no—
90C4×C173—42no—
91Dic173—42no—
92Dic173—42no—
93Dic173—81no—
94C2×C173—21no—
95C2×C173—22no—
96C2×C173—22no—
97C2×C173—21no—
98C2×C173#39211yescharacteristic
99C2×C173#45812yes—
100D173—82no—
101C173#202211yescharacteristic, complemented, Sylow
102#2022—1731nomaximal, Sylow
103#534—1731no—
104C2×#123—1732no—
105C2×#99—1732no—
106#381—1731no—
107#342D17311yescharacteristic, complemented
108C22×D4—1732no—
109C23×C4—1731no—
110#123—3464no—
111#123—3462no—
112#123—3464no—
113#121—3462no—
114#120—3462no—
115C2×C4⋊C4—3462no—
116C2×C22⋊C4—3462no—
117C2×C22⋊C4—1731no—
118C2×C22⋊C4C2×D17311yescharacteristic
119C2×C22⋊C4C2×D17312yes—
120#99—3464no—
121#95—3462no—
122C24—1732no—
123C24C22×D17311yescharacteristic
124C2×D4—3462no—
125C2×D4—3462no—
126C2×D4—3462no—
127C2×D4—3462no—
128C2×D4—3462no—
129C2×D4—3462no—
130C2×D4—3462no—
131C22×C4—22no—
132C22×C4—3462no—
133C22×C4—3461no—
134C22×C4—3462no—
135C22×C4—3461no—
136C22×C4—3461no—
137C22×C4—1732no—
138C22×C4—1732no—
139C22×C4D69212yes—
140C4⋊C4—3468no—
141C4⋊C4—3462no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3468no—
145C22⋊C4—3464no—
146C22⋊C4—3462no—
147C22⋊C4D69214yescomplemented
148C23—42no—
149C23—3462no—
150C23—3462no—
151C23—3462no—
152C23—6924no—
153C23D4×D17312yes—
154C23D4×D17311yescharacteristic
155C23C2×D69211yescharacteristic
156C23C2×D69212yes—
157C23#1458382711yescharacteristic
158D4—13842no—
159D4—6922no—
160D4—6922no—
161D4—6922no—
162D4—6922no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—3464no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—6922no—
173C2×C4—6921no—
174C2×C4—6921no—
175C2×C4—6922no—
176C2×C4—6921no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—13842no—
183C22—6924no—
184C22—6922no—
185C22—6924no—
186C22#1914964912yes—
187C22#1914964911yescharacteristic
188C22#1914965411yescharacteristic
189C22#1914965412yes—
190C22#1914967611yescharacteristic
191C4—44no—
192C4—42no—
193C4—13841no—
194C4—6922no—
195C4—6922no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—13842no—
201C2#2601140111yescharacteristic
202C2#2601168212yes—
subgroupG/H
centerC22#19149649
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#342D173
hypocenterC173#2022
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#19149663C2semidirect
3. C2×#19149593C2semidirect
4. #26011352C2semidirect
5. #342×C173C2semidirect
34. C22×Dic173D4semidirect
45. C22⋊C4×C173D4semidirect
101. C173#2022semidirect
107. #342D173semidirect
147. C22⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260120532
C2×#191496632
C2×#191495932
#260113522
#342×C1732
#2022173