Finite Group Databasebeta

← #36353636

Subgroups — #36353636

Order 22144.

504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊C4×D173C211yesmaximal, characteristic
2#26011346C212yesmaximal
3#26011258C211yesmaximal, characteristic, complemented
4#26011257C212yesmaximal, complemented
5#342×C173C211yesmaximal, characteristic, complemented
6C24×D173C2211yescharacteristic
7C22×C4×D173C412yescomplemented
8C2×C22⋊Dic173C2212yes—
9C2×C22⋊Dic173C2211yescharacteristic
10C2×#14583803—24no—
11C2×#14583803C412yescomplemented
12C22⋊C4×D173—24no—
13C2×C22⋊C4×C173C2212yes—
14C2×C22⋊C4×C173C2211yescharacteristic
15C23×D173—21no—
16C23×D173—22no—
17C23×D173—21no—
18C23×D173—44no—
19C23×D173C2×C411yescharacteristic
20C23×D173C2×C411yescharacteristic
21C23×D173D414yes—
22C22×Dic173—22no—
23C22×Dic173C2×C412yes—
24C2×C4×D173—22no—
25C2×C4×D173—22no—
26C2×C4×D173—44no—
27C22⋊Dic173—22no—
28C22⋊Dic173—22no—
29C22⋊Dic173—22no—
30#14583803—24no—
31#14583803—48no—
32C24×C173C2311yescharacteristic
33C22×C4×C173—22no—
34C22×C4×C173C2×C412yes—
35C22⋊C4×C173—22no—
36C22⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C22×D173—21no—
39C22×D173—21no—
40C22×D173—24no—
41C22×D173—21no—
42C22×D173—21no—
43C22×D173—24no—
44C22×D173—24no—
45C22×D173—28no—
46C22×D173—42no—
47C22×D173—44no—
48C22×D173—44no—
49C22×D173—48no—
50C22×D173—44no—
51C22×D173C22⋊C412yes—
52C22×D173C22⋊C412yes—
53C2×Dic173—22no—
54C2×Dic173—42no—
55C2×Dic173—44no—
56C2×Dic173—42no—
57C4×D173—44no—
58C4×D173—44no—
59C23×C173—42no—
60C23×C173C22×C411yescharacteristic
61C23×C173C2×D412yes—
62C23×C173#4111yescharacteristic
63C23×C173#4112yes—
64C23×C173#4111yescharacteristic
65C2×C4×C173—22no—
66C2×C4×C173—42no—
67C2×C4×C173—44no—
68C2×C4×C173—42no—
69C22×C173—22no—
70C22×C173—24no—
71C22×C173—41no—
72C22×C173—42no—
73C22×C173—41no—
74C22×C173C2×C22⋊C411yescharacteristic
75C22×C173#11711yescharacteristic
76C22×C173#11711yescharacteristic
77C22×C173#12312yes—
78C22×C173#12312yes—
79C2×D173—24no—
80C2×D173—24no—
81C2×D173—22no—
82C2×D173—22no—
83C2×D173—22no—
84C2×D173—24no—
85C2×D173—48no—
86C2×D173—48no—
87C2×D173—42no—
88C2×D173—42no—
89C2×D173#9914yes—
90C4×C173—42no—
91C4×C173—82no—
92Dic173—42no—
93Dic173—82no—
94C2×C173—22no—
95C2×C173—22no—
96C2×C173—21no—
97C2×C173—21no—
98C2×C173#38111yescharacteristic
99C2×C173C2×#9912yes—
100D173—24no—
101D173—42no—
102C173#181911yescharacteristic, complemented, Sylow
103#1819—1731nomaximal, Sylow
104C22×C22⋊C4—1731no—
105#379—1732no—
106#342—1732no—
107#342—1731no—
108#342D17311yescharacteristic, complemented
109C25—1731no—
110C23×C4—1732no—
111C2×C22⋊C4—3464no—
112C2×C22⋊C4—3464no—
113C2×C22⋊C4—1732no—
114C2×C22⋊C4—1731no—
115C2×C22⋊C4—1732no—
116C2×C22⋊C4C2×D17312yes—
117C2×C22⋊C4C2×D17311yescharacteristic
118C24—3461no—
119C24—3462no—
120C24—3461no—
121C24—1731no—
122C24—1734no—
123C24—1731no—
124C24—6924no—
125C24C22×D17311yescharacteristic
126C22×C4—22no—
127C22×C4—3462no—
128C22×C4—3462no—
129C22×C4—3462no—
130C22×C4—1732no—
131C22×C4—6924no—
132C22×C4C4×D17312yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—3464no—
137C22⋊C4—3462no—
138C22⋊C4—3462no—
139C22⋊C4—3462no—
140C22⋊C4—6928no—
141C23—42no—
142C23—3464no—
143C23—3468no—
144C23—3461no—
145C23—3461no—
146C23—3461no—
147C23—3461no—
148C23—3464no—
149C23—3464no—
150C23—1732no—
151C23—1732no—
152C23—6922no—
153C23—6924no—
154C23—6924no—
155C23—6928no—
156C23—6924no—
157C23C2×C4×D17311yescharacteristic
158C23D4×D17312yes—
159C23#1458382511yescharacteristic
160C23#1458382612yes—
161C23#1458382711yescharacteristic
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—3462no—
167C2×C4—6924no—
168C2×C4—6924no—
169C2×C4—6924no—
170C2×C4—6922no—
171C2×C4—6922no—
172C22—22no—
173C22—24no—
174C22—41no—
175C22—42no—
176C22—41no—
177C22—3464no—
178C22—3464no—
179C22—3462no—
180C22—3462no—
181C22—3464no—
182C22—3462no—
183C22—1734no—
184C22—6928no—
185C22—6928no—
186C22—6922no—
187C22—6922no—
188C22C22⋊C4×D17311yescharacteristic
189C22#1914963611yescharacteristic
190C22#1914963711yescharacteristic
191C22#1914966112yes—
192C22#1914966412yes—
193C4—42no—
194C4—82no—
195C4—13842no—
196C4—6922no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—21no—
201C2—3464no—
202C2—6922no—
203C2#2601135311yescharacteristic
204C2#99×D17312yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC23×C173C22×C4
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC24×D173C22
cosocleC24×C173C23
hypercenter#342D173
hypocenterC173#1819
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #26011258C2semidirect
4. #26011257C2semidirect
5. #342×C173C2semidirect
7. C22×C4×D173C4semidirect
11. C2×#14583803C4semidirect
102. C173#1819semidirect
108. #342D173semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1732
#260113462
#260112582
#260112572
#342×C1732
#1819173