Finite Group Databasebeta

← #36356726

Subgroups — #36356726

Order 22144.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#19149658C211yesmaximal, characteristic
2#26011395C212yesmaximal, complemented
3#26011386C212yesmaximal, complemented
4#26011385C212yesmaximal
5#26011379C212yesmaximal
6#26011374C212yesmaximal, complemented
7#26011372C212yesmaximal, complemented
8#26011354C211yesmaximal, characteristic
9#380×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11#19149658C418yescomplemented
12C2×C4⋊Dic173C2211yescharacteristic
13C2×#14583805C2212yes—
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×C22⋊Dic173C2212yes—
17C2×C22⋊Dic173C2212yes—
18C2×C4×Dic173C2212yes—
19C2×C4×Dic173C2212yes—
20#19149586C2212yes—
21#19149586C2212yes—
22#19149586C2212yes—
23#19149586C2211yescharacteristic
24#19149586C2211yescharacteristic
25#19149586C2212yes—
26#19149586C2212yes—
27C23×C4×C173C2211yescharacteristic
28C2×C4⋊C4×C173C2212yes—
29C2×C22⋊C4×C173C2211yescharacteristic
30C2×C22⋊C4×C173C2211yescharacteristic
31#95×C173C2212yes—
32C2×Dic346—42no—
33C2×Dic346C2×C414yes—
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C22×Dic173C2312yes—
37C22×Dic173C2312yes—
38C4⋊Dic173C2×C414yes—
39#14583805—42no—
40#14583805—42no—
41#14583805C2×C418yes—
42C22⋊Dic173—42no—
43C22⋊Dic173C2×C418yes—
44C4×Dic173—42no—
45C4×Dic173—42no—
46C24×C173C2311yescharacteristic
47C22×C4×C173—42no—
48C22×C4×C173C2×C414yes—
49C22×C4×C173C2312yes—
50C22×C4×C173C2312yes—
51C22×C4×C173C2311yescharacteristic
52C22×C4×C173C2311yescharacteristic
53C4⋊C4×C173—42no—
54C22⋊C4×C173—41no—
55C22⋊C4×C173—41no—
56Dic346—44no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—22no—
61C2×Dic173—22no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173C22×C414yes—
68C2×Dic173C22×C414yes—
69C23×C173—21no—
70C23×C173—22no—
71C23×C173—22no—
72C23×C173—21no—
73C23×C173C22×C412yes—
74C23×C173C2411yescharacteristic
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—22no—
80C2×C4×C173—22no—
81C2×C4×C173—22no—
82C2×C4×C173—22no—
83C2×C4×C173—44no—
84C2×C4×C173C22×C412yes—
85C2×C4×C173C22×C412yes—
86C22×C173—22no—
87C22×C173—22no—
88C22×C173—22no—
89C22×C173—42no—
90C22×C173—42no—
91C22×C173C23×C411yescharacteristic
92C22×C173H(5,𝔽2)12yes—
93C22×C173H(5,𝔽2)11yescharacteristic
94C22×C1732−1+412yes—
95C22×C1732−1+411yescharacteristic
96C4×C173—22no—
97C4×C173—22no—
98C4×C173—42no—
99C4×C173—42no—
100Dic173—24no—
101Dic173—24no—
102Dic173—42no—
103Dic173—42no—
104C2×C173—42no—
105C2×C173#51811yescharacteristic
106C2×C173#51911yescharacteristic
107C2×C173#52011yescharacteristic
108C2×C173#56212yes—
109C2×C173#56312yes—
110C173#237811yescharacteristic, complemented, Sylow
111#2378—1731nomaximal, Sylow
112C2×C22⋊Q8—1731no—
113#391—1732no—
114#389—1732no—
115#388—1732no—
116#388—1732no—
117#387—1732no—
118#387—1732no—
119#381—1731no—
120#380D17311yescharacteristic
121C22×Q8—1731no—
122C23×C4C2×D17311yescharacteristic
123C22⋊Q8—1738no—
124C2×C4⋊C4—1732no—
125C2×C4⋊C4—1731no—
126C2×C4⋊C4—1732no—
127C2×C4⋊C4—1732no—
128C2×C4⋊C4C2×D17312yes—
129C2×C22⋊C4—1732no—
130C2×C22⋊C4—1732no—
131C2×C22⋊C4C2×D17311yescharacteristic
132C2×C22⋊C4C2×D17311yescharacteristic
133C2×C42—1732no—
134C2×C42—1732no—
135#95—1731no—
136#95—1732no—
137#95—1731no—
138#95—1732no—
139#95—1732no—
140#95—1732no—
141#95—1732no—
142#95C2×D17312yes—
143C24C22×D17311yescharacteristic
144C2×Q8—1734no—
145C2×Q8—6922no—
146C22×C4—42no—
147C22×C4—1732no—
148C22×C4—1732no—
149C22×C4—1732no—
150C22×C4—1732no—
151C22×C4C4×D17314yes—
152C22×C4C22×D17312yes—
153C22×C4C22×D17312yes—
154C22×C4C22×D17311yescharacteristic
155C22×C4C22×D17311yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—1738no—
158C4⋊C4—1734no—
159C4⋊C4—6922no—
160C4⋊C4—6922no—
161C22⋊C4—41no—
162C22⋊C4—41no—
163C22⋊C4—1738no—
164C22⋊C4—6922no—
165C42—6922no—
166C42—6922no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23C2×C4×D17312yes—
172C23C23×D17311yescharacteristic
173Q8—6924no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—3462no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—3462no—
187C2×C4—3462no—
188C2×C4—3462no—
189C2×C4—3462no—
190C2×C4—3462no—
191C2×C4—3462no—
192C2×C4—3462no—
193C2×C4—1734no—
194C2×C4—1734no—
195C2×C4C2×C4×D17312yes—
196C2×C4C2×C4×D17312yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D17311yescharacteristic
203C22#1914972412yes—
204C22#1914972411yescharacteristic
205C22#1914972612yes—
206C22#1914972711yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—3464no—
212C4—3464no—
213C4—6922no—
214C4—6922no—
215C2—42no—
216C2#2601196411yescharacteristic
217C2#2601196811yescharacteristic
218C2#2601197211yescharacteristic
219C2#2601223612yes—
220C2#2601224212yes—
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#380×C173C2
socle, cosocleC23×C173C24
prime radicalC24×C173C23
hypercenter#380D173
hypocenterC173#2378
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #26011395C2semidirect
3. #26011386C2semidirect
6. #26011374C2semidirect
7. #26011372C2semidirect
11. #19149658C4semidirect
110. C173#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#191496582
#260113952
#260113862
#260113852
#260113792
#260113742
#260113722
#260113542
#380×C1732
#2378173