Finite Group Databasebeta

← #36361823

Subgroups — #36361823

Order 22144.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#26011431C212yesmaximal, complemented
2#26011427C211yesmaximal, characteristic
3#26011425C212yesmaximal, complemented
4#26011425C212yesmaximal, complemented
5#26011414C212yesmaximal
6#26011410C212yesmaximal, complemented
7#26011352C211yesmaximal, characteristic
8#26011349C212yesmaximal
9#393×C173C211yesmaximal, characteristic
10C23×Dic173C2211yescharacteristic
11C2×C4⋊Dic173C2212yes—
12C2×C4⋊Dic173C2211yescharacteristic
13C2×#14583805C2212yes—
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×#14583805C2212yes—
18C2×C22⋊Dic173C2211yescharacteristic
19C2×C22⋊Dic173C2212yes—
20#19149586C2212yes—
21#19149586C2212yes—
22#19149586C2212yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2211yescharacteristic
26C22×Q8×C173C2211yescharacteristic
27C2×C22⋊C4×C173C2212yes—
28C2×C22⋊C4×C173C2211yescharacteristic
29#95×C173C2211yescharacteristic
30#95×C173C2212yes—
31C22×Dic173—41no—
32C22×Dic173—41no—
33C22×Dic173Q814yescomplemented
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C22×Dic173C2312yes—
37C22×Dic173C2312yes—
38C4⋊Dic173—41no—
39C4⋊Dic173—42no—
40#14583805—42no—
41#14583805—42no—
42#14583805—42no—
43#14583805—42no—
44#14583805—42no—
45C22⋊Dic173—41no—
46C22⋊Dic173—42no—
47C24×C173C2311yescharacteristic
48C2×Q8×C173—41no—
49C2×Q8×C173—42no—
50C22×C4×C173C2312yes—
51C22×C4×C173C2311yescharacteristic
52C22×C4×C173C2311yescharacteristic
53C22×C4×C173C2312yes—
54C22⋊C4×C173—42no—
55C22⋊C4×C173—41no—
56C2×Dic173—22no—
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×Dic173—44no—
68C2×Dic173C2×Q814yes—
69C23×C173—21no—
70C23×C173—21no—
71C23×C173—22no—
72C23×C173—22no—
73C23×C173C2×Q812yes—
74C23×C173C2411yescharacteristic
75Q8×C173—44no—
76C2×C4×C173—21no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—21no—
80C2×C4×C173—22no—
81C2×C4×C173—22no—
82C2×C4×C173—22no—
83C2×C4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173—22no—
86C22×C173—22no—
87C22×C173—24no—
88C22×C173—41no—
89C22×C173—42no—
90C22×C173—41no—
91C22×C173C22×Q811yescharacteristic
92C22×C173H(5,𝔽2)11yescharacteristic
93C22×C173H(5,𝔽2)11yescharacteristic
94C22×C173H(5,𝔽2)12yes—
95C22×C1732−1+412yes—
96C4×C173—41no—
97C4×C173—42no—
98C4×C173—42no—
99C4×C173—41no—
100Dic173—24no—
101Dic173—42no—
102Dic173—42no—
103Dic173—42no—
104C2×C173—42no—
105C2×C173#54311yescharacteristic
106C2×C173#54412yes—
107C2×C173#56011yescharacteristic
108C2×C173#56212yes—
109C2×C173#56311yescharacteristic
110C173#284111yescharacteristic, complemented, Sylow
111#2841—1731nomaximal, Sylow
112#400—1732no—
113#399—1731no—
114#398—1732no—
115#398—1732no—
116#396—1732no—
117#395—1732no—
118#393D17311yescharacteristic
119#381—1731no—
120#380—1732no—
121C22×Q8C2×D17311yescharacteristic
122C23×C4—1731no—
123C2×C4⋊C4—1732no—
124C2×C4⋊C4—1732no—
125C2×C4⋊C4—1732no—
126C2×C4⋊C4—1732no—
127C2×C4⋊C4—1732no—
128C2×C4⋊C4—1732no—
129C2×C4⋊C4—1731no—
130C2×C22⋊C4—1732no—
131C2×C22⋊C4—1731no—
132C2×C22⋊C4C2×D17312yes—
133C2×C22⋊C4C2×D17311yescharacteristic
134#95—1732no—
135#95—1732no—
136#95—1732no—
137#95—1732no—
138#95—1732no—
139#95—1731no—
140#95C2×D17311yescharacteristic
141#95C2×D17312yes—
142C24C22×D17311yescharacteristic
143C2×Q8—42no—
144C2×Q8—41no—
145C22×C4—1732no—
146C22×C4—1732no—
147C22×C4—1732no—
148C22×C4—1732no—
149C22×C4—1734no—
150C22×C4—6921no—
151C22×C4—6921no—
152C22×C4C22×D17311yescharacteristic
153C22×C4C22×D17312yes—
154C22×C4C22×D17311yescharacteristic
155C22×C4C22×D17312yes—
156C4⋊C4—6922no—
157C4⋊C4—6922no—
158C4⋊C4—6922no—
159C4⋊C4—6922no—
160C4⋊C4—6922no—
161C4⋊C4—6921no—
162C4⋊C4—6922no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—6922no—
166C22⋊C4—6921no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D17312yes—
172C23C23×D17311yescharacteristic
173Q8—44no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
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—3462no—
194C2×C4—3462no—
195C2×C4—1734no—
196C2×C4—6924no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D17311yescharacteristic
203C22#1914972412yes—
204C22#1914972411yescharacteristic
205C22#1914972511yescharacteristic
206C22#1914972612yes—
207C4—42no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—3464no—
212C4—6922no—
213C4—6922no—
214C4—6922no—
215C2—42no—
216C2#2601211611yescharacteristic
217C2#2601212012yes—
218C2#2601223011yescharacteristic
219C2#2601223512yes—
220C2#2601224111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#393×C173C2
prime radicalC24×C173C23
hypercenter#393D173
hypocenterC173#2841
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #26011431C2semidirect
3. #26011425C2semidirect
4. #26011425C2semidirect
6. #26011410C2semidirect
33. C22×Dic173Q8semidirect
110. C173#2841semidirect

Maximal subgroups

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

subgroupindex
#260114312
#260114272
#260114252
#260114142
#260114102
#260113522
#260113492
#393×C1732
#2841173