Finite Group Databasebeta

← #36364576

Subgroups — #36364576

Order 22144.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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#26012048C211yesmaximal, characteristic, complemented
2#26012041C211yesmaximal, characteristic
3#26011885C212yesmaximal, complemented
4#26011851C212yesmaximal, complemented
5#26011702C212yesmaximal, complemented
6#26011692C212yesmaximal, complemented
7#26011655C214yesmaximal, complemented
8#432×C173C211yesmaximal, characteristic, complemented
9C2×#14583826C2212yes—
10C2×Dic692C2214yes—
11C2×#14583815C2214yes—
12#19149679C2211yescharacteristic
13#19149679C2211yescharacteristic
14#19149679C2211yescharacteristic
15#19149676C2211yescharacteristic
16#19149671—21no—
17#19149667C2212yes—
18#19149658—21no—
19#19149658C2212yes—
20#19149653—21no—
21#19149653—21no—
22C4×Dic346—21no—
23C4×Dic346C2212yes—
24C4×D692—21no—
25C4×D692—21no—
26#19149638—21no—
27#19149604C2214yes—
28#19149601C2214yes—
29C2×C42×C173C2211yescharacteristic
30C4⋊C8×C173C2212yes—
31C22⋊C8×C173C2212yes—
32#97×C173C2212yes—
33#14583826—44no—
34C2×Dic346—21no—
35C2×Dic346C2312yes—
36C2×Dic346C2312yes—
37C2×D692—21no—
38C2×D692—21no—
39C2×D692C2312yes—
40C2×C22⋊D173—22no—
41C2×C4×D173—22no—
42Dic692—44no—
43#14583815—44no—
44C4⋊Dic173—21no—
45C4⋊Dic173—21no—
46C4⋊Dic173—21no—
47C4⋊Dic173—21no—
48C4⋊Dic173—21no—
49C4⋊Dic173C2312yes—
50#14583805—22no—
51#14583805—22no—
52C22⋊Dic173—22no—
53#14583803—22no—
54#14583803—22no—
55C4×Dic173—22no—
56C22×C4×C173D411yescharacteristic
57C22×C4×C173D411yescharacteristic
58C22×C4×C173C2311yescharacteristic
59C2×C8×C173C2314yes—
60C42×C173D411yescharacteristic
61C42×C173D411yescharacteristic
62C42×C173C2311yescharacteristic
63C42×C173C2311yescharacteristic
64C22×D173—22no—
65Dic346—22no—
66Dic346—22no—
67Dic346—24no—
68Dic346—42no—
69D692—24no—
70D692—42no—
71D692—42no—
72C22⋊D173—44no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76C4×D173—44no—
77C23×C173C2×D411yescharacteristic
78C2×C4×C173—22no—
79C2×C4×C173—22no—
80C2×C4×C173—41no—
81C2×C4×C173C2×D411yescharacteristic
82C2×C4×C173C2×D411yescharacteristic
83C2×C4×C173C2×D411yescharacteristic
84C2×C4×C173C2×D411yescharacteristic
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2411yescharacteristic
87C8×C173—24no—
88C22×C173—21no—
89C22×C173—21no—
90C22×C173—21no—
91C22×C173C22×D411yescharacteristic
92C2×D173—42no—
93C2×D173—42no—
94C2×D173—42no—
95C4×C173—21no—
96C4×C173—21no—
97C4×C173—21no—
98C4×C173—21no—
99C4×C173#13712yes—
100C4×C173H(5,𝔽2)12yes—
101Dic173—42no—
102Dic173—42no—
103Dic173—42no—
104C2×C173—41no—
105C2×C173#53511yescharacteristic
106C2×C173C2×#13711yescharacteristic
107C2×C173#57511yescharacteristic
108D173—82no—
109C173#321611yescharacteristic, complemented, Sylow
110#3216—1731nomaximal, Sylow
111#533—1731no—
112#532—1731no—
113#497—1732no—
114#489—1732no—
115#462—1732no—
116#460—1732no—
117#452—1734no—
118#432D17311yescharacteristic, complemented
119C2×#41—1732no—
120C2×Q16—1734no—
121C2×SD16—1734no—
122C4⋊Q8—1731no—
123C4⋊Q8—1731no—
124C4⋊Q8—1731no—
125#127—1731no—
126#125—3461no—
127#124—1732no—
128C22⋊Q8—3461no—
129C22⋊Q8—1732no—
130#121—3461no—
131#121—3461no—
132C4×Q8—3461no—
133C4×Q8—1732no—
134C4×D4—3461no—
135C4×D4—3461no—
136#117—3461no—
137C2×C42C2×D17311yescharacteristic
138C4⋊C8C2×D17312yes—
139#103—1734no—
140#102—1734no—
141C22⋊C8C2×D17312yes—
142#97C2×D17312yes—
143#41—6924no—
144C2×Q8—3461no—
145C2×Q8—1732no—
146C2×Q8—1732no—
147C2×D4—3462no—
148C2×D4—3461no—
149C2×D4—3461no—
150C2×D4—1732no—
151C22×C4—3462no—
152C22×C4D69211yescharacteristic
153C22×C4D69211yescharacteristic
154C22×C4C22×D17311yescharacteristic
155Q16—6924no—
156SD16—6924no—
157C2×C8C22×D17314yes—
158C4⋊C4—3461no—
159C4⋊C4—3461no—
160C4⋊C4—3461no—
161C4⋊C4—3461no—
162C4⋊C4—3461no—
163C4⋊C4—3462no—
164C4⋊C4—3462no—
165C4⋊C4—1732no—
166C22⋊C4—3462no—
167C22⋊C4—3462no—
168C22⋊C4—3462no—
169C42—3462no—
170C42D69211yescharacteristic
171C42D69211yescharacteristic
172C42C22×D17311yescharacteristic
173C42C22×D17311yescharacteristic
174C23—3462no—
175C23C2×D69211yescharacteristic
176Q8—3462no—
177Q8—3462no—
178Q8—3464no—
179Q8—6922no—
180D4—3464no—
181D4—6922no—
182D4—6924no—
183D4—6922no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—3462no—
188C2×C4—3462no—
189C2×C4—3462no—
190C2×C4—6924no—
191C2×C4C2×D69211yescharacteristic
192C2×C4C2×D69211yescharacteristic
193C2×C4C2×D69211yescharacteristic
194C2×C4C2×D69211yescharacteristic
195C2×C4C2×D69211yescharacteristic
196C2×C4C23×D17311yescharacteristic
197C8—24no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—6922no—
202C22—6922no—
203C22—6922no—
204C22C22×D69211yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—6922no—
210C4—6922no—
211C4—6922no—
212C4#1914971212yes—
213C4#1914972512yes—
214C2—41no—
215C2—13842no—
216C2#2601206011yescharacteristic
217C2C2×#1914971211yescharacteristic
218C2#2601229111yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#432×C173C2
socleC22×C173C22×D4
prime radical#26012041C2
hypercenter#432D173
hypocenterC173#3216
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #26012048C2semidirect
3. #26011885C2semidirect
4. #26011851C2semidirect
5. #26011702C2semidirect
6. #26011692C2semidirect
7. #26011655C2semidirect
8. #432×C173C2semidirect
109. C173#3216semidirect
118. #432D173semidirect

Maximal subgroups

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

subgroupindex
#260120482
#260120412
#260118852
#260118512
#260117022
#260116922
#260116552
#432×C1732
#3216173