Finite Group Databasebeta

← #36353533

Subgroups — #36353533

Order 22144.

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

normal subgroups only

This group has 588 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#26011508C212yesmaximal, complemented
2#26011496C212yesmaximal, complemented
3#26011475C211yesmaximal, characteristic, complemented
4#26011465C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C173C211yesmaximal, characteristic
6C2×#14583810C2212yescomplemented
7#19149638C2212yescomplemented
8#19149606—24no—
9#19149600—24no—
10#19149598—21no—
11#19149596—21no—
12#19149594—22no—
13H(5,𝔽2)×C173—24no—
14H(5,𝔽2)×C173—24no—
15C2×#41×C173—21no—
16C2×#41×C173C412yescomplemented
17C2×#41×C173C2211yescharacteristic
18C2×#41×C173C2211yescharacteristic
19C22×D4×C173—22no—
20C22×D4×C173—21no—
21C22×D4×C173C412yescomplemented
22C22×D4×C173C2211yescharacteristic
23#14583810—22no—
24#14583810—22no—
25C2×#13063436—22no—
26C4⋊Dic173—22no—
27C22⋊Dic173—22no—
28C4×Dic173—22no—
29C24×C173—22no—
30C24×C173—21no—
31#41×C173—28no—
32#41×C173—24no—
33#41×C173—24no—
34#41×C173—22no—
35#41×C173—22no—
36#41×C173—22no—
37#41×C173D414yes—
38C2×Q8×C173C2×C411yescharacteristic
39C2×Q8×C173D411yescharacteristic
40C2×D4×C173—216no—
41C2×D4×C173—28no—
42C2×D4×C173—22no—
43C2×D4×C173—21no—
44C2×D4×C173—24no—
45C2×D4×C173—24no—
46C2×D4×C173—22no—
47C2×D4×C173—21no—
48C2×D4×C173—22no—
49C2×D4×C173C2×C411yescharacteristic
50C2×D4×C173C2×C412yes—
51C2×D4×C173D412yes—
52C2×D4×C173D411yescharacteristic
53C2×D4×C173D414yes—
54C22×C4×C173—22no—
55C22×C4×C173—21no—
56C22×C4×C173C2×C412yes—
57C22×C4×C173C2311yescharacteristic
58C2×Dic173—42no—
59#13063436—42no—
60C23×C173—28no—
61C23×C173—22no—
62C23×C173—21no—
63C23×C173—24no—
64C23×C173—82no—
65C23×C173—84no—
66C23×C173C22⋊C412yes—
67C23×C173C2×D411yescharacteristic
68Q8×C173—21no—
69Q8×C173—22no—
70Q8×C173C22⋊C412yes—
71D4×C173—28no—
72D4×C173—28no—
73D4×C173—24no—
74D4×C173—24no—
75D4×C173—22no—
76D4×C173—21no—
77D4×C173—22no—
78D4×C173—24no—
79D4×C173C22⋊C412yes—
80D4×C173C22⋊C414yes—
81C2×C4×C173—28no—
82C2×C4×C173—21no—
83C2×C4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173—24no—
86C2×C4×C173C22⋊C412yes—
87C2×C4×C173C22×C411yescharacteristic
88C2×C4×C173C2×D412yes—
89C2×C4×C173C2×D412yes—
90C2×C4×C173C2×D411yescharacteristic
91C22×C173—22no—
92C22×C173—24no—
93C22×C173—22no—
94C22×C173—41no—
95C22×C173—44no—
96C22×C173—42no—
97C22×C173—42no—
98C22×C173—84no—
99C22×C173—88no—
100C22×C173C2×C22⋊C411yescharacteristic
101C22×C173#12012yes—
102C4×C173—22no—
103C4×C173—22no—
104C4×C173C2×C22⋊C411yescharacteristic
105C4×C173C2×C22⋊C411yescharacteristic
106C4×C173#12012yes—
107Dic173—82no—
108C2×C173—22no—
109C2×C173—21no—
110C2×C173—44no—
111C2×C173—42no—
112C2×C173—42no—
113C2×C173#37911yescharacteristic
114C173#179511yescharacteristic, complemented, Sylow
115#1795—1731nomaximal, Sylow
116C2×H(5,𝔽2)D17311yescharacteristic
117#421—1732no—
118#418—1732no—
119#413—1731no—
120#410—1731no—
121H(5,𝔽2)—24no—
122H(5,𝔽2)—24no—
123C2×#41—21no—
124C2×#41Dic17312yescomplemented
125C2×#41C2×D17311yescharacteristic
126C2×#41C2×D17311yescharacteristic
127C22×D4—21no—
128C22×D4—22no—
129C22×D4Dic17312yescomplemented
130C22×D4C2×D17311yescharacteristic
131C2×M16—1732no—
132#117—1732no—
133#104—3464no—
134#102—3464no—
135#101—3461no—
136#100—3461no—
137#99—3462no—
138C24—21no—
139C24—22no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D17314yes—
147C2×Q8C2×Dic17311yescharacteristic
148C2×Q8C22⋊D17311yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—21no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4C2×Dic17311yescharacteristic
159C2×D4C2×Dic17312yes—
160C2×D4C22⋊D17311yescharacteristic
161C2×D4C22⋊D17314yes—
162C2×D4C22⋊D17312yes—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4C2×Dic17312yes—
166C22×C4C22×D17311yescharacteristic
167M16—3462no—
168M16—3462no—
169C2×C8—3462no—
170C4⋊C4—3462no—
171C22⋊C4—3462no—
172C42—3462no—
173C23—28no—
174C23—21no—
175C23—22no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic17312yes—
180C23C2×C22⋊D17311yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic17312yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic17312yes—
193D4C22⋊Dic17314yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—6922no—
200C2×C4C22⋊Dic17312yes—
201C2×C4C22×Dic17311yescharacteristic
202C2×C4C2×C22⋊D17312yes—
203C2×C4C2×C22⋊D17312yes—
204C2×C4C2×C22⋊D17311yescharacteristic
205C8—6922no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic17311yescharacteristic
216C22#1914964812yes—
217C4—22no—
218C4—22no—
219C4—13842no—
220C4C2×C22⋊Dic17311yescharacteristic
221C4C2×C22⋊Dic17311yescharacteristic
222C4#1914964812yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#2601134711yescharacteristic
subgroupG/H
centerC2#26011347
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radicalC2×H(5,𝔽2)×C173C2
socleC2×C173#379
cosocleC22×C4×C173C23
hypercenterC2×H(5,𝔽2)D173
hypocenterC173#1795
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26011508C2semidirect
2. #26011496C2semidirect
3. #26011475C2semidirect
4. #26011465C2semidirect
6. C2×#14583810C22semidirect
7. #19149638C22semidirect
16. C2×#41×C173C4semidirect
21. C22×D4×C173C4semidirect
114. C173#1795semidirect
124. C2×#41Dic173semidirect
129. C22×D4Dic173semidirect

Maximal subgroups

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

subgroupindex
#260115082
#260114962
#260114752
#260114652
C2×H(5,𝔽2)×C1732
#1795173