Finite Group Databasebeta

← #36352483

Subgroups — #36352483

Order 22144.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012053C211yesmaximal, characteristic, complemented
2#26011627C212yesmaximal, complemented
3#26011400C211yesmaximal, characteristic, complemented
4#26011226C212yesmaximal, complemented
5#323×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic, complemented
7C22×C22⋊D173—21no—
8C22×C22⋊D173—21no—
9C2×D1384—22no—
10#19149663—22no—
11#19149653—21no—
12#19149653C2212yes—
13#19149649—22no—
14C2×#14583803—21no—
15#19149601—22no—
16#19149586C2211yescharacteristic, complemented
17C2×C22⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yescomplemented
19C23×D173—21no—
20C23×D173—21no—
21C2×D692—21no—
22C2×D692—22no—
23C2×D692—42no—
24C2×D692D412yescomplemented
25C2×C22⋊D173—22no—
26C2×C22⋊D173—22no—
27C2×C22⋊D173—42no—
28C2×C22⋊D173—42no—
29C2×C22⋊D173—42no—
30C2×C22⋊D173—42no—
31C2×C22⋊D173—42no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34D1384—44no—
35C4⋊Dic173—21no—
36C4⋊Dic173D412yescomplemented
37#14583803—22no—
38#14583803—22no—
39#14583803—42no—
40C24×C173D411yescharacteristic
41C22×C4×C173D411yescharacteristic
42C22×C4×C173C2311yescharacteristic
43C2×C8×C173—22no—
44C22⋊C4×C173—22no—
45C22×D173—22no—
46C22×D173—22no—
47C22×D173—41no—
48C22×D173—42no—
49C22×D173—41no—
50C22×D173—41no—
51C22×D173—41no—
52C22×D173—41no—
53C22×D173—41no—
54C22×D173—81no—
55C22×D173—81no—
56D692—24no—
57D692—42no—
58D692—42no—
59D692—81no—
60C22⋊D173—44no—
61C22⋊D173—48no—
62C22⋊D173—42no—
63C22⋊D173—42no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—41no—
67C2×Dic173—42no—
68C2×Dic173—41no—
69C23×C173—21no—
70C23×C173—21no—
71C23×C173—41no—
72C23×C173—41no—
73C23×C173D812yescomplemented
74C23×C173C2×D411yescharacteristic
75C2×C4×C173—21no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173C2×D412yes—
79C8×C173—42no—
80C22×C173—21no—
81C22×C173—21no—
82C22×C173—22no—
83C22×C173—42no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—81no—
87C22×C173#12011yescharacteristic
88C22×C173C2×D811yescharacteristic
89C22×C173AGL(1,ℤ/8)11yescharacteristic
90C2×D173—42no—
91C2×D173—42no—
92C2×D173—42no—
93C2×D173—42no—
94C2×D173—42no—
95C2×D173—42no—
96C2×D173—81no—
97C2×D173—81no—
98C2×D173—84no—
99C2×D173—81no—
100C2×D173—81no—
101C4×C173—22no—
102C4×C173—41no—
103Dic173—42no—
104Dic173—42no—
105C2×C173—21no—
106C2×C173—21no—
107C2×C173—42no—
108C2×C173#44711yescharacteristic
109C2×C173#45311yescharacteristic
110C2×C173B(4,𝔽2)11yescharacteristic
111D173—82no—
112D173—82no—
113C173#159511yescharacteristic, complemented, Sylow
114#1595—1731nomaximal, Sylow
115#534—1731no—
116#447—1732no—
117#392—1731no—
118#327—1732no—
119#323D17311yescharacteristic, complemented
120C22×D4—3461no—
121C22×D4—3461no—
122C22×D4—1731no—
123C2×D8—3462no—
124#123—3462no—
125#121—3461no—
126#121—1732no—
127#120—3462no—
128C2×C22⋊C4—3461no—
129C2×C22⋊C4C2×D17311yescharacteristic
130#102—3462no—
131C22⋊C8C2×D17312yescomplemented
132#95—1731no—
133C24—3461no—
134C24—3461no—
135C24D69211yescharacteristic
136C2×D4—3461no—
137C2×D4—3462no—
138C2×D4—3462no—
139C2×D4—3462no—
140C2×D4—1732no—
141C2×D4—6922no—
142C2×D4—6922no—
143C2×D4—6922no—
144C2×D4—6922no—
145C2×D4—6922no—
146C2×D4—6922no—
147C22×C4—3461no—
148C22×C4—3461no—
149C22×C4D69211yescharacteristic
150C22×C4C22×D17311yescharacteristic
151D8—6924no—
152C2×C8—22no—
153C4⋊C4—3461no—
154C4⋊C4—1732no—
155C22⋊C4—22no—
156C22⋊C4—3462no—
157C22⋊C4—3462no—
158C22⋊C4—6922no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—3462no—
164C23—3462no—
165C23—13841no—
166C23—13841no—
167C23—6921no—
168C23—6921no—
169C23—6921no—
170C23—6922no—
171C23—6921no—
172C23—6921no—
173C23—6921no—
174C23D138412yescomplemented
175C23C2×D69211yescharacteristic
176D4—3464no—
177D4—13841no—
178D4—6922no—
179D4—6922no—
180D4—6924no—
181D4—6928no—
182D4—6922no—
183D4—6922no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—3462no—
188C2×C4—3462no—
189C2×C4—6921no—
190C2×C4—6922no—
191C2×C4—6921no—
192C2×C4D4×D17312yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—13841no—
202C22—13841no—
203C22—13844no—
204C22—13841no—
205C22—13841no—
206C22—6922no—
207C22—6922no—
208C22—6922no—
209C22—6922no—
210C22—6922no—
211C22—6922no—
212C22#1914964911yescharacteristic
213C22C2×D138411yescharacteristic
214C22#1914970711yescharacteristic
215C4—22no—
216C4—41no—
217C4—6922no—
218C4—6922no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—13842no—
223C2—13842no—
224C2#2601162711yescharacteristic
225C2#2601166111yescharacteristic
226C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
hypercenter#323D173
hypocenterC173#1595
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #26012053C2semidirect
2. #26011627C2semidirect
3. #26011400C2semidirect
4. #26011226C2semidirect
5. #323×C173C2semidirect
6. C22×D692C22semidirect
16. #19149586C22semidirect
18. C22⋊C8×C173C22semidirect
24. C2×D692D4semidirect
36. C4⋊Dic173D4semidirect
73. C23×C173D8semidirect
113. C173#1595semidirect
119. #323D173semidirect
131. C22⋊C8C2×D173semidirect
174. C23D1384semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260116272
#260114002
#260112262
#323×C1732
#1595173