Finite Group Databasebeta

← #36352545

Subgroups — #36352545

Order 22144.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012079C211yesmaximal, characteristic
2#26011649C211yesmaximal, characteristic, complemented
3#26011632C211yesmaximal, characteristic, complemented
4#26011403C211yesmaximal, characteristic, complemented
5#26011229C211yesmaximal, characteristic, complemented
6#26011226C211yesmaximal, characteristic, complemented
7#323×C173C211yesmaximal, characteristic, complemented
8C22×Dic346C2211yescharacteristic
9C2×Dic692—21no—
10C2×#14583815—21no—
11#19149672—21no—
12#19149665—21no—
13#19149661—21no—
14#19149658C2211yescharacteristic
15#19149653C2211yescharacteristic
16#19149652—21no—
17#19149648—21no—
18C4×C22⋊D173—21no—
19C2×C22⋊Dic173—21no—
20#19149604—21no—
21#19149604—21no—
22#19149586—21no—
23#19149586C2211yescharacteristic
24C2×C22⋊C4×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26C22⋊C8×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—42no—
30C2×Dic346D411yescharacteristic
31C2×D692D411yescharacteristic
32C2×C22⋊D173—21no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C22×Dic173—21no—
36C22×Dic173—21no—
37C2×C4×D173—21no—
38Dic692—42no—
39#14583815—42no—
40C4⋊Dic173D411yescharacteristic
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—42no—
48#14583803—21no—
49#14583803—21no—
50C4×Dic173—21no—
51C24×C173D411yescharacteristic
52C22×C4×C173D411yescharacteristic
53C22×C4×C173C2311yescharacteristic
54C2×C8×C173—21no—
55C2×C8×C173—21no—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—21no—
58C22×D173—21no—
59Dic346—22no—
60Dic346—22no—
61Dic346—41no—
62Dic346—41no—
63Dic346—41no—
64D692—41no—
65C22⋊D173—41no—
66C22⋊D173—41no—
67C22⋊D173—81no—
68C22⋊D173—81no—
69C2×Dic173—22no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173—41no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C4×D173—81no—
79C23×C173—21no—
80C23×C173—21no—
81C23×C173—21no—
82C23×C173—41no—
83C23×C173—41no—
84C23×C173C2×D411yescharacteristic
85C2×C4×C173—21no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2×D411yescharacteristic
90C8×C173—41no—
91C8×C173—41no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—42no—
95C22×C173—41no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—81no—
99C22×C173#12011yescharacteristic
100C22×C173#13511yescharacteristic
101C22×C173#13711yescharacteristic
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—41no—
105C4×C173—21no—
106C4×C173—21no—
107C4×C173—41no—
108Dic173—41no—
109Dic173—41no—
110Dic173—42no—
111Dic173—41no—
112Dic173—81no—
113C2×C173—21no—
114C2×C173—21no—
115C2×C173—81no—
116C2×C173#45211yescharacteristic
117C2×C173#45511yescharacteristic
118C2×C173B(4,𝔽2)11yescharacteristic
119D173—81no—
120C173#160611yescharacteristic, complemented, Sylow
121#1606—1731nomaximal, Sylow
122#538—1731no—
123#451—1731no—
124#448—1731no—
125#393—1731no—
126#328—1731no—
127#327—1731no—
128#323D17311yescharacteristic, complemented
129C22×Q8—1731no—
130C2×Q16—3461no—
131C2×SD16—3461no—
132#126—3461no—
133#124—3461no—
134#123—3461no—
135C22⋊Q8—1731no—
136#121—3461no—
137#121—1731no—
138#120—3461no—
139C4×D4—3461no—
140C2×C22⋊C4—3461no—
141C2×C22⋊C4C2×D17311yescharacteristic
142#103—3461no—
143#103—3461no—
144C22⋊C8C2×D17311yescharacteristic
145C22⋊C8C2×D17311yescharacteristic
146#95—3461no—
147#95—1731no—
148C24D69211yescharacteristic
149C2×Q8—3461no—
150C2×Q8—3461no—
151C2×Q8—1731no—
152C2×Q8—6922no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—1731no—
157C22×C4—3461no—
158C22×C4—3461no—
159C22×C4—3461no—
160C22×C4D69211yescharacteristic
161C22×C4C22×D17311yescharacteristic
162Q16—6922no—
163SD16—6922no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—1731no—
169C4⋊C4—1731no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—6922no—
178C42—3461no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3461no—
185C23C2×D69211yescharacteristic
186Q8—3462no—
187Q8—3462no—
188Q8—6921no—
189Q8—6921no—
190Q8—6921no—
191D4—13841no—
192D4—13841no—
193D4—6921no—
194D4—6921no—
195D4—6921no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3462no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—13841no—
204C2×C4—6921no—
205C2×C4—6921no—
206C2×C4—6921no—
207C2×C4—6921no—
208C2×C4—6921no—
209C2×C4D4×D17311yescharacteristic
210C2×C4D4×D17311yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—6921no—
221C22—6921no—
222C22—6921no—
223C22#1914964911yescharacteristic
224C22#1914970111yescharacteristic
225C22#1914971211yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—13841no—
230C4—6921no—
231C4—6921no—
232C4—6922no—
233C4—6921no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—13841no—
238C2#2601165511yescharacteristic
239C2#2601167111yescharacteristic
240C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#323×C173C2
socleC22×C173#120
prime radical#26012079C2
hypercenter#323D173
hypocenterC173#1606
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
2. #26011649C2semidirect
3. #26011632C2semidirect
4. #26011403C2semidirect
5. #26011229C2semidirect
6. #26011226C2semidirect
7. #323×C173C2semidirect
120. C173#1606semidirect
128. #323D173semidirect

Maximal subgroups

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

subgroupindex
#260120792
#260116492
#260116322
#260114032
#260112292
#260112262
#323×C1732
#1606173