Finite Group Databasebeta

← #36352546

Subgroups — #36352546

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#26012081C211yesmaximal, characteristic
2#26011646C211yesmaximal, characteristic, complemented
3#26011632C211yesmaximal, characteristic, complemented
4#26011404C211yesmaximal, characteristic, complemented
5#26011224C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#328×C173C211yesmaximal, characteristic, complemented
8C22×Dic346C2211yescharacteristic
9C2×#14583816—21no—
10C2×#14583815—21no—
11#19149674—21no—
12#19149668—21no—
13#19149664—21no—
14#19149654—21no—
15#19149653C2211yescharacteristic
16#19149649—21no—
17#19149639—21no—
18C2×#14583803—21no—
19C2×#14583803C2211yescharacteristic
20#19149604—21no—
21#19149602—21no—
22C22⋊#13063436C2211yescharacteristic
23#19149586—21no—
24C22⋊Q8×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—42no—
31C2×D692—21no—
32C2×D692—21no—
33C2×D692D411yescharacteristic
34C2×C22⋊D173—21no—
35C22×Dic173—21no—
36C22×Dic173D411yescharacteristic
37C2×C4×D173—21no—
38#14583816—42no—
39#14583815—42no—
40C2×#13063436—21no—
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46#14583803—21no—
47#14583803—21no—
48#14583803—21no—
49#14583803—42no—
50C4×Dic173—21no—
51C2×Q8×C173D411yescharacteristic
52C22×C4×C173—21no—
53C22×C4×C173C2311yescharacteristic
54C2×C8×C173—21no—
55C4⋊C4×C173—21no—
56C4⋊C4×C173D411yescharacteristic
57C22⋊C4×C173—21no—
58C22×D173—21no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—21no—
62C22×D173—41no—
63C22×D173—41no—
64Dic346—22no—
65Dic346—22no—
66Dic346—41no—
67Dic346—41no—
68D692—41no—
69D692—81no—
70D692—81no—
71C22⋊D173—41no—
72C22⋊D173—41no—
73C2×Dic173—22no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C4×D173—81no—
80#13063436—41no—
81C23×C173C2×D411yescharacteristic
82Q8×C173—41no—
83C2×C4×C173—21no—
84C2×C4×C173—21no—
85C2×C4×C173—41no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D411yescharacteristic
91C8×C173—41no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173#12011yescharacteristic
95C22×C173#13511yescharacteristic
96C22×C173#13711yescharacteristic
97C2×D173—42no—
98C2×D173—41no—
99C2×D173—41no—
100C2×D173—41no—
101C2×D173—41no—
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—81no—
105C4×C173—21no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—41no—
109C4×C173—81no—
110Dic173—41no—
111Dic173—41no—
112Dic173—42no—
113C2×C173—21no—
114C2×C173—21no—
115C2×C173#45211yescharacteristic
116C2×C173#45511yescharacteristic
117C2×C173B(4,𝔽2)11yescharacteristic
118D173—81no—
119D173—81no—
120C173#160611yescharacteristic, complemented, Sylow
121#1606—1731nomaximal, Sylow
122#538—1731no—
123#451—1731no—
124#448—1731no—
125#393—1731no—
126#328D17311yescharacteristic, complemented
127#327—1731no—
128#323—1731no—
129C22×Q8—1731no—
130C2×Q16—3461no—
131C2×SD16—3461no—
132#126—3461no—
133#124—3461no—
134#123—3461no—
135C22⋊Q8C2×D17311yescharacteristic
136#121—3461no—
137#121—1731no—
138#120—3461no—
139C4×D4—3461no—
140C2×C22⋊C4—3461no—
141C2×C22⋊C4—1731no—
142#103—3461no—
143#103—3461no—
144C22⋊C8—1731no—
145C22⋊C8C2×D17311yescharacteristic
146#95—3461no—
147#95C2×D17311yescharacteristic
148C24—1731no—
149C2×Q8—3461no—
150C2×Q8—3461no—
151C2×Q8—6922no—
152C2×Q8D69211yescharacteristic
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—1731no—
157C22×C4—21no—
158C22×C4—3461no—
159C22×C4—3461no—
160C22×C4—1731no—
161C22×C4C22×D17311yescharacteristic
162Q16—6922no—
163SD16—6922no—
164C2×C8—21no—
165C2×C8—3461no—
166C4⋊C4—21no—
167C4⋊C4—3461no—
168C4⋊C4—1731no—
169C4⋊C4D69211yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3461no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—6922no—
178C42—3461no—
179C23—3461no—
180C23—3461no—
181C23—3461no—
182C23—3461no—
183C23—6921no—
184C23—6921no—
185C23D4×D17311yescharacteristic
186Q8—41no—
187Q8—3462no—
188Q8—3462no—
189Q8—6921no—
190Q8—6921no—
191D4—13841no—
192D4—13841no—
193D4—6921no—
194D4—6921no—
195D4—6921no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3462no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—13841no—
206C2×C4—6921no—
207C2×C4—6921no—
208C2×C4—6921no—
209C2×C4D4×D17311yescharacteristic
210C2×C4C2×D69211yescharacteristic
211C8—41no—
212C8—6921no—
213C22—21no—
214C22—21no—
215C22—13841no—
216C22—6922no—
217C22—6921no—
218C22—6921no—
219C22—6921no—
220C22—6921no—
221C22—6921no—
222C22—6921no—
223C22#1914964911yescharacteristic
224C22#1914970011yescharacteristic
225C22#1914971111yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—6921no—
232C4—6921no—
233C4—6922no—
234C2—21no—
235C2—21no—
236C2—13841no—
237C2—13841no—
238C2#2601165611yescharacteristic
239C2#2601167011yescharacteristic
240C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#328×C173C2
socleC22×C173#120
prime radical#26012081C2
hypercenter#328D173
hypocenterC173#1606
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26011646C2semidirect
3. #26011632C2semidirect
4. #26011404C2semidirect
5. #26011224C2semidirect
6. #26011214C2semidirect
7. #328×C173C2semidirect
120. C173#1606semidirect
126. #328D173semidirect

Maximal subgroups

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

subgroupindex
#260120812
#260116462
#260116322
#260114042
#260112242
#260112142
#328×C1732
#1606173