Finite Group Databasebeta

← #36352510

Subgroups — #36352510

Order 22144.

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

normal subgroups only

This group has 320 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#26012078C211yesmaximal, characteristic, complemented
2#26011644C211yesmaximal, characteristic, complemented
3#26011624C211yesmaximal, characteristic, complemented
4#26011401C211yesmaximal, characteristic, complemented
5#26011227C211yesmaximal, characteristic, complemented
6#26011214C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C22×D692—21no—
9C22×D692C2211yescharacteristic
10C2×#14583815—21no—
11C2×#14583811—21no—
12#19149674—21no—
13#19149665—21no—
14#19149663—21no—
15#19149658C2211yescharacteristic
16#19149651—21no—
17#19149647—21no—
18D4×Dic173—21no—
19C2×#14583803—21no—
20C2×#14583803C2211yescharacteristic
21#19149601—21no—
22#19149599—21no—
23C22⋊#13063436C2211yescharacteristic
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173—21no—
28C23×D173D411yescharacteristic
29C2×Dic346D411yescharacteristic, complemented
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—42no—
33C2×D692—42no—
34C2×D692—42no—
35C2×D692—42no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C22×Dic173—21no—
39C22×Dic173D411yescharacteristic
40#14583815—42no—
41#14583811—42no—
42C2×#13063436—21no—
43C4⋊Dic173—21no—
44C4⋊Dic173D411yescharacteristic, complemented
45#14583805—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—21no—
51#14583803—21no—
52#14583803—42no—
53C4×Dic173—21no—
54C2×D4×C173—21no—
55C2×D4×C173D411yescharacteristic, complemented
56C22×C4×C173—21no—
57C22×C4×C173C2311yescharacteristic
58C2×C8×C173—21no—
59C4⋊C4×C173D411yescharacteristic, complemented
60C22⋊C4×C173—21no—
61C22×D173—22no—
62C22×D173—21no—
63C22×D173—21no—
64C22×D173—21no—
65C22×D173—41no—
66C22×D173—41no—
67C22×D173—41no—
68C22×D173—41no—
69C22×D173—41no—
70C22×D173—41no—
71C22×D173—41no—
72C22×D173—41no—
73Dic346—41no—
74D692—22no—
75D692—22no—
76D692—41no—
77D692—41no—
78D692—48no—
79C22⋊D173—81no—
80C22⋊D173—81no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—21no—
84C2×Dic173—41no—
85C2×Dic173—81no—
86#13063436—41no—
87C23×C173—21no—
88C23×C173C2×D411yescharacteristic
89D4×C173—41no—
90D4×C173—41no—
91D4×C173—41no—
92C2×C4×C173—22no—
93C2×C4×C173—21no—
94C2×C4×C173—41no—
95C2×C4×C173—41no—
96C2×C4×C173—41no—
97C2×C4×C173C2×D411yescharacteristic
98C2×C4×C173C2×D411yescharacteristic
99C8×C173—41no—
100C22×C173—21no—
101C22×C173—21no—
102C22×C173—41no—
103C22×C173—41no—
104C22×C173—41no—
105C22×C173#12011yescharacteristic
106C22×C173#13511yescharacteristic
107C22×C173AGL(1,ℤ/8)11yescharacteristic
108C2×D173—42no—
109C2×D173—42no—
110C2×D173—42no—
111C2×D173—41no—
112C2×D173—41no—
113C2×D173—41no—
114C2×D173—42no—
115C2×D173—81no—
116C2×D173—81no—
117C2×D173—81no—
118C2×D173—81no—
119C2×D173—81no—
120C4×C173—21no—
121C4×C173—21no—
122C4×C173—42no—
123C4×C173—41no—
124Dic173—41no—
125Dic173—41no—
126Dic173—41no—
127C2×C173—21no—
128C2×C173—21no—
129C2×C173—81no—
130C2×C173#44911yescharacteristic
131C2×C173#45311yescharacteristic
132C2×C173B(4,𝔽2)11yescharacteristic
133D173—81no—
134D173—82no—
135C173#160011yescharacteristic, complemented, Sylow
136#1600—1731nomaximal, Sylow
137#538—1731no—
138#450—1731no—
139#447—1731no—
140#392—1731no—
141#328—1731no—
142#327D17311yescharacteristic, complemented
143#323—1731no—
144C22×D4—3461no—
145C22×D4—1731no—
146C2×SD16—3461no—
147C2×D8—3461no—
148#126—3461no—
149#124—3461no—
150#123—3461no—
151C22⋊Q8—1731no—
152#121—3461no—
153#121C2×D17311yescharacteristic
154#120—3461no—
155C4×D4—3461no—
156C2×C22⋊C4—3461no—
157C2×C22⋊C4—1731no—
158#102—3461no—
159#102—3461no—
160C22⋊C8—1731no—
161C22⋊C8C2×D17311yescharacteristic
162#95C2×D17311yescharacteristic
163C24—3461no—
164C24—1731no—
165C2×Q8—1731no—
166C2×D4—21no—
167C2×D4—3461no—
168C2×D4—3461no—
169C2×D4—3461no—
170C2×D4—3461no—
171C2×D4—6922no—
172C2×D4—6922no—
173C2×D4—6922no—
174C2×D4—6922no—
175C2×D4D69211yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—3461no—
178C22×C4—1731no—
179C22×C4C22×D17311yescharacteristic
180SD16—6922no—
181D8—6922no—
182C2×C8—21no—
183C2×C8—3461no—
184C4⋊C4—3461no—
185C4⋊C4—3461no—
186C4⋊C4—1731no—
187C4⋊C4D69211yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—3461no—
190C22⋊C4—3461no—
191C22⋊C4—3461no—
192C22⋊C4—3461no—
193C22⋊C4—3461no—
194C22⋊C4—3461no—
195C22⋊C4—6922no—
196C42—3461no—
197C23—21no—
198C23—3462no—
199C23—3461no—
200C23—3461no—
201C23—3461no—
202C23—6921no—
203C23—6921no—
204C23—6921no—
205C23—6921no—
206C23—6921no—
207C23—6921no—
208C23—6921no—
209C23—6921no—
210C23D4×D17311yescharacteristic
211Q8—6921no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—3462no—
216D4—3462no—
217D4—13841no—
218D4—13841no—
219D4—6921no—
220D4—6921no—
221D4—6928no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—3461no—
228C2×C4—3461no—
229C2×C4—3461no—
230C2×C4—13841no—
231C2×C4—6921no—
232C2×C4D4×D17311yescharacteristic
233C2×C4C2×D69211yescharacteristic
234C8—41no—
235C8—6921no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—13841no—
242C22—13841no—
243C22—13841no—
244C22—13841no—
245C22—13841no—
246C22—6922no—
247C22—6922no—
248C22—6922no—
249C22—6921no—
250C22—6922no—
251C22—6921no—
252C22—6921no—
253C22#1914964911yescharacteristic
254C22#1914970011yescharacteristic
255C22#1914970311yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—6921no—
261C4—6921no—
262C4—6921no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—13842no—
267C2—13841no—
268C2#2601163711yescharacteristic
269C2#2601166011yescharacteristic
270C2#2601167911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173#120
hypercenter#327D173
hypocenterC173#1600
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #26012078C2semidirect
2. #26011644C2semidirect
3. #26011624C2semidirect
4. #26011401C2semidirect
5. #26011227C2semidirect
6. #26011214C2semidirect
7. #327×C173C2semidirect
29. C2×Dic346D4semidirect
44. C4⋊Dic173D4semidirect
55. C2×D4×C173D4semidirect
59. C4⋊C4×C173D4semidirect
135. C173#1600semidirect
142. #327D173semidirect
175. C2×D4D692semidirect
187. C4⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260120782
#260116442
#260116242
#260114012
#260112272
#260112142
#327×C1732
#1600173