Finite Group Databasebeta

← #36352103

Subgroups — #36352103

Order 22144.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#26012001C211yesmaximal, characteristic
2#26011453C211yesmaximal, characteristic, complemented
3#26011350C211yesmaximal, characteristic, complemented
4#26011229C211yesmaximal, characteristic, complemented
5#26011227C211yesmaximal, characteristic, complemented
6#26011225C211yesmaximal, characteristic, complemented
7#327×C173C211yesmaximal, characteristic, complemented
8C2×#14583825C411yescharacteristic, complemented
9C22×C4×D173C2211yescharacteristic
10C2×#14583809—21no—
11#19149661—21no—
12#19149659C411yescharacteristic, complemented
13#19149658C2211yescharacteristic
14#19149650C411yescharacteristic, complemented
15#19149647—21no—
16D4×Dic173—21no—
17#19149639—21no—
18#19149637C411yescharacteristic, complemented
19C2×#14583805—21no—
20C2×#14583803—21no—
21C22⋊#13063436C2211yescharacteristic
22#19149591—21no—
23#19149586C2211yescharacteristic
24#121×C173C2211yescharacteristic
25C22⋊C8×C173C2211yescharacteristic
26#95×C173C2211yescharacteristic
27C23×D173D411yescharacteristic
28#14583825—42no—
29C2×Dic346C2×C411yescharacteristic
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C22×Dic173—21no—
33C22×Dic173—21no—
34C22×Dic173D411yescharacteristic
35C2×C4×D173—42no—
36C2×C4×D173C2×C411yescharacteristic
37C2×C4×D173C2×C411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C2×C4×D173D411yescharacteristic
40#14583809—42no—
41C2×#13063436—21no—
42C4⋊Dic173—21no—
43C4⋊Dic173C2×C411yescharacteristic
44#14583805—21no—
45#14583805—42no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48#14583803—21no—
49#14583803—21no—
50#14583803—42no—
51C4×Dic173—21no—
52C2×D4×C173—21no—
53C2×D4×C173C2×C411yescharacteristic
54C22×C4×C173—21no—
55C22×C4×C173C2311yescharacteristic
56C2×C8×C173—21no—
57C4⋊C4×C173C2×C411yescharacteristic
58C22⋊C4×C173—21no—
59C22×D173—21no—
60C22×D173—21no—
61C22×D173—41no—
62C22×D173—41no—
63C22×D173C22⋊C411yescharacteristic
64C22×D173C22⋊C411yescharacteristic
65Dic346—41no—
66C22⋊D173—42no—
67C22⋊D173—42no—
68C2×Dic173—22no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—42no—
72C2×Dic173—41no—
73C2×Dic173—41no—
74C2×Dic173—41no—
75C2×Dic173C22⋊C411yescharacteristic
76C2×Dic173C22⋊C411yescharacteristic
77C4×D173—22no—
78C4×D173—22no—
79C4×D173—41no—
80C4×D173—41no—
81#13063436—41no—
82C23×C173—21no—
83C23×C173C2×D411yescharacteristic
84D4×C173—41no—
85D4×C173—41no—
86D4×C173—41no—
87C2×C4×C173—21no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173—41no—
92C2×C4×C173C22×C411yescharacteristic
93C2×C4×C173C2×D411yescharacteristic
94C8×C173—41no—
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173C2×C22⋊C411yescharacteristic
101C22×C173AGL(1,ℤ/8)11yescharacteristic
102C22×C173#13711yescharacteristic
103C2×D173—21no—
104C2×D173—21no—
105C2×D173—21no—
106C2×D173—21no—
107C2×D173—21no—
108C2×D173—21no—
109C2×D173—42no—
110C2×D173—42no—
111C4×C173—21no—
112C4×C173—21no—
113C4×C173—41no—
114C4×C173—81no—
115Dic173—21no—
116Dic173—41no—
117Dic173—41no—
118Dic173—81no—
119Dic173#9912yes—
120C2×C173—21no—
121C2×C173—21no—
122C2×C173—81no—
123C2×C173C2×#9911yescharacteristic
124C2×C173#41711yescharacteristic
125C2×C173#42111yescharacteristic
126D173—41no—
127D173—41no—
128C173#151311yescharacteristic, complemented, Sylow
129#1513—1731nomaximal, Sylow
130#525—1731no—
131#407—1731no—
132#380—1731no—
133#328—1731no—
134#328—1731no—
135#327—1731no—
136#327D17311yescharacteristic, complemented
137C2×#41—1731no—
138C23×C4—1731no—
139C2×M16—3461no—
140#123—3461no—
141C22⋊Q8—1731no—
142C22⋊Q8—1731no—
143#121—1731no—
144#121C2×D17311yescharacteristic
145#120—3461no—
146C4×D4—3461no—
147C4×D4—3461no—
148#117—1731no—
149C2×C4⋊C4—3461no—
150C2×C22⋊C4—3461no—
151C22⋊C8—3461no—
152C22⋊C8—1731no—
153C22⋊C8C2×D17311yescharacteristic
154#95—1731no—
155#95C2×D17311yescharacteristic
156C24—1731no—
157#41—6922no—
158C2×Q8—1731no—
159C2×D4—21no—
160C2×D4—3461no—
161C2×D4—3461no—
162C2×D4C4×D17311yescharacteristic
163C22×C4—21no—
164C22×C4—3461no—
165C22×C4—3461no—
166C22×C4—1731no—
167C22×C4—1731no—
168C22×C4—1731no—
169C22×C4—1731no—
170C22×C4—1731no—
171C22×C4—6922no—
172C22×C4C22×D17311yescharacteristic
173M16—6922no—
174C2×C8—21no—
175C2×C8—3461no—
176C4⋊C4—3461no—
177C4⋊C4—3461no—
178C4⋊C4—1731no—
179C4⋊C4—6922no—
180C4⋊C4C4×D17311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—3461no—
183C22⋊C4—3461no—
184C22⋊C4—3461no—
185C22⋊C4—3461no—
186C22⋊C4—6922no—
187C42—3461no—
188C23—21no—
189C23—3461no—
190C23—3461no—
191C23—1731no—
192C23—1731no—
193C23—6921no—
194C23—6921no—
195C23D4×D17311yescharacteristic
196Q8—6921no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—6922no—
201D4—6922no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3462no—
208C2×C4—3462no—
209C2×C4—3462no—
210C2×C4—3461no—
211C2×C4—3461no—
212C2×C4—1731no—
213C2×C4—1731no—
214C2×C4—6921no—
215C2×C4—6921no—
216C2×C4—6921no—
217C2×C4—6921no—
218C2×C4—6921no—
219C2×C4—6922no—
220C2×C4C2×C4×D17311yescharacteristic
221C2×C4D4×D17311yescharacteristic
222C8—41no—
223C8—6921no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—3461no—
230C22—3461no—
231C22—3461no—
232C22—3461no—
233C22—3461no—
234C22—3461no—
235C22—6922no—
236C22—6922no—
237C22C22⋊C4×D17311yescharacteristic
238C22#1914970311yescharacteristic
239C22#1914970811yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—81no—
244C4—3461no—
245C4—13841no—
246C4—1732no—
247C4—6921no—
248C4—6921no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—6921no—
253C2—6921no—
254C2#99×D17311yescharacteristic
255C2#2601148911yescharacteristic
256C2#2601150611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#327×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012001C2
cosocleC22×C4×C173C23
hypercenter#327D173
hypocenterC173#1513
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26011453C2semidirect
3. #26011350C2semidirect
4. #26011229C2semidirect
5. #26011227C2semidirect
6. #26011225C2semidirect
7. #327×C173C2semidirect
8. C2×#14583825C4semidirect
12. #19149659C4semidirect
14. #19149650C4semidirect
18. #19149637C4semidirect
128. C173#1513semidirect
136. #327D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
#260114532
#260113502
#260112292
#260112272
#260112252
#327×C1732
#1513173