Finite Group Databasebeta

← #36353582

Subgroups — #36353582

Order 22144.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#26011959C211yesmaximal, characteristic
2#26011514C211yesmaximal, characteristic, complemented
3C2×#19149607C211yesmaximal, characteristic
4C2×#19149605C211yesmaximal, characteristic
5C4×#14583805C211yesmaximal, characteristic, complemented
6#26011250C211yesmaximal, characteristic, complemented
7#339×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C2×#14583810C2211yescharacteristic
10#19149644C411yescharacteristic, complemented
11#19149644C411yescharacteristic, complemented
12#19149641C411yescharacteristic, complemented
13#19149641C411yescharacteristic, complemented
14#19149639—21no—
15#19149639—21no—
16#19149636—21no—
17C2×#14583805—21no—
18C2×C4×Dic173C2211yescharacteristic
19C2×C4×Dic173C2211yescharacteristic
20C42×D173—21no—
21#19149608—21no—
22#19149608—21no—
23#19149607—22no—
24#19149605—22no—
25#19149586—21no—
26C2×M16×C173C2211yescharacteristic
27#117×C173C2211yescharacteristic
28C2×C42×C173C2211yescharacteristic
29#14583826—21no—
30#14583826—21no—
31#14583826D412yes—
32#14583826Q812yes—
33C2×Dic346C2×C411yescharacteristic
34C2×D692C2×C411yescharacteristic
35C2×C22⋊D173—21no—
36C22×Dic173—21no—
37C22×Dic173D411yescharacteristic
38C22×Dic173D411yescharacteristic
39C2×C4×D173—21no—
40C2×C4×D173—21no—
41C2×C4×D173—21no—
42#14583810—21no—
43#14583810—21no—
44C2×#13063436—21no—
45#14583805—24no—
46#14583805—21no—
47#14583805—21no—
48#14583803—21no—
49#14583803—21no—
50C4×Dic173—21no—
51C4×Dic173—21no—
52C4×Dic173—21no—
53C4×Dic173—21no—
54C4×Dic173C2×C411yescharacteristic
55C4×Dic173C2×C411yescharacteristic
56C4×Dic173D411yescharacteristic
57C4×Dic173D411yescharacteristic
58C22×C4×C173—21no—
59C22×C4×C173C2311yescharacteristic
60M16×C173—21no—
61M16×C173—21no—
62C2×C8×C173—21no—
63C4⋊C4×C173C2×C411yescharacteristic
64C4⋊C4×C173C2×C411yescharacteristic
65C22⋊C4×C173—21no—
66C42×C173—21no—
67C42×C173—21no—
68C42×C173—21no—
69C22×D173—21no—
70Dic346—21no—
71Dic346C4⋊C412yescomplemented
72D692—21no—
73D692C4⋊C412yescomplemented
74C22⋊D173—22no—
75C22⋊D173—22no—
76C2×Dic173—21no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—41no—
80C2×Dic173—41no—
81C2×Dic173—41no—
82C2×Dic173C22⋊C411yescharacteristic
83C2×Dic173C22⋊C411yescharacteristic
84C2×Dic173C22⋊C411yescharacteristic
85C2×Dic173C22⋊C411yescharacteristic
86C4×D173—22no—
87C4×D173—22no—
88C4×D173—42no—
89C4×D173—42no—
90#13063436—41no—
91C23×C173C2×D411yescharacteristic
92C2×C4×C173—22no—
93C2×C4×C173—21no—
94C2×C4×C173—21no—
95C2×C4×C173—41no—
96C2×C4×C173—41no—
97C2×C4×C173C22×C411yescharacteristic
98C2×C4×C173C2×D411yescharacteristic
99C2×C4×C173C2×D411yescharacteristic
100C2×C4×C173C2×Q811yescharacteristic
101C2×C4×C173#4111yescharacteristic
102C2×C4×C173#4111yescharacteristic
103C8×C173—41no—
104C22×C173—21no—
105C22×C173—21no—
106C22×C173C2×C22⋊C411yescharacteristic
107C22×C173#12111yescharacteristic
108C22×C173C22⋊Q811yescharacteristic
109C2×D173—22no—
110C2×D173—41no—
111C2×D173—41no—
112C4×C173—42no—
113C4×C173—41no—
114C4×C173—41no—
115C4×C173C2×C4⋊C411yescharacteristic
116C4×C173#11711yescharacteristic
117C4×C173#12111yescharacteristic
118C4×C173C22⋊Q811yescharacteristic
119Dic173—22no—
120Dic173—21no—
121Dic173—21no—
122Dic173—21no—
123Dic173—42no—
124Dic173#10412yescomplemented
125C2×C173—21no—
126C2×C173—21no—
127C2×C173#38011yescharacteristic
128C2×C173C2×#10411yescharacteristic
129C2×C173#42111yescharacteristic
130D173—42no—
131C173#180611yescharacteristic, complemented, Sylow
132#1806—1731nomaximal, Sylow
133C4×#41—1731no—
134#423—1731no—
135C2×#104—1731no—
136C2×#104—1731no—
137C4×C4⋊C4—1731no—
138#339—1731no—
139#339D17311yescharacteristic, complemented
140C2×#41—1731no—
141C2×M16—1731no—
142C2×M16C2×D17311yescharacteristic
143C4×Q8—1731no—
144C4×Q8—1731no—
145C4×D4—3461no—
146C4×D4—3461no—
147C4×D4—1731no—
148C4×D4—1731no—
149#117—3461no—
150#117C2×D17311yescharacteristic
151C2×C4⋊C4—3461no—
152C2×C42—3461no—
153C2×C42—1731no—
154C2×C42—1731no—
155C2×C42C2×D17311yescharacteristic
156C4⋊C8—3461no—
157C4⋊C8—3461no—
158#104—3462no—
159#104—3462no—
160#95—3461no—
161#41—3461no—
162#41—3461no—
163#41—1732no—
164#41—1732no—
165C2×Q8—1731no—
166C2×D4—3461no—
167C2×D4—1731no—
168C22×C4—21no—
169C22×C4—3461no—
170C22×C4—3461no—
171C22×C4—3461no—
172C22×C4—3461no—
173C22×C4—1731no—
174C22×C4—1731no—
175C22×C4C22×D17311yescharacteristic
176M16—21no—
177M16—21no—
178M16—3461no—
179M16—3461no—
180C2×C8—21no—
181C2×C8—3461no—
182C4⋊C4—3464no—
183C4⋊C4—3461no—
184C4⋊C4—3461no—
185C4⋊C4C4×D17311yescharacteristic
186C4⋊C4C4×D17311yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—3461no—
189C22⋊C4—3461no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—3461no—
194C42—3461no—
195C42—3461no—
196C42—3461no—
197C42—1731no—
198C42—1731no—
199C42—1731no—
200C42—1731no—
201C23—3461no—
202C23D4×D17311yescharacteristic
203Q8—3461no—
204Q8—1732no—
205D4—3462no—
206D4—3462no—
207D4—3461no—
208D4—1732no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3461no—
215C2×C4—3462no—
216C2×C4—3462no—
217C2×C4—3462no—
218C2×C4—3462no—
219C2×C4—1731no—
220C2×C4—1731no—
221C2×C4—1731no—
222C2×C4—1731no—
223C2×C4—6922no—
224C2×C4—6922no—
225C2×C4—6921no—
226C2×C4—6921no—
227C2×C4—6921no—
228C2×C4C2×C4×D17311yescharacteristic
229C2×C4D4×D17311yescharacteristic
230C2×C4D4×D17311yescharacteristic
231C2×C4Q8×D17311yescharacteristic
232C2×C4#1458382611yescharacteristic
233C2×C4#1458382611yescharacteristic
234C8—41no—
235C8—6921no—
236C22—21no—
237C22—21no—
238C22—3462no—
239C22—6921no—
240C22—6921no—
241C22C22⋊C4×D17311yescharacteristic
242C22#1914965211yescharacteristic
243C22#1914965711yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—3461no—
248C4—3461no—
249C4—3461no—
250C4—3462no—
251C4—1732no—
252C4—6922no—
253C4C4⋊C4×D17311yescharacteristic
254C4#1914963611yescharacteristic
255C4#1914965211yescharacteristic
256C4#1914965711yescharacteristic
257C2—21no—
258C2—21no—
259C2—6922no—
260C2#2601135011yescharacteristic
261C2#104×D17311yescharacteristic
262C2#2601150611yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#339×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC2×#14583826C22
cosocleC22×C4×C173C23
hypercenter#339D173
hypocenterC173#1806
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #26011514C2semidirect
5. C4×#14583805C2semidirect
6. #26011250C2semidirect
7. #339×C173C2semidirect
10. #19149644C4semidirect
11. #19149644C4semidirect
12. #19149641C4semidirect
13. #19149641C4semidirect
71. Dic346C4⋊C4semidirect
73. D692C4⋊C4semidirect
124. Dic173#104semidirect
131. C173#1806semidirect
139. #339D173semidirect

Maximal subgroups

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

subgroupindex
#260119592
#260115142
C2×#191496072
C2×#191496052
C4×#145838052
#260112502
#339×C1732
#1806173