Finite Group Databasebeta

← #22619409

Subgroups — #22619409

Order 9344.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#14556616C211yesmaximal, characteristic, complemented
2C2×#14556587C211yesmaximal, characteristic, complemented
3#41×Dic73C211yesmaximal, characteristic, complemented
4#17988785C214yesmaximal, complemented
5#17988780C214yesmaximal, complemented
6#17988610C211yesmaximal, characteristic, complemented
7#17988593C211yesmaximal, characteristic, complemented
8#17988592C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C73C211yesmaximal, characteristic, complemented
10C22×D292C2211yescharacteristic
11C22×C22⋊D73—22no—
12#14556616—24no—
13C2×#11924015C2212yescomplemented
14C2×#11924013C2212yescomplemented
15C2×#11924012C2211yescharacteristic
16#14556587—21no—
17#14556587—21no—
18#14556587C2214yes—
19Q8×Dic73C2212yescomplemented
20D4×Dic73—22no—
21D4×Dic73C2212yescomplemented
22#14556550—21no—
23#14556550C2211yescharacteristic
24C2×C4×Dic73—21no—
25C2×C4×Dic73C2211yescharacteristic
26#14556520C2214yescomplemented
27#14556513C2214yescomplemented
28#14556512C2214yescomplemented
29C2×#41×C73C2211yescharacteristic
30C22×D4×C73C2211yescharacteristic
31AGL(1,ℤ/8)×C73—24no—
32C2×SD16×C73C2212yescomplemented
33C2×D8×C73C2212yescomplemented
34C2×M16×C73C2211yescharacteristic
35C23×D73—21no—
36C2×D292—22no—
37C2×D292—22no—
38C2×D292—21no—
39C2×D292C2312yes—
40C2×C22⋊D73—24no—
41C2×C22⋊D73—24no—
42C2×C22⋊D73—48no—
43C22×Dic73—22no—
44C22×Dic73D412yes—
45#11924015—24no—
46#11924013—24no—
47#11924012—22no—
48C2×#10596831C2312yes—
49C4⋊Dic73—22no—
50C4⋊Dic73C2312yes—
51C22⋊Dic73—21no—
52C22⋊Dic73—22no—
53C4×Dic73—21no—
54C4×Dic73—22no—
55C4×Dic73D411yescharacteristic
56C4×Dic73D411yescharacteristic
57C4×Dic73C2312yes—
58C24×C73—21no—
59#41×C73—22no—
60#41×C73D414yescomplemented
61C2×Q8×C73C2311yescharacteristic
62C2×D4×C73—21no—
63C2×D4×C73—22no—
64C2×D4×C73—21no—
65C2×D4×C73—22no—
66C2×D4×C73C2312yes—
67C2×D4×C73C2311yescharacteristic
68C22×C4×C73—21no—
69C22×C4×C73C2311yescharacteristic
70SD16×C73—24no—
71D8×C73—24no—
72M16×C73—22no—
73C2×C8×C73C2312yes—
74C23×C73—21no—
75C23×C73—22no—
76C23×C73—44no—
77C23×C73—42no—
78C23×C73C2×D411yescharacteristic
79C22×D73—22no—
80C22×D73—44no—
81C22×D73—42no—
82Q8×C73—21no—
83Q8×C73C2×D412yescomplemented
84D4×C73—24no—
85D4×C73—22no—
86D4×C73—21no—
87D4×C73—22no—
88D4×C73—42no—
89D4×C73C2×D412yescomplemented
90C2×C4×C73—21no—
91C2×C4×C73—24no—
92C2×C4×C73C2×D411yescharacteristic
93C2×C4×C73C2×D412yes—
94C2×C4×C73#4112yes—
95C2×C4×C73C2411yescharacteristic
96C22⋊D73—48no—
97C22⋊D73—48no—
98C2×Dic73—24no—
99C2×Dic73—22no—
100C2×Dic73—44no—
101C2×Dic73C2×D412yes—
102C2×Dic73C2×D412yes—
103D292—22no—
104D292—22no—
105D292—42no—
106C8×C73—22no—
107#10596831—22no—
108C22×C73—22no—
109C22×C73—22no—
110C22×C73—42no—
111C22×C73—42no—
112C22×C73—44no—
113C22×C73—84no—
114C22×C73#12112yes—
115C22×C73C22×D411yescharacteristic
116C2×D73—42no—
117C2×D73—44no—
118C2×D73—84no—
119C4×C73—22no—
120C4×C73#12112yes—
121C4×C73C22×D411yescharacteristic
122C4×C73C2×#4111yescharacteristic
123Dic73—22no—
124Dic73—42no—
125Dic73AGL(1,ℤ/8)14yescomplemented
126C2×C73—22no—
127C2×C73—42no—
128C2×C73—82no—
129C2×C73C2×#12111yescharacteristic
130C2×C73C2×AGL(1,ℤ/8)12yes—
131D73—82no—
132#3038—731nomaximal, Sylow
133C73#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—731no—
135C2×AGL(1,ℤ/8)D7311yescharacteristic, complemented
136C2×#127—731no—
137C4×#41—731no—
138#460—734no—
139#459—734no—
140#423—731no—
141#418—731no—
142#418—731no—
143C2×#41C2×D7311yescharacteristic
144C22×D4—731no—
145C22×D4—1462no—
146C22×D4C2×D7311yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—1464no—
149C2×SD16—732no—
150C2×SD16C2×D7312yescomplemented
151C2×D8—732no—
152C2×D8C2×D7312yescomplemented
153C2×M16—731no—
154C2×M16C2×D7311yescharacteristic
155#127—734no—
156#127—1461no—
157#127—1461no—
158C4×Q8—732no—
159C4×D4—732no—
160C4×D4—1462no—
161#117—731no—
162#117—1461no—
163C2×C42—731no—
164C2×C42—1461no—
165C4⋊C8—734no—
166#102—734no—
167#102—734no—
168C24—21no—
169C24—1461no—
170#41—22no—
171#41C22⋊D7314yescomplemented
172C2×Q8C22×D7311yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—2928no—
178C2×D4—732no—
179C2×D4—1464no—
180C2×D4—1464no—
181C2×D4—1461no—
182C2×D4—1462no—
183C2×D4—1462no—
184C2×D4C22×D7312yes—
185C2×D4C22×D7311yescharacteristic
186C22×C4—21no—
187C22×C4—732no—
188C22×C4—1462no—
189C22×C4C22×D7311yescharacteristic
190SD16—24no—
191SD16—1464no—
192D8—24no—
193D8—1464no—
194M16—22no—
195M16—1462no—
196C2×C8—732no—
197C2×C8C22×D7312yes—
198C4⋊C4—732no—
199C4⋊C4—1462no—
200C22⋊C4—1462no—
201C22⋊C4—1461no—
202C42—731no—
203C42—731no—
204C42—732no—
205C42—1462no—
206C42—1461no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—2924no—
212C23—2922no—
213C23—1462no—
214C23D4×D7311yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D7312yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—2928no—
223D4—2928no—
224D4—2922no—
225D4—1462no—
226D4—1462no—
227D4C2×C22⋊D7312yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—2924no—
231C2×C4—732no—
232C2×C4—732no—
233C2×C4—1462no—
234C2×C4—1464no—
235C2×C4D4×D7311yescharacteristic
236C2×C4C2×C22⋊D7312yes—
237C2×C4#1192402712yes—
238C2×C4C23×D7311yescharacteristic
239C8—22no—
240C8—1462no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—2922no—
248C22—2924no—
249C22—5844no—
250C22#1455656312yes—
251C22C2×D4×D7311yescharacteristic
252C4—22no—
253C4—2922no—
254C4—734no—
255C4—1462no—
256C4#1455656312yes—
257C4C22×C22⋊D7311yescharacteristic
258C4C2×#1192402711yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—5842no—
263C2C2×#1455656311yescharacteristic
264C2AGL(1,ℤ/8)×D7312yes—
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
FittingC2×AGL(1,ℤ/8)×C73C2
socleC22×C73C22×D4
hypercenterC2×AGL(1,ℤ/8)D73
hypocenterC73#3038
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14556616C2semidirect
2. C2×#14556587C2semidirect
3. #41×Dic73C2semidirect
4. #17988785C2semidirect
5. #17988780C2semidirect
6. #17988610C2semidirect
7. #17988593C2semidirect
8. #17988592C2semidirect
9. C2×AGL(1,ℤ/8)×C73C2semidirect
13. C2×#11924015C22semidirect
14. C2×#11924013C22semidirect
19. Q8×Dic73C22semidirect
21. D4×Dic73C22semidirect
26. #14556520C22semidirect
27. #14556513C22semidirect
28. #14556512C22semidirect
32. C2×SD16×C73C22semidirect
33. C2×D8×C73C22semidirect
60. #41×C73D4semidirect
83. Q8×C73C2×D4semidirect
89. D4×C73C2×D4semidirect
125. Dic73AGL(1,ℤ/8)semidirect
133. C73#3038semidirect
135. C2×AGL(1,ℤ/8)D73semidirect
150. C2×SD16C2×D73semidirect
152. C2×D8C2×D73semidirect
171. #41C22⋊D73semidirect
216. Q8C2×C22⋊D73semidirect
227. D4C2×C22⋊D73semidirect