Finite Group Databasebeta

← #22621150

Subgroups — #22621150

Order 9344.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14556590C211yesmaximal, characteristic, complemented
2#17989118C211yesmaximal, characteristic
3#17988995C212yesmaximal, complemented
4#17988945C212yesmaximal, complemented
5#17988918C211yesmaximal, characteristic, complemented
6#17988917C211yesmaximal, characteristic, complemented
7#17988898C211yesmaximal, characteristic
8#17988881C212yesmaximal, complemented
9#17988875C212yesmaximal, complemented
10#17988641C211yesmaximal, characteristic, complemented
11#480×C73C211yesmaximal, characteristic
12C22×Dic146—21no—
13#14556590—21no—
14#14556590—21no—
15#14556590C2212yes—
16#14556590C2212yes—
17#14556584—21no—
18#14556582C2211yescharacteristic
19#14556578C2211yescharacteristic
20#14556572—21no—
21#14556570C2211yescharacteristic
22#14556556C2211yescharacteristic
23D4×Dic73—21no—
24D4×Dic73C2211yescharacteristic
25#14556547—21no—
26C2×C4⋊Dic73C2211yescharacteristic
27C2×#11924007—21no—
28C2×C4×Dic73C2211yescharacteristic
29#14556524C2212yes—
30#14556523C2212yes—
31#14556522C2212yes—
32#14556520C2212yes—
33#14556516C2212yes—
34#14556514C2212yes—
35#14556512C2212yes—
36C22⋊#10596831C2211yescharacteristic
37#14556501C2212yes—
38#121×C73C2211yescharacteristic
39C2×C4⋊C4×C73C2211yescharacteristic
40#107×C73C2212yes—
41#102×C73C2212yes—
42C22⋊C8×C73C2211yescharacteristic
43C2×Dic146—22no—
44C2×Dic146—22no—
45C2×Dic146—44no—
46C2×Dic146C2311yescharacteristic
47C22×Dic73—21no—
48C22×Dic73—21no—
49C22×Dic73D411yescharacteristic
50C22×Dic73D411yescharacteristic
51C2×#10596831C2312yes—
52C4⋊Dic73—21no—
53C4⋊Dic73—21no—
54C4⋊Dic73C2312yes—
55C4⋊Dic73C2311yescharacteristic
56#11924007—21no—
57#11924007—22no—
58#11924007—22no—
59#11924007—21no—
60#11924007—21no—
61C22⋊Dic73—21no—
62C22⋊Dic73—21no—
63C22⋊Dic73—21no—
64C22⋊Dic73—21no—
65C4×Dic73—21no—
66C4×Dic73D411yescharacteristic
67C4×Dic73D411yescharacteristic
68C4×Dic73C2311yescharacteristic
69C4×Dic73C2311yescharacteristic
70C2×D4×C73—21no—
71C2×D4×C73C2311yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73C2311yescharacteristic
74C2×C8×C73C2312yes—
75C4⋊C4×C73—21no—
76C4⋊C4×C73C2312yes—
77C4⋊C4×C73C2311yescharacteristic
78C22⋊C4×C73—21no—
79C23×C73—21no—
80C23×C73C2×D411yescharacteristic
81D4×C73—22no—
82D4×C73—42no—
83C2×C4×C73—22no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—42no—
87C2×C4×C73C2×D411yescharacteristic
88C2×C4×C73C2411yescharacteristic
89C2×Dic73—22no—
90C2×Dic73—22no—
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—22no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73C2×D411yescharacteristic
97C2×Dic73C2×D411yescharacteristic
98C2×Dic73C2×D411yescharacteristic
99C2×Dic73C2×D411yescharacteristic
100Dic146—22no—
101Dic146—44no—
102Dic146—44no—
103C8×C73—22no—
104#10596831—22no—
105C22×C73—21no—
106C22×C73—21no—
107C22×C73—41no—
108C22×C73—41no—
109C22×C73—41no—
110C22×C73AGL(1,ℤ/8)12yes—
111C22×C73C22×D411yescharacteristic
112C4×C73—21no—
113C4×C73—42no—
114C4×C73—41no—
115C4×C732−1+412yes—
116Dic73—21no—
117Dic73—21no—
118Dic73—21no—
119Dic73—41no—
120Dic73—42no—
121Dic73—41no—
122Dic73#13712yes—
123C2×C73—22no—
124C2×C73—81no—
125C2×C73#53611yescharacteristic
126C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C73C2×#13711yescharacteristic
128#3238—731nomaximal, Sylow
129C73#323811yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—731no—
131#529—731no—
132#501—732no—
133#489—732no—
134#483—731no—
135#483—731no—
136#480—731no—
137#480D7311yescharacteristic
138#477—732no—
139#476—732no—
140#432—731no—
141C22×Q8—1461no—
142C4⋊Q8—732no—
143C4⋊Q8—732no—
144C4⋊Q8—1461no—
145C4⋊Q8—1461no—
146#126—1461no—
147#125—731no—
148#124—731no—
149#123—1461no—
150C22⋊Q8—731no—
151#121C2×D7311yescharacteristic
152C4×Q8—731no—
153C4×D4—731no—
154C4×D4—1461no—
155#117—1461no—
156C2×C4⋊C4—731no—
157C2×C4⋊C4—1461no—
158C2×C4⋊C4C2×D7311yescharacteristic
159C2×C42—731no—
160#107—732no—
161#107C2×D7312yes—
162#106—732no—
163#106—732no—
164C4⋊C8—732no—
165#103—732no—
166#103—732no—
167#102—732no—
168#102C2×D7312yes—
169C22⋊C8—731no—
170C22⋊C8C2×D7311yescharacteristic
171#97—732no—
172C2×Q8—2924no—
173C2×Q8—731no—
174C2×Q8—1462no—
175C2×Q8—1462no—
176C2×D4—21no—
177C2×D4C22×D7311yescharacteristic
178C22×C4—21no—
179C22×C4—731no—
180C22×C4—731no—
181C22×C4—1461no—
182C22×C4—1461no—
183C22×C4C22×D7311yescharacteristic
184C2×C8—732no—
185C2×C8C22×D7312yes—
186C4⋊C4—21no—
187C4⋊C4—732no—
188C4⋊C4—731no—
189C4⋊C4—1462no—
190C4⋊C4—1462no—
191C4⋊C4—1461no—
192C4⋊C4—1461no—
193C4⋊C4—1461no—
194C4⋊C4—1461no—
195C4⋊C4—1461no—
196C4⋊C4C22×D7312yes—
197C4⋊C4C22×D7311yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—1461no—
200C22⋊C4—1461no—
201C22⋊C4—1461no—
202C22⋊C4—1461no—
203C42—731no—
204C42—731no—
205C42—731no—
206C42—731no—
207C42—1461no—
208C23—21no—
209C23D4×D7311yescharacteristic
210Q8—2924no—
211Q8—2924no—
212Q8—1462no—
213D4—22no—
214D4—42no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—2922no—
220C2×C4—2922no—
221C2×C4—731no—
222C2×C4—731no—
223C2×C4—731no—
224C2×C4—731no—
225C2×C4—1461no—
226C2×C4—1461no—
227C2×C4—1462no—
228C2×C4—1462no—
229C2×C4—1462no—
230C2×C4D4×D7311yescharacteristic
231C2×C4C23×D7311yescharacteristic
232C8—22no—
233C8—1462no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22#1455661512yes—
240C22C2×D4×D7311yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—2921no—
245C4—2921no—
246C4—2922no—
247C4—732no—
248C4—1461no—
249C4—1461no—
250C4—1461no—
251C4#1455663812yes—
252C2—22no—
253C2—81no—
254C2#1798915911yescharacteristic
255C2C2×#1455661511yescharacteristic
256C2#137×D7311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#480×C73C2
socleC22×C73C22×D4
hypercenter#480D73
hypocenterC73#3238
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. C2×#14556590C2semidirect
3. #17988995C2semidirect
4. #17988945C2semidirect
5. #17988918C2semidirect
6. #17988917C2semidirect
8. #17988881C2semidirect
9. #17988875C2semidirect
10. #17988641C2semidirect
129. C73#3238semidirect