Finite Group Databasebeta

← #22620956

Subgroups — #22620956

Order 9344.

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

normal subgroups only

This group has 386 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×#14556589C211yesmaximal, characteristic, complemented
2#17989118C211yesmaximal, characteristic
3#17988978C212yesmaximal, complemented
4#17988954C212yesmaximal, complemented
5#17988796C212yesmaximal, complemented
6#17988790C212yesmaximal, complemented
7#17988745C211yesmaximal, characteristic, complemented
8#17988743C211yesmaximal, characteristic, complemented
9#17988725C211yesmaximal, characteristic
10#17988641C211yesmaximal, characteristic, complemented
11#448×C73C211yesmaximal, characteristic
12C22×Dic146C2211yescharacteristic
13C2×Dic292C2212yes—
14C2×#11924018C2212yes—
15C2×#11924016C2212yes—
16#14556589—21no—
17#14556589—21no—
18#14556589C2212yes—
19#14556589C2212yes—
20#14556584—21no—
21#14556582C2211yescharacteristic
22#14556578C2211yescharacteristic
23#14556572—21no—
24#14556570C2211yescharacteristic
25#14556556C2211yescharacteristic
26D4×Dic73—21no—
27D4×Dic73C2211yescharacteristic
28#14556547—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×C4×Dic73C2211yescharacteristic
32#14556520C2212yes—
33#14556516C2212yes—
34#14556515C2212yes—
35#14556514C2212yes—
36C22⋊#10596831C2211yescharacteristic
37#14556501C2212yes—
38C22×Q8×C73C2211yescharacteristic
39C2×SD16×C73C2212yes—
40#121×C73C2211yescharacteristic
41#103×C73C2212yes—
42C22⋊C8×C73C2211yescharacteristic
43C2×Dic146—21no—
44C2×Dic146—42no—
45C2×Dic146C2312yes—
46C2×Dic146C2311yescharacteristic
47C22×Dic73—21no—
48C22×Dic73—21no—
49C22×Dic73D411yescharacteristic
50C22×Dic73D411yescharacteristic
51Dic292—42no—
52#11924018—42no—
53#11924016—42no—
54C2×#10596831C2312yes—
55C4⋊Dic73—21no—
56C4⋊Dic73C2311yescharacteristic
57#11924007—21no—
58#11924007—22no—
59#11924007—22no—
60#11924007—22no—
61#11924007—22no—
62#11924007—21no—
63#11924007—21no—
64C22⋊Dic73—21no—
65C22⋊Dic73—21no—
66C22⋊Dic73—21no—
67C22⋊Dic73—21no—
68C4×Dic73—21no—
69C4×Dic73D411yescharacteristic
70C4×Dic73D411yescharacteristic
71C4×Dic73C2311yescharacteristic
72C4×Dic73C2311yescharacteristic
73C2×Q8×C73—21no—
74C2×Q8×C73—42no—
75C2×Q8×C73C2312yes—
76C2×D4×C73—21no—
77C2×D4×C73C2311yescharacteristic
78C22×C4×C73—21no—
79C22×C4×C73C2311yescharacteristic
80SD16×C73—42no—
81C2×C8×C73C2312yes—
82C4⋊C4×C73C2311yescharacteristic
83C22⋊C4×C73—21no—
84C23×C73—21no—
85C23×C73C2×D411yescharacteristic
86Q8×C73—22no—
87Q8×C73—22no—
88Q8×C73—42no—
89D4×C73—22no—
90D4×C73—42no—
91C2×C4×C73—22no—
92C2×C4×C73—21no—
93C2×C4×C73—22no—
94C2×C4×C73—42no—
95C2×C4×C73C2×D411yescharacteristic
96C2×C4×C73C2411yescharacteristic
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—22no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104C2×Dic73C2×D411yescharacteristic
105C2×Dic73C2×D411yescharacteristic
106C2×Dic73C2×D411yescharacteristic
107C2×Dic73C2×D411yescharacteristic
108Dic146—22no—
109Dic146—22no—
110Dic146—22no—
111Dic146—42no—
112C8×C73—22no—
113#10596831—22no—
114C22×C73—21no—
115C22×C73—21no—
116C22×C73—41no—
117C22×C73—41no—
118C22×C73—41no—
119C22×C73#13712yes—
120C22×C73C22×D411yescharacteristic
121C4×C73—21no—
122C4×C73—42no—
123C4×C73—41no—
124C4×C73H(5,𝔽2)12yes—
125Dic73—21no—
126Dic73—21no—
127Dic73—21no—
128Dic73—42no—
129Dic73—41no—
130Dic73—41no—
131Dic73#13712yes—
132C2×C73—22no—
133C2×C73—81no—
134C2×C73#53511yescharacteristic
135C2×C73C2×#13711yescharacteristic
136C2×C73C2×#13711yescharacteristic
137#3218—731nomaximal, Sylow
138C73#321811yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—731no—
140#529—731no—
141#497—732no—
142#491—732no—
143#462—732no—
144#461—732no—
145#451—731no—
146#451—731no—
147#448—731no—
148#448D7311yescharacteristic
149#432—731no—
150C22×Q8—731no—
151C22×Q8C2×D7311yescharacteristic
152C2×Q16—732no—
153C2×Q16—732no—
154C2×SD16—732no—
155C2×SD16C2×D7312yes—
156C4⋊Q8—732no—
157C4⋊Q8—732no—
158C4⋊Q8—1461no—
159C4⋊Q8—1461no—
160#126—1461no—
161#125—731no—
162#124—731no—
163#123—1461no—
164C22⋊Q8—731no—
165#121C2×D7311yescharacteristic
166C4×Q8—731no—
167C4×D4—731no—
168C4×D4—1461no—
169#117—1461no—
170C2×C4⋊C4—1461no—
171C2×C4⋊C4—1461no—
172C2×C42—731no—
173C4⋊C8—732no—
174#103—732no—
175#103—732no—
176#103—732no—
177#103C2×D7312yes—
178C22⋊C8—731no—
179C22⋊C8C2×D7311yescharacteristic
180#97—732no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8—2922no—
184C2×Q8—732no—
185C2×Q8—731no—
186C2×Q8—1461no—
187C2×Q8C22×D7312yes—
188C2×D4—21no—
189C2×D4C22×D7311yescharacteristic
190C22×C4—21no—
191C22×C4—731no—
192C22×C4—731no—
193C22×C4—1461no—
194C22×C4—1461no—
195C22×C4C22×D7311yescharacteristic
196Q16—2922no—
197Q16—2922no—
198SD16—42no—
199SD16—2922no—
200C2×C8—732no—
201C2×C8C22×D7312yes—
202C4⋊C4—731no—
203C4⋊C4—1462no—
204C4⋊C4—1462no—
205C4⋊C4—1462no—
206C4⋊C4—1462no—
207C4⋊C4—1461no—
208C4⋊C4—1461no—
209C4⋊C4—1461no—
210C4⋊C4—1461no—
211C4⋊C4C22×D7311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1461no—
214C22⋊C4—1461no—
215C22⋊C4—1461no—
216C22⋊C4—1461no—
217C42—731no—
218C42—731no—
219C42—731no—
220C42—731no—
221C42—1461no—
222C23—21no—
223C23D4×D7311yescharacteristic
224Q8—22no—
225Q8—22no—
226Q8—42no—
227Q8—2922no—
228Q8—1462no—
229Q8—1462no—
230Q8—1462no—
231D4—22no—
232D4—42no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—2922no—
238C2×C4—2922no—
239C2×C4—731no—
240C2×C4—731no—
241C2×C4—731no—
242C2×C4—731no—
243C2×C4—1461no—
244C2×C4—1461no—
245C2×C4—1462no—
246C2×C4—1462no—
247C2×C4—1462no—
248C2×C4D4×D7311yescharacteristic
249C2×C4C23×D7311yescharacteristic
250C8—22no—
251C8—1462no—
252C22—21no—
253C22—21no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22#1455662012yes—
258C22C2×D4×D7311yescharacteristic
259C4—21no—
260C4—41no—
261C4—42no—
262C4—2921no—
263C4—2922no—
264C4—2921no—
265C4—732no—
266C4—1461no—
267C4—1461no—
268C4—1461no—
269C4#1455663612yes—
270C2—22no—
271C2—81no—
272C2#1798915211yescharacteristic
273C2#137×D7311yescharacteristic
274C2C2×#1455662011yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#448×C73C2
socleC22×C73C22×D4
hypercenter#448D73
hypocenterC73#3218
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#14556589C2semidirect
3. #17988978C2semidirect
4. #17988954C2semidirect
5. #17988796C2semidirect
6. #17988790C2semidirect
7. #17988745C2semidirect
8. #17988743C2semidirect
10. #17988641C2semidirect
138. C73#3218semidirect