Finite Group Databasebeta

← #22620702

Subgroups — #22620702

Order 9344.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D73C212yesmaximal
2#17988924C212yesmaximal, complemented
3#17988921C212yesmaximal, complemented
4#17988828C211yesmaximal, characteristic, complemented
5#17988826C212yesmaximal, complemented
6#17988744C212yesmaximal, complemented
7#17988741C212yesmaximal, complemented
8C2×#14556503C211yesmaximal, characteristic
9#467×C73C211yesmaximal, characteristic, complemented
10C2×Q8×D73C2212yes—
11C22×C4×D73C2211yescharacteristic
12C2×Dic292C2211yescharacteristic, complemented
13C2×#11924018C2212yescomplemented
14C22×#10596831C2211yescharacteristic
15#14556571C2212yes—
16#14556570C2212yes—
17#14556569—22no—
18#14556568—22no—
19#14556567C2212yes—
20C4⋊C4×D73—22no—
21C4⋊C4×D73C2212yes—
22C22⋊C4×D73—22no—
23#14556525C2211yescharacteristic, complemented
24#14556524C2212yescomplemented
25#14556516C2212yescomplemented
26#14556515C2212yescomplemented
27#14556514C2212yescomplemented
28#14556503C2212yes—
29#14556503C2212yes—
30C2×Q16×C73C2211yescharacteristic, complemented
31C22×C8×C73C2211yescharacteristic
32C22⋊Q8×C73C2212yes—
33#107×C73C2211yescharacteristic, complemented
34#103×C73C2212yescomplemented
35C23×D73D411yescharacteristic
36C2×Dic146—22no—
37C2×Dic146C2312yes—
38Q8×D73—42no—
39Q8×D73—42no—
40C22×Dic73D411yescharacteristic
41C2×C4×D73—22no—
42C2×C4×D73—22no—
43C2×C4×D73—41no—
44C2×C4×D73—41no—
45C2×C4×D73D412yes—
46C2×C4×D73C2312yes—
47Dic292—41no—
48#11924018—42no—
49C2×#10596831—41no—
50C2×#10596831C2312yes—
51C4⋊Dic73—22no—
52C4⋊Dic73C2312yes—
53#11924007—22no—
54#11924007—22no—
55#11924007—22no—
56C22⋊Dic73—22no—
57#11924005—22no—
58#11924005—22no—
59C2×Q8×C73C2312yes—
60C22×C4×C73C2311yescharacteristic
61Q16×C73—41no—
62C2×C8×C73—41no—
63C2×C8×C73C2311yescharacteristic
64C2×C8×C73C2311yescharacteristic
65C4⋊C4×C73—22no—
66C4⋊C4×C73C2312yes—
67C22⋊C4×C73—22no—
68C23×C73C2×D411yescharacteristic
69C22×D73—41no—
70C22×D73—41no—
71C22×D73Q1614yescomplemented
72C22×D73C2×D412yes—
73Q8×C73—22no—
74Q8×C73—22no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73C2×D411yescharacteristic
80C2×C4×C73C2411yescharacteristic
81C2×Dic73—22no—
82C2×Dic73—22no—
83C2×Dic73—41no—
84C2×Dic73C2×D412yes—
85C4×D73—22no—
86C4×D73—22no—
87C4×D73—42no—
88C4×D73—44no—
89C4×D73—44no—
90Dic146—22no—
91Dic146—22no—
92Dic146—44no—
93C8×C73—21no—
94C8×C73—21no—
95#10596831—22no—
96C22×C73—21no—
97C22×C73—21no—
98C22×C73C2×Q1612yes—
99C22×C73C22×D411yescharacteristic
100C2×D73—22no—
101C2×D73—22no—
102C2×D73—44no—
103C2×D73C2×Q1614yes—
104C4×C73—21no—
105C4×C73—42no—
106C4×C73—42no—
107C4×C73H(5,𝔽2)11yescharacteristic
108C4×C73H(5,𝔽2)11yescharacteristic
109Dic73—22no—
110Dic73—42no—
111Dic73—42no—
112C2×C73—22no—
113C2×C73#53411yescharacteristic
114C2×C73C22×Q1611yescharacteristic
115C2×C73#57511yescharacteristic
116D73—24no—
117#3189—731nomaximal, Sylow
118C73#318911yescharacteristic, complemented, Sylow
119C2×C22⋊Q8—732no—
120#484—732no—
121#484—732no—
122#467—731no—
123#467—732no—
124#467D7311yescharacteristic, complemented
125#451—732no—
126#451—732no—
127C2×C22⋊C8—731no—
128C22×Q8—732no—
129C23×C4—731no—
130C2×Q16—732no—
131C2×Q16—731no—
132C2×Q16C2×D7311yescharacteristic, complemented
133C22×C8—731no—
134C22×C8C2×D7311yescharacteristic
135C22⋊Q8—732no—
136C22⋊Q8—732no—
137C22⋊Q8—732no—
138C22⋊Q8—1462no—
139C22⋊Q8—1462no—
140C22⋊Q8C2×D7312yes—
141C2×C4⋊C4—732no—
142C2×C4⋊C4—1462no—
143C2×C22⋊C4—1462no—
144#107—731no—
145#107—732no—
146#107C2×D7311yescharacteristic, complemented
147#103—732no—
148#103—732no—
149#103—732no—
150#103C2×D7312yescomplemented
151C22⋊C8—732no—
152C22⋊C8—732no—
153C24—731no—
154C2×Q8—2922no—
155C2×Q8—2922no—
156C2×Q8—732no—
157C2×Q8—1462no—
158C2×Q8C22×D7312yes—
159C22×C4—2921no—
160C22×C4—2921no—
161C22×C4—731no—
162C22×C4—732no—
163C22×C4—732no—
164C22×C4—1462no—
165C22×C4—1462no—
166C22×C4C22×D7311yescharacteristic
167Q16—41no—
168Q16—2921no—
169Q16—2922no—
170C2×C8—41no—
171C2×C8—2921no—
172C2×C8—732no—
173C2×C8C22×D7311yescharacteristic
174C2×C8C22×D7311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—732no—
177C4⋊C4—1462no—
178C4⋊C4—1462no—
179C4⋊C4—1462no—
180C4⋊C4—1462no—
181C4⋊C4C22×D7312yes—
182C22⋊C4—22no—
183C22⋊C4—1462no—
184C22⋊C4—1462no—
185C22⋊C4—1462no—
186C23—2921no—
187C23—2921no—
188C23—734no—
189C23—732no—
190C23D4×D7311yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—2924no—
194Q8—1462no—
195Q8—1462no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2922no—
201C2×C4—2924no—
202C2×C4—2924no—
203C2×C4—2921no—
204C2×C4—732no—
205C2×C4—1462no—
206C2×C4—1462no—
207C2×C4—1462no—
208C2×C4—1462no—
209C2×C4D4×D7311yescharacteristic
210C2×C4C23×D7311yescharacteristic
211C8—21no—
212C8—21no—
213C8—1462no—
214C22—21no—
215C22—21no—
216C22—2924no—
217C22—734no—
218C22—1462no—
219C22—1462no—
220C22Q16×D7312yes—
221C22C2×D4×D7311yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—2922no—
226C4—2922no—
227C4—1462no—
228C4#1455663611yescharacteristic
229C4#1455663711yescharacteristic
230C2—22no—
231C2—1464no—
232C2#1798914511yescharacteristic
233C2C2×Q16×D7311yescharacteristic
234C2#1798938411yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#467×C73C2
socleC22×C73C22×D4
hypercenter#467D73
hypocenterC73#3189
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #17988924C2semidirect
3. #17988921C2semidirect
4. #17988828C2semidirect
5. #17988826C2semidirect
6. #17988744C2semidirect
7. #17988741C2semidirect
9. #467×C73C2semidirect
12. C2×Dic292C22semidirect
13. C2×#11924018C22semidirect
23. #14556525C22semidirect
24. #14556524C22semidirect
25. #14556516C22semidirect
26. #14556515C22semidirect
27. #14556514C22semidirect
30. C2×Q16×C73C22semidirect
33. #107×C73C22semidirect
34. #103×C73C22semidirect
71. C22×D73Q16semidirect
118. C73#3189semidirect
124. #467D73semidirect
132. C2×Q16C2×D73semidirect
146. #107C2×D73semidirect
150. #103C2×D73semidirect