Finite Group Databasebeta

← #22620751

Subgroups — #22620751

Order 9344.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#17988919C212yesmaximal, complemented
3#17988916C212yesmaximal, complemented
4#17988845C211yesmaximal, characteristic, complemented
5#17988843C212yesmaximal, complemented
6#17988744C212yesmaximal, complemented
7#17988741C212yesmaximal, complemented
8#17988550C211yesmaximal, characteristic
9#470×C73C211yesmaximal, characteristic, complemented
10C2×Q8×D73C2212yes—
11C22×C4×D73C2211yescharacteristic
12C2×Dic292C2211yescharacteristic, complemented
13C2×#11924018C2212yescomplemented
14C2×#11924012C2211yescharacteristic
15#14556571C2212yes—
16#14556570C2212yes—
17#14556569—22no—
18#14556568—22no—
19#14556567C2212yes—
20C4⋊C4×D73—22no—
21C4⋊C4×D73C2212yes—
22C22⋊C4×D73—22no—
23#14556523C2211yescharacteristic, complemented
24#14556522C2212yescomplemented
25#14556516C2212yescomplemented
26#14556515C2212yescomplemented
27#14556514C2212yescomplemented
28#14556503C2212yes—
29#14556503C2212yes—
30C2×Q16×C73C2211yescharacteristic, complemented
31C2×M16×C73C2211yescharacteristic
32C22⋊Q8×C73C2212yes—
33#106×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—
49#11924012—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—
62M16×C73—41no—
63C2×C8×C73C2311yescharacteristic
64C2×C8×C73C2311yescharacteristic
65C4⋊C4×C73—22no—
66C4⋊C4×C73C2312yes—
67C22⋊C4×C73—22no—
68C23×C73C2×D411yescharacteristic
69C22×D73—24no—
70C22×D73—41no—
71C22×D73C2×D412yes—
72Q8×C73—22no—
73Q8×C73—22no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73C2×D411yescharacteristic
79C2×C4×C73C2411yescharacteristic
80C2×Dic73—22no—
81C2×Dic73—22no—
82C2×Dic73—41no—
83C2×Dic73C2×D412yes—
84C4×D73—22no—
85C4×D73—22no—
86C4×D73—42no—
87C4×D73—44no—
88C4×D73—44no—
89Dic146—22no—
90Dic146—22no—
91Dic146—44no—
92C8×C73—21no—
93C8×C73—21no—
94#10596831—22no—
95C22×C73—21no—
96C22×C73—22no—
97C22×C73C22×D411yescharacteristic
98C2×D73—22no—
99C2×D73—22no—
100C2×D73—44no—
101C2×D73#13714yes—
102C4×C73—21no—
103C4×C73—42no—
104C4×C73—42no—
105C4×C73H(5,𝔽2)11yescharacteristic
106C4×C73H(5,𝔽2)11yescharacteristic
107Dic73—22no—
108Dic73—42no—
109Dic73—42no—
110C2×C73—41no—
111C2×C73#53411yescharacteristic
112C2×C73C2×#13712yes—
113D73—24no—
114#3195—731nomaximal, Sylow
115C73#319511yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—732no—
117#483—732no—
118#483—732no—
119#470—731no—
120#470—732no—
121#470D7311yescharacteristic, complemented
122#451—732no—
123#451—732no—
124#407—731no—
125C22×Q8—732no—
126C23×C4—731no—
127C2×Q16—732no—
128C2×Q16—731no—
129C2×Q16C2×D7311yescharacteristic, complemented
130C2×M16—731no—
131C2×M16C2×D7311yescharacteristic
132C22⋊Q8—732no—
133C22⋊Q8—732no—
134C22⋊Q8—732no—
135C22⋊Q8—1462no—
136C22⋊Q8—1462no—
137C22⋊Q8C2×D7312yes—
138C2×C4⋊C4—732no—
139C2×C4⋊C4—1462no—
140C2×C22⋊C4—1462no—
141#106—731no—
142#106—732no—
143#106C2×D7311yescharacteristic, complemented
144#103—732no—
145#103—732no—
146#103—732no—
147#103C2×D7312yescomplemented
148C22⋊C8—732no—
149C22⋊C8—732no—
150C24—731no—
151C2×Q8—2922no—
152C2×Q8—2922no—
153C2×Q8—732no—
154C2×Q8—1462no—
155C2×Q8C22×D7312yes—
156C22×C4—2921no—
157C22×C4—2921no—
158C22×C4—731no—
159C22×C4—732no—
160C22×C4—732no—
161C22×C4—1462no—
162C22×C4—1462no—
163C22×C4C22×D7311yescharacteristic
164Q16—41no—
165Q16—2921no—
166Q16—2922no—
167M16—41no—
168M16—2921no—
169C2×C8—732no—
170C2×C8C22×D7311yescharacteristic
171C2×C8C22×D7311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—732no—
174C4⋊C4—1462no—
175C4⋊C4—1462no—
176C4⋊C4—1462no—
177C4⋊C4—1462no—
178C4⋊C4C22×D7312yes—
179C22⋊C4—22no—
180C22⋊C4—1462no—
181C22⋊C4—1462no—
182C22⋊C4—1462no—
183C23—2921no—
184C23—732no—
185C23—1464no—
186C23D4×D7311yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—2924no—
190Q8—1462no—
191Q8—1462no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—2922no—
197C2×C4—2924no—
198C2×C4—2924no—
199C2×C4—2921no—
200C2×C4—732no—
201C2×C4—1462no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1462no—
205C2×C4D4×D7311yescharacteristic
206C2×C4C23×D7311yescharacteristic
207C8—21no—
208C8—21no—
209C8—1462no—
210C22—22no—
211C22—21no—
212C22—2924no—
213C22—734no—
214C22—1462no—
215C22—1462no—
216C22C2×D4×D7311yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—2922no—
221C4—2922no—
222C4—1462no—
223C4#1455663611yescharacteristic
224C4#1455663711yescharacteristic
225C2—41no—
226C2—1464no—
227C2#1798914511yescharacteristic
228C2#137×D7312yes—
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#470×C73C2
socleC22×C73C22×D4
hypercenter#470D73
hypocenterC73#3195
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #17988919C2semidirect
3. #17988916C2semidirect
4. #17988845C2semidirect
5. #17988843C2semidirect
6. #17988744C2semidirect
7. #17988741C2semidirect
9. #470×C73C2semidirect
12. C2×Dic292C22semidirect
13. C2×#11924018C22semidirect
23. #14556523C22semidirect
24. #14556522C22semidirect
25. #14556516C22semidirect
26. #14556515C22semidirect
27. #14556514C22semidirect
30. C2×Q16×C73C22semidirect
33. #106×C73C22semidirect
34. #103×C73C22semidirect
115. C73#3195semidirect
121. #470D73semidirect
129. C2×Q16C2×D73semidirect
143. #106C2×D73semidirect
147. #103C2×D73semidirect