Finite Group Databasebeta

← #22599369

Subgroups — #22599369

Order 9088.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556128C211yesmaximal, characteristic, complemented
2#17986533C211yesmaximal, characteristic
3#17986393C212yesmaximal, complemented
4#17986368C212yesmaximal, complemented
5#17986210C212yesmaximal, complemented
6#17986205C212yesmaximal, complemented
7#17986157C212yesmaximal, complemented
8#17986140C212yesmaximal
9#432×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2212yes—
11C2×Dic284C2214yes—
12C2×#11765854C2214yes—
13#14556128—21no—
14#14556128—21no—
15#14556128C2212yes—
16#14556128C2212yes—
17#14556122—21no—
18#14556120C2211yescharacteristic
19#14556116C2211yescharacteristic
20#14556111—21no—
21#14556107C2211yescharacteristic
22#14556102C2211yescharacteristic
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25C4×C22⋊D71—21no—
26#14556085—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29#14556053C2214yes—
30#14556053C2214yes—
31C2×C42×C71C2211yescharacteristic
32C4⋊C8×C71C2212yes—
33C22⋊C8×C71C2212yes—
34#97×C71C2212yes—
35C2×Dic142—22no—
36C2×Dic142—44no—
37C2×Dic142C2314yes—
38C2×Dic142C2311yescharacteristic
39C2×D284C2311yescharacteristic
40C2×C22⋊D71—21no—
41C22×Dic71—22no—
42C2×C4×D71—21no—
43Dic284—44no—
44#11765854—44no—
45C4⋊Dic71—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49C4⋊Dic71C2311yescharacteristic
50C4⋊Dic71C2311yescharacteristic
51Dic71⋊C4—21no—
52Dic71⋊C4—21no—
53Dic71⋊C4—21no—
54Dic71⋊C4—21no—
55C22⋊Dic71—21no—
56(C2×D71)⋊C4—21no—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—21no—
59(C2×D71)⋊C4—21no—
60C4×Dic71—21no—
61C22×C4×C71D411yescharacteristic
62C22×C4×C71D411yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C2×C8×C71C2314yes—
65C42×C71D411yescharacteristic
66C42×C71D411yescharacteristic
67C42×C71C2311yescharacteristic
68C42×C71C2311yescharacteristic
69C23×C71C2×D411yescharacteristic
70C22×D71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2411yescharacteristic
80C22⋊D71—42no—
81C2×Dic71—21no—
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71—24no—
85C2×Dic71—44no—
86D284—22no—
87C4×D71—42no—
88Dic142—24no—
89Dic142—22no—
90Dic142—24no—
91Dic142—44no—
92C8×C71—24no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71#13712yes—
96C22×C71C22×D411yescharacteristic
97C2×D71—41no—
98C2×D71—41no—
99C2×D71—41no—
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—21no—
104C4×C71#13712yes—
105C4×C71H(5,𝔽2)12yes—
106Dic71—44no—
107Dic71—41no—
108Dic71—41no—
109Dic71—41no—
110C2×C71—22no—
111C2×C71#53511yescharacteristic
112C2×C71C2×#13711yescharacteristic
113C2×C71C2×#13711yescharacteristic
114D71—81no—
115#3218—711nomaximal, Sylow
116C71#321811yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—711no—
118#529—711no—
119#497—712no—
120#491—712no—
121#462—712no—
122#461—712no—
123#451—712no—
124#448—712no—
125#432D7111yescharacteristic, complemented
126C22×Q8—712no—
127C2×Q16—714no—
128C2×SD16—714no—
129C4⋊Q8—712no—
130C4⋊Q8—712no—
131C4⋊Q8—1421no—
132C4⋊Q8—1421no—
133#126—1421no—
134#125—711no—
135#124—711no—
136#123—1421no—
137C22⋊Q8—711no—
138#121—711no—
139C4×Q8—711no—
140C4×D4—711no—
141C4×D4—1421no—
142#117—1421no—
143C2×C4⋊C4—1421no—
144C2×C4⋊C4—1421no—
145C2×C42C2×D7111yescharacteristic
146C4⋊C8C2×D7112yes—
147#103—714no—
148#103—714no—
149C22⋊C8C2×D7112yes—
150#97C2×D7112yes—
151C2×Q8—2844no—
152C2×Q8—714no—
153C2×Q8—711no—
154C2×Q8—1422no—
155C2×D4—711no—
156C2×D4—1421no—
157C22×C4—1422no—
158C22×C4—1421no—
159C22×C4D28411yescharacteristic
160C22×C4D28411yescharacteristic
161C22×C4C22×D7111yescharacteristic
162Q16—2844no—
163SD16—2844no—
164C2×C8C22×D7114yes—
165C4⋊C4—711no—
166C4⋊C4—711no—
167C4⋊C4—1421no—
168C4⋊C4—1421no—
169C4⋊C4—1421no—
170C4⋊C4—1421no—
171C4⋊C4—1422no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C22⋊C4—1421no—
176C22⋊C4—1421no—
177C22⋊C4—1421no—
178C22⋊C4—1421no—
179C22⋊C4—1421no—
180C42—1421no—
181C42D28411yescharacteristic
182C42D28411yescharacteristic
183C42C22×D7111yescharacteristic
184C42C22×D7111yescharacteristic
185C23—1421no—
186C23C2×D28411yescharacteristic
187Q8—2844no—
188Q8—1424no—
189Q8—1422no—
190Q8—1424no—
191D4—2842no—
192D4—1422no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—2844no—
197C2×C4—2842no—
198C2×C4—1421no—
199C2×C4—1421no—
200C2×C4—1421no—
201C2×C4—1424no—
202C2×C4C2×D28411yescharacteristic
203C2×C4C2×D28411yescharacteristic
204C2×C4C2×D28411yescharacteristic
205C2×C4C2×D28411yescharacteristic
206C2×C4C2×D28411yescharacteristic
207C2×C4C23×D7111yescharacteristic
208C8—24no—
209C22—21no—
210C22—21no—
211C22—2841no—
212C22—2841no—
213C22—2841no—
214C22#1455616112yes—
215C22C22×D28411yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—2844no—
221C4—2841no—
222C4—2841no—
223C4—2841no—
224C4#1455616112yes—
225C4#1455617412yes—
226C2—22no—
227C2—5681no—
228C2#1798656811yescharacteristic
229C2C2×#1455616111yescharacteristic
230C2C2×#1455616111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3218
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#14556128C2semidirect
3. #17986393C2semidirect
4. #17986368C2semidirect
5. #17986210C2semidirect
6. #17986205C2semidirect
7. #17986157C2semidirect
9. #432×C71C2semidirect
116. C71#3218semidirect
125. #432D71semidirect