Finite Group Databasebeta

← #22599353

Subgroups — #22599353

Order 9088.

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

normal subgroups only

This group has 390 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
1#17986556C211yesmaximal, characteristic, complemented
2#17986549C211yesmaximal, characteristic
3#17986393C212yesmaximal, complemented
4#17986359C212yesmaximal, complemented
5#17986210C212yesmaximal, complemented
6#17986200C212yesmaximal, complemented
7#17986163C214yesmaximal, complemented
8#432×C71C211yesmaximal, characteristic, complemented
9C2×#11765865C2212yes—
10C2×Dic284C2214yes—
11C2×#11765854C2214yes—
12#14556128C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556128C2211yescharacteristic
15#14556125C2211yescharacteristic
16#14556120—21no—
17#14556116C2212yes—
18#14556107—21no—
19#14556107C2212yes—
20#14556102—21no—
21#14556102—21no—
22C4×Dic142—21no—
23C4×Dic142C2212yes—
24C4×D284—21no—
25C4×D284—21no—
26#14556087—21no—
27#14556053C2214yes—
28#14556050C2214yes—
29C2×C42×C71C2211yescharacteristic
30C4⋊C8×C71C2212yes—
31C22⋊C8×C71C2212yes—
32#97×C71C2212yes—
33#11765865—44no—
34C2×Dic142—21no—
35C2×Dic142C2312yes—
36C2×Dic142C2312yes—
37C2×D284—21no—
38C2×D284—21no—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C4×D71—22no—
42Dic284—44no—
43#11765854—44no—
44C4⋊Dic71—21no—
45C4⋊Dic71—21no—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71—21no—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52C22⋊Dic71—22no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55C4×Dic71—22no—
56C22×C4×C71D411yescharacteristic
57C22×C4×C71D411yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C2×C8×C71C2314yes—
60C42×C71D411yescharacteristic
61C42×C71D411yescharacteristic
62C42×C71C2311yescharacteristic
63C42×C71C2311yescharacteristic
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—41no—
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2411yescharacteristic
75C22⋊D71—44no—
76C2×Dic71—22no—
77C2×Dic71—22no—
78C2×Dic71—22no—
79D284—24no—
80D284—42no—
81D284—42no—
82C4×D71—44no—
83Dic142—22no—
84Dic142—22no—
85Dic142—24no—
86Dic142—42no—
87C8×C71—24no—
88C22×C71—21no—
89C22×C71—21no—
90C22×C71—21no—
91C22×C71C22×D411yescharacteristic
92C2×D71—42no—
93C2×D71—42no—
94C2×D71—42no—
95C4×C71—21no—
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71#13712yes—
100C4×C71H(5,𝔽2)12yes—
101Dic71—42no—
102Dic71—42no—
103Dic71—42no—
104C2×C71—41no—
105C2×C71#53511yescharacteristic
106C2×C71C2×#13711yescharacteristic
107C2×C71#57511yescharacteristic
108D71—82no—
109#3216—711nomaximal, Sylow
110C71#321611yescharacteristic, complemented, Sylow
111#533—711no—
112#532—711no—
113#497—712no—
114#489—712no—
115#462—712no—
116#460—712no—
117#452—714no—
118#432D7111yescharacteristic, complemented
119C2×#41—712no—
120C2×Q16—714no—
121C2×SD16—714no—
122C4⋊Q8—711no—
123C4⋊Q8—711no—
124C4⋊Q8—711no—
125#127—711no—
126#125—1421no—
127#124—712no—
128C22⋊Q8—712no—
129C22⋊Q8—1421no—
130#121—1421no—
131#121—1421no—
132C4×Q8—712no—
133C4×Q8—1421no—
134C4×D4—1421no—
135C4×D4—1421no—
136#117—1421no—
137C2×C42C2×D7111yescharacteristic
138C4⋊C8C2×D7112yes—
139#103—714no—
140#102—714no—
141C22⋊C8C2×D7112yes—
142#97C2×D7112yes—
143#41—2844no—
144C2×Q8—712no—
145C2×Q8—712no—
146C2×Q8—1421no—
147C2×D4—712no—
148C2×D4—1422no—
149C2×D4—1421no—
150C2×D4—1421no—
151C22×C4—1422no—
152C22×C4D28411yescharacteristic
153C22×C4D28411yescharacteristic
154C22×C4C22×D7111yescharacteristic
155Q16—2844no—
156SD16—2844no—
157C2×C8C22×D7114yes—
158C4⋊C4—712no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1421no—
162C4⋊C4—1421no—
163C4⋊C4—1421no—
164C4⋊C4—1421no—
165C4⋊C4—1421no—
166C22⋊C4—1422no—
167C22⋊C4—1422no—
168C22⋊C4—1422no—
169C42—1422no—
170C42D28411yescharacteristic
171C42D28411yescharacteristic
172C42C22×D7111yescharacteristic
173C42C22×D7111yescharacteristic
174C23—1422no—
175C23C2×D28411yescharacteristic
176Q8—2842no—
177Q8—1422no—
178Q8—1422no—
179Q8—1424no—
180D4—2842no—
181D4—2844no—
182D4—2842no—
183D4—1424no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—2844no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4C2×D28411yescharacteristic
192C2×C4C2×D28411yescharacteristic
193C2×C4C2×D28411yescharacteristic
194C2×C4C2×D28411yescharacteristic
195C2×C4C2×D28411yescharacteristic
196C2×C4C23×D7111yescharacteristic
197C8—24no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—2842no—
202C22—2842no—
203C22—2842no—
204C22C22×D28411yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—2842no—
210C4—2842no—
211C4—2842no—
212C4#1455616112yes—
213C4#1455617412yes—
214C2—41no—
215C2—5682no—
216C2#1798656811yescharacteristic
217C2C2×#1455616111yescharacteristic
218C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3216
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. #17986556C2semidirect
3. #17986393C2semidirect
4. #17986359C2semidirect
5. #17986210C2semidirect
6. #17986200C2semidirect
7. #17986163C2semidirect
8. #432×C71C2semidirect
110. C71#3216semidirect
118. #432D71semidirect