Finite Group Databasebeta

← #22599347

Subgroups — #22599347

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#17986549C211yesmaximal, characteristic
2#17986391C212yesmaximal, complemented
3#17986361C212yesmaximal, complemented
4#17986209C212yesmaximal, complemented
5#17986199C212yesmaximal, complemented
6#17986159C214yesmaximal, complemented
7#17986056C211yesmaximal, characteristic, complemented
8#533×C71C211yesmaximal, characteristic, complemented
9C2×#11765865C2212yes—
10C2×#11765855C2214yes—
11C2×#11765852C2214yes—
12#14556128C2211yescharacteristic
13#14556125C2211yescharacteristic
14#14556116C2212yes—
15#14556102—21no—
16#14556102—21no—
17C4×D284—21no—
18C4×D284—21no—
19C4⋊(C71⋊C8)C2212yes—
20#14556051C2214yes—
21#14556048C2214yes—
22C22⋊(C71⋊C8)C2212yes—
23#14556039C2212yes—
24C4⋊Q8×C71C2211yescharacteristic
25C4⋊Q8×C71C2211yescharacteristic
26#125×C71—21no—
27C22⋊Q8×C71—21no—
28C22⋊Q8×C71C2212yes—
29C4×Q8×C71—21no—
30C4×Q8×C71C2212yes—
31#117×C71—21no—
32C2×C42×C71C2211yescharacteristic
33#11765865—44no—
34C2×Dic142C2312yes—
35C2×D284—21no—
36C2×D284—21no—
37C2×D284C2312yes—
38C2×C22⋊D71—22no—
39C2×C4×D71—22no—
40#11765855—44no—
41#11765852—44no—
42C2×C71⋊C8C2314yes—
43C4⋊Dic71—21no—
44C4⋊Dic71—21no—
45(C2×D71)⋊C4—22no—
46(C2×D71)⋊C4—22no—
47C2×Q8×C71—21no—
48C2×Q8×C71C2312yes—
49C22×C4×C71D411yescharacteristic
50C22×C4×C71D411yescharacteristic
51C22×C4×C71C2311yescharacteristic
52C4⋊C4×C71—21no—
53C4⋊C4×C71—21no—
54C4⋊C4×C71—21no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71C2312yes—
58C22⋊C4×C71—22no—
59C42×C71—22no—
60C42×C71D411yescharacteristic
61C42×C71D411yescharacteristic
62C42×C71C2311yescharacteristic
63C42×C71C2311yescharacteristic
64C23×C71C2×D411yescharacteristic
65C22×D71—22no—
66Q8×C71—22no—
67Q8×C71—22no—
68Q8×C71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—41no—
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—44no—
81C2×Dic71—22no—
82D284—24no—
83D284—42no—
84D284—42no—
85C4×D71—44no—
86Dic142—24no—
87C71⋊C8—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—42no—
100C4×C71—42no—
101C4×C71#13712yes—
102C4×C71H(5,𝔽2)12yes—
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#533D7111yescharacteristic, complemented
112#532—711no—
113#497—712no—
114#489—712no—
115#462—712no—
116#460—712no—
117#452—714no—
118#432—711no—
119C2×#41—712no—
120C2×Q16—714no—
121C2×SD16—714no—
122C4⋊Q8—711no—
123C4⋊Q8C2×D7111yescharacteristic
124C4⋊Q8C2×D7111yescharacteristic
125#127—711no—
126#125—21no—
127#124—712no—
128C22⋊Q8—21no—
129C22⋊Q8C2×D7112yes—
130#121—1421no—
131#121—1421no—
132C4×Q8—21no—
133C4×Q8C2×D7112yes—
134C4×D4—1421no—
135C4×D4—1421no—
136#117—21no—
137C2×C42C2×D7111yescharacteristic
138C4⋊C8—712no—
139#103—714no—
140#102—714no—
141C22⋊C8—712no—
142#97—712no—
143#41—2844no—
144C2×Q8—21no—
145C2×Q8—712no—
146C2×Q8C22×D7112yes—
147C2×D4—712no—
148C2×D4—1422no—
149C2×D4—1421no—
150C2×D4—1421no—
151C22×C4—1422no—
152C22×C4C22⋊D7111yescharacteristic
153C22×C4C22⋊D7111yescharacteristic
154C22×C4C22×D7111yescharacteristic
155Q16—2844no—
156SD16—2844no—
157C2×C8—714no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—1421no—
164C4⋊C4—1421no—
165C4⋊C4C22×D7112yes—
166C22⋊C4—22no—
167C22⋊C4—1422no—
168C22⋊C4—1422no—
169C42—22no—
170C42C22⋊D7111yescharacteristic
171C42C22⋊D7111yescharacteristic
172C42C22×D7111yescharacteristic
173C42C22×D7111yescharacteristic
174C23—1422no—
175C23C2×C22⋊D7111yescharacteristic
176Q8—22no—
177Q8—22no—
178Q8—42no—
179Q8—1424no—
180D4—2842no—
181D4—2844no—
182D4—2842no—
183D4—1424no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—2844no—
190C2×C4—1422no—
191C2×C4C2×C22⋊D7111yescharacteristic
192C2×C4C2×C22⋊D7111yescharacteristic
193C2×C4C2×C22⋊D7111yescharacteristic
194C2×C4C2×C22⋊D7111yescharacteristic
195C2×C4C2×C22⋊D7111yescharacteristic
196C2×C4C23×D7111yescharacteristic
197C8—1424no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—2842no—
202C22—2842no—
203C22—2842no—
204C22C22×C22⋊D7111yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—2842no—
212C4#1455615912yes—
213C4#1455617412yes—
214C2—41no—
215C2—5682no—
216C2#1798656311yescharacteristic
217C2C2×#1455615911yescharacteristic
218C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3216
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #17986391C2semidirect
3. #17986361C2semidirect
4. #17986209C2semidirect
5. #17986199C2semidirect
6. #17986159C2semidirect
7. #17986056C2semidirect
8. #533×C71C2semidirect
110. C71#3216semidirect
111. #533D71semidirect