Finite Group Databasebeta

← #22599059

Subgroups — #22599059

Order 9088.

402 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 129 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#17986402C211yesmaximal, characteristic, complemented
3#17986373C211yesmaximal, characteristic, complemented
4#17986349C211yesmaximal, characteristic, complemented
5#17986347C211yesmaximal, characteristic, complemented
6#17986247C214yesmaximal, complemented
7#17986097C214yesmaximal, complemented
8C4×#11765849C211yesmaximal, characteristic, complemented
9#533×C71C211yesmaximal, characteristic, complemented
10C2×#11765865—21no—
11C2×#11765852C2214yes—
12C2×#11765849C2212yes—
13#14556128C2211yescharacteristic
14#14556125C2211yescharacteristic
15#14556116—21no—
16#14556102—21no—
17#14556102C2212yes—
18C4×D284—21no—
19C4×D284C2212yes—
20#14556061C2214yes—
21#14556052C2214yes—
22#14556048C2214yes—
23#14556039C2212yes—
24C4×C71⋊C8C2211yescharacteristic
25C4×C71⋊C8C2211yescharacteristic
26C4⋊Q8×C71C2211yescharacteristic
27C4⋊Q8×C71C2211yescharacteristic
28#125×C71—21no—
29C22⋊Q8×C71—21no—
30C22⋊Q8×C71C2212yes—
31C4×Q8×C71—21no—
32C4×Q8×C71C2212yes—
33#117×C71—21no—
34C2×C42×C71C2211yescharacteristic
35#11765865—44no—
36C2×Dic142—21no—
37C2×D284—21no—
38C2×D284—21no—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C4×D71—22no—
42#11765852—28no—
43#11765849—24no—
44C2×C71⋊C8C2312yes—
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—21no—
47C4⋊Dic71C2312yes—
48(C2×D71)⋊C4—22no—
49(C2×D71)⋊C4—22no—
50C2×Q8×C71—21no—
51C2×Q8×C71C2312yes—
52C22×C4×C71D411yescharacteristic
53C22×C4×C71D411yescharacteristic
54C22×C4×C71C2311yescharacteristic
55C4⋊C4×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71—21no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71C2312yes—
61C22⋊C4×C71—22no—
62C42×C71—22no—
63C42×C71D411yescharacteristic
64C42×C71D411yescharacteristic
65C42×C71C2311yescharacteristic
66C42×C71C2311yescharacteristic
67C23×C71C2×D411yescharacteristic
68C22×D71—22no—
69Q8×C71—24no—
70Q8×C71—42no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C22⋊D71—44no—
83C2×Dic71—22no—
84D284—24no—
85D284—42no—
86D284—42no—
87C4×D71—44no—
88Dic142—42no—
89C71⋊C8—22no—
90C71⋊C8#4114yescomplemented
91C22×C71—21no—
92C22×C71—21no—
93C22×C71—21no—
94C22×C71C22×D411yescharacteristic
95C2×D71—42no—
96C2×D71—42no—
97C2×D71—42no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71AGL(1,ℤ/8)12yes—
104C4×C71#13712yes—
105C4×C71C2×#4112yes—
106Dic71—42no—
107C2×C71—41no—
108C2×C71#53211yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C71C2×#13711yescharacteristic
111D71—82no—
112#3182—711nomaximal, Sylow
113C71#318211yescharacteristic, complemented, Sylow
114#533D7111yescharacteristic, complemented
115#532—711no—
116#500—711no—
117#492—711no—
118#487—711no—
119#486—711no—
120#468—714no—
121#440—714no—
122C4×M16—711no—
123C2×#41—1421no—
124C2×SD16—714no—
125C2×M16—712no—
126C4⋊Q8—711no—
127C4⋊Q8C2×D7111yescharacteristic
128C4⋊Q8C2×D7111yescharacteristic
129#127—711no—
130#125—21no—
131#124—1421no—
132C22⋊Q8—21no—
133C22⋊Q8C2×D7112yes—
134#121—712no—
135#121—1421no—
136C4×Q8—21no—
137C4×Q8C2×D7112yes—
138C4×D4—712no—
139C4×D4—1421no—
140#117—21no—
141C2×C42C2×D7111yescharacteristic
142#107—714no—
143#103—714no—
144#102—714no—
145#97—712no—
146C4×C8—711no—
147C4×C8—711no—
148#41—2844no—
149C2×Q8—21no—
150C2×Q8—1421no—
151C2×Q8C22×D7112yes—
152C2×D4—712no—
153C2×D4—1422no—
154C2×D4—1421no—
155C2×D4—1421no—
156C22×C4—1422no—
157C22×C4C22⋊D7111yescharacteristic
158C22×C4C22⋊D7111yescharacteristic
159C22×C4C22×D7111yescharacteristic
160SD16—1428no—
161M16—1424no—
162C2×C8—712no—
163C2×C8—712no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—712no—
170C4⋊C4—1421no—
171C4⋊C4C22×D7112yes—
172C22⋊C4—22no—
173C22⋊C4—1422no—
174C22⋊C4—1422no—
175C42—22no—
176C42C22⋊D7111yescharacteristic
177C42C22⋊D7111yescharacteristic
178C42C22×D7111yescharacteristic
179C42C22×D7111yescharacteristic
180C23—1422no—
181C23C2×C22⋊D7111yescharacteristic
182Q8—24no—
183Q8—42no—
184Q8—2842no—
185D4—2844no—
186D4—2842no—
187D4—2842no—
188D4—1424no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—2844no—
195C2×C4—1422no—
196C2×C4C2×C22⋊D7111yescharacteristic
197C2×C4C2×C22⋊D7111yescharacteristic
198C2×C4C2×C22⋊D7111yescharacteristic
199C2×C4C2×C22⋊D7111yescharacteristic
200C2×C4C2×C22⋊D7111yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—714no—
203C8—1422no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—2842no—
208C22—2842no—
209C22—2842no—
210C22C22×C22⋊D7111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—2842no—
217C4#1455615312yes—
218C4#1455615912yes—
219C4#41×D7112yes—
220C2—41no—
221C2—5682no—
222C2#1798654511yescharacteristic
223C2C2×#1455615311yescharacteristic
224C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3182
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. #17986402C2semidirect
3. #17986373C2semidirect
4. #17986349C2semidirect
5. #17986347C2semidirect
6. #17986247C2semidirect
7. #17986097C2semidirect
8. C4×#11765849C2semidirect
9. #533×C71C2semidirect
90. C71⋊C8#41semidirect
113. C71#3182semidirect
114. #533D71semidirect