Finite Group Databasebeta

← #22599476

Subgroups — #22599476

Order 9088.

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

normal subgroups only

This group has 370 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
1#17986549C211yesmaximal, characteristic
2#17986402C211yesmaximal, characteristic, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986347C211yesmaximal, characteristic, complemented
5#17986347C211yesmaximal, characteristic, complemented
6#17986323C214yesmaximal, complemented
7#17986296C214yesmaximal, complemented
8#17986052C211yesmaximal, characteristic, complemented
9#533×C71C211yesmaximal, characteristic, complemented
10C2×#11765865—21no—
11#14556128C2211yescharacteristic
12#14556125C2211yescharacteristic
13#14556116—21no—
14#14556102—21no—
15#14556102C2212yes—
16C4×D284—21no—
17C4×D284C2212yes—
18#14556061C2214yes—
19#14556059C2214yes—
20C4⋊(C71⋊C8)C2212yes—
21#14556048C2214yes—
22#14556048C2214yes—
23C22⋊(C71⋊C8)C2212yes—
24C4×C71⋊C8C2211yescharacteristic
25C4×C71⋊C8C2211yescharacteristic
26C4⋊Q8×C71C2211yescharacteristic
27C4⋊Q8×C71C2211yescharacteristic
28#125×C71C2212yes—
29C22⋊Q8×C71—21no—
30C22⋊Q8×C71—21no—
31C4×Q8×C71—21no—
32C4×Q8×C71—21no—
33#117×C71C2212yes—
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—
42C2×C71⋊C8C2312yes—
43C2×C71⋊C8C2312yes—
44C4⋊Dic71—21no—
45C4⋊Dic71C2312yes—
46(C2×D71)⋊C4—22no—
47(C2×D71)⋊C4—22no—
48C2×Q8×C71—21no—
49C2×Q8×C71—21no—
50C22×C4×C71D411yescharacteristic
51C22×C4×C71D411yescharacteristic
52C22×C4×C71C2311yescharacteristic
53C4⋊C4×C71—22no—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71C2312yes—
58C4⋊C4×C71C2312yes—
59C22⋊C4×C71—22no—
60C42×C71—22no—
61C42×C71D411yescharacteristic
62C42×C71D411yescharacteristic
63C42×C71C2311yescharacteristic
64C42×C71C2311yescharacteristic
65C23×C71C2×D411yescharacteristic
66C22×D71—22no—
67Q8×C71—42no—
68Q8×C71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
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—44no—
81C2×Dic71—22no—
82D284—24no—
83D284—42no—
84D284—42no—
85C4×D71—44no—
86Dic142—42no—
87C71⋊C8—22no—
88C71⋊C8—22no—
89C22×C71—21no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71C22×D411yescharacteristic
93C2×D71—42no—
94C2×D71—42no—
95C2×D71—42no—
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—42no—
100C4×C71—42no—
101C4×C71#13512yes—
102C4×C71AGL(1,ℤ/8)12yes—
103C4×C712−1+412yes—
104Dic71—42no—
105C2×C71—41no—
106C2×C71#53611yescharacteristic
107C2×C71C2×#13511yescharacteristic
108C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
109D71—82no—
110#3228—711nomaximal, Sylow
111C71#322811yescharacteristic, complemented, Sylow
112#533D7111yescharacteristic, complemented
113#532—711no—
114#500—711no—
115#498—711no—
116#486—711no—
117#486—711no—
118#482—714no—
119#478—714no—
120#431—711no—
121C2×#41—1421no—
122C4⋊Q8—711no—
123C4⋊Q8C2×D7111yescharacteristic
124C4⋊Q8C2×D7111yescharacteristic
125#127—711no—
126#125C2×D7112yes—
127#124—1421no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130#121—712no—
131#121—1421no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×D4—712no—
135C4×D4—1421no—
136#117C2×D7112yes—
137C2×C42C2×D7111yescharacteristic
138#107—714no—
139#106—714no—
140C4⋊C8—712no—
141#102—714no—
142#102—714no—
143C22⋊C8—712no—
144C4×C8—711no—
145C4×C8—711no—
146#41—2844no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1421no—
150C2×D4—712no—
151C2×D4—1422no—
152C2×D4—1421no—
153C2×D4—1421no—
154C22×C4—1422no—
155C22×C4C22⋊D7111yescharacteristic
156C22×C4C22⋊D7111yescharacteristic
157C22×C4C22×D7111yescharacteristic
158C2×C8—712no—
159C2×C8—712no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—712no—
165C4⋊C4—1421no—
166C4⋊C4C22×D7112yes—
167C4⋊C4C22×D7112yes—
168C22⋊C4—22no—
169C22⋊C4—1422no—
170C22⋊C4—1422no—
171C42—22no—
172C42C22⋊D7111yescharacteristic
173C42C22⋊D7111yescharacteristic
174C42C22×D7111yescharacteristic
175C42C22×D7111yescharacteristic
176C23—1422no—
177C23C2×C22⋊D7111yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—2842no—
181D4—2844no—
182D4—2842no—
183D4—2842no—
184D4—1424no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—2844no—
191C2×C4—1422no—
192C2×C4C2×C22⋊D7111yescharacteristic
193C2×C4C2×C22⋊D7111yescharacteristic
194C2×C4C2×C22⋊D7111yescharacteristic
195C2×C4C2×C22⋊D7111yescharacteristic
196C2×C4C2×C22⋊D7111yescharacteristic
197C2×C4C23×D7111yescharacteristic
198C8—1422no—
199C8—1422no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—2842no—
204C22—2842no—
205C22—2842no—
206C22C22×C22⋊D7111yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—2842no—
213C4#1455614812yes—
214C4#1455615312yes—
215C4#1455617612yes—
216C2—41no—
217C2—5682no—
218C2#1798657311yescharacteristic
219C2C2×#1455614811yescharacteristic
220C2C2×#1455615311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3228
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. #17986395C2semidirect
4. #17986347C2semidirect
5. #17986347C2semidirect
6. #17986323C2semidirect
7. #17986296C2semidirect
8. #17986052C2semidirect
9. #533×C71C2semidirect
111. C71#3228semidirect
112. #533D71semidirect