Finite Group Databasebeta

← #22599563

Subgroups — #22599563

Order 9088.

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

normal subgroups only

This group has 362 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#17986408C212yesmaximal, complemented
4#17986359C212yesmaximal, complemented
5#17986330C212yesmaximal, complemented
6#17986313C212yesmaximal
7#17986295C212yesmaximal, complemented
8#17986290C212yesmaximal, complemented
9#432×C71C211yesmaximal, characteristic, complemented
10C22×Dic142—21no—
11#14556128—21no—
12#14556128—21no—
13#14556128C2212yes—
14#14556128C2212yes—
15#14556122—21no—
16#14556120C2211yescharacteristic
17#14556116C2211yescharacteristic
18#14556111—21no—
19#14556107C2211yescharacteristic
20#14556102C2211yescharacteristic
21C4×Dic142C2211yescharacteristic
22C4×D284C2211yescharacteristic
23C4×C22⋊D71—21no—
24#14556085—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2212yes—
27#14556062C2214yes—
28#14556060C2214yes—
29#14556053C2214yes—
30#14556050C2214yes—
31C2×C42×C71C2211yescharacteristic
32C4⋊C8×C71C2212yes—
33C22⋊C8×C71C2212yes—
34#97×C71C2212yes—
35C2×Dic142—22no—
36C2×Dic142—22no—
37C2×Dic142—44no—
38C2×Dic142C2311yescharacteristic
39C2×D284C2311yescharacteristic
40C2×C22⋊D71—21no—
41C22×Dic71—22no—
42C2×C4×D71—21no—
43C4⋊Dic71—22no—
44C4⋊Dic71—22no—
45C4⋊Dic71—22no—
46C4⋊Dic71C2314yes—
47C4⋊Dic71C2311yescharacteristic
48C4⋊Dic71C2311yescharacteristic
49Dic71⋊C4—21no—
50Dic71⋊C4—21no—
51Dic71⋊C4—21no—
52Dic71⋊C4—21no—
53C22⋊Dic71—21no—
54(C2×D71)⋊C4—21no—
55(C2×D71)⋊C4—21no—
56(C2×D71)⋊C4—21no—
57(C2×D71)⋊C4—21no—
58C4×Dic71—21no—
59C22×C4×C71D411yescharacteristic
60C22×C4×C71D411yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C2×C8×C71C2314yes—
63C42×C71D411yescharacteristic
64C42×C71D411yescharacteristic
65C42×C71C2311yescharacteristic
66C42×C71C2311yescharacteristic
67C23×C71C2×D411yescharacteristic
68C22×D71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2411yescharacteristic
78C22⋊D71—42no—
79C2×Dic71—21no—
80C2×Dic71—21no—
81C2×Dic71—21no—
82C2×Dic71—24no—
83C2×Dic71—44no—
84D284—22no—
85C4×D71—42no—
86Dic142—22no—
87Dic142—44no—
88Dic142—44no—
89C8×C71—24no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71AGL(1,ℤ/8)12yes—
93C22×C71C22×D411yescharacteristic
94C2×D71—41no—
95C2×D71—41no—
96C2×D71—41no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—21no—
100C4×C71—21no—
101C4×C71#13712yes—
102C4×C712−1+412yes—
103Dic71—41no—
104Dic71—44no—
105Dic71—41no—
106Dic71—41no—
107C2×C71—22no—
108C2×C71#53611yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C71C2×#13711yescharacteristic
111D71—81no—
112#3238—711nomaximal, Sylow
113C71#323811yescharacteristic, complemented, Sylow
114C2×C4⋊Q8—711no—
115#529—711no—
116#501—712no—
117#489—712no—
118#483—712no—
119#480—712no—
120#477—712no—
121#476—712no—
122#432D7111yescharacteristic, complemented
123C22×Q8—1421no—
124C4⋊Q8—712no—
125C4⋊Q8—712no—
126C4⋊Q8—1421no—
127C4⋊Q8—1421no—
128#126—1421no—
129#125—711no—
130#124—711no—
131#123—1421no—
132C22⋊Q8—711no—
133#121—711no—
134C4×Q8—711no—
135C4×D4—711no—
136C4×D4—1421no—
137#117—1421no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—1421no—
140C2×C42C2×D7111yescharacteristic
141#107—714no—
142#106—714no—
143C4⋊C8C2×D7112yes—
144#103—714no—
145#102—714no—
146C22⋊C8C2×D7112yes—
147#97C2×D7112yes—
148C2×Q8—2844no—
149C2×Q8—711no—
150C2×Q8—1422no—
151C2×Q8—1422no—
152C2×D4—711no—
153C2×D4—1421no—
154C22×C4—1422no—
155C22×C4—1421no—
156C22×C4D28411yescharacteristic
157C22×C4D28411yescharacteristic
158C22×C4C22×D7111yescharacteristic
159C2×C8C22×D7114yes—
160C4⋊C4—714no—
161C4⋊C4—711no—
162C4⋊C4—711no—
163C4⋊C4—1421no—
164C4⋊C4—1421no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C22⋊C4—1421no—
171C22⋊C4—1421no—
172C22⋊C4—1421no—
173C22⋊C4—1421no—
174C22⋊C4—1421no—
175C42—1421no—
176C42D28411yescharacteristic
177C42D28411yescharacteristic
178C42C22×D7111yescharacteristic
179C42C22×D7111yescharacteristic
180C23—1421no—
181C23C2×D28411yescharacteristic
182Q8—2844no—
183Q8—2844no—
184Q8—1422no—
185D4—2842no—
186D4—1422no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—2844no—
191C2×C4—2842no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4—1421no—
195C2×C4—1424no—
196C2×C4C2×D28411yescharacteristic
197C2×C4C2×D28411yescharacteristic
198C2×C4C2×D28411yescharacteristic
199C2×C4C2×D28411yescharacteristic
200C2×C4C2×D28411yescharacteristic
201C2×C4C23×D7111yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—2841no—
206C22—2841no—
207C22—2841no—
208C22#1455615612yes—
209C22C22×D28411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—2841no—
215C4—2844no—
216C4—2841no—
217C4—2841no—
218C4#1455616112yes—
219C4#1455617512yes—
220C2—22no—
221C2—5681no—
222C2#1798657511yescharacteristic
223C2C2×#1455615611yescharacteristic
224C2C2×#1455616111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#432×C71C2
socleC22×C71C22×D4
hypercenter#432D71
hypocenterC71#3238
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. #17986408C2semidirect
4. #17986359C2semidirect
5. #17986330C2semidirect
7. #17986295C2semidirect
8. #17986290C2semidirect
9. #432×C71C2semidirect
113. C71#3238semidirect
122. #432D71semidirect