Finite Group Databasebeta

← #22587429

Subgroups — #22587429

Order 9088.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#17986347C211yesmaximal, characteristic, complemented
2#17986194C211yesmaximal, characteristic, complemented
3#17986194C211yesmaximal, characteristic, complemented
4#17985745C211yesmaximal, characteristic, complemented
5D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
7#547×C71C211yesmaximal, characteristic, complemented
8C2×#11765850—21no—
9C2×#11765850—21no—
10#14556125C2211yescharacteristic
11C4⋊(C71⋊C8)C2211yescharacteristic
12C4⋊(C71⋊C8)C2211yescharacteristic
13#14556048—22no—
14#14556048—21no—
15#14556048—21no—
16C4×C71⋊C8C2211yescharacteristic
17C2×#41×C71—21no—
18C4⋊Q8×C71C2211yescharacteristic
19#123×C71—22no—
20C22⋊Q8×C71—21no—
21#121×C71—21no—
22C4×D4×C71C2211yescharacteristic
23C4×D4×C71C2211yescharacteristic
24C2×C4⋊C4×C71—21no—
25C2×D284—21no—
26C2×D284—21no—
27C2×D284—21no—
28#11765850—42no—
29#11765850—42no—
30C2×C71⋊C8—21no—
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33#41×C71—42no—
34#41×C71—42no—
35C2×Q8×C71—21no—
36C2×D4×C71—22no—
37C2×D4×C71D411yescharacteristic
38C2×D4×C71D411yescharacteristic
39C22×C4×C71—22no—
40C22×C4×C71—21no—
41C22×C4×C71—21no—
42C4⋊C4×C71—22no—
43C4⋊C4×C71—21no—
44C4⋊C4×C71D411yescharacteristic
45C4⋊C4×C71D411yescharacteristic
46C4⋊C4×C71D412yes—
47C22⋊C4×C71—22no—
48C22⋊C4×C71—21no—
49C22⋊C4×C71—21no—
50C42×C71C2311yescharacteristic
51C23×C71—21no—
52C23×C71—21no—
53C22×D71—41no—
54Q8×C71—41no—
55Q8×C71—41no—
56D4×C71—21no—
57D4×C71—21no—
58D4×C71—21no—
59D4×C71—44no—
60D4×C71D812yescomplemented
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—22no—
64C2×C4×C71—44no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71—41no—
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72D284—42no—
73D284—41no—
74D284—41no—
75D284—41no—
76D284—41no—
77C71⋊C8—41no—
78C71⋊C8—41no—
79C71⋊C8—41no—
80C22×C71—22no—
81C22×C71—41no—
82C22×C71—41no—
83C22×C71—41no—
84C22×C71—41no—
85C22×C71—41no—
86C22×C71#12011yescharacteristic
87C2×D71—81no—
88C2×D71—81no—
89C2×D71—81no—
90C4×C71—21no—
91C4×C71—41no—
92C4×C71—42no—
93C4×C71—41no—
94C4×C71C2×D811yescharacteristic
95C4×C71#13511yescharacteristic
96C4×C71AGL(1,ℤ/8)11yescharacteristic
97C4×C71AGL(1,ℤ/8)11yescharacteristic
98C2×C71—42no—
99C2×C71—81no—
100C2×C71#44711yescharacteristic
101C2×C71#44911yescharacteristic
102C2×C71#45411yescharacteristic
103D71—161no—
104#1625—711nomaximal, Sylow
105C71#162511yescharacteristic, complemented, Sylow
106#547D7111yescharacteristic, complemented
107#486—711no—
108#459—711no—
109#459—711no—
110#332—711no—
111D4⋊C8—711no—
112D4⋊C8—711no—
113C2×#41—21no—
114C2×D8—1421no—
115C2×D8—1421no—
116C4⋊Q8C2×D7111yescharacteristic
117#127—711no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D7111yescharacteristic
122C4×D4C2×D7111yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—711no—
125C4⋊C8—711no—
126#102—1421no—
127#102—1421no—
128#102—1422no—
129C4×C8—711no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—1421no—
135C2×D4—1421no—
136C2×D4—1421no—
137C2×D4C22⋊D7111yescharacteristic
138C2×D4C22⋊D7111yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—2842no—
143D8—2842no—
144C2×C8—1421no—
145C2×C8—1421no—
146C2×C8—1421no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D7111yescharacteristic
150C4⋊C4C22⋊D7111yescharacteristic
151C4⋊C4C22⋊D7112yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D7111yescharacteristic
156C23—21no—
157C23—21no—
158C23—2841no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—2842no—
166D4—2841no—
167D4—2841no—
168D4—2841no—
169D4—2841no—
170D4#1176585012yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D7111yescharacteristic
180C2×C4C2×C22⋊D7111yescharacteristic
181C2×C4C2×C22⋊D7111yescharacteristic
182C8—2841no—
183C8—2841no—
184C8—2841no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—5681no—
192C22—5681no—
193C22—5681no—
194C22#1455609711yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1176585011yescharacteristic
200C4#1455614811yescharacteristic
201C4#1455615311yescharacteristic
202C4#1455615411yescharacteristic
203C2—42no—
204C2—81no—
205C2—11361no—
206C2#1798613311yescharacteristic
207C2#1798614411yescharacteristic
208C2#1798617211yescharacteristic
subgroupG/H
centerC22#14556097
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#547×C71C2
socleC22×C71#120
hypercenter#547D71
hypocenterC71#1625
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #17986347C2semidirect
2. #17986194C2semidirect
3. #17986194C2semidirect
4. #17985745C2semidirect
5. D4⋊(C71⋊C8)C2semidirect
6. D4⋊(C71⋊C8)C2semidirect
7. #547×C71C2semidirect
60. D4×C71D8semidirect
105. C71#1625semidirect
106. #547D71semidirect
170. D4#11765850semidirect