Finite Group Databasebeta

← #22599775

Subgroups — #22599775

Order 9088.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986604C211yesmaximal, characteristic, complemented
2#17986299C212yesmaximal, complemented
3#17986296C212yesmaximal, complemented
4#17986250C212yesmaximal, complemented
5#17986233C212yesmaximal, complemented
6#17986194C212yesmaximal, complemented
7#17986191C212yesmaximal, complemented
8#17986026C211yesmaximal, characteristic, complemented
9#540×C71C211yesmaximal, characteristic, complemented
10C2×#11765850C2212yes—
11C2×#11765850C2212yes—
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8C2211yescharacteristic
14#14556125C2211yescharacteristic
15#14556120C2211yescharacteristic
16#14556113—22no—
17#14556102C2212yes—
18#14556101—22no—
19C4×D284C2212yes—
20#14556061C2212yes—
21#14556059C2212yes—
22C4⋊(C71⋊C8)C2212yes—
23C4⋊(C71⋊C8)C2212yes—
24#14556049C2212yes—
25#14556049C2212yes—
26#14556048C2212yes—
27#14556048C2212yes—
28#127×C71C2211yescharacteristic
29#125×C71C2211yescharacteristic
30#123×C71—22no—
31#121×C71—22no—
32#121×C71C2212yes—
33C4×D4×C71C2212yes—
34#117×C71C2211yescharacteristic
35C2×D284—22no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C4×D71—22no—
39#11765850—42no—
40#11765850—42no—
41#11765849—41no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8C2312yes—
44C2×C71⋊C8C2312yes—
45C4⋊Dic71C2312yes—
46Dic71⋊C4—22no—
47(C2×D71)⋊C4—22no—
48(C2×D71)⋊C4—22no—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71C2312yes—
52C22×C4×C71—22no—
53C22×C4×C71C2311yescharacteristic
54C4⋊C4×C71—22no—
55C4⋊C4×C71D412yes—
56C4⋊C4×C71C2312yes—
57C22⋊C4×C71—22no—
58C22⋊C4×C71—22no—
59C22⋊C4×C71D412yes—
60C42×C71C2311yescharacteristic
61C42×C71C2311yescharacteristic
62C23×C71—22no—
63C23×C71C2×D411yescharacteristic
64C22×D71—22no—
65D4×C71—22no—
66D4×C71—22no—
67D4×C71—82no—
68D4×C71—82no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—82no—
73C2×C4×C71C2×D412yes—
74C2×C4×C71C2×D412yes—
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2411yescharacteristic
77C22⋊D71—82no—
78C2×Dic71—22no—
79D284—22no—
80D284—22no—
81D284—82no—
82C4×D71—82no—
83C71⋊C8—22no—
84C71⋊C8—22no—
85C22×C71—21no—
86C22×C71—22no—
87C22×C71—42no—
88C22×C71—42no—
89C22×C71—42no—
90C22×C71C22×D411yescharacteristic
91C2×D71—42no—
92C2×D71—42no—
93C2×D71—42no—
94C4×C71—21no—
95C4×C71—22no—
96C4×C71—22no—
97C4×C71—42no—
98C4×C71H(5,𝔽2)11yescharacteristic
99C4×C712−1+411yescharacteristic
100Dic71—42no—
101C2×C71—41no—
102C2×C71—82no—
103C2×C71#53711yescharacteristic
104C2×C71#57612yes—
105D71—82no—
106#3259—711nomaximal, Sylow
107C71#325911yescharacteristic, complemented, Sylow
108#540—711no—
109#540D7111yescharacteristic, complemented
110#478—712no—
111#478—712no—
112#469—712no—
113#466—712no—
114#459—712no—
115#459—712no—
116#424—711no—
117C2×D8—712no—
118C2×D8—712no—
119C2×M16—711no—
120C22×C8—711no—
121#127—711no—
122#127C2×D7111yescharacteristic
123#125—711no—
124#125C2×D7111yescharacteristic
125#123—22no—
126#123—1422no—
127#121—22no—
128#121—712no—
129#121—1422no—
130#121C2×D7112yes—
131C4×D4—712no—
132C4×D4C2×D7112yes—
133#117C2×D7111yescharacteristic
134#107—712no—
135#106—712no—
136C4⋊C8—712no—
137C4⋊C8—712no—
138#102—712no—
139#102—712no—
140#102—712no—
141#102—712no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—712no—
145C2×D4—1422no—
146C2×D4—1422no—
147C2×D4C22×D7112yes—
148C22×C4—22no—
149C22×C4—1422no—
150C22×C4C22×D7111yescharacteristic
151D8—2842no—
152D8—2842no—
153M16—2841no—
154C2×C8—2841no—
155C2×C8—712no—
156C2×C8—712no—
157C4⋊C4—22no—
158C4⋊C4—712no—
159C4⋊C4—1422no—
160C4⋊C4C22⋊D7112yes—
161C4⋊C4C22×D7112yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—1422no—
165C22⋊C4—1422no—
166C22⋊C4C22⋊D7112yes—
167C42C22×D7111yescharacteristic
168C42C22×D7111yescharacteristic
169C23—22no—
170C23—1422no—
171C23C2×C22⋊D7111yescharacteristic
172D4—22no—
173D4—22no—
174D4—82no—
175D4—82no—
176D4—5682no—
177D4—5682no—
178D4—1422no—
179D4—1422no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—5682no—
185C2×C4—1422no—
186C2×C4C2×C22⋊D7112yes—
187C2×C4C2×C22⋊D7112yes—
188C2×C4C2×C22⋊D7111yescharacteristic
189C2×C4C23×D7111yescharacteristic
190C8—1422no—
191C8—1422no—
192C22—21no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—2842no—
198C22—2842no—
199C22—2842no—
200C22C22×C22⋊D7111yescharacteristic
201C4—21no—
202C4—22no—
203C4—22no—
204C4—42no—
205C4—2842no—
206C4#1455617411yescharacteristic
207C4#1455617511yescharacteristic
208C2—41no—
209C2—82no—
210C2—5682no—
211C2#1798657711yescharacteristic
212C2#1798680212yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#540×C71C2
socleC22×C71C22×D4
hypercenter#540D71
hypocenterC71#3259
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #17986604C2semidirect
2. #17986299C2semidirect
3. #17986296C2semidirect
4. #17986250C2semidirect
5. #17986233C2semidirect
6. #17986194C2semidirect
7. #17986191C2semidirect
8. #17986026C2semidirect
9. #540×C71C2semidirect
107. C71#3259semidirect
109. #540D71semidirect