Finite Group Databasebeta

← #22599538

Subgroups — #22599538

Order 9088.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986409C212yesmaximal, complemented
3#17986365C212yesmaximal, complemented
4#17986323C214yesmaximal, complemented
5#17986286C212yesmaximal, complemented
6#17986276C212yesmaximal, complemented
7#17986056C211yesmaximal, characteristic, complemented
8#533×C71C211yesmaximal, characteristic, complemented
9C2×#11765865—21no—
10#14556128C2211yescharacteristic
11#14556125C2211yescharacteristic
12#14556116—21no—
13#14556102—21no—
14#14556102C2212yes—
15C4×D284—21no—
16C4×D284C2212yes—
17#14556061C2214yes—
18#14556059C2214yes—
19C4⋊(C71⋊C8)C2212yes—
20#14556048C2214yes—
21#14556048C2214yes—
22C22⋊(C71⋊C8)C2212yes—
23#14556039C2212yes—
24C4⋊Q8×C71C2211yescharacteristic
25C4⋊Q8×C71C2211yescharacteristic
26#125×C71C2212yes—
27C22⋊Q8×C71—21no—
28C22⋊Q8×C71—21no—
29C4×Q8×C71—21no—
30C4×Q8×C71—21no—
31#117×C71C2212yes—
32C2×C42×C71C2211yescharacteristic
33#11765865—44no—
34C2×Dic142—21no—
35C2×D284—21no—
36C2×D284—21no—
37C2×D284C2312yes—
38C2×C22⋊D71—22no—
39C2×C4×D71—22no—
40C2×C71⋊C8C2314yes—
41C4⋊Dic71—21no—
42C4⋊Dic71C2312yes—
43(C2×D71)⋊C4—22no—
44(C2×D71)⋊C4—22no—
45C2×Q8×C71—21no—
46C2×Q8×C71—21no—
47C22×C4×C71D411yescharacteristic
48C22×C4×C71D411yescharacteristic
49C22×C4×C71C2311yescharacteristic
50C4⋊C4×C71—22no—
51C4⋊C4×C71—22no—
52C4⋊C4×C71—21no—
53C4⋊C4×C71—21no—
54C4⋊C4×C71C2312yes—
55C4⋊C4×C71C2312yes—
56C22⋊C4×C71—22no—
57C42×C71—22no—
58C42×C71D411yescharacteristic
59C42×C71D411yescharacteristic
60C42×C71C2311yescharacteristic
61C42×C71C2311yescharacteristic
62C23×C71C2×D411yescharacteristic
63C22×D71—22no—
64Q8×C71—42no—
65Q8×C71—42no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—22no—
70C2×C4×C71—41no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2411yescharacteristic
77C22⋊D71—44no—
78C2×Dic71—22no—
79D284—24no—
80D284—42no—
81D284—42no—
82C4×D71—44no—
83Dic142—42no—
84C71⋊C8—24no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71—21no—
88C22×C71C22×D411yescharacteristic
89C2×D71—42no—
90C2×D71—42no—
91C2×D71—42no—
92C4×C71—21no—
93C4×C71—21no—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—42no—
97C4×C71—42no—
98C4×C71AGL(1,ℤ/8)12yes—
99C4×C712−1+412yes—
100Dic71—42no—
101C2×C71—41no—
102C2×C71#53611yescharacteristic
103C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C71#57511yescharacteristic
105D71—82no—
106#3235—711nomaximal, Sylow
107C71#323511yescharacteristic, complemented, Sylow
108#533D7111yescharacteristic, complemented
109#532—711no—
110#501—712no—
111#490—712no—
112#482—714no—
113#476—712no—
114#474—712no—
115#432—711no—
116C2×#41—1421no—
117C4⋊Q8—711no—
118C4⋊Q8C2×D7111yescharacteristic
119C4⋊Q8C2×D7111yescharacteristic
120#127—711no—
121#125C2×D7112yes—
122#124—1421no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125#121—712no—
126#121—1421no—
127C4×Q8—21no—
128C4×Q8—21no—
129C4×D4—712no—
130C4×D4—1421no—
131#117C2×D7112yes—
132C2×C42C2×D7111yescharacteristic
133#107—714no—
134#106—714no—
135C4⋊C8—712no—
136#102—714no—
137#102—714no—
138C22⋊C8—712no—
139#97—712no—
140#41—2844no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—1421no—
144C2×D4—712no—
145C2×D4—1422no—
146C2×D4—1421no—
147C2×D4—1421no—
148C22×C4—1422no—
149C22×C4C22⋊D7111yescharacteristic
150C22×C4C22⋊D7111yescharacteristic
151C22×C4C22×D7111yescharacteristic
152C2×C8—714no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—712no—
158C4⋊C4—1421no—
159C4⋊C4C22×D7112yes—
160C4⋊C4C22×D7112yes—
161C22⋊C4—22no—
162C22⋊C4—1422no—
163C22⋊C4—1422no—
164C42—22no—
165C42C22⋊D7111yescharacteristic
166C42C22⋊D7111yescharacteristic
167C42C22×D7111yescharacteristic
168C42C22×D7111yescharacteristic
169C23—1422no—
170C23C2×C22⋊D7111yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—2842no—
174D4—2842no—
175D4—2842no—
176D4—2844no—
177D4—1424no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—2844no—
184C2×C4—1422no—
185C2×C4C2×C22⋊D7111yescharacteristic
186C2×C4C2×C22⋊D7111yescharacteristic
187C2×C4C2×C22⋊D7111yescharacteristic
188C2×C4C2×C22⋊D7111yescharacteristic
189C2×C4C2×C22⋊D7111yescharacteristic
190C2×C4C23×D7111yescharacteristic
191C8—1424no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—2842no—
196C22—2842no—
197C22—2842no—
198C22C22×C22⋊D7111yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—42no—
205C4—2842no—
206C4#1455615312yes—
207C4#1455617612yes—
208C2—41no—
209C2—5682no—
210C2#1798657311yescharacteristic
211C2C2×#1455615311yescharacteristic
212C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3235
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. #17986409C2semidirect
3. #17986365C2semidirect
4. #17986323C2semidirect
5. #17986286C2semidirect
6. #17986276C2semidirect
7. #17986056C2semidirect
8. #533×C71C2semidirect
107. C71#3235semidirect
108. #533D71semidirect