Finite Group Databasebeta

← #22599542

Subgroups — #22599542

Order 9088.

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

normal subgroups only

This group has 324 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
1C2×#14556120C211yesmaximal, characteristic
2#17986405C212yesmaximal, complemented
3#17986367C212yesmaximal, complemented
4#17986332C212yesmaximal
5#17986327C212yesmaximal
6#17986306C214yesmaximal, complemented
7#17986056C211yesmaximal, characteristic
8#533×C71C211yesmaximal, characteristic
9#14556120—21no—
10#14556120—21no—
11#14556120C2212yes—
12#14556120C2212yes—
13C2×C4⋊Dic71C2211yescharacteristic
14C2×C4⋊Dic71C2211yescharacteristic
15C2×Dic71⋊C4—21no—
16C2×Dic71⋊C4—21no—
17#14556061C2214yes—
18#14556059C2214yes—
19C4⋊(C71⋊C8)C2212yes—
20#14556052C2214yes—
21#14556052C2214yes—
22C22⋊(C71⋊C8)C2212yes—
23#14556039C2212yes—
24C4⋊Q8×C71C2211yescharacteristic
25C4⋊Q8×C71C2211yescharacteristic
26#125×C71—21no—
27C22⋊Q8×C71—21no—
28C22⋊Q8×C71C2212yes—
29C4×Q8×C71—21no—
30C4×Q8×C71C2212yes—
31#117×C71—21no—
32C2×C42×C71C2211yescharacteristic
33C22×Dic71—21no—
34C22×Dic71—21no—
35C2×C71⋊C8C2314yes—
36C4⋊Dic71—21no—
37C4⋊Dic71—21no—
38C4⋊Dic71C2312yes—
39C4⋊Dic71C2312yes—
40Dic71⋊C4—22no—
41Dic71⋊C4—22no—
42Dic71⋊C4—22no—
43Dic71⋊C4—22no—
44C2×Q8×C71—21no—
45C2×Q8×C71C2312yes—
46C22×C4×C71D411yescharacteristic
47C22×C4×C71D411yescharacteristic
48C22×C4×C71C2311yescharacteristic
49C4⋊C4×C71—21no—
50C4⋊C4×C71—21no—
51C4⋊C4×C71—21no—
52C4⋊C4×C71—22no—
53C4⋊C4×C71—22no—
54C4⋊C4×C71C2312yes—
55C22⋊C4×C71—22no—
56C42×C71—22no—
57C42×C71D411yescharacteristic
58C42×C71D411yescharacteristic
59C42×C71C2311yescharacteristic
60C42×C71C2311yescharacteristic
61C23×C71C2×D411yescharacteristic
62Q8×C71—24no—
63Q8×C71—42no—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—41no—
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2411yescharacteristic
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71—42no—
78C2×Dic71—42no—
79C71⋊C8—24no—
80C22×C71—21no—
81C22×C71—21no—
82C22×C71#13712yes—
83C22×C71C22×D411yescharacteristic
84C4×C71—21no—
85C4×C71—21no—
86C4×C71—21no—
87C4×C71—21no—
88C4×C71—21no—
89C4×C71—42no—
90C4×C71—42no—
91C4×C712−1+412yes—
92Dic71—42no—
93Dic71—42no—
94C2×C71—22no—
95C2×C71#53611yescharacteristic
96C2×C71C2×#13711yescharacteristic
97C2×C71#57511yescharacteristic
98#3236—711nomaximal, Sylow
99C71#323611yescharacteristic, complemented, Sylow
100#533D7111yescharacteristic
101C2×#125—711no—
102#501—712no—
103#491—712no—
104#484—712no—
105#483—712no—
106#479—714no—
107#432—711no—
108C4⋊Q8C2×D7111yescharacteristic
109C4⋊Q8C2×D7111yescharacteristic
110#125—21no—
111#125—712no—
112#125—712no—
113#125—1421no—
114#125—1421no—
115C22⋊Q8—21no—
116C22⋊Q8C2×D7112yes—
117C4×Q8—21no—
118C4×Q8C2×D7112yes—
119#117—21no—
120C2×C4⋊C4—711no—
121C2×C4⋊C4—711no—
122C2×C4⋊C4—1421no—
123C2×C4⋊C4—1421no—
124C2×C42C2×D7111yescharacteristic
125#107—714no—
126#106—714no—
127C4⋊C8—712no—
128#103—714no—
129#103—714no—
130C22⋊C8—712no—
131#97—712no—
132C2×Q8—21no—
133C2×Q8C22×D7112yes—
134C22×C4—1421no—
135C22×C4—1421no—
136C22×C4C22⋊D7111yescharacteristic
137C22×C4C22⋊D7111yescharacteristic
138C22×C4C22×D7111yescharacteristic
139C2×C8—714no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—712no—
146C4⋊C4—712no—
147C4⋊C4—1422no—
148C4⋊C4—1422no—
149C4⋊C4—1422no—
150C4⋊C4—1422no—
151C4⋊C4—1421no—
152C4⋊C4—1421no—
153C4⋊C4C22×D7112yes—
154C22⋊C4—22no—
155C42—22no—
156C42C22⋊D7111yescharacteristic
157C42C22⋊D7111yescharacteristic
158C42C22×D7111yescharacteristic
159C42C22×D7111yescharacteristic
160C23C2×C22⋊D7111yescharacteristic
161Q8—24no—
162Q8—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—2842no—
169C2×C4—2842no—
170C2×C4—1422no—
171C2×C4—1422no—
172C2×C4C2×C22⋊D7111yescharacteristic
173C2×C4C2×C22⋊D7111yescharacteristic
174C2×C4C2×C22⋊D7111yescharacteristic
175C2×C4C2×C22⋊D7111yescharacteristic
176C2×C4C2×C22⋊D7111yescharacteristic
177C2×C4C23×D7111yescharacteristic
178C8—1424no—
179C22—21no—
180C22—21no—
181C22#1455615912yes—
182C22C22×C22⋊D7111yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—42no—
189C4—42no—
190C4—2842no—
191C4—2842no—
192C4#1455617512yes—
193C2—22no—
194C2#1798657011yescharacteristic
195C2C2×#1455615911yescharacteristic
196C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3236
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. #17986405C2semidirect
3. #17986367C2semidirect
6. #17986306C2semidirect
99. C71#3236semidirect