Finite Group Databasebeta

← #22599495

Subgroups — #22599495

Order 9088.

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

normal subgroups only

This group has 326 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
1#17986556C211yesmaximal, characteristic
2#17986402C211yesmaximal, characteristic, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986349C211yesmaximal, characteristic, complemented
5#17986349C211yesmaximal, characteristic, complemented
6#17986337C214yesmaximal
7#17986306C214yesmaximal, complemented
8#17986052C211yesmaximal, characteristic
9#533×C71C211yesmaximal, characteristic
10#14556128C2211yescharacteristic
11#14556128C2211yescharacteristic
12#14556120C2212yes—
13#14556107—21no—
14#14556107—21no—
15C4×Dic142—21no—
16C4×Dic142—21no—
17#14556087C2212yes—
18#14556061C2214yes—
19#14556059C2214yes—
20C4⋊(C71⋊C8)C2212yes—
21#14556052C2214yes—
22#14556052C2214yes—
23C22⋊(C71⋊C8)C2212yes—
24C4×C71⋊C8C2211yescharacteristic
25C4×C71⋊C8C2211yescharacteristic
26C4⋊Q8×C71C2211yescharacteristic
27C4⋊Q8×C71C2211yescharacteristic
28#125×C71—21no—
29C22⋊Q8×C71—21no—
30C22⋊Q8×C71C2212yes—
31C4×Q8×C71—21no—
32C4×Q8×C71C2212yes—
33#117×C71—21no—
34C2×C42×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142—21no—
37C2×C71⋊C8C2312yes—
38C2×C71⋊C8C2312yes—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41C4⋊Dic71C2312yes—
42C4⋊Dic71C2312yes—
43Dic71⋊C4—22no—
44Dic71⋊C4—22no—
45C22⋊Dic71—22no—
46C4×Dic71—22no—
47C2×Q8×C71—21no—
48C2×Q8×C71C2312yes—
49C22×C4×C71D411yescharacteristic
50C22×C4×C71D411yescharacteristic
51C22×C4×C71C2311yescharacteristic
52C4⋊C4×C71—21no—
53C4⋊C4×C71—21no—
54C4⋊C4×C71—21no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71C2312yes—
58C22⋊C4×C71—22no—
59C42×C71—22no—
60C42×C71D411yescharacteristic
61C42×C71D411yescharacteristic
62C42×C71C2311yescharacteristic
63C42×C71C2311yescharacteristic
64C23×C71C2×D411yescharacteristic
65Q8×C71—24no—
66Q8×C71—42no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
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
78C2×Dic71—22no—
79C2×Dic71—22no—
80Dic142—42no—
81Dic142—42no—
82C71⋊C8—22no—
83C71⋊C8—22no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71C22×D411yescharacteristic
88C4×C71—21no—
89C4×C71—21no—
90C4×C71—21no—
91C4×C71—42no—
92C4×C71—42no—
93C4×C71#13512yes—
94C4×C71#13712yes—
95C4×C712−1+412yes—
96Dic71—42no—
97Dic71—42no—
98C2×C71—41no—
99C2×C71#53611yescharacteristic
100C2×C71C2×#13511yescharacteristic
101C2×C71C2×#13711yescharacteristic
102#3231—711nomaximal, Sylow
103C71#323111yescharacteristic, complemented, Sylow
104#533—711no—
105#533D7111yescharacteristic
106#500—711no—
107#498—711no—
108#487—711no—
109#487—711no—
110#485—714no—
111#479—714no—
112#431—711no—
113C4⋊Q8—711no—
114C4⋊Q8—711no—
115C4⋊Q8C2×D7111yescharacteristic
116C4⋊Q8C2×D7111yescharacteristic
117#125—21no—
118#125—712no—
119C22⋊Q8—21no—
120C22⋊Q8—1421no—
121C22⋊Q8—1421no—
122C22⋊Q8C2×D7112yes—
123C4×Q8—21no—
124C4×Q8—1421no—
125C4×Q8—1421no—
126C4×Q8C2×D7112yes—
127#117—21no—
128#117—712no—
129C2×C42C2×D7111yescharacteristic
130#107—714no—
131#106—714no—
132C4⋊C8—712no—
133#103—714no—
134#103—714no—
135C22⋊C8—712no—
136C4×C8—711no—
137C4×C8—711no—
138C2×Q8—21no—
139C2×Q8—1421no—
140C2×Q8—1421no—
141C2×Q8C22×D7112yes—
142C22×C4C22⋊D7111yescharacteristic
143C22×C4C22⋊D7111yescharacteristic
144C22×C4C22×D7111yescharacteristic
145C2×C8—712no—
146C2×C8—712no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—712no—
153C4⋊C4—712no—
154C4⋊C4—1422no—
155C4⋊C4—1422no—
156C4⋊C4—1421no—
157C4⋊C4—1421no—
158C4⋊C4C22×D7112yes—
159C22⋊C4—22no—
160C22⋊C4—1422no—
161C42—22no—
162C42—1422no—
163C42C22⋊D7111yescharacteristic
164C42C22⋊D7111yescharacteristic
165C42C22×D7111yescharacteristic
166C42C22×D7111yescharacteristic
167C23C2×C22⋊D7111yescharacteristic
168Q8—24no—
169Q8—42no—
170Q8—2842no—
171Q8—2842no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—1422no—
178C2×C4—1422no—
179C2×C4C2×C22⋊D7111yescharacteristic
180C2×C4C2×C22⋊D7111yescharacteristic
181C2×C4C2×C22⋊D7111yescharacteristic
182C2×C4C2×C22⋊D7111yescharacteristic
183C2×C4C2×C22⋊D7111yescharacteristic
184C2×C4C23×D7111yescharacteristic
185C8—1422no—
186C8—1422no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22C22×C22⋊D7111yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—2842no—
197C4—2842no—
198C4#1455614812yes—
199C4#1455615912yes—
200C4#1455617512yes—
201C2—41no—
202C2#1798657011yescharacteristic
203C2C2×#1455614811yescharacteristic
204C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3231
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. #17986402C2semidirect
3. #17986395C2semidirect
4. #17986349C2semidirect
5. #17986349C2semidirect
7. #17986306C2semidirect
103. C71#3231semidirect