Finite Group Databasebeta

← #22599472

Subgroups — #22599472

Order 9088.

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

normal subgroups only

This group has 370 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, complemented
2#17986402C211yesmaximal, characteristic, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986345C211yesmaximal, characteristic, complemented
5#17986345C211yesmaximal, characteristic, complemented
6#17986320C214yesmaximal, complemented
7#17986299C214yesmaximal, complemented
8#17986052C211yesmaximal, characteristic, complemented
9#532×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#14556049C2214yes—
22#14556049C2214yes—
23C22⋊(C71⋊C8)C2212yes—
24C4×C71⋊C8C2211yescharacteristic
25C4×C71⋊C8C2211yescharacteristic
26C2×#41×C71—21no—
27C4⋊Q8×C71C2211yescharacteristic
28#127×C71C2211yescharacteristic
29#124×C71—21no—
30#121×C71—21no—
31#121×C71C2212yes—
32C4×D4×C71—21no—
33C4×D4×C71C2212yes—
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—
47#41×C71—44no—
48C2×Q8×C71—21no—
49C2×D4×C71—22no—
50C2×D4×C71—21no—
51C2×D4×C71—21no—
52C2×D4×C71C2312yes—
53C22×C4×C71—22no—
54C22×C4×C71D411yescharacteristic
55C22×C4×C71D411yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C4⋊C4×C71—21no—
58C4⋊C4×C71C2312yes—
59C22⋊C4×C71—22no—
60C22⋊C4×C71—22no—
61C42×C71D411yescharacteristic
62C42×C71D411yescharacteristic
63C42×C71C2311yescharacteristic
64C42×C71C2311yescharacteristic
65C23×C71—22no—
66C23×C71C2×D411yescharacteristic
67Q8×C71—42no—
68D4×C71—24no—
69D4×C71—44no—
70D4×C71—42no—
71D4×C71—42no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—44no—
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2411yescharacteristic
83C2×Dic71—22no—
84C2×Dic71—22no—
85Dic142—42no—
86Dic142—42no—
87C71⋊C8—22no—
88C71⋊C8—22no—
89C22×C71—21no—
90C22×C71—21no—
91C22×C71—21no—
92C22×C71—42no—
93C22×C71—42no—
94C22×C71—42no—
95C22×C71C22×D411yescharacteristic
96C4×C71—21no—
97C4×C71—21no—
98C4×C71—21no—
99C4×C71—42no—
100C4×C71#13512yes—
101C4×C71AGL(1,ℤ/8)12yes—
102C4×C712−1+412yes—
103Dic71—42no—
104Dic71—42no—
105C2×C71—41no—
106C2×C71—82no—
107C2×C71#53611yescharacteristic
108C2×C71C2×#13511yescharacteristic
109C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
110#3228—711nomaximal, Sylow
111C71#322811yescharacteristic, complemented, Sylow
112#533—711no—
113#532D7111yescharacteristic
114#500—711no—
115#498—711no—
116#486—711no—
117#486—711no—
118#482—714no—
119#478—714no—
120#431—711no—
121C2×#41—21no—
122C4⋊Q8—711no—
123C4⋊Q8—711no—
124C4⋊Q8C2×D7111yescharacteristic
125#127C2×D7111yescharacteristic
126#125—712no—
127#124—21no—
128C22⋊Q8—1421no—
129C22⋊Q8—1421no—
130#121—21no—
131#121C2×D7112yes—
132C4×Q8—1421no—
133C4×Q8—1421no—
134C4×D4—21no—
135C4×D4C2×D7112yes—
136#117—712no—
137C2×C42C2×D7111yescharacteristic
138#107—714no—
139#106—714no—
140C4⋊C8—712no—
141#102—714no—
142#102—714no—
143C22⋊C8—712no—
144C4×C8—711no—
145C4×C8—711no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—1421no—
149C2×Q8—1421no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D7112yes—
154C22×C4—22no—
155C22×C4C22⋊D7111yescharacteristic
156C22×C4C22⋊D7111yescharacteristic
157C22×C4C22×D7111yescharacteristic
158C2×C8—712no—
159C2×C8—712no—
160C4⋊C4—21no—
161C4⋊C4—712no—
162C4⋊C4—712no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4—1421no—
166C4⋊C4—1421no—
167C4⋊C4C22×D7112yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—1422no—
171C42—1422no—
172C42C22⋊D7111yescharacteristic
173C42C22⋊D7111yescharacteristic
174C42C22×D7111yescharacteristic
175C42C22×D7111yescharacteristic
176C23—22no—
177C23C2×C22⋊D7111yescharacteristic
178Q8—42no—
179Q8—2842no—
180Q8—2842no—
181D4—24no—
182D4—44no—
183D4—42no—
184D4—42no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—1422no—
191C2×C4—1422no—
192C2×C4C2×C22⋊D7111yescharacteristic
193C2×C4C2×C22⋊D7111yescharacteristic
194C2×C4C2×C22⋊D7111yescharacteristic
195C2×C4C2×C22⋊D7111yescharacteristic
196C2×C4C2×C22⋊D7111yescharacteristic
197C2×C4C23×D7111yescharacteristic
198C8—1422no—
199C8—1422no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C22×C22⋊D7111yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—2842no—
212C4—2842no—
213C4#1455614812yes—
214C4#1455615412yes—
215C4#1455617512yes—
216C2—41no—
217C2—82no—
218C2#1798657011yescharacteristic
219C2C2×#1455614811yescharacteristic
220C2C2×#1455615411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#532×C71C2
socleC22×C71C22×D4
hypercenter#532D71
hypocenterC71#3228
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #17986556C2semidirect
2. #17986402C2semidirect
3. #17986395C2semidirect
4. #17986345C2semidirect
5. #17986345C2semidirect
6. #17986320C2semidirect
7. #17986299C2semidirect
8. #17986052C2semidirect
111. C71#3228semidirect