Finite Group Databasebeta

← #22599458

Subgroups — #22599458

Order 9088.

404 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556128C211yesmaximal, characteristic, complemented
2#17986402C212yesmaximal, complemented
3#17986345C212yesmaximal, complemented
4#17986310C214yesmaximal, complemented
5#17986279C214yesmaximal, complemented
6#17986052C211yesmaximal, characteristic, complemented
7#532×C71C211yesmaximal, characteristic
8C22×Dic142—21no—
9#14556128—21no—
10#14556128—21no—
11#14556128C2212yes—
12#14556128C2212yes—
13C2×C4⋊Dic71—21no—
14C2×C4⋊Dic71C2212yes—
15#14556061C2218yes—
16C4⋊(C71⋊C8)C2212yes—
17#14556049C2218yes—
18C22⋊(C71⋊C8)C2212yes—
19C4×C71⋊C8C2212yes—
20C2×#41×C71—21no—
21C4⋊Q8×C71C2211yescharacteristic
22#127×C71C2211yescharacteristic
23#124×C71—21no—
24#121×C71—21no—
25#121×C71C2212yes—
26C4×D4×C71—21no—
27C4×D4×C71C2212yes—
28C2×C42×C71C2211yescharacteristic
29C2×Dic142—22no—
30C2×Dic142—22no—
31C2×Dic142—44no—
32C22×Dic71—22no—
33C2×C71⋊C8C2314yes—
34C4⋊Dic71—22no—
35C4⋊Dic71—22no—
36C4⋊Dic71—22no—
37C4⋊Dic71C2314yes—
38#41×C71—44no—
39C2×Q8×C71—21no—
40C2×D4×C71—22no—
41C2×D4×C71—21no—
42C2×D4×C71—21no—
43C2×D4×C71C2312yes—
44C22×C4×C71—22no—
45C22×C4×C71D411yescharacteristic
46C22×C4×C71D411yescharacteristic
47C22×C4×C71C2311yescharacteristic
48C4⋊C4×C71—21no—
49C4⋊C4×C71C2312yes—
50C22⋊C4×C71—22no—
51C22⋊C4×C71—22no—
52C42×C71D411yescharacteristic
53C42×C71D411yescharacteristic
54C42×C71C2311yescharacteristic
55C42×C71C2311yescharacteristic
56C23×C71—22no—
57C23×C71C2×D411yescharacteristic
58Q8×C71—42no—
59D4×C71—24no—
60D4×C71—44no—
61D4×C71—42no—
62D4×C71—42no—
63C2×C4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—22no—
66C2×C4×C71—44no—
67C2×C4×C71D814yes—
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2411yescharacteristic
74C2×Dic71—24no—
75C2×Dic71—44no—
76Dic142—44no—
77Dic142—44no—
78C71⋊C8—24no—
79C22×C71—21no—
80C22×C71—21no—
81C22×C71—42no—
82C22×C71—42no—
83C22×C71—42no—
84C22×C71C2×D812yes—
85C22×C71C22×D411yescharacteristic
86C4×C71—21no—
87C4×C71—21no—
88C4×C71—42no—
89C4×C71C2×D812yes—
90C4×C71C2×D812yes—
91C4×C71#13712yes—
92C4×C712−1+412yes—
93Dic71—44no—
94C2×C71—22no—
95C2×C71—82no—
96C2×C71#53611yescharacteristic
97C2×C71C22×D811yescharacteristic
98C2×C71C2×#13711yescharacteristic
99#3226—711nomaximal, Sylow
100C71#322611yescharacteristic, complemented, Sylow
101#532D7111yescharacteristic
102C2×C4⋊Q8—711no—
103#500—712no—
104#486—712no—
105#480—714no—
106#474—714no—
107#431—711no—
108C2×#41—21no—
109C22×Q8—1421no—
110C4⋊Q8—712no—
111C4⋊Q8—712no—
112C4⋊Q8—1421no—
113C4⋊Q8—1421no—
114C4⋊Q8C2×D7111yescharacteristic
115#127C2×D7111yescharacteristic
116#124—21no—
117#121—21no—
118#121C2×D7112yes—
119C4×D4—21no—
120C4×D4C2×D7112yes—
121C2×C4⋊C4—712no—
122C2×C4⋊C4—1421no—
123C2×C42C2×D7111yescharacteristic
124#107—718no—
125C4⋊C8—712no—
126#102—718no—
127C22⋊C8—712no—
128C4×C8—712no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—2844no—
132C2×Q8—1422no—
133C2×Q8—1422no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C22×D7112yes—
138C22×C4—22no—
139C22×C4—1422no—
140C22×C4C22⋊D7111yescharacteristic
141C22×C4C22⋊D7111yescharacteristic
142C22×C4C22×D7111yescharacteristic
143C2×C8—714no—
144C4⋊C4—21no—
145C4⋊C4—714no—
146C4⋊C4—1422no—
147C4⋊C4—1422no—
148C4⋊C4—1422no—
149C4⋊C4C22×D7112yes—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C42C22⋊D7111yescharacteristic
153C42C22⋊D7111yescharacteristic
154C42C22×D7111yescharacteristic
155C42C22×D7111yescharacteristic
156C23—22no—
157C23C2×C22⋊D7111yescharacteristic
158Q8—42no—
159Q8—2844no—
160Q8—2844no—
161D4—24no—
162D4—44no—
163D4—42no—
164D4—42no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—44no—
169C2×C4—2844no—
170C2×C4—1424no—
171C2×C4#1176585014yes—
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—42no—
182C22—42no—
183C22—42no—
184C22C2×#1176585012yes—
185C22C22×C22⋊D7111yescharacteristic
186C4—21no—
187C4—21no—
188C4—42no—
189C4—2844no—
190C4C2×#1176585012yes—
191C4C2×#1176585012yes—
192C4#1455615812yes—
193C4#1455617512yes—
194C2—22no—
195C2—82no—
196C2#1798657011yescharacteristic
197C2C22×#1176585011yescharacteristic
198C2C2×#1455615811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#532×C71C2
socleC22×C71C22×D4
hypercenter#532D71
hypocenterC71#3226
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. C2×#14556128C2semidirect
2. #17986402C2semidirect
3. #17986345C2semidirect
4. #17986310C2semidirect
5. #17986279C2semidirect
6. #17986052C2semidirect
100. C71#3226semidirect