Finite Group Databasebeta

← #22599484

Subgroups — #22599484

Order 9088.

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

normal subgroups only

This group has 360 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
1#17986556C211yesmaximal, characteristic
2#17986395C212yesmaximal, complemented
3#17986351C212yesmaximal, complemented
4#17986329C214yesmaximal
5#17986281C214yesmaximal, complemented
6#17986052C211yesmaximal, characteristic
7C2×C4⋊Q8×C71C211yesmaximal, characteristic
8#14556128C2211yescharacteristic
9#14556128C2211yescharacteristic
10#14556120—21no—
11#14556107—21no—
12#14556107C2212yes—
13C4×Dic142—21no—
14C4×Dic142C2212yes—
15#14556087—21no—
16#14556059C2218yes—
17C4⋊(C71⋊C8)C2212yes—
18#14556051C2218yes—
19C22⋊(C71⋊C8)C2212yes—
20C4×C71⋊C8C2212yes—
21C22×Q8×C71—21no—
22C4⋊Q8×C71—21no—
23C4⋊Q8×C71—21no—
24C4⋊Q8×C71C2212yes—
25C4⋊Q8×C71C2212yes—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71C2212yes—
28C2×C42×C71C2211yescharacteristic
29C2×Dic142—21no—
30C2×Dic142C2312yes—
31C2×C71⋊C8C2314yes—
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71C2312yes—
36Dic71⋊C4—22no—
37Dic71⋊C4—22no—
38C22⋊Dic71—22no—
39C4×Dic71—22no—
40C2×Q8×C71—22no—
41C2×Q8×C71—22no—
42C2×Q8×C71—44no—
43C22×C4×C71—22no—
44C22×C4×C71D411yescharacteristic
45C22×C4×C71D411yescharacteristic
46C22×C4×C71C2311yescharacteristic
47C4⋊C4×C71—22no—
48C4⋊C4×C71—22no—
49C4⋊C4×C71—22no—
50C4⋊C4×C71C2314yes—
51C42×C71D411yescharacteristic
52C42×C71D411yescharacteristic
53C42×C71C2311yescharacteristic
54C42×C71C2311yescharacteristic
55C23×C71C2×D411yescharacteristic
56Q8×C71—44no—
57Q8×C71—44no—
58C2×C4×C71—22no—
59C2×C4×C71—22no—
60C2×C4×C71—24no—
61C2×C4×C71—44no—
62C2×C4×C71SD1614yes—
63C2×C4×C71C2×D411yescharacteristic
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2411yescharacteristic
69C2×Dic71—22no—
70C2×Dic71—22no—
71Dic142—24no—
72Dic142—42no—
73C71⋊C8—24no—
74C22×C71—21no—
75C22×C71—21no—
76C22×C71C2×SD1612yes—
77C22×C71C22×D411yescharacteristic
78C4×C71—21no—
79C4×C71—21no—
80C4×C71—44no—
81C4×C71C2×SD1612yes—
82C4×C71C2×SD1612yes—
83C4×C71#13712yes—
84C4×C712−1+412yes—
85Dic71—42no—
86Dic71—42no—
87C2×C71—22no—
88C2×C71#53611yescharacteristic
89C2×C71C22×SD1611yescharacteristic
90C2×C71C2×#13711yescharacteristic
91#3229—711nomaximal, Sylow
92C71#322911yescharacteristic, complemented, Sylow
93#533—711no—
94C2×C4⋊Q8D7111yescharacteristic
95#498—712no—
96#487—712no—
97#483—714no—
98#475—714no—
99#431—711no—
100C22×Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—711no—
104C4⋊Q8—711no—
105C4⋊Q8C2×D7112yes—
106C4⋊Q8C2×D7112yes—
107#125—1421no—
108C22⋊Q8—712no—
109C22⋊Q8—1421no—
110C4×Q8—712no—
111C4×Q8—1421no—
112#117—1421no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4C2×D7112yes—
115C2×C42C2×D7111yescharacteristic
116#106—718no—
117C4⋊C8—712no—
118#103—718no—
119C22⋊C8—712no—
120C4×C8—712no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—44no—
124C2×Q8—712no—
125C2×Q8—1421no—
126C22×C4—22no—
127C22×C4C22⋊D7111yescharacteristic
128C22×C4C22⋊D7111yescharacteristic
129C22×C4C22×D7111yescharacteristic
130C2×C8—714no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—712no—
135C4⋊C4—1422no—
136C4⋊C4—1422no—
137C4⋊C4—1421no—
138C4⋊C4—1421no—
139C4⋊C4—1421no—
140C4⋊C4C22×D7114yes—
141C22⋊C4—1422no—
142C42—1422no—
143C42C22⋊D7111yescharacteristic
144C42C22⋊D7111yescharacteristic
145C42C22×D7111yescharacteristic
146C42C22×D7111yescharacteristic
147C23C2×C22⋊D7111yescharacteristic
148Q8—44no—
149Q8—44no—
150Q8—2842no—
151Q8—1424no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—44no—
156C2×C4—1422no—
157C2×C4—1422no—
158C2×C4#1176585314yes—
159C2×C4C2×C22⋊D7111yescharacteristic
160C2×C4C2×C22⋊D7111yescharacteristic
161C2×C4C2×C22⋊D7111yescharacteristic
162C2×C4C2×C22⋊D7111yescharacteristic
163C2×C4C2×C22⋊D7111yescharacteristic
164C2×C4C23×D7111yescharacteristic
165C8—1424no—
166C22—21no—
167C22—21no—
168C22C2×#1176585312yes—
169C22C22×C22⋊D7111yescharacteristic
170C4—21no—
171C4—21no—
172C4—44no—
173C4—2842no—
174C4—2842no—
175C4C2×#1176585312yes—
176C4C2×#1176585312yes—
177C4#1455615912yes—
178C4#1455617612yes—
179C2—22no—
180C2#1798657311yescharacteristic
181C2C22×#1176585311yescharacteristic
182C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C4⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C4⋊Q8D71
hypocenterC71#3229
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #17986395C2semidirect
3. #17986351C2semidirect
5. #17986281C2semidirect
92. C71#3229semidirect