Finite Group Databasebeta

← #22599288

Subgroups — #22599288

Order 9088.

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

normal subgroups only

This group has 384 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#17986378C212yesmaximal, complemented
3#17986349C212yesmaximal, complemented
4#17986206C214yesmaximal, complemented
5#17986155C214yesmaximal
6#17986052C211yesmaximal, characteristic
7C2×C4⋊Q8×C71C211yesmaximal, characteristic
8C2×#11765855C2218yes—
9#14556128C2211yescharacteristic
10#14556128C2211yescharacteristic
11#14556120—21no—
12#14556107—21no—
13#14556107C2212yes—
14C4×Dic142—21no—
15C4×Dic142C2212yes—
16#14556087—21no—
17C4⋊(C71⋊C8)C2212yes—
18#14556052C2218yes—
19C22⋊(C71⋊C8)C2212yes—
20C4×C71⋊C8C2212yes—
21C22×Q8×C71C2212yes—
22C4⋊Q8×C71—21no—
23C4⋊Q8×C71—21no—
24C4⋊Q8×C71C2212yes—
25C4⋊Q8×C71C2212yes—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—21no—
28C2×C42×C71C2211yescharacteristic
29C2×Dic142—21no—
30C2×Dic142C2312yes—
31#11765855—48no—
32C2×C71⋊C8C2314yes—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71—21no—
36C4⋊Dic71C2312yes—
37Dic71⋊C4—22no—
38Dic71⋊C4—22no—
39C22⋊Dic71—22no—
40C4×Dic71—22no—
41C2×Q8×C71—22no—
42C2×Q8×C71—44no—
43C2×Q8×C71C2314yes—
44C22×C4×C71—22no—
45C22×C4×C71D411yescharacteristic
46C22×C4×C71D411yescharacteristic
47C22×C4×C71C2311yescharacteristic
48C4⋊C4×C71—22no—
49C4⋊C4×C71—22no—
50C4⋊C4×C71—22no—
51C4⋊C4×C71—22no—
52C42×C71D411yescharacteristic
53C42×C71D411yescharacteristic
54C42×C71C2311yescharacteristic
55C42×C71C2311yescharacteristic
56C23×C71C2×D411yescharacteristic
57Q8×C71—28no—
58Q8×C71—44no—
59C2×C4×C71—22no—
60C2×C4×C71—22no—
61C2×C4×C71—24no—
62C2×C4×C71—44no—
63C2×C4×C71Q1614yescomplemented
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2×D411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2411yescharacteristic
70C2×Dic71—22no—
71C2×Dic71—22no—
72Dic142—24no—
73Dic142—42no—
74C71⋊C8—24no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71C2×Q1612yes—
78C22×C71C22×D411yescharacteristic
79C4×C71—21no—
80C4×C71—21no—
81C4×C71—44no—
82C4×C71C2×Q1612yes—
83C4×C71C2×Q1612yes—
84C4×C71#13712yes—
85C4×C71H(5,𝔽2)12yes—
86Dic71—42no—
87Dic71—42no—
88C2×C71—22no—
89C2×C71#53511yescharacteristic
90C2×C71C22×Q1611yescharacteristic
91C2×C71C2×#13711yescharacteristic
92#3209—711nomaximal, Sylow
93C71#320911yescharacteristic, complemented, Sylow
94#533—711no—
95C2×C4⋊Q8D7111yescharacteristic
96#494—712no—
97#487—712no—
98#462—714no—
99#451—714no—
100#431—711no—
101C22×Q8C2×D7112yes—
102C2×Q16—718no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—711no—
106C4⋊Q8—711no—
107C4⋊Q8C2×D7112yes—
108C4⋊Q8C2×D7112yes—
109#125—1421no—
110C22⋊Q8—712no—
111C22⋊Q8—1421no—
112C4×Q8—712no—
113C4×Q8—1421no—
114#117—1421no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C42C2×D7111yescharacteristic
118C4⋊C8—712no—
119#103—718no—
120C22⋊C8—712no—
121C4×C8—712no—
122C2×Q8—22no—
123C2×Q8—44no—
124C2×Q8—712no—
125C2×Q8—1421no—
126C2×Q8C22×D7114yes—
127C22×C4—22no—
128C22×C4C22⋊D7111yescharacteristic
129C22×C4C22⋊D7111yescharacteristic
130C22×C4C22×D7111yescharacteristic
131Q16—2848no—
132C2×C8—714no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—712no—
138C4⋊C4—1422no—
139C4⋊C4—1422no—
140C4⋊C4—1421no—
141C4⋊C4—1421no—
142C4⋊C4—1421no—
143C22⋊C4—1422no—
144C42—1422no—
145C42C22⋊D7111yescharacteristic
146C42C22⋊D7111yescharacteristic
147C42C22×D7111yescharacteristic
148C42C22×D7111yescharacteristic
149C23C2×C22⋊D7111yescharacteristic
150Q8—28no—
151Q8—44no—
152Q8—2842no—
153Q8—1424no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—1422no—
159C2×C4—1422no—
160C2×C4#1176585514yescomplemented
161C2×C4C2×C22⋊D7111yescharacteristic
162C2×C4C2×C22⋊D7111yescharacteristic
163C2×C4C2×C22⋊D7111yescharacteristic
164C2×C4C2×C22⋊D7111yescharacteristic
165C2×C4C2×C22⋊D7111yescharacteristic
166C2×C4C23×D7111yescharacteristic
167C8—1424no—
168C22—21no—
169C22—21no—
170C22C2×#1176585512yes—
171C22C22×C22⋊D7111yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—2842no—
176C4—2842no—
177C4C2×#1176585512yes—
178C4C2×#1176585512yes—
179C4#1455615912yes—
180C4#1455617312yes—
181C2—22no—
182C2#1798656611yescharacteristic
183C2C22×#1176585511yescharacteristic
184C2C2×#1455615911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×C4⋊Q8×C71C2
socleC22×C71C22×D4
hypercenterC2×C4⋊Q8D71
hypocenterC71#3209
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. #17986378C2semidirect
3. #17986349C2semidirect
4. #17986206C2semidirect
63. C2×C4×C71Q16semidirect
93. C71#3209semidirect
160. C2×C4#11765855semidirect