Finite Group Databasebeta

← #22599285

Subgroups — #22599285

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
1C2×#14556128C211yesmaximal, characteristic
2#17986378C212yesmaximal, complemented
3#17986351C212yesmaximal, complemented
4#17986209C214yesmaximal, complemented
5#17986154C214yesmaximal
6#17986052C211yesmaximal, characteristic
7#533×C71C211yesmaximal, characteristic
8C22×Dic142C2212yes—
9C2×#11765855C2218yes—
10#14556128—21no—
11#14556128—21no—
12#14556128C2212yes—
13#14556128C2212yes—
14C2×C4⋊Dic71—21no—
15C2×C4⋊Dic71—21no—
16C4⋊(C71⋊C8)C2212yes—
17#14556051C2218yes—
18C22⋊(C71⋊C8)C2212yes—
19C4×C71⋊C8C2212yes—
20C4⋊Q8×C71C2211yescharacteristic
21C4⋊Q8×C71C2211yescharacteristic
22#125×C71—21no—
23C22⋊Q8×C71—21no—
24C22⋊Q8×C71C2212yes—
25C4×Q8×C71—21no—
26C4×Q8×C71C2212yes—
27#117×C71—21no—
28C2×C42×C71C2211yescharacteristic
29C2×Dic142—22no—
30C2×Dic142—44no—
31C2×Dic142C2314yes—
32C22×Dic71—22no—
33#11765855—48no—
34C2×C71⋊C8C2314yes—
35C4⋊Dic71—22no—
36C4⋊Dic71—22no—
37C4⋊Dic71—22no—
38C4⋊Dic71—22no—
39C2×Q8×C71—21no—
40C2×Q8×C71C2312yes—
41C22×C4×C71D411yescharacteristic
42C22×C4×C71D411yescharacteristic
43C22×C4×C71C2311yescharacteristic
44C4⋊C4×C71—21no—
45C4⋊C4×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—22no—
48C4⋊C4×C71—22no—
49C4⋊C4×C71C2312yes—
50C22⋊C4×C71—22no—
51C42×C71—22no—
52C42×C71D411yescharacteristic
53C42×C71D411yescharacteristic
54C42×C71C2311yescharacteristic
55C42×C71C2311yescharacteristic
56C23×C71C2×D411yescharacteristic
57Q8×C71—24no—
58Q8×C71—42no—
59C2×C4×C71—22no—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71—22no—
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—24no—
71C2×Dic71—44no—
72Dic142—28no—
73Dic142—44no—
74C71⋊C8—24no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71C2×Q1612yes—
78C22×C71C22×D411yescharacteristic
79C4×C71—21no—
80C4×C71—21no—
81C4×C71—42no—
82C4×C71—42no—
83C4×C71C2×Q1612yes—
84C4×C71C2×Q1612yes—
85C4×C71#13712yes—
86C4×C71H(5,𝔽2)12yes—
87Dic71—44no—
88C2×C71—22no—
89C2×C71#53511yescharacteristic
90C2×C71C22×Q1611yescharacteristic
91C2×C71C2×#13711yescharacteristic
92#3209—711nomaximal, Sylow
93C71#320911yescharacteristic, complemented, Sylow
94#533D7111yescharacteristic
95C2×C4⋊Q8—711no—
96#494—712no—
97#487—712no—
98#462—714no—
99#451—714no—
100#431—711no—
101C22×Q8—712no—
102C2×Q16—718no—
103C4⋊Q8—712no—
104C4⋊Q8—712no—
105C4⋊Q8—1421no—
106C4⋊Q8—1421no—
107C4⋊Q8C2×D7111yescharacteristic
108C4⋊Q8C2×D7111yescharacteristic
109#125—21no—
110C22⋊Q8—21no—
111C22⋊Q8C2×D7112yes—
112C4×Q8—21no—
113C4×Q8C2×D7112yes—
114#117—21no—
115C2×C4⋊C4—1421no—
116C2×C4⋊C4—1421no—
117C2×C42C2×D7111yescharacteristic
118C4⋊C8—712no—
119#103—718no—
120C22⋊C8—712no—
121C4×C8—712no—
122C2×Q8—21no—
123C2×Q8—2844no—
124C2×Q8—714no—
125C2×Q8—1422no—
126C2×Q8C22×D7112yes—
127C22×C4—1422no—
128C22×C4C22⋊D7111yescharacteristic
129C22×C4C22⋊D7111yescharacteristic
130C22×C4C22×D7111yescharacteristic
131Q16—2848no—
132C2×C8—714no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—1422no—
139C4⋊C4—1422no—
140C4⋊C4—1422no—
141C4⋊C4—1422no—
142C4⋊C4C22×D7112yes—
143C22⋊C4—22no—
144C42—22no—
145C42C22⋊D7111yescharacteristic
146C42C22⋊D7111yescharacteristic
147C42C22×D7111yescharacteristic
148C42C22×D7111yescharacteristic
149C23C2×C22⋊D7111yescharacteristic
150Q8—24no—
151Q8—42no—
152Q8—2844no—
153Q8—1428no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—2844no—
159C2×C4—1424no—
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—42no—
175C4—42no—
176C4—2844no—
177C4C2×#1176585512yes—
178C4C2×#1176585512yes—
179C4#1455615812yes—
180C4#1455617412yes—
181C2—22no—
182C2#1798656311yescharacteristic
183C2C22×#1176585511yescharacteristic
184C2C2×#1455615811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3209
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. #17986378C2semidirect
3. #17986351C2semidirect
4. #17986209C2semidirect
63. C2×C4×C71Q16semidirect
93. C71#3209semidirect
160. C2×C4#11765855semidirect