Finite Group Databasebeta

← #22599682

Subgroups — #22599682

Order 9088.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#17986294C212yesmaximal, complemented
3#17986291C212yesmaximal, complemented
4#17986238C214yesmaximal
5#17986209C212yesmaximal, complemented
6#17986206C212yesmaximal, complemented
7C2×C4⋊(C71⋊C8)C211yesmaximal, characteristic
8#533×C71C211yesmaximal, characteristic
9C2×#11765855C2214yes—
10C22×C71⋊C8C2212yes—
11#14556128C2211yescharacteristic
12#14556128C2211yescharacteristic
13#14556120—21no—
14#14556107—21no—
15#14556107C2212yes—
16C4×Dic142—21no—
17C4×Dic142C2212yes—
18#14556087—21no—
19#14556061C2214yes—
20C4⋊(C71⋊C8)C2212yes—
21C4⋊(C71⋊C8)C2212yes—
22#14556052C2214yes—
23#14556051C2214yes—
24C4⋊Q8×C71C2211yescharacteristic
25C4⋊Q8×C71C2211yescharacteristic
26#125×C71—21no—
27C22⋊Q8×C71—21no—
28C22⋊Q8×C71C2212yes—
29C4×Q8×C71—21no—
30C4×Q8×C71C2212yes—
31#117×C71—21no—
32C2×C42×C71C2211yescharacteristic
33C2×Dic142—21no—
34C2×Dic142C2312yes—
35#11765855—44no—
36C2×C71⋊C8—42no—
37C2×C71⋊C8C2314yes—
38C4⋊Dic71—21no—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41C4⋊Dic71C2312yes—
42Dic71⋊C4—22no—
43Dic71⋊C4—22no—
44C22⋊Dic71—22no—
45C4×Dic71—22no—
46C2×Q8×C71—21no—
47C2×Q8×C71C2312yes—
48C22×C4×C71D411yescharacteristic
49C22×C4×C71D411yescharacteristic
50C22×C4×C71C2311yescharacteristic
51C4⋊C4×C71—21no—
52C4⋊C4×C71—21no—
53C4⋊C4×C71—21no—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71C2312yes—
57C22⋊C4×C71—22no—
58C42×C71—22no—
59C42×C71D411yescharacteristic
60C42×C71D411yescharacteristic
61C42×C71C2311yescharacteristic
62C42×C71C2311yescharacteristic
63C23×C71C2×D411yescharacteristic
64Q8×C71—22no—
65Q8×C71—22no—
66Q8×C71—42no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—41no—
72C2×C4×C71Q1614yescomplemented
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D411yescharacteristic
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C2×Dic71—22no—
80C2×Dic71—22no—
81Dic142—22no—
82Dic142—22no—
83Dic142—42no—
84C71⋊C8—24no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71C2×Q1612yes—
88C22×C71C22×D411yescharacteristic
89C4×C71—21no—
90C4×C71—21no—
91C4×C71—21no—
92C4×C71—42no—
93C4×C71—42no—
94C4×C71C2×Q1612yes—
95C4×C71C2×Q1612yes—
96C4×C71H(5,𝔽2)11yescharacteristic
97C4×C712−1+411yescharacteristic
98Dic71—42no—
99Dic71—42no—
100C2×C71—22no—
101C2×C71#53711yescharacteristic
102C2×C71C22×Q1611yescharacteristic
103C2×C71#57511yescharacteristic
104#3249—711nomaximal, Sylow
105C71#324911yescharacteristic, complemented, Sylow
106#533—711no—
107#533D7111yescharacteristic
108#477—712no—
109#477—712no—
110#467—714no—
111#462—712no—
112#462—712no—
113C2×C4⋊C8—711no—
114C2×Q16—714no—
115C22×C8—712no—
116C4⋊Q8—711no—
117C4⋊Q8—711no—
118C4⋊Q8C2×D7111yescharacteristic
119C4⋊Q8C2×D7111yescharacteristic
120#125—21no—
121#125—1421no—
122C22⋊Q8—21no—
123C22⋊Q8—712no—
124C22⋊Q8—1421no—
125C22⋊Q8C2×D7112yes—
126C4×Q8—21no—
127C4×Q8—712no—
128C4×Q8—1421no—
129C4×Q8C2×D7112yes—
130#117—21no—
131#117—1421no—
132C2×C42C2×D7111yescharacteristic
133#107—714no—
134C4⋊C8—712no—
135C4⋊C8—712no—
136#103—714no—
137#103—714no—
138C2×Q8—21no—
139C2×Q8—712no—
140C2×Q8—1421no—
141C2×Q8C22×D7112yes—
142C22×C4C22⋊D7111yescharacteristic
143C22×C4C22⋊D7111yescharacteristic
144C22×C4C22×D7111yescharacteristic
145Q16—2844no—
146C2×C8—2842no—
147C2×C8—714no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—712no—
154C4⋊C4—1422no—
155C4⋊C4—1422no—
156C4⋊C4—1421no—
157C4⋊C4—1421no—
158C4⋊C4—1421no—
159C4⋊C4C22×D7112yes—
160C22⋊C4—22no—
161C22⋊C4—1422no—
162C42—22no—
163C42—1422no—
164C42C22⋊D7111yescharacteristic
165C42C22⋊D7111yescharacteristic
166C42C22×D7111yescharacteristic
167C42C22×D7111yescharacteristic
168C23C2×C22⋊D7111yescharacteristic
169Q8—22no—
170Q8—22no—
171Q8—42no—
172Q8—2842no—
173Q8—1422no—
174Q8—1422no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—1422no—
181C2×C4—1422no—
182C2×C4#1176585514yescomplemented
183C2×C4C2×C22⋊D7111yescharacteristic
184C2×C4C2×C22⋊D7111yescharacteristic
185C2×C4C2×C22⋊D7111yescharacteristic
186C2×C4C2×C22⋊D7111yescharacteristic
187C2×C4C2×C22⋊D7111yescharacteristic
188C2×C4C23×D7111yescharacteristic
189C8—1424no—
190C22—21no—
191C22—21no—
192C22C2×#1176585512yes—
193C22C22×C22⋊D7111yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—2842no—
200C4—2842no—
201C4C2×#1176585512yes—
202C4C2×#1176585512yes—
203C4#1455617411yescharacteristic
204C4#1455617511yescharacteristic
205C2—22no—
206C2#1798657711yescharacteristic
207C2C22×#1176585511yescharacteristic
208C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3249
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. #17986294C2semidirect
3. #17986291C2semidirect
5. #17986209C2semidirect
6. #17986206C2semidirect
72. C2×C4×C71Q16semidirect
105. C71#3249semidirect
182. C2×C4#11765855semidirect