Finite Group Databasebeta

← #22599730

Subgroups — #22599730

Order 9088.

342 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 113 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#17986551C212yesmaximal
2#17986283C212yesmaximal, complemented
3#17986282C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic
5#17986255C212yesmaximal
6#17986208C212yesmaximal, complemented
7#17986207C212yesmaximal, complemented
8#17986022C211yesmaximal, characteristic
9#470×C71C211yesmaximal, characteristic
10C2×Dic284C2211yescharacteristic
11C2×#11765855C2212yes—
12C2×#11765849C2211yescharacteristic
13#14556127C2212yes—
14#14556126C2212yes—
15#14556118—22no—
16#14556107C2212yes—
17#14556105—22no—
18Q8×Dic71C2212yes—
19#14556093—22no—
20#14556093C2212yes—
21#14556084—22no—
22C2×C4×Dic71C2211yescharacteristic
23#14556060C2211yescharacteristic
24#14556059C2212yes—
25Dic71⋊C8C2212yes—
26Dic71⋊C8C2212yes—
27#14556053C2212yes—
28#14556052C2212yes—
29#14556051C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C2×M16×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#106×C71C2211yescharacteristic
34#103×C71C2212yes—
35C2×Dic142—22no—
36C2×Dic142C2312yes—
37C22×Dic71D412yes—
38Dic284—41no—
39#11765855—42no—
40#11765849—41no—
41C2×C71⋊C8C2312yes—
42C4⋊Dic71—22no—
43C4⋊Dic71C2312yes—
44Dic71⋊C4—22no—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49C22⋊Dic71—22no—
50C4×Dic71—22no—
51C4×Dic71—22no—
52C4×Dic71D411yescharacteristic
53C4×Dic71D411yescharacteristic
54C4×Dic71C2312yes—
55C2×Q8×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57Q16×C71—41no—
58M16×C71—41no—
59C2×C8×C71C2311yescharacteristic
60C2×C8×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C23×C71C2×D411yescharacteristic
65Q8×C71—22no—
66Q8×C71—22no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2411yescharacteristic
73C2×Dic71—24no—
74C2×Dic71—22no—
75C2×Dic71—22no—
76C2×Dic71C2×D412yes—
77C2×Dic71C2×D412yes—
78Dic142—22no—
79Dic142—22no—
80Dic142—44no—
81C8×C71—21no—
82C8×C71—21no—
83C71⋊C8—22no—
84C22×C71—21no—
85C22×C71—22no—
86C22×C71C22×D411yescharacteristic
87C4×C71—21no—
88C4×C71—42no—
89C4×C71—42no—
90C4×C71H(5,𝔽2)11yescharacteristic
91C4×C712−1+411yescharacteristic
92Dic71—22no—
93Dic71—42no—
94Dic71—42no—
95Dic71#13714yes—
96C2×C71—41no—
97C2×C71#53711yescharacteristic
98C2×C71C2×#13712yes—
99#3255—711nomaximal, Sylow
100C71#325511yescharacteristic, complemented, Sylow
101#533—712no—
102#475—712no—
103#475—712no—
104#470—711no—
105#470—712no—
106#470D7111yescharacteristic
107#462—712no—
108#462—712no—
109#423—711no—
110C2×Q16—712no—
111C2×Q16—711no—
112C2×Q16C2×D7111yescharacteristic
113C2×M16—711no—
114C2×M16C2×D7111yescharacteristic
115C4⋊Q8—712no—
116C4⋊Q8—712no—
117#125—1422no—
118C22⋊Q8—712no—
119C22⋊Q8—1422no—
120C22⋊Q8C2×D7112yes—
121C4×Q8—712no—
122C4×Q8—712no—
123C4×Q8—1422no—
124#117—1422no—
125C2×C42—711no—
126#106—711no—
127#106—712no—
128#106C2×D7111yescharacteristic
129C4⋊C8—712no—
130C4⋊C8—712no—
131#103—712no—
132#103—712no—
133#103—712no—
134#103C2×D7112yes—
135C2×Q8—712no—
136C2×Q8—1422no—
137C2×Q8C22×D7112yes—
138C22×C4—712no—
139C22×C4C22×D7111yescharacteristic
140Q16—41no—
141Q16—2841no—
142Q16—2842no—
143M16—41no—
144M16—2841no—
145C2×C8—712no—
146C2×C8C22×D7111yescharacteristic
147C2×C8C22×D7111yescharacteristic
148C4⋊C4—22no—
149C4⋊C4—712no—
150C4⋊C4—1422no—
151C4⋊C4—1422no—
152C4⋊C4—1422no—
153C4⋊C4—1422no—
154C4⋊C4—1422no—
155C4⋊C4—1422no—
156C4⋊C4C22×D7112yes—
157C22⋊C4—22no—
158C22⋊C4—1422no—
159C42—711no—
160C42—711no—
161C42—712no—
162C42—1422no—
163C42—1422no—
164C23D4×D7111yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—2844no—
168Q8—1422no—
169Q8—1422no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—712no—
175C2×C4—712no—
176C2×C4—1422no—
177C2×C4—1422no—
178C2×C4—1424no—
179C2×C4D4×D7111yescharacteristic
180C2×C4C23×D7111yescharacteristic
181C8—21no—
182C8—21no—
183C8—1422no—
184C22—21no—
185C22—22no—
186C22C2×D4×D7111yescharacteristic
187C4—21no—
188C4—42no—
189C4—42no—
190C4—2842no—
191C4—2842no—
192C4—714no—
193C4—1422no—
194C4#1455617311yescharacteristic
195C4#1455617511yescharacteristic
196C2—41no—
197C2#1798657611yescharacteristic
198C2#137×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3255
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #17986283C2semidirect
3. #17986282C2semidirect
6. #17986208C2semidirect
7. #17986207C2semidirect
100. C71#3255semidirect