Finite Group Databasebeta

← #22599791

Subgroups — #22599791

Order 9088.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986609C212yesmaximal
2#17986309C212yesmaximal, complemented
3#17986304C212yesmaximal, complemented
4#17986257C211yesmaximal, characteristic
5#17986238C212yesmaximal
6#17986208C212yesmaximal, complemented
7#17986207C212yesmaximal, complemented
8#17986027C211yesmaximal, characteristic
9#470×C71C211yesmaximal, characteristic
10C2×Dic284C2211yescharacteristic
11C2×#11765855C2212yes—
12C22×C71⋊C8C2211yescharacteristic
13#14556126C2212yes—
14#14556121—22no—
15#14556121—22no—
16#14556119—22no—
17#14556119C2212yes—
18#14556118—22no—
19#14556107C2212yes—
20Q8×Dic71C2212yes—
21#14556093C2212yes—
22#14556087C2211yescharacteristic
23#14556061C2212yes—
24#14556060C2211yescharacteristic
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×Dic142C2312yes—
36Dic284—41no—
37#11765855—42no—
38C2×C71⋊C8—41no—
39C2×C71⋊C8C2312yes—
40C4⋊Dic71—22no—
41C4⋊Dic71—22no—
42C4⋊Dic71D412yes—
43C4⋊Dic71C2312yes—
44Dic71⋊C4—22no—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49C22⋊Dic71—22no—
50C22⋊Dic71D411yescharacteristic
51C22⋊Dic71D411yescharacteristic
52C4×Dic71—22no—
53C4×Dic71—22no—
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—22no—
74C2×Dic71—22no—
75C2×Dic71C2×D412yes—
76C2×Dic71C2×D412yes—
77Dic142—22no—
78Dic142—22no—
79C8×C71—21no—
80C8×C71—21no—
81C71⋊C8—22no—
82C22×C71—21no—
83C22×C71—22no—
84C22×C71C22×D411yescharacteristic
85C4×C71—21no—
86C4×C71—42no—
87C4×C71—42no—
88C4×C71H(5,𝔽2)11yescharacteristic
89C4×C712−1+411yescharacteristic
90Dic71—22no—
91Dic71—22no—
92Dic71—42no—
93Dic71—42no—
94C2×C71—41no—
95C2×C71#53711yescharacteristic
96C2×C71#57812yes—
97#3261—711nomaximal, Sylow
98C71#326111yescharacteristic, complemented, Sylow
99#541—712no—
100#479—712no—
101#479—712no—
102#470—711no—
103#470D7111yescharacteristic
104#467—712no—
105#462—712no—
106#462—712no—
107#424—711no—
108C2×Q16—712no—
109C2×Q16—711no—
110C2×Q16C2×D7111yescharacteristic
111C2×M16C2×D7111yescharacteristic
112C22×C8—711no—
113C4⋊Q8—712no—
114#126—1422no—
115#126—1422no—
116#125—712no—
117#125—1422no—
118#125—1422no—
119C22⋊Q8—712no—
120C22⋊Q8C2×D7112yes—
121C4×Q8—712no—
122C4×Q8—712no—
123#117—711no—
124#107—712no—
125#106—711no—
126#106C2×D7111yescharacteristic
127C4⋊C8—712no—
128C4⋊C8—712no—
129#103—712no—
130#103—712no—
131#103—712no—
132#103C2×D7112yes—
133C2×Q8—712no—
134C2×Q8C22×D7112yes—
135C22×C4C22×D7111yescharacteristic
136Q16—41no—
137Q16—2841no—
138Q16—2842no—
139M16—41no—
140C2×C8—2841no—
141C2×C8—712no—
142C2×C8C22×D7111yescharacteristic
143C2×C8C22×D7111yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—712no—
146C4⋊C4—712no—
147C4⋊C4—1422no—
148C4⋊C4—1422no—
149C4⋊C4—1422no—
150C4⋊C4—1422no—
151C4⋊C4—1422no—
152C4⋊C4—1422no—
153C4⋊C4—1422no—
154C4⋊C4C22×D7112yes—
155C22⋊C4—22no—
156C22⋊C4—711no—
157C22⋊C4—711no—
158C22⋊C4—1422no—
159C42—712no—
160C42—1422no—
161C42—1422no—
162C23D4×D7111yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—1422no—
166Q8—1422no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—712no—
172C2×C4—712no—
173C2×C4—1422no—
174C2×C4—1422no—
175C2×C4D4×D7111yescharacteristic
176C2×C4C23×D7111yescharacteristic
177C8—21no—
178C8—21no—
179C8—1422no—
180C22—21no—
181C22—22no—
182C22C2×D4×D7111yescharacteristic
183C4—21no—
184C4—42no—
185C4—42no—
186C4—2842no—
187C4—2842no—
188C4—1422no—
189C4—1422no—
190C4#1455617311yescharacteristic
191C4#1455617511yescharacteristic
192C2—41no—
193C2#1798657611yescharacteristic
194C2#1798681512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3261
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. #17986309C2semidirect
3. #17986304C2semidirect
6. #17986208C2semidirect
7. #17986207C2semidirect
98. C71#3261semidirect