Finite Group Databasebeta

← #22588582

Subgroups — #22588582

Order 9088.

262 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 93 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#17986766C211yesmaximal, characteristic
2#17986048C211yesmaximal, characteristic, complemented
3#17986023C211yesmaximal, characteristic, complemented
4#17986016C211yesmaximal, characteristic
5#17985771C211yesmaximal, characteristic, complemented
6#17985759C211yesmaximal, characteristic, complemented
7C2×#104×C71C211yesmaximal, characteristic
8#14556135—22no—
9#14556135—22no—
10C2×#11765849—22no—
11C2×#11765849C2211yescharacteristic
12C2×#11765849C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556087C2211yescharacteristic
15#14556064—22no—
16C4⋊(C71⋊C8)—22no—
17#14556055—22no—
18C2×#41×C71C2211yescharacteristic
19C2×M16×C71C2211yescharacteristic
20C2×C42×C71C2211yescharacteristic
21#104×C71C412yescomplemented
22#104×C71C412yescomplemented
23#11765849—24no—
24#11765849—21no—
25#11765849—21no—
26#11765849—21no—
27#11765849—21no—
28#11765849—21no—
29#11765849D412yes—
30C2×C71⋊C8—24no—
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33C2×C71⋊C8D412yes—
34C4⋊Dic71—21no—
35C22⋊Dic71—21no—
36C4×Dic71—21no—
37#41×C71—21no—
38#41×C71—21no—
39#41×C71C2×C412yes—
40#41×C71D412yes—
41C2×Q8×C71Q811yescharacteristic
42C2×D4×C71—21no—
43C2×D4×C71Q811yescharacteristic
44C22×C4×C71—21no—
45C22×C4×C71—21no—
46C22×C4×C71C2311yescharacteristic
47M16×C71—21no—
48M16×C71C2×C412yes—
49C2×C8×C71—21no—
50C42×C71—21no—
51C42×C71C2×C411yescharacteristic
52C42×C71C2×C411yescharacteristic
53C23×C71—21no—
54C23×C71#4111yescharacteristic
55Q8×C71—21no—
56Q8×C71C4⋊C412yes—
57D4×C71—22no—
58D4×C71—22no—
59D4×C71—21no—
60D4×C71C4⋊C412yes—
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—22no—
64C2×C4×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—42no—
67C2×C4×C71C22×C411yescharacteristic
68C2×C4×C71C2×D411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×Q811yescharacteristic
72C2×C4×C71#4111yescharacteristic
73C2×Dic71—41no—
74C8×C71—22no—
75C71⋊C8—22no—
76C71⋊C8—22no—
77C71⋊C8—41no—
78C22×C71—22no—
79C22×C71—41no—
80C22×C71—81no—
81C22×C71C4×D411yescharacteristic
82C22×C71C22⋊Q811yescharacteristic
83C22×C71#12311yescharacteristic
84C4×C71—22no—
85C4×C71—22no—
86C4×C71—41no—
87C4×C71C2×C4⋊C411yescharacteristic
88C4×C71C4×D411yescharacteristic
89C4×C71#12011yescharacteristic
90C4×C71C22⋊Q811yescharacteristic
91Dic71—81no—
92C2×C71—21no—
93C2×C71—21no—
94C2×C71—21no—
95C2×C71—42no—
96C2×C71#38511yescharacteristic
97#1861—711nomaximal, Sylow
98C71#186111yescharacteristic, complemented, Sylow
99#568—711no—
100#430—711no—
101#423—711no—
102#421—711no—
103C2×#104D7111yescharacteristic
104#344—711no—
105#339—711no—
106C2×#41C2×D7111yescharacteristic
107#131—1422no—
108#131—1422no—
109C2×M16—711no—
110C2×M16—711no—
111C2×M16—711no—
112C2×M16—1422no—
113C2×M16C2×D7111yescharacteristic
114#117—711no—
115C2×C42C2×D7111yescharacteristic
116#108—1422no—
117C4⋊C8—1422no—
118#104—1422no—
119#104Dic7112yescomplemented
120#104Dic7112yescomplemented
121#41—21no—
122#41—21no—
123#41D28412yes—
124#41C2×Dic7112yes—
125C2×Q8Dic14211yescharacteristic
126C2×D4—21no—
127C2×D4Dic14211yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22×D7111yescharacteristic
131M16—21no—
132M16—712no—
133M16—1421no—
134M16—1421no—
135M16—1424no—
136M16—1421no—
137M16—1421no—
138M16—1421no—
139M16C2×Dic7112yes—
140C2×C8—21no—
141C2×C8—712no—
142C2×C8—1421no—
143C2×C8—1424no—
144C2×C8—1421no—
145C4⋊C4—1421no—
146C22⋊C4—1421no—
147C42—21no—
148C42—1421no—
149C42C2×Dic7111yescharacteristic
150C42C2×Dic7111yescharacteristic
151C23—21no—
152C23#1176586411yescharacteristic
153Q8—21no—
154Q8C4⋊Dic7112yes—
155D4—22no—
156D4—22no—
157D4—21no—
158D4C4⋊Dic7112yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—2841no—
166C2×C4C22×Dic7111yescharacteristic
167C2×C4D4×D7111yescharacteristic
168C2×C4D4×D7111yescharacteristic
169C2×C4C2×D28411yescharacteristic
170C2×C4C2×Dic14211yescharacteristic
171C2×C4#1176586411yescharacteristic
172C8—22no—
173C8—2841no—
174C8—1422no—
175C8—1422no—
176C22—22no—
177C22—41no—
178C22—81no—
179C22D4×Dic7111yescharacteristic
180C22#1455610511yescharacteristic
181C22#1455611211yescharacteristic
182C4—22no—
183C4—22no—
184C4—41no—
185C4—5681no—
186C4C2×C4⋊Dic7111yescharacteristic
187C4D4×Dic7111yescharacteristic
188C4#1455609811yescharacteristic
189C4#1455610511yescharacteristic
190C2—21no—
191C2—21no—
192C2—21no—
193C2—42no—
194C2#1798587411yescharacteristic
subgroupG/H
centerC4#14556098
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
FittingC2×#104×C71C2
socleC2×C71#385
hypercenterC2×#104D71
hypocenterC71#1861
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #17986048C2semidirect
3. #17986023C2semidirect
5. #17985771C2semidirect
6. #17985759C2semidirect
21. #104×C71C4semidirect
22. #104×C71C4semidirect
98. C71#1861semidirect
119. #104Dic71semidirect
120. #104Dic71semidirect