Finite Group Databasebeta

← #22587450

Subgroups — #22587450

Order 9088.

216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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#17986347C211yesmaximal, characteristic, complemented
2#17986199C211yesmaximal, characteristic, complemented
3#17986194C211yesmaximal, characteristic
4#17985745C211yesmaximal, characteristic, complemented
5Q8⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
7#556×C71C211yesmaximal, characteristic, complemented
8C2×#11765852—21no—
9C2×#11765850—21no—
10#14556125C2211yescharacteristic
11C4⋊(C71⋊C8)C2211yescharacteristic
12C4⋊(C71⋊C8)C2211yescharacteristic
13#14556048—22no—
14#14556048—21no—
15#14556048—21no—
16C4×C71⋊C8C2211yescharacteristic
17C4⋊Q8×C71C2211yescharacteristic
18#126×C71—22no—
19#124×C71—21no—
20C22⋊Q8×C71—21no—
21C4×Q8×C71—21no—
22C4×Q8×C71C2211yescharacteristic
23C4×D4×C71C2211yescharacteristic
24#117×C71—21no—
25C2×D284—21no—
26C2×D284—21no—
27C2×D284—21no—
28#11765852—42no—
29#11765850—42no—
30C2×C71⋊C8—21no—
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33C2×Q8×C71—21no—
34C2×Q8×C71D411yescharacteristic
35C2×D4×C71D411yescharacteristic
36C22×C4×C71—21no—
37C4⋊C4×C71—22no—
38C4⋊C4×C71—21no—
39C4⋊C4×C71—21no—
40C4⋊C4×C71D411yescharacteristic
41C4⋊C4×C71D411yescharacteristic
42C4⋊C4×C71D412yes—
43C22⋊C4×C71—22no—
44C22⋊C4×C71—22no—
45C22⋊C4×C71—21no—
46C42×C71—22no—
47C42×C71—21no—
48C42×C71C2311yescharacteristic
49C23×C71—21no—
50C22×D71—41no—
51Q8×C71—21no—
52Q8×C71—21no—
53Q8×C71—41no—
54Q8×C71—41no—
55D4×C71—21no—
56D4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—21no—
59C2×C4×C71—21no—
60C2×C4×C71—22no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71C2×D411yescharacteristic
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66D284—42no—
67D284—41no—
68D284—41no—
69D284—41no—
70D284—41no—
71C71⋊C8—41no—
72C71⋊C8—41no—
73C71⋊C8—41no—
74C22×C71—41no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71#12011yescharacteristic
78C2×D71—81no—
79C2×D71—81no—
80C2×D71—81no—
81C4×C71—21no—
82C4×C71—41no—
83C4×C71—41no—
84C4×C71—42no—
85C4×C71—41no—
86C4×C71#13511yescharacteristic
87C4×C71#13511yescharacteristic
88C4×C71AGL(1,ℤ/8)11yescharacteristic
89C4×C71AGL(1,ℤ/8)11yescharacteristic
90C2×C71—81no—
91C2×C71#44911yescharacteristic
92C2×C71#44911yescharacteristic
93C2×C71#45411yescharacteristic
94D71—161no—
95#1628—711nomaximal, Sylow
96C71#162811yescharacteristic, complemented, Sylow
97#556D7111yescharacteristic, complemented
98#486—711no—
99#460—711no—
100#459—711no—
101#332—711no—
102Q8⋊C8—711no—
103D4⋊C8—711no—
104C2×SD16—1421no—
105C2×D8—1421no—
106C4⋊Q8C2×D7111yescharacteristic
107#127—711no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D7111yescharacteristic
113C4×D4C2×D7111yescharacteristic
114#117—21no—
115C4⋊C8—711no—
116C4⋊C8—711no—
117#102—1421no—
118#102—1421no—
119#102—1422no—
120C4×C8—711no—
121C2×Q8—21no—
122C2×Q8C22⋊D7111yescharacteristic
123C2×D4—1421no—
124C2×D4—1421no—
125C2×D4—1421no—
126C2×D4C22⋊D7111yescharacteristic
127C22×C4—21no—
128SD16—2842no—
129D8—2842no—
130C2×C8—1421no—
131C2×C8—1421no—
132C2×C8—1421no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D7111yescharacteristic
137C4⋊C4C22⋊D7111yescharacteristic
138C4⋊C4C22⋊D7112yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D7111yescharacteristic
145C23—21no—
146C23—2841no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—2842no—
154D4—2841no—
155D4—2841no—
156D4—2841no—
157D4—2841no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D7111yescharacteristic
165C2×C4C2×C22⋊D7111yescharacteristic
166C2×C4C2×C22⋊D7111yescharacteristic
167C8—2841no—
168C8—2841no—
169C8—2841no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—5681no—
174C22—5681no—
175C22—5681no—
176C22#1455609711yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1455614811yescharacteristic
183C4#1455614811yescharacteristic
184C4#1455615311yescharacteristic
185C4#1455615311yescharacteristic
186C2—81no—
187C2—11361no—
188C2#1798614411yescharacteristic
189C2#1798614411yescharacteristic
190C2#1798617211yescharacteristic
subgroupG/H
centerC22#14556097
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#556×C71C2
socleC22×C71#120
hypercenter#556D71
hypocenterC71#1628
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #17986347C2semidirect
2. #17986199C2semidirect
4. #17985745C2semidirect
5. Q8⋊(C71⋊C8)C2semidirect
6. D4⋊(C71⋊C8)C2semidirect
7. #556×C71C2semidirect
96. C71#1628semidirect
97. #556D71semidirect