Finite Group Databasebeta

← #22587568

Subgroups — #22587568

Order 9088.

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

normal subgroups only

This group has 212 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#17986349C211yesmaximal, characteristic, complemented
2#17986279C211yesmaximal, characteristic, complemented
3#17986204C211yesmaximal, characteristic, complemented
4#17985748C211yesmaximal, characteristic, complemented
5D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
7#555×C71C211yesmaximal, characteristic
8C2×#11765853—21no—
9#14556128C2211yescharacteristic
10#14556061—21no—
11C4⋊(C71⋊C8)C2211yescharacteristic
12C4⋊(C71⋊C8)C2211yescharacteristic
13#14556052—22no—
14#14556051—21no—
15#14556049—21no—
16C4×C71⋊C8C2211yescharacteristic
17C4⋊Q8×C71C2211yescharacteristic
18#124×C71—22no—
19#121×C71—21no—
20#121×C71—21no—
21C4×D4×C71C2211yescharacteristic
22C4×D4×C71C2211yescharacteristic
23#117×C71—21no—
24#117×C71—21no—
25C2×Dic142—21no—
26#11765853—42no—
27C2×C71⋊C8—21no—
28C2×C71⋊C8—21no—
29C2×C71⋊C8—21no—
30C4⋊Dic71—21no—
31C4⋊Dic71—21no—
32C2×Q8×C71D412yes—
33C2×D4×C71—22no—
34C2×D4×C71D411yescharacteristic
35C2×D4×C71D411yescharacteristic
36C22×C4×C71—21no—
37C22×C4×C71—21no—
38C4⋊C4×C71—21no—
39C4⋊C4×C71—21no—
40C4⋊C4×C71D411yescharacteristic
41C4⋊C4×C71D411yescharacteristic
42C22⋊C4×C71—22no—
43C22⋊C4×C71—22no—
44C22⋊C4×C71—21no—
45C22⋊C4×C71—21no—
46C42×C71—22no—
47C42×C71C2311yescharacteristic
48C23×C71—21no—
49C23×C71—21no—
50Q8×C71—42no—
51D4×C71—21no—
52D4×C71—21no—
53D4×C71—21no—
54D4×C71—21no—
55D4×C71—82no—
56C2×C4×C71—21no—
57C2×C4×C71—21no—
58C2×C4×C71—22no—
59C2×C4×C71—41no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71C2×D411yescharacteristic
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71C2×D411yescharacteristic
66C2×Dic71—41no—
67Dic142—41no—
68Dic142—41no—
69C71⋊C8—41no—
70C71⋊C8—41no—
71C71⋊C8—41no—
72C22×C71—41no—
73C22×C71—41no—
74C22×C71—41no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71—41no—
78C22×C71#12011yescharacteristic
79C4×C71—21no—
80C4×C71—41no—
81C4×C71—41no—
82C4×C71—42no—
83C4×C71#13511yescharacteristic
84C4×C71#13511yescharacteristic
85C4×C71AGL(1,ℤ/8)11yescharacteristic
86C4×C71#13711yescharacteristic
87Dic71—81no—
88C2×C71—81no—
89C2×C71—81no—
90C2×C71#44911yescharacteristic
91C2×C71#45211yescharacteristic
92C2×C71#45511yescharacteristic
93#1646—711nomaximal, Sylow
94C71#164611yescharacteristic, complemented, Sylow
95#555D7111yescharacteristic
96#487—711no—
97#474—711no—
98#461—711no—
99#333—711no—
100D4⋊C8—711no—
101D4⋊C8—711no—
102C2×SD16—1421no—
103C4⋊Q8—711no—
104C4⋊Q8C2×D7111yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D7111yescharacteristic
109C4×D4C2×D7111yescharacteristic
110#117—21no—
111#117—21no—
112#107—1421no—
113C4⋊C8—711no—
114C4⋊C8—711no—
115#103—1422no—
116#103—1421no—
117#102—1421no—
118C4×C8—711no—
119C2×Q8—1421no—
120C2×Q8C22⋊D7112yes—
121C2×D4—22no—
122C2×D4C22⋊D7111yescharacteristic
123C2×D4C22⋊D7111yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—2842no—
127C2×C8—1421no—
128C2×C8—1421no—
129C2×C8—1421no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—1421no—
133C4⋊C4—1421no—
134C4⋊C4C22⋊D7111yescharacteristic
135C4⋊C4C22⋊D7111yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D7111yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—2841no—
146Q8—2841no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—2841no—
160C2×C4C2×C22⋊D7111yescharacteristic
161C2×C4C2×C22⋊D7111yescharacteristic
162C2×C4C2×C22⋊D7111yescharacteristic
163C8—2841no—
164C8—2841no—
165C8—2841no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1455609711yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—5681no—
178C4#1455614811yescharacteristic
179C4#1455614811yescharacteristic
180C4#1455615311yescharacteristic
181C4#1455615811yescharacteristic
182C2—81no—
183C2—81no—
184C2#1798614411yescharacteristic
185C2#1798616111yescharacteristic
186C2#1798617711yescharacteristic
subgroupG/H
centerC22#14556097
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#555×C71C2
socleC22×C71#120
hypercenter#555D71
hypocenterC71#1646
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #17986349C2semidirect
2. #17986279C2semidirect
3. #17986204C2semidirect
4. #17985748C2semidirect
5. D4⋊(C71⋊C8)C2semidirect
6. D4⋊(C71⋊C8)C2semidirect
94. C71#1646semidirect