Finite Group Databasebeta

← #22587474

Subgroups — #22587474

Order 9088.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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#17986378C211yesmaximal, characteristic, complemented
2#17986294C211yesmaximal, characteristic, complemented
3#17986289C211yesmaximal, characteristic
4#17985748C211yesmaximal, characteristic, complemented
5Q8⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C71⋊C8)C211yesmaximal, characteristic
7#554×C71C211yesmaximal, characteristic
8C2×#11765855—22no—
9#14556128C2211yescharacteristic
10#14556061—21no—
11#14556059—21no—
12C4⋊(C71⋊C8)C2211yescharacteristic
13C4⋊(C71⋊C8)C2211yescharacteristic
14#14556052—21no—
15#14556049—21no—
16C4×C71⋊C8C2211yescharacteristic
17C4⋊Q8×C71C2211yescharacteristic
18#125×C71—21no—
19C22⋊Q8×C71—21no—
20C22⋊Q8×C71—22no—
21C4×Q8×C71C2211yescharacteristic
22C4×D4×C71—21no—
23C4×D4×C71C2211yescharacteristic
24C2×C4⋊C4×C71—21no—
25C2×Dic142—21no—
26#11765855—44no—
27C2×C71⋊C8—21no—
28C2×C71⋊C8—21no—
29C2×C71⋊C8—21no—
30C4⋊Dic71—21no—
31C4⋊Dic71—21no—
32C2×Q8×C71D411yescharacteristic
33C2×Q8×C71D412yes—
34C2×D4×C71D411yescharacteristic
35C22×C4×C71—22no—
36C22×C4×C71—21no—
37C4⋊C4×C71—22no—
38C4⋊C4×C71—22no—
39C4⋊C4×C71—21no—
40C4⋊C4×C71—21no—
41C4⋊C4×C71—21no—
42C4⋊C4×C71D411yescharacteristic
43C4⋊C4×C71D411yescharacteristic
44C22⋊C4×C71—22no—
45C22⋊C4×C71—21no—
46C42×C71—21no—
47C42×C71C2311yescharacteristic
48C23×C71—21no—
49Q8×C71—21no—
50Q8×C71—21no—
51Q8×C71—42no—
52D4×C71—21no—
53D4×C71Q1612yescomplemented
54C2×C4×C71—21no—
55C2×C4×C71—21no—
56C2×C4×C71—21no—
57C2×C4×C71—22no—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71—82no—
61C2×C4×C71C2×D411yescharacteristic
62C2×C4×C71C2×D411yescharacteristic
63C2×C4×C71C2×D411yescharacteristic
64C2×Dic71—41no—
65Dic142—42no—
66C71⋊C8—22no—
67C71⋊C8—41no—
68C71⋊C8—41no—
69C22×C71—22no—
70C22×C71—41no—
71C22×C71—41no—
72C22×C71#12011yescharacteristic
73C4×C71—21no—
74C4×C71—41no—
75C4×C71—41no—
76C4×C71—41no—
77C4×C71—42no—
78C4×C71C2×Q1611yescharacteristic
79C4×C71#13511yescharacteristic
80C4×C71#13711yescharacteristic
81C4×C71#13711yescharacteristic
82Dic71—81no—
83C2×C71—42no—
84C2×C71#45111yescharacteristic
85C2×C71#45211yescharacteristic
86C2×C71#45311yescharacteristic
87#1632—711nomaximal, Sylow
88C71#163211yescharacteristic, complemented, Sylow
89#554D7111yescharacteristic
90#494—711no—
91#477—711no—
92#476—711no—
93#333—711no—
94Q8⋊C8—711no—
95D4⋊C8—711no—
96C2×Q16—1422no—
97C4⋊Q8—711no—
98C4⋊Q8C2×D7111yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D7111yescharacteristic
103C4×D4—21no—
104C4×D4C2×D7111yescharacteristic
105C2×C4⋊C4—21no—
106#107—1421no—
107#106—1421no—
108C4⋊C8—711no—
109C4⋊C8—711no—
110#103—1421no—
111#102—1421no—
112C4×C8—711no—
113C2×Q8—1421no—
114C2×Q8C22⋊D7111yescharacteristic
115C2×Q8C22⋊D7112yes—
116C2×D4C22⋊D7111yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—2844no—
120C2×C8—1421no—
121C2×C8—1421no—
122C2×C8—1421no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—1421no—
129C4⋊C4—1421no—
130C4⋊C4C22⋊D7111yescharacteristic
131C4⋊C4C22⋊D7111yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D7111yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—2842no—
141D4—21no—
142D4#1176585512yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—2841no—
151C2×C4C2×C22⋊D7111yescharacteristic
152C2×C4C2×C22⋊D7111yescharacteristic
153C2×C4C2×C22⋊D7111yescharacteristic
154C8—2841no—
155C8—2841no—
156C8—1422no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1455609711yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—5681no—
167C4C2×#1176585511yescharacteristic
168C4#1455614811yescharacteristic
169C4#1455615811yescharacteristic
170C4#1455615911yescharacteristic
171C2—42no—
172C2#1798615511yescharacteristic
173C2#1798616111yescharacteristic
174C2#1798616711yescharacteristic
subgroupG/H
centerC22#14556097
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#554×C71C2
socleC22×C71#120
hypercenter#554D71
hypocenterC71#1632
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #17986378C2semidirect
2. #17986294C2semidirect
4. #17985748C2semidirect
5. Q8⋊(C71⋊C8)C2semidirect
53. D4×C71Q16semidirect
88. C71#1632semidirect
142. D4#11765855semidirect