Finite Group Databasebeta

← #22587586

Subgroups — #22587586

Order 9088.

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

normal subgroups only

This group has 234 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#17986664C211yesmaximal, characteristic
2#17986276C211yesmaximal, characteristic
3#17986206C211yesmaximal, characteristic, complemented
4#17985745C211yesmaximal, characteristic, complemented
5#17985731C211yesmaximal, characteristic, complemented
6#17985728C211yesmaximal, characteristic
7#500×C71C211yesmaximal, characteristic, complemented
8C2×Q8×D71—21no—
9C2×#11765855—21no—
10#14556127—21no—
11#14556106—22no—
12#14556104—21no—
13C4×Dic142C2211yescharacteristic
14C4×D284C2211yescharacteristic
15#14556090—21no—
16#14556061—21no—
17C4⋊(C71⋊C8)C2211yescharacteristic
18C4⋊(C71⋊C8)C2211yescharacteristic
19#14556052—21no—
20#14556048—21no—
21C4⋊Q8×C71C2211yescharacteristic
22C4⋊Q8×C71C2211yescharacteristic
23#107×C71—22no—
24C4×C8×C71C2211yescharacteristic
25C2×Dic142—22no—
26C2×Dic142D411yescharacteristic
27Q8×D71—42no—
28Q8×D71—42no—
29C2×D284D411yescharacteristic
30C2×C4×D71—22no—
31C2×C4×D71—21no—
32#11765855—42no—
33C2×C71⋊C8—21no—
34C2×C71⋊C8—21no—
35C4⋊Dic71D411yescharacteristic
36C4⋊Dic71D411yescharacteristic
37Dic71⋊C4—22no—
38Dic71⋊C4—21no—
39(C2×D71)⋊C4—22no—
40(C2×D71)⋊C4—21no—
41C4×Dic71—21no—
42C2×Q8×C71—21no—
43C2×Q8×C71—21no—
44C2×C8×C71—21no—
45C4⋊C4×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—21no—
48C4⋊C4×C71D412yes—
49C42×C71C2311yescharacteristic
50C22×D71—21no—
51Q8×C71—41no—
52Q8×C71—41no—
53Q8×C71—41no—
54Q8×C71—41no—
55C2×C4×C71—22no—
56C2×C4×C71—41no—
57C2×C4×C71C2×D411yescharacteristic
58C2×C4×C71C2×D411yescharacteristic
59C2×C4×C71C2×D411yescharacteristic
60C2×Dic71—21no—
61C2×Dic71—21no—
62C2×Dic71—21no—
63D284—21no—
64D284Q1612yescomplemented
65C4×D71—44no—
66C4×D71—41no—
67C4×D71—41no—
68Dic142—21no—
69Dic142—44no—
70Dic142D812yes—
71C8×C71—22no—
72C71⋊C8—41no—
73C71⋊C8—41no—
74C22×C71#12011yescharacteristic
75C2×D71—22no—
76C2×D71—41no—
77C2×D71—41no—
78C4×C71—21no—
79C4×C71—42no—
80C4×C71—81no—
81C4×C71C2×D811yescharacteristic
82C4×C71C2×Q1611yescharacteristic
83C4×C71AGL(1,ℤ/8)11yescharacteristic
84C4×C71#13711yescharacteristic
85Dic71—22no—
86Dic71—41no—
87Dic71—41no—
88C2×C71#44711yescharacteristic
89C2×C71#45111yescharacteristic
90C2×C71#45611yescharacteristic
91D71—42no—
92#1649—711nomaximal, Sylow
93C71#164911yescharacteristic, complemented, Sylow
94D4×Q8—711no—
95#500D7111yescharacteristic, complemented
96#474—711no—
97#462—711no—
98#332—711no—
99Q8⋊C8—711no—
100D4⋊C8—711no—
101C22×Q8—1421no—
102C2×Q16—1421no—
103C4⋊Q8—1421no—
104C4⋊Q8C2×D7111yescharacteristic
105C4⋊Q8C2×D7111yescharacteristic
106C22⋊Q8—1421no—
107C22⋊Q8—1422no—
108C4×Q8—711no—
109C4×D4—711no—
110C4×D4—1421no—
111#107—22no—
112#107—1421no—
113C4⋊C8—711no—
114C4⋊C8—711no—
115#103—1421no—
116#102—1421no—
117C4×C8C2×D7111yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—2842no—
121C2×Q8—2842no—
122C2×Q8—711no—
123C2×Q8—1422no—
124C2×D4—711no—
125C22×C4—1422no—
126C22×C4—1421no—
127Q16—2842no—
128C2×C8—21no—
129C2×C8—1421no—
130C2×C8—1421no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—711no—
135C4⋊C4—711no—
136C4⋊C4—1422no—
137C4⋊C4—1421no—
138C4⋊C4C22⋊D7112yes—
139C22⋊C4—1422no—
140C22⋊C4—1421no—
141C42—1421no—
142C42C22×D7111yescharacteristic
143C23—1421no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—2844no—
149Q8—712no—
150Q8—1421no—
151D4—712no—
152D4—1421no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—2844no—
156C2×C4—2841no—
157C2×C4—2841no—
158C2×C4—1421no—
159C2×C4—1421no—
160C2×C4—1421no—
161C2×C4D4×D7111yescharacteristic
162C2×C4D4×D7111yescharacteristic
163C2×C4C2×C22⋊D7111yescharacteristic
164C8—22no—
165C8—2841no—
166C8—2841no—
167C22—2841no—
168C22—2841no—
169C22—1422no—
170C22#1455609611yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—2841no—
175C4—2841no—
176C4—1422no—
177C4D8×D7111yescharacteristic
178C4Q16×D7111yescharacteristic
179C4#1455615211yescharacteristic
180C4#1455616011yescharacteristic
181C2—2842no—
182C2#1798613111yescharacteristic
183C2#1798615311yescharacteristic
184C2#1798618111yescharacteristic
subgroupG/H
centerC22#14556096
commutatorC42×C71C23
FrattiniC42C22×D71
Fitting#500×C71C2
socleC22×C71#120
hypercenter#500D71
hypocenterC71#1649
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #17986206C2semidirect
4. #17985745C2semidirect
5. #17985731C2semidirect
7. #500×C71C2semidirect
64. D284Q16semidirect
93. C71#1649semidirect
95. #500D71semidirect