Finite Group Databasebeta

← #22588397

Subgroups — #22588397

Order 9088.

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

normal subgroups only

This group has 264 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
1C2×#14556135C211yesmaximal, characteristic
2C2×#14556063C211yesmaximal, characteristic, complemented
3C2×#14556059C211yesmaximal, characteristic, complemented
4#17986000C211yesmaximal, characteristic
5#17985768C211yesmaximal, characteristic, complemented
6#17985767C211yesmaximal, characteristic, complemented
7#417×C71C211yesmaximal, characteristic
8#14556135—22no—
9#14556135C414yescomplemented
10C2×#11765849—21no—
11C2×#11765849C2211yescharacteristic
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8—21no—
14C22×C71⋊C8C2211yescharacteristic
15C2×C4⋊Dic71C2211yescharacteristic
16#14556063—22no—
17#14556059—21no—
18#14556059—21no—
19#14556052—21no—
20#14556049—21no—
21C2×#41×C71C2211yescharacteristic
22C2×M16×C71C2211yescharacteristic
23C2×C4⋊C4×C71C2211yescharacteristic
24#103×C71—21no—
25#102×C71—21no—
26C22×Dic71—21no—
27#11765849—21no—
28#11765849—24no—
29#11765849—21no—
30#11765849—21no—
31#11765849C2×C412yes—
32C2×C71⋊C8—21no—
33C2×C71⋊C8—24no—
34C2×C71⋊C8—21no—
35C2×C71⋊C8C2×C412yes—
36C2×C71⋊C8D411yescharacteristic
37C2×C71⋊C8D411yescharacteristic
38C2×C71⋊C8D412yes—
39C4⋊Dic71—21no—
40C4⋊Dic71—21no—
41#41×C71—21no—
42#41×C71—21no—
43#41×C71C2×C412yes—
44#41×C71Q812yes—
45C2×Q8×C71D411yescharacteristic
46C2×D4×C71—21no—
47C2×D4×C71D411yescharacteristic
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71C2311yescharacteristic
51M16×C71—21no—
52M16×C71—21no—
53C2×C8×C71—21no—
54C4⋊C4×C71—21no—
55C4⋊C4×C71—21no—
56C23×C71—21no—
57C23×C71#4111yescharacteristic
58Q8×C71—21no—
59Q8×C71C4⋊C412yes—
60D4×C71—22no—
61D4×C71—22no—
62D4×C71—21no—
63D4×C71C4⋊C412yes—
64C2×C4×C71—21no—
65C2×C4×C71—22no—
66C2×C4×C71—22no—
67C2×C4×C71—41no—
68C2×C4×C71—81no—
69C2×C4×C71C22×C411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×Q811yescharacteristic
74C2×C4×C71#4111yescharacteristic
75C2×Dic71—41no—
76C2×Dic71—81no—
77C8×C71—41no—
78C71⋊C8—22no—
79C71⋊C8—41no—
80C71⋊C8C22⋊C412yes—
81C71⋊C8C22⋊C412yes—
82C22×C71—21no—
83C22×C71—21no—
84C22×C71—22no—
85C22×C71—41no—
86C22×C71—41no—
87C22×C71#11711yescharacteristic
88C22×C71#12111yescharacteristic
89C22×C71C22⋊Q811yescharacteristic
90C4×C71—22no—
91C4×C71—81no—
92C4×C71C2×C22⋊C411yescharacteristic
93C4×C71C2×C4⋊C411yescharacteristic
94C4×C71#12111yescharacteristic
95C4×C71C22⋊Q811yescharacteristic
96Dic71—81no—
97C2×C71—21no—
98C2×C71—21no—
99C2×C71—42no—
100C2×C71#38011yescharacteristic
101C2×C71#47111yescharacteristic
102C2×C71#47111yescharacteristic
103#1815—711nomaximal, Sylow
104C71#181511yescharacteristic, complemented, Sylow
105C2×#131—711no—
106C2×#108—711no—
107C2×#106—711no—
108#417—711no—
109#417D7111yescharacteristic
110#343—711no—
111#343—711no—
112C2×#41C2×D7111yescharacteristic
113#131—714no—
114#131—1422no—
115C2×M16—711no—
116C2×M16—711no—
117C2×M16—1421no—
118C2×M16C2×D7111yescharacteristic
119C22×C8—711no—
120C22×C8—1421no—
121C2×C4⋊C4—711no—
122C2×C4⋊C4C2×D7111yescharacteristic
123#108—1422no—
124#106—1421no—
125#106—1421no—
126#103—21no—
127#103—1421no—
128#102—21no—
129#102—1421no—
130#41—21no—
131#41—21no—
132#41Dic14212yes—
133#41C4×D7112yes—
134C2×Q8C22⋊D7111yescharacteristic
135C2×D4—21no—
136C2×D4C22⋊D7111yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—1421no—
140C22×C4C22×D7111yescharacteristic
141M16—21no—
142M16—21no—
143M16—712no—
144M16—1421no—
145M16—1421no—
146M16—1424no—
147M16—1421no—
148C2×C8—21no—
149C2×C8—712no—
150C2×C8—711no—
151C2×C8—711no—
152C2×C8—712no—
153C2×C8—1421no—
154C2×C8—1424no—
155C2×C8—1421no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—1421no—
159C4⋊C4—1421no—
160C23—21no—
161C23#1176586411yescharacteristic
162Q8—21no—
163Q8Dic71⋊C412yes—
164D4—22no—
165D4—22no—
166D4—21no—
167D4Dic71⋊C412yes—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—2841no—
174C2×C4—5681no—
175C2×C4C2×C4×D7111yescharacteristic
176C2×C4D4×D7111yescharacteristic
177C2×C4D4×D7111yescharacteristic
178C2×C4C2×C22⋊D7111yescharacteristic
179C2×C4C2×Dic14211yescharacteristic
180C2×C4#1176586411yescharacteristic
181C8—41no—
182C8—2841no—
183C8—712no—
184C8—712no—
185C8—1422no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22#1455608411yescharacteristic
192C22#1455610011yescharacteristic
193C22#1455610511yescharacteristic
194C4—22no—
195C4—81no—
196C4—5681no—
197C4C22⋊C4×D7111yescharacteristic
198C4C2×Dic71⋊C411yescharacteristic
199C4#1455610011yescharacteristic
200C4#1455610511yescharacteristic
201C2—21no—
202C2—21no—
203C2—42no—
204C2#1798585711yescharacteristic
205C2#1798626011yescharacteristic
206C2#1798626411yescharacteristic
subgroupG/H
centerC22#14556100
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#417×C71C2
socleC22×C71#121
hypercenter#417D71
hypocenterC71#1815
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#14556063C2semidirect
3. C2×#14556059C2semidirect
5. #17985768C2semidirect
6. #17985767C2semidirect
9. #14556135C4semidirect
104. C71#1815semidirect