Finite Group Databasebeta

← #22598970

Subgroups — #22598970

Order 9088.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986612C211yesmaximal, characteristic, complemented
2#17986605C211yesmaximal, characteristic
3#17986372C211yesmaximal, characteristic, complemented
4#17986369C211yesmaximal, characteristic, complemented
5#17986365C211yesmaximal, characteristic, complemented
6#17986246C212yesmaximal, complemented
7#17986229C212yesmaximal, complemented
8#17986098C212yesmaximal, complemented
9#17986086C212yesmaximal, complemented
10#17985953C211yesmaximal, characteristic, complemented
11#499×C71C211yesmaximal, characteristic, complemented
12C2×#11765854C2212yes—
13C2×#11765854C2212yes—
14C2×#11765848C2211yescharacteristic
15C2×C8×D71C2211yescharacteristic
16#14556128C2211yescharacteristic
17#14556125C2211yescharacteristic
18#14556123—22no—
19#14556119—22no—
20#14556113—22no—
21#14556108C2212yes—
22#14556103—22no—
23#14556103C2212yes—
24#14556093C2212yes—
25#14556090C2212yes—
26#14556085C2211yescharacteristic
27#14556051C2212yes—
28#14556051C2212yes—
29#14556048C2212yes—
30#14556048C2212yes—
31#14556039C2211yescharacteristic
32#14556038C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34#125×C71C2211yescharacteristic
35#125×C71C2211yescharacteristic
36#107×C71C2212yes—
37#106×C71C2212yes—
38C4×C8×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40C2×D284—22no—
41C2×D284—22no—
42C2×D284C2312yes—
43C2×C4×D71—22no—
44C2×C4×D71C2311yescharacteristic
45#11765854—24no—
46#11765854—24no—
47#11765848—22no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2311yescharacteristic
50C8×D71—22no—
51C4⋊Dic71—22no—
52C4⋊Dic71—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4D412yes—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4D412yes—
59C4×Dic71—22no—
60C4×Dic71C2311yescharacteristic
61C2×C8×C71C2311yescharacteristic
62C2×C8×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71C2312yes—
66C4⋊C4×C71C2312yes—
67C42×C71C2311yescharacteristic
68C22×D71—22no—
69C22×D71C2×D411yescharacteristic
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D412yes—
73C2×C4×C71C2411yescharacteristic
74C2×Dic71—22no—
75C2×Dic71C2×D412yes—
76C2×Dic71C2×D411yescharacteristic
77D284—22no—
78D284—22no—
79D284—82no—
80D284—82no—
81C4×D71—21no—
82C4×D71—21no—
83C4×D71—82no—
84Dic142—22no—
85Dic142—22no—
86C8×C71#4112yes—
87C8×C71#4112yes—
88C71⋊C8—21no—
89C71⋊C8—21no—
90C22×C71C22×D411yescharacteristic
91C2×D71—21no—
92C2×D71—22no—
93C2×D71—42no—
94C2×D71—42no—
95C2×D71—42no—
96C4×C71—22no—
97C4×C71—42no—
98C4×C71—42no—
99C4×C71C2×#4111yescharacteristic
100C4×C71C2×#4111yescharacteristic
101Dic71—21no—
102Dic71—22no—
103Dic71—42no—
104C2×C71#53211yescharacteristic
105C2×C71#57712yes—
106D71—41no—
107D71—82no—
108#3174—711nomaximal, Sylow
109C71#317411yescharacteristic, complemented, Sylow
110#541—711no—
111#540—711no—
112#499D7111yescharacteristic, complemented
113#492—711no—
114#491—711no—
115#490—711no—
116#468—712no—
117#465—712no—
118#440—712no—
119C4×SD16—712no—
120#405—711no—
121C2×SD16—712no—
122C2×SD16—712no—
123C2×M16—711no—
124C22×C8—711no—
125C4⋊Q8—711no—
126#127—711no—
127#126—1422no—
128#125—1422no—
129#125C2×D7111yescharacteristic
130#125C2×D7111yescharacteristic
131#123—1422no—
132C22⋊Q8—712no—
133#121—712no—
134#121—1422no—
135C4×Q8—712no—
136C4×D4—712no—
137#117—711no—
138#107C2×D7112yes—
139#106C2×D7112yes—
140#103—712no—
141#103—712no—
142#102—712no—
143#102—712no—
144#97—711no—
145#97—711no—
146C4×C8—711no—
147C4×C8C2×D7111yescharacteristic
148C2×Q8—712no—
149C2×D4—712no—
150C2×D4—1422no—
151C2×D4—1422no—
152C22×C4—711no—
153C22×C4—1422no—
154SD16—1424no—
155SD16—1424no—
156M16—1422no—
157C2×C8—711no—
158C2×C8—711no—
159C2×C8—1422no—
160C2×C8C22×D7111yescharacteristic
161C2×C8C22×D7111yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—712no—
165C4⋊C4—1422no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4C22×D7112yes—
169C4⋊C4C22×D7112yes—
170C22⋊C4—712no—
171C22⋊C4—1422no—
172C22⋊C4—1422no—
173C22⋊C4—1422no—
174C42—711no—
175C42—1422no—
176C42C22×D7111yescharacteristic
177C23—711no—
178C23—1422no—
179Q8—1422no—
180Q8—1422no—
181D4—5682no—
182D4—5682no—
183D4—1422no—
184D4—1422no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—5682no—
188C2×C4—712no—
189C2×C4—711no—
190C2×C4—1422no—
191C2×C4—1421no—
192C2×C4—1421no—
193C2×C4D4×D7112yes—
194C2×C4C23×D7111yescharacteristic
195C8—1421no—
196C8—1421no—
197C8#1176586612yes—
198C8#1176586612yes—
199C22—2842no—
200C22—2842no—
201C22—2842no—
202C22—1421no—
203C22—1422no—
204C22C2×D4×D7111yescharacteristic
205C4—22no—
206C4—42no—
207C4—42no—
208C4—2842no—
209C4—1421no—
210C4—1422no—
211C4C2×#1176586611yescharacteristic
212C4C2×#1176586611yescharacteristic
213C2—2841no—
214C2—5682no—
215C2#1798655011yescharacteristic
216C2#1798680612yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#499×C71C2
socleC22×C71C22×D4
hypercenter#499D71
hypocenterC71#3174
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #17986612C2semidirect
3. #17986372C2semidirect
4. #17986369C2semidirect
5. #17986365C2semidirect
6. #17986246C2semidirect
7. #17986229C2semidirect
8. #17986098C2semidirect
9. #17986086C2semidirect
10. #17985953C2semidirect
11. #499×C71C2semidirect
109. C71#3174semidirect
112. #499D71semidirect