Finite Group Databasebeta

← #22599394

Subgroups — #22599394

Order 9088.

388 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986603C211yesmaximal, characteristic, complemented
2#17986574C211yesmaximal, characteristic, complemented
3#17986386C211yesmaximal, characteristic, complemented
4#17986367C211yesmaximal, characteristic, complemented
5#17986354C211yesmaximal, characteristic, complemented
6#17986216C212yesmaximal, complemented
7#17986202C212yesmaximal, complemented
8#17986143C212yesmaximal, complemented
9#17986136C212yesmaximal, complemented
10#17986058C211yesmaximal, characteristic, complemented
11#492×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×Q8×D71C2211yescharacteristic
14C2×#11765853C2212yes—
15C2×#11765850C2212yes—
16#14556126C2211yescharacteristic
17#14556120C2211yescharacteristic
18#14556117C2211yescharacteristic
19#14556116C2211yescharacteristic
20#14556113—22no—
21#14556109—22no—
22#14556106—22no—
23#14556100—22no—
24#14556099C2212yes—
25D4×Dic71C2212yes—
26#14556085C2211yescharacteristic
27Dic71⋊C8C2212yes—
28#14556053C2212yes—
29#14556052C2212yes—
30#14556052C2212yes—
31#14556050C2212yes—
32(C2×D71)⋊C8C2212yes—
33#14556039C2211yescharacteristic
34C2×SD16×C71C2212yes—
35C2×SD16×C71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38C4×C8×C71C2211yescharacteristic
39#11765866—42no—
40C2×Dic142—21no—
41C2×Dic142C2311yescharacteristic
42Q8×D71—42no—
43C2×D284—21no—
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—22no—
46C22×Dic71—22no—
47C2×C4×D71—21no—
48C2×C4×D71—21no—
49C2×C4×D71C2311yescharacteristic
50#11765853—42no—
51#11765850—42no—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71C2312yes—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56Dic71⋊C4D412yes—
57C22⋊Dic71—22no—
58C22⋊Dic71—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4D412yes—
62C4×Dic71C2311yescharacteristic
63C2×Q8×C71C2311yescharacteristic
64C2×Q8×C71C2311yescharacteristic
65C2×D4×C71—22no—
66C2×D4×C71C2312yes—
67SD16×C71—42no—
68SD16×C71—42no—
69C2×C8×C71C2312yes—
70C4⋊C4×C71—22no—
71C42×C71C2311yescharacteristic
72C23×C71—22no—
73C22×D71—21no—
74C22×D71C2×D411yescharacteristic
75Q8×C71—22no—
76Q8×C71—22no—
77D4×C71—22no—
78D4×C71—22no—
79D4×C71—82no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2411yescharacteristic
84C22⋊D71—82no—
85C2×Dic71—21no—
86C2×Dic71—22no—
87C2×Dic71—82no—
88C2×Dic71C2×D412yes—
89C2×Dic71C2×D411yescharacteristic
90D284—22no—
91D284—42no—
92C4×D71—22no—
93C4×D71—42no—
94C4×D71—42no—
95Dic142—22no—
96Dic142—42no—
97C8×C71—22no—
98C71⋊C8—22no—
99C22×C71—42no—
100C22×C71—42no—
101C22×C71—42no—
102C22×C71C22×D411yescharacteristic
103C2×D71—21no—
104C2×D71—22no—
105C2×D71—41no—
106C2×D71—42no—
107C4×C71—22no—
108C4×C71—41no—
109C4×C71—41no—
110C4×C71H(5,𝔽2)12yes—
111Dic71—21no—
112Dic71—22no—
113Dic71—41no—
114Dic71—42no—
115C2×C71—82no—
116C2×C71#53511yescharacteristic
117C2×C71#57712yes—
118D71—41no—
119D71—81no—
120#3221—711nomaximal, Sylow
121C71#322111yescharacteristic, complemented, Sylow
122#540—711no—
123#536—711no—
124#496—711no—
125#492D7111yescharacteristic, complemented
126#491—711no—
127#488—711no—
128#463—712no—
129#461—712no—
130#449—712no—
131#448—712no—
132#433—711no—
133C2×#41—711no—
134C22×Q8—711no—
135C2×SD16—712no—
136C2×SD16C2×D7112yes—
137C2×SD16C2×D7112yes—
138C2×D8—712no—
139C4⋊Q8—711no—
140C4⋊Q8C2×D7111yescharacteristic
141#127C2×D7111yescharacteristic
142#125—711no—
143#124—711no—
144#124—711no—
145#123—1422no—
146#123—1422no—
147C22⋊Q8—1422no—
148#121—712no—
149#121—1422no—
150C4×D4—712no—
151#117—711no—
152C4⋊C8—712no—
153#103—712no—
154#103—712no—
155#103—712no—
156#102—712no—
157C22⋊C8—712no—
158#97—711no—
159C4×C8C2×D7111yescharacteristic
160#41—2842no—
161C2×Q8—2842no—
162C2×Q8—711no—
163C2×Q8—1421no—
164C2×Q8C22×D7111yescharacteristic
165C2×Q8C22×D7111yescharacteristic
166C2×D4—22no—
167C2×D4—711no—
168C2×D4—1421no—
169C2×D4—1422no—
170C2×D4C22×D7112yes—
171C22×C4—711no—
172C22×C4—1421no—
173C22×C4—1421no—
174C22×C4—1422no—
175SD16—42no—
176SD16—42no—
177SD16—2842no—
178D8—2842no—
179C2×C8—712no—
180C2×C8C22×D7112yes—
181C4⋊C4—22no—
182C4⋊C4—712no—
183C4⋊C4—712no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C22⋊C4—712no—
187C22⋊C4—1422no—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C22⋊C4—1422no—
191C42—711no—
192C42C22×D7111yescharacteristic
193C23—22no—
194C23—711no—
195C23—1421no—
196Q8—22no—
197Q8—22no—
198Q8—2842no—
199Q8—1422no—
200D4—22no—
201D4—22no—
202D4—82no—
203D4—2842no—
204D4—5682no—
205D4—1422no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—2842no—
209C2×C4—2842no—
210C2×C4—5682no—
211C2×C4—712no—
212C2×C4—711no—
213C2×C4—1422no—
214C2×C4—1421no—
215C2×C4—1422no—
216C2×C4D4×D7112yes—
217C2×C4C23×D7111yescharacteristic
218C8—22no—
219C8—1422no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—2841no—
224C22—2842no—
225C22—1421no—
226C22—1422no—
227C22C2×D4×D7111yescharacteristic
228C4—22no—
229C4—41no—
230C4—41no—
231C4—2841no—
232C4—2842no—
233C4—1421no—
234C4—1422no—
235C4#1455617312yes—
236C2—82no—
237C2—2841no—
238C2—5681no—
239C2#1798656211yescharacteristic
240C2#1798680512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#492×C71C2
socleC22×C71C22×D4
hypercenter#492D71
hypocenterC71#3221
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #17986603C2semidirect
2. #17986574C2semidirect
3. #17986386C2semidirect
4. #17986367C2semidirect
5. #17986354C2semidirect
6. #17986216C2semidirect
7. #17986202C2semidirect
8. #17986143C2semidirect
9. #17986136C2semidirect
10. #17986058C2semidirect
11. #492×C71C2semidirect
121. C71#3221semidirect
125. #492D71semidirect