Finite Group Databasebeta

← #22601030

Subgroups — #22601030

Order 9088.

336 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986724C211yesmaximal, characteristic, complemented
2#17986723C211yesmaximal, characteristic
3#17986355C212yesmaximal
4#17986304C212yesmaximal, complemented
5#17986276C211yesmaximal, characteristic
6#17986224C212yesmaximal
7#17986206C211yesmaximal, characteristic, complemented
8C8×Dic142C211yesmaximal, characteristic, complemented
9#17986091C212yesmaximal, complemented
10C4×#11765854C211yesmaximal, characteristic
11#487×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2212yes—
13C2×#11765854C2211yescharacteristic
14#14556126—21no—
15#14556123—22no—
16#14556119C2212yes—
17#14556118—22no—
18#14556117C2212yes—
19#14556108—21no—
20Q8×Dic71—21no—
21Q8×Dic71C2212yes—
22C4×Dic142C2211yescharacteristic
23C4×Dic142C2211yescharacteristic
24#14556093C2212yes—
25C4×D284C2211yescharacteristic
26#14556086—21no—
27#14556061C2212yes—
28#14556060C2211yescharacteristic
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556052C2212yes—
33#14556050C2211yescharacteristic
34#14556048C2212yes—
35C8×Dic71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142D411yescharacteristic
42C2×Dic142C2311yescharacteristic
43C2×D284C2311yescharacteristic
44C2×C4×D71—21no—
45#11765855—24no—
46#11765854—22no—
47C2×C71⋊C8C2312yes—
48C4⋊Dic71—22no—
49C4⋊Dic71—22no—
50C4⋊Dic71D411yescharacteristic
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53Dic71⋊C4—21no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56Dic71⋊C4D412yes—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60C4×Dic71—21no—
61C4×Dic71—22no—
62C4×Dic71—22no—
63C4×Dic71C2312yes—
64C2×Q8×C71—21no—
65C2×Q8×C71C2312yes—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71—21no—
71C4⋊C4×C71C2312yes—
72C42×C71C2311yescharacteristic
73C22×D71—21no—
74Q8×C71—22no—
75Q8×C71—22no—
76Q8×C71—42no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2×D411yescharacteristic
81C2×C4×C71C2411yescharacteristic
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71C2×D412yes—
86C2×Dic71C2×D412yes—
87D284—21no—
88D284—21no—
89C4×D71—42no—
90Dic142—22no—
91Dic142—21no—
92Dic142#4112yes—
93C8×C71—21no—
94C8×C71—21no—
95C71⋊C8—22no—
96C22×C71C22×D411yescharacteristic
97C2×D71—41no—
98C2×D71—41no—
99C2×D71—41no—
100C4×C71—21no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71#13512yes—
104C4×C71C2×#4111yescharacteristic
105C4×C712−1+411yescharacteristic
106Dic71—22no—
107Dic71—22no—
108Dic71—22no—
109Dic71—41no—
110Dic71—41no—
111C2×C71#54811yescharacteristic
112C2×C71C2×#13511yescharacteristic
113C2×C71#57811yescharacteristic
114D71—81no—
115#3364—711nomaximal, Sylow
116C71#336411yescharacteristic, complemented, Sylow
117#557—711no—
118#556—711no—
119#488—712no—
120#487D7111yescharacteristic, complemented
121#479—712no—
122#474—711no—
123#464—712no—
124#462—711no—
125C8×Q8—711no—
126C4×Q16—712no—
127C4×SD16—711no—
128C2×Q16—712no—
129C2×SD16—711no—
130C4⋊Q8—1421no—
131C4⋊Q8C2×D7111yescharacteristic
132C4⋊Q8C2×D7111yescharacteristic
133#126—1422no—
134#125—712no—
135#125—1422no—
136#124—712no—
137C22⋊Q8—1421no—
138C4×Q8—711no—
139C4×Q8—712no—
140C4×Q8—712no—
141C4×Q8—711no—
142C4×Q8—1421no—
143C4×D4—711no—
144#117—1421no—
145#107—712no—
146#106—711no—
147C4⋊C8—712no—
148C4⋊C8—711no—
149#103—712no—
150#103—711no—
151#103C2×D7112yes—
152#103C2×D7112yes—
153#102—711no—
154#102—712no—
155C4×C8—712no—
156C4×C8C2×D7111yescharacteristic
157C2×Q8—21no—
158C2×Q8—711no—
159C2×Q8—711no—
160C2×Q8C22×D7112yes—
161C2×D4—711no—
162C22×C4—1421no—
163Q16—1424no—
164SD16—1422no—
165C2×C8—712no—
166C2×C8C22×D7111yescharacteristic
167C2×C8C22×D7111yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—712no—
172C4⋊C4—711no—
173C4⋊C4—711no—
174C4⋊C4—711no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4—1421no—
180C4⋊C4C22×D7112yes—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4—1421no—
184C42—712no—
185C42—1422no—
186C42—1422no—
187C42—1421no—
188C42C22×D7111yescharacteristic
189C23—1421no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—712no—
194Q8—1421no—
195Q8—1422no—
196D4—1421no—
197D4—1421no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2842no—
201C2×C4—712no—
202C2×C4—712no—
203C2×C4—1421no—
204C2×C4—1421no—
205C2×C4—1421no—
206C2×C4D4×D7111yescharacteristic
207C2×C4D4×D7111yescharacteristic
208C2×C4C23×D7111yescharacteristic
209C8—21no—
210C8—21no—
211C8—1422no—
212C22—2841no—
213C22—2841no—
214C22—2841no—
215C22C2×D4×D7111yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—2841no—
220C4—2841no—
221C4—1422no—
222C4—1422no—
223C4—1422no—
224C4#1455614912yes—
225C4#41×D7111yescharacteristic
226C4#1455617511yescharacteristic
227C2—5681no—
228C2#1798665711yescharacteristic
229C2C2×#1455614911yescharacteristic
230C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3364
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986724C2semidirect
4. #17986304C2semidirect
7. #17986206C2semidirect
8. C8×Dic142C2semidirect
9. #17986091C2semidirect
11. #487×C71C2semidirect
116. C71#3364semidirect
120. #487D71semidirect