Finite Group Databasebeta

← #22599424

Subgroups — #22599424

Order 9088.

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

normal subgroups only

This group has 364 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#17986612C211yesmaximal, characteristic, complemented
2#17986574C211yesmaximal, characteristic
3#17986393C211yesmaximal, characteristic, complemented
4#17986378C211yesmaximal, characteristic, complemented
5#17986356C211yesmaximal, characteristic, complemented
6#17986224C212yesmaximal, complemented
7#17986208C212yesmaximal, complemented
8#17986164C212yesmaximal
9#17986156C212yesmaximal, complemented
10#17986061C211yesmaximal, characteristic, complemented
11#491×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×Q8×D71C2211yescharacteristic
14C2×Dic284C2212yes—
15C2×#11765855C2212yes—
16C2×#11765855C2212yes—
17C2×#11765854C2212yes—
18#14556128C2211yescharacteristic
19#14556126C2211yescharacteristic
20#14556123—22no—
21#14556119—22no—
22#14556117C2211yescharacteristic
23#14556116C2211yescharacteristic
24#14556113—22no—
25#14556108C2212yes—
26#14556106—22no—
27#14556093C2212yes—
28#14556085C2211yescharacteristic
29Dic71⋊C8C2212yes—
30#14556051C2212yes—
31#14556048C2212yes—
32(C2×D71)⋊C8C2212yes—
33C4×C71⋊C8C2211yescharacteristic
34C4⋊Q8×C71C2211yescharacteristic
35#125×C71C2211yescharacteristic
36#103×C71C2212yes—
37#103×C71C2212yes—
38#97×C71C2211yescharacteristic
39#11765866—42no—
40C2×Dic142—21no—
41C2×Dic142C2311yescharacteristic
42C2×Dic142C2312yes—
43Q8×D71—42no—
44C2×D284—21no—
45C2×D284C2311yescharacteristic
46C2×C4×D71—21no—
47C2×C4×D71—21no—
48C2×C4×D71C2311yescharacteristic
49Dic284—42no—
50#11765855—42no—
51#11765855—42no—
52#11765854—42no—
53C2×C71⋊C8C2312yes—
54C4⋊Dic71—22no—
55C4⋊Dic71—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4D412yes—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4D412yes—
63C4×Dic71—22no—
64C4×Dic71C2311yescharacteristic
65C2×Q8×C71C2311yescharacteristic
66C2×Q8×C71C2311yescharacteristic
67C2×C8×C71C2312yes—
68C4⋊C4×C71—22no—
69C4⋊C4×C71—22no—
70C4⋊C4×C71C2312yes—
71C42×C71C2311yescharacteristic
72C22×D71—21no—
73C22×D71C2×D411yescharacteristic
74Q8×C71—22no—
75Q8×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71C2411yescharacteristic
81C2×Dic71—22no—
82C2×Dic71—21no—
83C2×Dic71C2×D412yes—
84C2×Dic71C2×D411yescharacteristic
85D284—22no—
86D284—42no—
87C4×D71—22no—
88C4×D71—42no—
89C4×D71—42no—
90Dic142—22no—
91Dic142—22no—
92Dic142—22no—
93Dic142—42no—
94C8×C71—22no—
95C71⋊C8—22no—
96C22×C71C22×D411yescharacteristic
97C2×D71—21no—
98C2×D71—22no—
99C2×D71—41no—
100C2×D71—42no—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—42no—
105C4×C71H(5,𝔽2)12yes—
106Dic71—21no—
107Dic71—22no—
108Dic71—41no—
109Dic71—42no—
110C2×C71#53511yescharacteristic
111C2×C71#57812yes—
112D71—41no—
113D71—81no—
114#3223—711nomaximal, Sylow
115C71#322311yescharacteristic, complemented, Sylow
116#541—711no—
117#536—711no—
118#497—711no—
119#494—711no—
120#491D7111yescharacteristic, complemented
121#488—711no—
122#464—712no—
123#462—712no—
124#452—712no—
125#451—712no—
126#433—711no—
127C2×#41—711no—
128C22×Q8—711no—
129C2×Q16—712no—
130C2×Q16—712no—
131C2×Q16—712no—
132C2×SD16—712no—
133C4⋊Q8—711no—
134C4⋊Q8—711no—
135C4⋊Q8C2×D7111yescharacteristic
136#126—1422no—
137#125—1422no—
138#125C2×D7111yescharacteristic
139#124—711no—
140#124—711no—
141#123—1422no—
142C22⋊Q8—712no—
143C22⋊Q8—1422no—
144C4×Q8—712no—
145#117—711no—
146C4⋊C8—712no—
147#103—712no—
148#103C2×D7112yes—
149#103C2×D7112yes—
150#102—712no—
151C22⋊C8—712no—
152#97C2×D7111yescharacteristic
153C4×C8—711no—
154#41—2842no—
155C2×Q8—2842no—
156C2×Q8—711no—
157C2×Q8—712no—
158C2×Q8—1421no—
159C2×Q8C22×D7111yescharacteristic
160C2×Q8C22×D7111yescharacteristic
161C2×D4—711no—
162C2×D4—1421no—
163C22×C4—711no—
164C22×C4—1421no—
165C22×C4—1421no—
166Q16—2842no—
167Q16—2842no—
168Q16—2842no—
169SD16—2842no—
170C2×C8—712no—
171C2×C8C22×D7112yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—712no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4—1422no—
179C4⋊C4C22×D7112yes—
180C22⋊C4—712no—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C42—711no—
185C42—1422no—
186C42C22×D7111yescharacteristic
187C23—711no—
188C23—1421no—
189Q8—22no—
190Q8—22no—
191Q8—2842no—
192Q8—1422no—
193Q8—1422no—
194Q8—1422no—
195D4—2842no—
196D4—1422no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—2842no—
201C2×C4—2842no—
202C2×C4—712no—
203C2×C4—711no—
204C2×C4—1422no—
205C2×C4—1421no—
206C2×C4—1422no—
207C2×C4D4×D7112yes—
208C2×C4C23×D7111yescharacteristic
209C8—22no—
210C8—1422no—
211C22—2841no—
212C22—2842no—
213C22—1421no—
214C22—1422no—
215C22C2×D4×D7111yescharacteristic
216C4—22no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—2841no—
221C4—2842no—
222C4—1421no—
223C4—1422no—
224C4#1455617412yes—
225C2—2841no—
226C2—5681no—
227C2#1798656711yescharacteristic
228C2#1798681512yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#491×C71C2
socleC22×C71C22×D4
hypercenter#491D71
hypocenterC71#3223
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #17986612C2semidirect
3. #17986393C2semidirect
4. #17986378C2semidirect
5. #17986356C2semidirect
6. #17986224C2semidirect
7. #17986208C2semidirect
9. #17986156C2semidirect
10. #17986061C2semidirect
11. #491×C71C2semidirect
115. C71#3223semidirect
120. #491D71semidirect