Finite Group Databasebeta

← #22599580

Subgroups — #22599580

Order 9088.

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

normal subgroups only

This group has 328 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#17986533C211yesmaximal, characteristic
2#17986405C211yesmaximal, characteristic, complemented
3#17986405C211yesmaximal, characteristic, complemented
4#17986361C211yesmaximal, characteristic, complemented
5#17986361C211yesmaximal, characteristic, complemented
6#17986340C212yesmaximal, complemented
7#17986323C212yesmaximal
8#17986303C212yesmaximal, complemented
9#17986296C212yesmaximal, complemented
10#17986056C211yesmaximal, characteristic, complemented
11#533×C71C211yesmaximal, characteristic, complemented
12#14556122—21no—
13#14556120C2211yescharacteristic
14#14556116C2211yescharacteristic
15#14556111—21no—
16#14556107C2211yescharacteristic
17#14556102C2211yescharacteristic
18C4×Dic142C2211yescharacteristic
19C4×D284C2211yescharacteristic
20C4×C22⋊D71—21no—
21#14556085—21no—
22#14556061C2212yes—
23#14556061C2212yes—
24#14556059C2212yes—
25#14556059C2212yes—
26C4⋊(C71⋊C8)C2212yes—
27#14556051C2212yes—
28#14556051C2212yes—
29#14556048C2212yes—
30#14556048C2212yes—
31C22⋊(C71⋊C8)C2212yes—
32#14556039C2211yescharacteristic
33#14556039C2211yescharacteristic
34C4⋊Q8×C71C2211yescharacteristic
35C4⋊Q8×C71C2211yescharacteristic
36#125×C71C2212yes—
37C22⋊Q8×C71—21no—
38C22⋊Q8×C71—21no—
39C4×Q8×C71—21no—
40C4×Q8×C71—21no—
41#117×C71C2212yes—
42C2×C42×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71—21no—
47C2×C71⋊C8C2312yes—
48C2×C71⋊C8C2312yes—
49C4⋊Dic71C2311yescharacteristic
50C4⋊Dic71C2311yescharacteristic
51Dic71⋊C4—21no—
52Dic71⋊C4—21no—
53Dic71⋊C4—21no—
54Dic71⋊C4—21no—
55C22⋊Dic71—21no—
56(C2×D71)⋊C4—21no—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—21no—
59(C2×D71)⋊C4—21no—
60C4×Dic71—21no—
61C2×Q8×C71—21no—
62C2×Q8×C71—21no—
63C22×C4×C71D411yescharacteristic
64C22×C4×C71D411yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71C2312yes—
71C4⋊C4×C71C2312yes—
72C22⋊C4×C71—22no—
73C42×C71—22no—
74C42×C71D411yescharacteristic
75C42×C71D411yescharacteristic
76C42×C71C2311yescharacteristic
77C42×C71C2311yescharacteristic
78C23×C71C2×D411yescharacteristic
79C22×D71—21no—
80Q8×C71—42no—
81Q8×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2×D411yescharacteristic
92C2×C4×C71C2411yescharacteristic
93C22⋊D71—42no—
94C2×Dic71—21no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97D284—22no—
98C4×D71—42no—
99Dic142—22no—
100C71⋊C8—22no—
101C71⋊C8—22no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—21no—
105C22×C71C22×D411yescharacteristic
106C2×D71—41no—
107C2×D71—41no—
108C2×D71—41no—
109C4×C71—21no—
110C4×C71—21no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—21no—
114C4×C71—42no—
115C4×C71—42no—
116C4×C712−1+412yes—
117Dic71—41no—
118Dic71—41no—
119Dic71—41no—
120C2×C71—41no—
121C2×C71#53611yescharacteristic
122C2×C71#57511yescharacteristic
123C2×C71#57511yescharacteristic
124D71—81no—
125#3240—711nomaximal, Sylow
126C71#324011yescharacteristic, complemented, Sylow
127#533D7111yescharacteristic, complemented
128#529—711no—
129#501—711no—
130#501—711no—
131#489—711no—
132#489—711no—
133#485—712no—
134#482—712no—
135#479—712no—
136#478—712no—
137#432—711no—
138C4⋊Q8C2×D7111yescharacteristic
139C4⋊Q8C2×D7111yescharacteristic
140#126—1421no—
141#125—711no—
142#125C2×D7112yes—
143#124—711no—
144#123—1421no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—711no—
148#121—711no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—711no—
152C4×D4—711no—
153C4×D4—1421no—
154#117—1421no—
155#117C2×D7112yes—
156C2×C42C2×D7111yescharacteristic
157#107—712no—
158#107—712no—
159#106—712no—
160#106—712no—
161C4⋊C8—712no—
162#103—712no—
163#103—712no—
164#102—712no—
165#102—712no—
166C22⋊C8—712no—
167#97—711no—
168#97—711no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—711no—
172C2×D4—711no—
173C2×D4—1421no—
174C22×C4—1421no—
175C22×C4C22⋊D7111yescharacteristic
176C22×C4C22⋊D7111yescharacteristic
177C22×C4C22×D7111yescharacteristic
178C2×C8—712no—
179C2×C8—712no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—711no—
185C4⋊C4—711no—
186C4⋊C4—1421no—
187C4⋊C4—1421no—
188C4⋊C4—1421no—
189C4⋊C4—1421no—
190C4⋊C4C22×D7112yes—
191C4⋊C4C22×D7112yes—
192C22⋊C4—22no—
193C22⋊C4—1421no—
194C22⋊C4—1421no—
195C22⋊C4—1421no—
196C22⋊C4—1421no—
197C22⋊C4—1421no—
198C42—22no—
199C42—1421no—
200C42C22⋊D7111yescharacteristic
201C42C22⋊D7111yescharacteristic
202C42C22×D7111yescharacteristic
203C42C22×D7111yescharacteristic
204C23—1421no—
205C23C2×C22⋊D7111yescharacteristic
206Q8—42no—
207Q8—42no—
208Q8—1422no—
209D4—2842no—
210D4—1422no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—2842no—
217C2×C4—1421no—
218C2×C4—1421no—
219C2×C4—1421no—
220C2×C4C2×C22⋊D7111yescharacteristic
221C2×C4C2×C22⋊D7111yescharacteristic
222C2×C4C2×C22⋊D7111yescharacteristic
223C2×C4C2×C22⋊D7111yescharacteristic
224C2×C4C2×C22⋊D7111yescharacteristic
225C2×C4C23×D7111yescharacteristic
226C8—1422no—
227C8—1422no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—2841no—
232C22—2841no—
233C22—2841no—
234C22C22×C22⋊D7111yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—2841no—
243C4—2841no—
244C4—2841no—
245C4#1455617612yes—
246C2—41no—
247C2—5681no—
248C2#1798657311yescharacteristic
249C2#1798679711yescharacteristic
250C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#533×C71C2
socleC22×C71C22×D4
hypercenter#533D71
hypocenterC71#3240
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #17986405C2semidirect
3. #17986405C2semidirect
4. #17986361C2semidirect
5. #17986361C2semidirect
6. #17986340C2semidirect
8. #17986303C2semidirect
9. #17986296C2semidirect
10. #17986056C2semidirect
11. #533×C71C2semidirect
126. C71#3240semidirect
127. #533D71semidirect