Finite Group Databasebeta

← #22599231

Subgroups — #22599231

Order 9088.

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

normal subgroups only

This group has 378 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#17986570C211yesmaximal, characteristic
2#17986340C212yesmaximal, complemented
3#17986337C212yesmaximal, complemented
4#17986255C212yesmaximal, complemented
5#17986238C212yesmaximal, complemented
6#17986155C212yesmaximal, complemented
7#17986154C212yesmaximal, complemented
8#17985966C211yesmaximal, characteristic
9#536×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic
11C2×#11765855C2212yes—
12C2×#11765855C2212yes—
13C2×#11765849C2211yescharacteristic
14C22×C71⋊C8C2211yescharacteristic
15#14556126—21no—
16#14556115—21no—
17#14556109—21no—
18#14556109—21no—
19#14556107C2212yes—
20#14556107C2212yes—
21#14556087C2211yescharacteristic
22#14556061C2212yes—
23#14556059C2212yes—
24#14556052C2212yes—
25#14556052C2212yes—
26#14556051C2212yes—
27#14556051C2212yes—
28C22⋊(C71⋊C8)C2212yes—
29C22⋊(C71⋊C8)C2212yes—
30C2×#41×C71C2211yescharacteristic
31C22×Q8×C71C2211yescharacteristic
32C4⋊Q8×C71—21no—
33#124×C71—21no—
34#123×C71—21no—
35#123×C71—21no—
36C22⋊Q8×C71C2212yes—
37C22⋊Q8×C71C2212yes—
38#117×C71C2211yescharacteristic
39C2×Dic142—21no—
40C2×Dic142—41no—
41C2×Dic142—41no—
42C2×Dic142C2312yes—
43C22×Dic71—21no—
44#11765855—42no—
45#11765855—42no—
46#11765849—41no—
47C2×C71⋊C8—41no—
48C2×C71⋊C8C2312yes—
49C2×C71⋊C8C2312yes—
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53C22⋊Dic71—21no—
54C22⋊Dic71—22no—
55C22⋊Dic71—22no—
56C4×Dic71—21no—
57#41×C71—41no—
58#41×C71—41no—
59C2×Q8×C71—21no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C2×Q8×C71D411yescharacteristic
63C2×Q8×C71C2312yes—
64C2×D4×C71D411yescharacteristic
65C2×D4×C71D411yescharacteristic
66C2×D4×C71D411yescharacteristic
67C22×C4×C71—21no—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C22×C4×C71C2311yescharacteristic
71C4⋊C4×C71—22no—
72C4⋊C4×C71—22no—
73C4⋊C4×C71C2312yes—
74C22⋊C4×C71—21no—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—22no—
77C42×C71—21no—
78C23×C71C2×D411yescharacteristic
79C23×C71C2×D411yescharacteristic
80C23×C71C2×D411yescharacteristic
81Q8×C71—22no—
82Q8×C71—22no—
83Q8×C71—41no—
84Q8×C71—42no—
85D4×C71—41no—
86D4×C71—41no—
87D4×C71—41no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2411yescharacteristic
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—42no—
104Dic142—22no—
105Dic142—22no—
106Dic142—42no—
107C71⋊C8—22no—
108C71⋊C8—22no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71—21no—
112C22×C71—22no—
113C22×C71—22no—
114C22×C71—22no—
115C22×C71C22×D411yescharacteristic
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71H(5,𝔽2)11yescharacteristic
122C4×C71H(5,𝔽2)11yescharacteristic
123Dic71—42no—
124Dic71—42no—
125C2×C71—41no—
126C2×C71—41no—
127C2×C71—41no—
128C2×C71#53411yescharacteristic
129C2×C71#57812yes—
130#3202—711nomaximal, Sylow
131C71#320211yescharacteristic, complemented, Sylow
132#536—711no—
133#536D7111yescharacteristic
134#485—712no—
135#485—712no—
136#470—712no—
137#467—712no—
138#451—712no—
139#451—712no—
140#408—711no—
141C2×#41C2×D7111yescharacteristic
142C22×Q8—711no—
143C22×Q8C2×D7111yescharacteristic
144C2×Q16—712no—
145C2×Q16—712no—
146C2×M16—711no—
147C22×C8—711no—
148C4⋊Q8—21no—
149C4⋊Q8—1421no—
150#124—21no—
151#124—1421no—
152#123—21no—
153#123—21no—
154#123—1421no—
155#123—1421no—
156C22⋊Q8—712no—
157C22⋊Q8—712no—
158C22⋊Q8C2×D7112yes—
159C22⋊Q8C2×D7112yes—
160#117—711no—
161#117C2×D7111yescharacteristic
162#107—712no—
163#106—712no—
164#103—712no—
165#103—712no—
166#103—712no—
167#103—712no—
168C22⋊C8—712no—
169C22⋊C8—712no—
170#41—41no—
171#41—41no—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—2841no—
176C2×Q8—2841no—
177C2×Q8—712no—
178C2×Q8—1421no—
179C2×Q8C22⋊D7111yescharacteristic
180C2×Q8C22×D7112yes—
181C2×D4C22⋊D7111yescharacteristic
182C2×D4C22⋊D7111yescharacteristic
183C2×D4C22⋊D7111yescharacteristic
184C22×C4—21no—
185C22×C4—1421no—
186C22×C4C22×D7111yescharacteristic
187C22×C4C22×D7111yescharacteristic
188C22×C4C22×D7111yescharacteristic
189Q16—2842no—
190Q16—2842no—
191M16—2841no—
192C2×C8—2841no—
193C2×C8—712no—
194C2×C8—712no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—712no—
198C4⋊C4—1422no—
199C4⋊C4—1422no—
200C4⋊C4C22×D7112yes—
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—1421no—
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C42—21no—
208C42—1421no—
209C23C2×C22⋊D7111yescharacteristic
210C23C2×C22⋊D7111yescharacteristic
211C23C2×C22⋊D7111yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—41no—
215Q8—42no—
216Q8—2842no—
217Q8—1422no—
218Q8—1422no—
219D4—41no—
220D4—41no—
221D4—41no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—2842no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4C2×C22⋊D7111yescharacteristic
235C2×C4C2×C22⋊D7111yescharacteristic
236C2×C4C2×C22⋊D7111yescharacteristic
237C2×C4C23×D7111yescharacteristic
238C8—1422no—
239C8—1422no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22C22×C22⋊D7111yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—2842no—
253C4—2842no—
254C4#1455617311yescharacteristic
255C4#1455617311yescharacteristic
256C2—41no—
257C2—41no—
258C2—41no—
259C2#1798655811yescharacteristic
260C2#1798681412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#536×C71C2
socleC22×C71C22×D4
hypercenter#536D71
hypocenterC71#3202
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #17986340C2semidirect
3. #17986337C2semidirect
4. #17986255C2semidirect
5. #17986238C2semidirect
6. #17986155C2semidirect
7. #17986154C2semidirect
131. C71#3202semidirect