Finite Group Databasebeta

← #22598705

Subgroups — #22598705

Order 9088.

372 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 133 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#17986627C211yesmaximal, characteristic, complemented
2#17986367C212yesmaximal, complemented
3#17986358C212yesmaximal, complemented
4#17986353C212yesmaximal, complemented
5#17986349C212yesmaximal, complemented
6#17986008C211yesmaximal, characteristic, complemented
7#17986000C211yesmaximal, characteristic
8#17985995C211yesmaximal, characteristic
9C2×#14556052C211yesmaximal, characteristic, complemented
10#17985955C211yesmaximal, characteristic, complemented
11#536×C71C211yesmaximal, characteristic
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8C2211yescharacteristic
14#14556128C2212yes—
15#14556127—21no—
16#14556120C2212yes—
17#14556119—21no—
18#14556105—22no—
19#14556087C2211yescharacteristic
20C2×C4⋊Dic71C2211yescharacteristic
21#14556052C2211yescharacteristic
22#14556052C2212yes—
23#14556052C2212yes—
24#14556052C2212yes—
25#14556052C2212yes—
26#14556052C2212yes—
27#14556052C2211yescharacteristic
28#14556049C2211yescharacteristic
29#14556049C2212yes—
30#14556049C2211yescharacteristic
31#14556039C2212yes—
32C4×C71⋊C8C2212yes—
33C2×#41×C71C2211yescharacteristic
34C22×Q8×C71C2211yescharacteristic
35C4⋊Q8×C71C2212yes—
36#124×C71C2212yes—
37#123×C71—22no—
38C22⋊Q8×C71—21no—
39C22⋊Q8×C71—21no—
40#117×C71C2211yescharacteristic
41C2×Dic142—22no—
42C22×Dic71—21no—
43#11765849—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2312yes—
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—21no—
49C4⋊Dic71—22no—
50C4⋊Dic71C2312yes—
51C4⋊Dic71C2311yescharacteristic
52C4⋊Dic71C2311yescharacteristic
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55C22⋊Dic71—21no—
56C4×Dic71—21no—
57#41×C71—22no—
58#41×C71—22no—
59C2×Q8×C71—21no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71C2312yes—
63C2×Q8×C71C2311yescharacteristic
64C2×D4×C71—21no—
65C2×D4×C71C2311yescharacteristic
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C4⋊C4×C71D411yescharacteristic
72C4⋊C4×C71D411yescharacteristic
73C22⋊C4×C71—22no—
74C22⋊C4×C71—22no—
75C22⋊C4×C71D412yes—
76C42×C71C2312yes—
77C23×C71—21no—
78C23×C71C2×D411yescharacteristic
79Q8×C71—22no—
80Q8×C71—22no—
81Q8×C71—21no—
82Q8×C71—21no—
83Q8×C71—42no—
84D4×C71—21no—
85D4×C71—21no—
86D4×C71—42no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71C2×D412yes—
92C2×C4×C71C2×D412yes—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71#4112yes—
95C2×C4×C71#4112yes—
96C2×C4×C71C2411yescharacteristic
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—22no—
100C2×Dic71—81no—
101Dic142—82no—
102C71⋊C8—21no—
103C71⋊C8—21no—
104C71⋊C8—22no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—41no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71#12412yes—
111C22×C71C22×D411yescharacteristic
112C4×C71—22no—
113C4×C71—22no—
114C4×C71—41no—
115C4×C71—42no—
116C4×C71#12412yes—
117C4×C71C2×#4111yescharacteristic
118C4×C71C2×#4111yescharacteristic
119Dic71—41no—
120Dic71—42no—
121Dic71—41no—
122C2×C71—22no—
123C2×C71—81no—
124C2×C71C2×#12411yescharacteristic
125C2×C71#57711yescharacteristic
126C2×C71#57811yescharacteristic
127#3137—711nomaximal, Sylow
128C71#313711yescharacteristic, complemented, Sylow
129#544—711no—
130#536D7111yescharacteristic
131#491—712no—
132#489—712no—
133#488—712no—
134#487—712no—
135#419—711no—
136#417—711no—
137#416—711no—
138C2×#103—711no—
139#405—711no—
140C2×#41C2×D7111yescharacteristic
141C22×Q8C2×D7111yescharacteristic
142C2×M16—711no—
143C22×C8—711no—
144C4⋊Q8—712no—
145C4⋊Q8—1421no—
146C4⋊Q8C2×D7112yes—
147#125—712no—
148#125—1421no—
149#124C2×D7112yes—
150#123—22no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153C22⋊Q8—1422no—
154#117—711no—
155#117C2×D7111yescharacteristic
156C2×C4⋊C4—711no—
157#103—712no—
158#103—712no—
159#103—711no—
160#103—711no—
161#103—712no—
162#103—712no—
163#103—712no—
164#102—711no—
165#102—711no—
166#102—712no—
167#97—712no—
168C4×C8—712no—
169#41—22no—
170#41—22no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—1422no—
175C2×Q8C22×D7112yes—
176C2×Q8C22×D7111yescharacteristic
177C2×D4—21no—
178C2×D4C22×D7111yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1421no—
182C22×C4C22×D7111yescharacteristic
183M16—1422no—
184C2×C8—711no—
185C2×C8—712no—
186C2×C8—711no—
187C2×C8—1422no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—712no—
191C4⋊C4—711no—
192C4⋊C4—711no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1421no—
196C4⋊C4—1422no—
197C4⋊C4C22⋊D7111yescharacteristic
198C4⋊C4C22⋊D7111yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—1421no—
202C22⋊C4C22⋊D7112yes—
203C42—1421no—
204C42C22×D7112yes—
205C23—21no—
206C23C2×C22⋊D7111yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—5682no—
213D4—21no—
214D4—21no—
215D4—42no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—5681no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4—1422no—
224C2×C4C2×C22⋊D7112yes—
225C2×C4C2×C22⋊D7112yes—
226C2×C4C2×C22⋊D7111yescharacteristic
227C2×C4#1176586412yes—
228C2×C4#1176586412yes—
229C2×C4C23×D7111yescharacteristic
230C8—1421no—
231C8—1421no—
232C8—1422no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22#1455611512yes—
239C22C22×C22⋊D7111yescharacteristic
240C4—22no—
241C4—22no—
242C4—41no—
243C4—42no—
244C4—2842no—
245C4—2841no—
246C4—2841no—
247C4#1455611512yes—
248C4C2×#1176586411yescharacteristic
249C4C2×#1176586411yescharacteristic
250C2—22no—
251C2—81no—
252C2C2×#1455611511yescharacteristic
253C2#1798680711yescharacteristic
254C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#536×C71C2
socleC22×C71C22×D4
hypercenter#536D71
hypocenterC71#3137
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #17986627C2semidirect
2. #17986367C2semidirect
3. #17986358C2semidirect
4. #17986353C2semidirect
5. #17986349C2semidirect
6. #17986008C2semidirect
9. C2×#14556052C2semidirect
10. #17985955C2semidirect
128. C71#3137semidirect