Finite Group Databasebeta

← #22598303

Subgroups — #22598303

Order 9088.

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

normal subgroups only

This group has 386 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#17986482C211yesmaximal, characteristic
2#17986337C212yesmaximal, complemented
3#17986337C212yesmaximal, complemented
4#17986332C212yesmaximal, complemented
5#17986327C212yesmaximal, complemented
6#17986042C211yesmaximal, characteristic, complemented
7#17986035C211yesmaximal, characteristic, complemented
8#17986008C211yesmaximal, characteristic, complemented
9C2×#14556052C211yesmaximal, characteristic, complemented
10#17985966C211yesmaximal, characteristic
11#536×C71C211yesmaximal, characteristic
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8C2211yescharacteristic
14Q8×Dic71—21no—
15D4×Dic71—22no—
16D4×Dic71—21no—
17#14556087C2212yes—
18#14556087C2211yescharacteristic
19C2×C4⋊Dic71C2212yes—
20C2×C4⋊Dic71C2211yescharacteristic
21#14556061C2211yescharacteristic
22#14556061C2212yes—
23#14556061C2211yescharacteristic
24#14556059C2211yescharacteristic
25#14556059C2212yes—
26#14556059C2211yescharacteristic
27#14556052C2212yes—
28#14556052C2212yes—
29#14556052C2212yes—
30#14556052C2212yes—
31C22⋊(C71⋊C8)C2212yes—
32C22⋊(C71⋊C8)C2212yes—
33C2×#41×C71C2211yescharacteristic
34C22×Q8×C71C2211yescharacteristic
35C4⋊Q8×C71—21no—
36#124×C71—21no—
37#123×C71—22no—
38C22⋊Q8×C71C2212yes—
39C22⋊Q8×C71C2212yes—
40#117×C71C2211yescharacteristic
41C22×Dic71—22no—
42C22×Dic71—21no—
43#11765849—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2312yes—
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—22no—
49C4⋊Dic71—21no—
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2312yes—
52C4⋊Dic71C2311yescharacteristic
53C22⋊Dic71—22no—
54C22⋊Dic71—21no—
55C4×Dic71—22no—
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×C71D411yescharacteristic
63C2×Q8×C71C2312yes—
64C2×D4×C71D412yes—
65C2×D4×C71D411yescharacteristic
66C22×C4×C71—21no—
67C22×C4×C71C2312yes—
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C4⋊C4×C71C2311yescharacteristic
72C4⋊C4×C71C2311yescharacteristic
73C22⋊C4×C71—21no—
74C22⋊C4×C71—22no—
75C22⋊C4×C71—22no—
76C42×C71—21no—
77C23×C71C2×D412yes—
78C23×C71C2×D411yescharacteristic
79Q8×C71—22no—
80Q8×C71—22no—
81Q8×C71—22no—
82Q8×C71—42no—
83D4×C71—24no—
84D4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—42no—
91C2×C4×C71C2×D412yes—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71#4112yes—
94C2×C4×C71#4112yes—
95C2×C4×C71C2411yescharacteristic
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—81no—
100C2×Dic71—82no—
101C71⋊C8—21no—
102C71⋊C8—21no—
103C71⋊C8—22no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—22no—
107C22×C71—22no—
108C22×C71—22no—
109C22×C71#12312yes—
110C22×C71C22×D411yescharacteristic
111C4×C71—22no—
112C4×C71—42no—
113C4×C71—41no—
114C4×C71—41no—
115C4×C71#12312yes—
116C4×C71C2×#4111yescharacteristic
117C4×C71C2×#4111yescharacteristic
118Dic71—42no—
119Dic71—41no—
120Dic71—41no—
121C2×C71—22no—
122C2×C71—42no—
123C2×C71C2×#12311yescharacteristic
124C2×C71#57711yescharacteristic
125C2×C71#57811yescharacteristic
126#3097—711nomaximal, Sylow
127C71#309711yescharacteristic, complemented, Sylow
128#536D7111yescharacteristic
129#520—711no—
130#485—712no—
131#485—712no—
132#484—712no—
133#483—712no—
134#428—711no—
135#427—711no—
136#419—711no—
137C2×#103—711no—
138#408—711no—
139C2×#41C2×D7111yescharacteristic
140C22×Q8C2×D7111yescharacteristic
141C2×M16—711no—
142C22×C8—711no—
143C4⋊Q8—21no—
144#124—21no—
145#123—22no—
146C22⋊Q8C2×D7112yes—
147C22⋊Q8C2×D7112yes—
148C4×Q8—1421no—
149C4×D4—1422no—
150C4×D4—1421no—
151#117—711no—
152#117—712no—
153#117C2×D7111yescharacteristic
154C2×C4⋊C4—711no—
155C2×C4⋊C4—712no—
156#107—711no—
157#107—711no—
158#107—712no—
159#106—711no—
160#106—711no—
161#106—712no—
162#103—712no—
163#103—712no—
164#103—712no—
165#103—712no—
166C22⋊C8—712no—
167C22⋊C8—712no—
168#41—22no—
169#41—22no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8C22⋊D7111yescharacteristic
174C2×Q8C22×D7112yes—
175C2×D4C22⋊D7112yes—
176C2×D4C22⋊D7111yescharacteristic
177C22×C4—21no—
178C22×C4—1422no—
179C22×C4—1421no—
180C22×C4C22×D7112yes—
181C22×C4C22×D7111yescharacteristic
182M16—1422no—
183C2×C8—711no—
184C2×C8—712no—
185C2×C8—711no—
186C2×C8—1422no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—712no—
190C4⋊C4—711no—
191C4⋊C4—711no—
192C4⋊C4—1422no—
193C4⋊C4—1421no—
194C4⋊C4C22×D7111yescharacteristic
195C4⋊C4C22×D7111yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—1422no—
200C22⋊C4—1421no—
201C42—21no—
202C42—1422no—
203C42—1421no—
204C23C2×C22⋊D7112yes—
205C23C2×C22⋊D7111yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—22no—
209Q8—42no—
210D4—24no—
211D4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—5681no—
219C2×C4—5682no—
220C2×C4—1421no—
221C2×C4—1422no—
222C2×C4—1421no—
223C2×C4C2×C22⋊D7112yes—
224C2×C4C2×C22⋊D7111yescharacteristic
225C2×C4#1176586412yes—
226C2×C4#1176586412yes—
227C2×C4C23×D7111yescharacteristic
228C8—1421no—
229C8—1421no—
230C8—1422no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22#1455610912yes—
237C22C22×C22⋊D7111yescharacteristic
238C4—22no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—2842no—
243C4—2841no—
244C4—2841no—
245C4#1455610912yes—
246C4C2×#1176586411yescharacteristic
247C4C2×#1176586411yescharacteristic
248C2—22no—
249C2—42no—
250C2C2×#1455610911yescharacteristic
251C2#1798680711yescharacteristic
252C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#536×C71C2
socleC22×C71C22×D4
hypercenter#536D71
hypocenterC71#3097
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. #17986337C2semidirect
3. #17986337C2semidirect
4. #17986332C2semidirect
5. #17986327C2semidirect
6. #17986042C2semidirect
7. #17986035C2semidirect
8. #17986008C2semidirect
9. C2×#14556052C2semidirect
127. C71#3097semidirect