Finite Group Databasebeta

← #22601371

Subgroups — #22601371

Order 9088.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986728C211yesmaximal, characteristic, complemented
2#17986394C212yesmaximal
3#17986378C211yesmaximal, characteristic, complemented
4#17986218C212yesmaximal
5#17986209C212yesmaximal, complemented
6#17986206C211yesmaximal, characteristic, complemented
7#17986204C211yesmaximal, characteristic
8#17986130C211yesmaximal, characteristic, complemented
9#17986104C212yesmaximal, complemented
10C4×#11765853C211yesmaximal, characteristic
11#556×C71C211yesmaximal, characteristic
12C2×#11765855C2212yes—
13C2×#11765855C2212yes—
14C2×#11765855C2212yes—
15C2×#11765853C2211yescharacteristic
16C2×#11765853C2212yes—
17#14556128C2211yescharacteristic
18#14556128C2212yes—
19#14556119—21no—
20#14556119—22no—
21Q8×Dic71—21no—
22C4×Dic142C2211yescharacteristic
23C4×Dic142C2212yes—
24#14556059C2211yescharacteristic
25C4⋊(C71⋊C8)C2211yescharacteristic
26C4⋊(C71⋊C8)C2212yes—
27#14556052C2212yes—
28#14556051C2211yescharacteristic
29#14556051C2212yes—
30#14556049C2211yescharacteristic
31#14556039C2212yes—
32C4×C71⋊C8C2211yescharacteristic
33C4⋊Q8×C71C2211yescharacteristic
34#126×C71—22no—
35#124×C71C2212yes—
36C22⋊Q8×C71—21no—
37C4×Q8×C71C2212yes—
38C4×Q8×C71C2211yescharacteristic
39C4×D4×C71C2211yescharacteristic
40#117×C71—21no—
41C2×Dic142C2311yescharacteristic
42C2×Dic142C2312yes—
43#11765855—24no—
44#11765855—42no—
45#11765855—42no—
46#11765853—22no—
47#11765853—42no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2312yes—
50C2×C71⋊C8C2311yescharacteristic
51C4⋊Dic71—22no—
52C4⋊Dic71—21no—
53C4⋊Dic71—22no—
54C4⋊Dic71—21no—
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—22no—
57Dic71⋊C4—21no—
58C4×Dic71—22no—
59C4×Dic71—21no—
60C2×Q8×C71D411yescharacteristic
61C2×Q8×C71C2312yes—
62C2×D4×C71C2311yescharacteristic
63C22×C4×C71—21no—
64C4⋊C4×C71—21no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—21no—
67C4⋊C4×C71D411yescharacteristic
68C4⋊C4×C71D412yes—
69C4⋊C4×C71C2311yescharacteristic
70C22⋊C4×C71—22no—
71C22⋊C4×C71—21no—
72C22⋊C4×C71—22no—
73C42×C71—22no—
74C42×C71C2311yescharacteristic
75C42×C71C2312yes—
76C23×C71—21no—
77Q8×C71—22no—
78Q8×C71—22no—
79Q8×C71—22no—
80D4×C71—21no—
81D4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—42no—
85C2×C4×C71C2×D412yes—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D412yes—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71C2411yescharacteristic
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93Dic142—21no—
94Dic142—22no—
95Dic142—22no—
96Dic142—21no—
97C71⋊C8—22no—
98C71⋊C8—21no—
99C71⋊C8#4112yes—
100C22×C71—41no—
101C22×C71—41no—
102C22×C71—41no—
103C22×C71C22×D411yescharacteristic
104C4×C71—22no—
105C4×C71—22no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71—41no—
109C4×C71#13712yes—
110C4×C71C2×#4111yescharacteristic
111C4×C71H(5,𝔽2)11yescharacteristic
112Dic71—41no—
113Dic71—42no—
114Dic71—41no—
115C2×C71—81no—
116C2×C71#55011yescharacteristic
117C2×C71C2×#13711yescharacteristic
118C2×C71#57811yescharacteristic
119#3398—711nomaximal, Sylow
120C71#339811yescharacteristic, complemented, Sylow
121#557—711no—
122#556D7111yescharacteristic
123#497—712no—
124#494—711no—
125#464—712no—
126#462—711no—
127#462—712no—
128#461—711no—
129#446—711no—
130#441—712no—
131C4×SD16—711no—
132C2×Q16—712no—
133C2×Q16—712no—
134C2×Q16—712no—
135C2×SD16—712no—
136C2×SD16—711no—
137C4⋊Q8—711no—
138C4⋊Q8—712no—
139C4⋊Q8C2×D7111yescharacteristic
140#126—22no—
141#125—1421no—
142#125—1422no—
143#124C2×D7112yes—
144C22⋊Q8—21no—
145C4×Q8—711no—
146C4×Q8—712no—
147C4×Q8—1421no—
148C4×Q8C2×D7112yes—
149C4×Q8C2×D7111yescharacteristic
150C4×D4C2×D7111yescharacteristic
151#117—21no—
152#106—711no—
153C4⋊C8—711no—
154C4⋊C8—712no—
155#103—712no—
156#103—711no—
157#103—712no—
158#102—711no—
159#97—712no—
160C4×C8—711no—
161C2×Q8—711no—
162C2×Q8—712no—
163C2×Q8C22⋊D7111yescharacteristic
164C2×Q8C22×D7112yes—
165C2×D4C22×D7111yescharacteristic
166C22×C4—21no—
167Q16—2842no—
168Q16—2842no—
169Q16—1424no—
170SD16—2842no—
171SD16—1422no—
172C2×C8—711no—
173C2×C8—711no—
174C2×C8—712no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—711no—
179C4⋊C4—1422no—
180C4⋊C4—1421no—
181C4⋊C4—1422no—
182C4⋊C4—1421no—
183C4⋊C4—1422no—
184C4⋊C4—1421no—
185C4⋊C4C22⋊D7111yescharacteristic
186C4⋊C4C22⋊D7112yes—
187C4⋊C4C22×D7111yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—22no—
192C42—1422no—
193C42—1421no—
194C42C22×D7111yescharacteristic
195C42C22×D7112yes—
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—1421no—
201Q8—1422no—
202Q8—1422no—
203Q8—1421no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1421no—
210C2×C4—1421no—
211C2×C4—1422no—
212C2×C4C2×C22⋊D7112yes—
213C2×C4C2×C22⋊D7112yes—
214C2×C4C2×C22⋊D7111yescharacteristic
215C2×C4C2×C22⋊D7111yescharacteristic
216C2×C4C23×D7111yescharacteristic
217C8—712no—
218C8—1422no—
219C8—1421no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C22×C22⋊D7111yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—42no—
228C4—41no—
229C4—2841no—
230C4—2842no—
231C4—2841no—
232C4#1455615812yes—
233C4#41×D7111yescharacteristic
234C4#1455617411yescharacteristic
235C2—81no—
236C2#1798666811yescharacteristic
237C2C2×#1455615811yescharacteristic
238C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#556×C71C2
socleC22×C71C22×D4
hypercenter#556D71
hypocenterC71#3398
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #17986728C2semidirect
3. #17986378C2semidirect
5. #17986209C2semidirect
6. #17986206C2semidirect
8. #17986130C2semidirect
9. #17986104C2semidirect
120. C71#3398semidirect