Finite Group Databasebeta

← #22600766

Subgroups — #22600766

Order 9088.

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

normal subgroups only

This group has 340 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#17986719C211yesmaximal, characteristic
2#17986402C211yesmaximal, characteristic, complemented
3#17986340C212yesmaximal, complemented
4#17986337C212yesmaximal, complemented
5#17986284C211yesmaximal, characteristic, complemented
6#17986281C211yesmaximal, characteristic, complemented
7#17986238C212yesmaximal, complemented
8C4×#11765855C211yesmaximal, characteristic, complemented
9D4×C71⋊C8C211yesmaximal, characteristic
10#17986035C212yesmaximal, complemented
11#556×C71C211yesmaximal, characteristic
12C2×#11765855C2211yescharacteristic
13C22×C71⋊C8C2212yes—
14#14556128C2211yescharacteristic
15#14556121—22no—
16#14556115—21no—
17#14556107C2212yes—
18C4×Dic142C2211yescharacteristic
19#14556093—21no—
20#14556087C2212yes—
21#14556061C2212yes—
22#14556061C2211yescharacteristic
23#14556061C2212yes—
24#14556059C2212yes—
25#14556059C2212yes—
26C4⋊(C71⋊C8)C2211yescharacteristic
27#14556052C2211yescharacteristic
28#14556052C2212yes—
29#14556051C2211yescharacteristic
30#14556051C2212yes—
31C22⋊(C71⋊C8)C2212yes—
32C4×C71⋊C8C2211yescharacteristic
33C4⋊Q8×C71C2211yescharacteristic
34#126×C71—22no—
35#124×C71—21no—
36C22⋊Q8×C71C2212yes—
37C4×Q8×C71—21no—
38C4×Q8×C71C2211yescharacteristic
39C4×D4×C71C2211yescharacteristic
40#117×C71C2212yes—
41C2×Dic142—21no—
42C2×Dic142C2311yescharacteristic
43#11765855—22no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2312yes—
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—21no—
49C4⋊Dic71C2311yescharacteristic
50C4⋊Dic71C2312yes—
51Dic71⋊C4—22no—
52Dic71⋊C4—21no—
53C22⋊Dic71—22no—
54C22⋊Dic71—22no—
55C4×Dic71—22no—
56C4×Dic71—21no—
57C2×Q8×C71—21no—
58C2×Q8×C71C2311yescharacteristic
59C2×D4×C71D411yescharacteristic
60C22×C4×C71C2312yes—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71D411yescharacteristic
65C4⋊C4×C71C2311yescharacteristic
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C22⋊C4×C71—22no—
69C22⋊C4×C71D412yes—
70C42×C71—22no—
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73C23×C71C2×D412yes—
74Q8×C71—21no—
75Q8×C71—21no—
76Q8×C71—42no—
77D4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D412yes—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—22no—
90Dic142—21no—
91Dic142—21no—
92Dic142—42no—
93C71⋊C8—22no—
94C71⋊C8—21no—
95C71⋊C8#4112yes—
96C22×C71—22no—
97C22×C71—22no—
98C22×C71—22no—
99C22×C71C22×D411yescharacteristic
100C4×C71—22no—
101C4×C71—21no—
102C4×C71—41no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71#13512yes—
106C4×C71C2×#4111yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—41no—
109Dic71—42no—
110Dic71—41no—
111C2×C71—42no—
112C2×C71#54711yescharacteristic
113C2×C71C2×#13511yescharacteristic
114C2×C71#57811yescharacteristic
115#3342—711nomaximal, Sylow
116C71#334211yescharacteristic, complemented, Sylow
117#556—711no—
118#556D7111yescharacteristic
119#500—711no—
120#485—712no—
121#485—712no—
122#475—711no—
123#475—711no—
124#467—712no—
125C4×Q16—711no—
126C8×D4—711no—
127#427—712no—
128C2×Q16—711no—
129C22×C8—712no—
130C4⋊Q8—711no—
131C4⋊Q8C2×D7111yescharacteristic
132#126—22no—
133#126—1422no—
134#124—21no—
135#124—1421no—
136C22⋊Q8—712no—
137C22⋊Q8C2×D7112yes—
138C4×Q8—21no—
139C4×Q8—711no—
140C4×Q8—1421no—
141C4×Q8C2×D7111yescharacteristic
142C4×D4C2×D7111yescharacteristic
143#117—712no—
144#117C2×D7112yes—
145#107—711no—
146#107—712no—
147#107—712no—
148#106—712no—
149#106—712no—
150C4⋊C8—711no—
151#103—711no—
152#103—712no—
153#103—711no—
154#103—712no—
155C22⋊C8—712no—
156C4×C8—711no—
157C2×Q8—21no—
158C2×Q8—711no—
159C2×Q8—1421no—
160C2×Q8C22×D7111yescharacteristic
161C2×D4C22⋊D7111yescharacteristic
162C22×C4C22×D7112yes—
163Q16—1422no—
164C2×C8—711no—
165C2×C8—711no—
166C2×C8—712no—
167C2×C8—1424no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—711no—
172C4⋊C4—712no—
173C4⋊C4—1421no—
174C4⋊C4—1422no—
175C4⋊C4—1421no—
176C4⋊C4C22⋊D7111yescharacteristic
177C4⋊C4C22×D7111yescharacteristic
178C4⋊C4C22×D7112yes—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C22⋊C4C22⋊D7112yes—
184C42—22no—
185C42—21no—
186C42—1422no—
187C42—1421no—
188C42C22×D7111yescharacteristic
189C23C2×C22⋊D7112yes—
190Q8—21no—
191Q8—21no—
192Q8—42no—
193Q8—2842no—
194Q8—1421no—
195Q8—1421no—
196D4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—1421no—
203C2×C4—1422no—
204C2×C4—1421no—
205C2×C4C2×C22⋊D7112yes—
206C2×C4C2×C22⋊D7111yescharacteristic
207C2×C4C2×C22⋊D7111yescharacteristic
208C2×C4C23×D7111yescharacteristic
209C8—712no—
210C8—1422no—
211C8—1421no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22C22×C22⋊D7111yescharacteristic
216C4—22no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—2841no—
222C4—2842no—
223C4—2841no—
224C4#1455614812yes—
225C4#41×D7111yescharacteristic
226C4#1455617511yescharacteristic
227C2—42no—
228C2#1798664811yescharacteristic
229C2C2×#1455614811yescharacteristic
230C2#1798681411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#556×C71C2
socleC22×C71C22×D4
hypercenter#556D71
hypocenterC71#3342
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #17986402C2semidirect
3. #17986340C2semidirect
4. #17986337C2semidirect
5. #17986284C2semidirect
6. #17986281C2semidirect
7. #17986238C2semidirect
8. C4×#11765855C2semidirect
10. #17986035C2semidirect
116. C71#3342semidirect