Finite Group Databasebeta

← #22601211

Subgroups — #22601211

Order 9088.

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

normal subgroups only

This group has 338 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#17986709C211yesmaximal, characteristic
2#17986398C212yesmaximal, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986306C212yesmaximal, complemented
5#17986296C212yesmaximal
6#17986294C211yesmaximal, characteristic, complemented
7#17986276C211yesmaximal, characteristic
8Q8×C71⋊C8C211yesmaximal, characteristic, complemented
9C4×#11765852C212yesmaximal
10C4×#11765852C211yesmaximal, characteristic
11#557×C71C211yesmaximal, characteristic, complemented
12C2×#11765852C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556120C2212yes—
15#14556106—22no—
16#14556104—21no—
17C4×D284C2211yescharacteristic
18C4×D284C2212yes—
19C4⋊C4×D71—21no—
20#14556061C2212yes—
21#14556061C2212yes—
22#14556059C2212yes—
23#14556059C2212yes—
24#14556059C2211yescharacteristic
25C4⋊(C71⋊C8)C2212yes—
26C4⋊(C71⋊C8)C2211yescharacteristic
27#14556052C2211yescharacteristic
28#14556052C2212yes—
29#14556048C2212yes—
30#14556048C2211yescharacteristic
31C4×C71⋊C8C2211yescharacteristic
32C4×C71⋊C8C2212yes—
33C4⋊Q8×C71—21no—
34C4⋊Q8×C71C2211yescharacteristic
35#125×C71—22no—
36#125×C71C2212yes—
37C4×Q8×C71—21no—
38C4×Q8×C71C2212yes—
39C4×Q8×C71C2211yescharacteristic
40C4×Q8×C71C2211yescharacteristic
41C2×Dic142—21no—
42C2×D284C2311yescharacteristic
43C2×C4×D71—21no—
44C2×C4×D71—22no—
45#11765852Q814yescomplemented
46C2×C71⋊C8C2311yescharacteristic
47C2×C71⋊C8C2312yes—
48C2×C71⋊C8C2311yescharacteristic
49C4⋊Dic71—21no—
50C4⋊Dic71C2312yes—
51C4⋊Dic71C2311yescharacteristic
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—21no—
56C2×Q8×C71—21no—
57C2×Q8×C71D411yescharacteristic
58C2×Q8×C71C2311yescharacteristic
59C4⋊C4×C71—22no—
60C4⋊C4×C71—21no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71D411yescharacteristic
65C4⋊C4×C71D412yes—
66C4⋊C4×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C42×C71—22no—
69C42×C71—21no—
70C42×C71C2312yes—
71C42×C71C2311yescharacteristic
72C22×D71—21no—
73Q8×C71—22no—
74Q8×C71—21no—
75Q8×C71—42no—
76Q8×C71C2×Q812yes—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71C2×D412yes—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C2×Dic71—21no—
86C2×Dic71—22no—
87D284—21no—
88D284C2×Q812yes—
89C4×D71—42no—
90C4×D71—82no—
91Dic142—81no—
92C71⋊C8—21no—
93C71⋊C8—22no—
94C71⋊C8C2×Q812yes—
95C22×C71C22×D411yescharacteristic
96C2×D71—22no—
97C2×D71—41no—
98C2×D71—41no—
99C4×C71—22no—
100C4×C71—22no—
101C4×C71—21no—
102C4×C71—22no—
103C4×C71—42no—
104C4×C71—41no—
105C4×C71#13512yes—
106C4×C71C22×Q811yescharacteristic
107C4×C712−1+411yescharacteristic
108Dic71—41no—
109Dic71—42no—
110C2×C71D4×Q811yescharacteristic
111C2×C71C2×#13511yescharacteristic
112C2×C71#57711yescharacteristic
113D71—42no—
114#3381—711nomaximal, Sylow
115C71#338111yescharacteristic, complemented, Sylow
116#557D7111yescharacteristic, complemented
117#554—711no—
118#499—712no—
119#498—711no—
120#479—712no—
121#478—712no—
122#477—711no—
123#474—711no—
124C8×Q8—711no—
125C4×SD16—711no—
126C4×SD16—712no—
127C2×SD16—711no—
128C4⋊Q8—21no—
129C4⋊Q8—711no—
130C4⋊Q8C2×D7111yescharacteristic
131#125—22no—
132#125—712no—
133#125C2×D7112yes—
134C22⋊Q8—1421no—
135C22⋊Q8—1422no—
136C4×Q8—21no—
137C4×Q8C2×D7111yescharacteristic
138C4×Q8C2×D7112yes—
139C4×Q8C2×D7111yescharacteristic
140C4×D4—711no—
141C4×D4—712no—
142C2×C4⋊C4—1421no—
143#107—712no—
144#107—712no—
145#106—712no—
146#106—711no—
147#106—712no—
148C4⋊C8—711no—
149C4⋊C8—712no—
150#103—712no—
151#103—711no—
152#102—712no—
153#102—711no—
154C4×C8—711no—
155C4×C8—712no—
156C2×Q8—21no—
157C2×Q8—1421no—
158C2×Q8C22⋊D7111yescharacteristic
159C2×Q8C22×D7111yescharacteristic
160C2×D4—711no—
161C22×C4—1421no—
162C22×C4—1422no—
163SD16—714no—
164C2×C8—711no—
165C2×C8—711no—
166C2×C8—712no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—712no—
173C4⋊C4—711no—
174C4⋊C4—1422no—
175C4⋊C4—1422no—
176C4⋊C4—1421no—
177C4⋊C4C22⋊D7111yescharacteristic
178C4⋊C4C22⋊D7112yes—
179C4⋊C4C22×D7112yes—
180C4⋊C4C22×D7111yescharacteristic
181C22⋊C4—1421no—
182C22⋊C4—1422no—
183C42—22no—
184C42—21no—
185C42C22×D7112yes—
186C42C22×D7111yescharacteristic
187C23—1421no—
188Q8—21no—
189Q8—22no—
190Q8—42no—
191Q8—5681no—
192Q8Q8×D7112yes—
193D4—712no—
194D4—1421no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—2842no—
199C2×C4—5682no—
200C2×C4—1421no—
201C2×C4—1422no—
202C2×C4C2×C22⋊D7112yes—
203C2×C4C2×C22⋊D7112yes—
204C2×C4C2×C22⋊D7111yescharacteristic
205C2×C4C2×C22⋊D7111yescharacteristic
206C2×C4C23×D7111yescharacteristic
207C8—712no—
208C8—1421no—
209C8—1422no—
210C22—2841no—
211C22—2841no—
212C22—1422no—
213C22C22×C22⋊D7111yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—2841no—
221C4—2842no—
222C4#1455614812yes—
223C4C2×Q8×D7111yescharacteristic
224C4#1455617611yescharacteristic
225C2—2842no—
226C2Q8×C22⋊D7111yescharacteristic
227C2C2×#1455614811yescharacteristic
228C2#1798680711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#557×C71C2
socleC22×C71C22×D4
hypercenter#557D71
hypocenterC71#3381
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #17986398C2semidirect
3. #17986395C2semidirect
4. #17986306C2semidirect
6. #17986294C2semidirect
8. Q8×C71⋊C8C2semidirect
11. #557×C71C2semidirect
45. #11765852Q8semidirect
115. C71#3381semidirect
116. #557D71semidirect