Finite Group Databasebeta

← #22599477

Subgroups — #22599477

Order 9088.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986554C211yesmaximal, characteristic, complemented
2#17986550C211yesmaximal, characteristic
3#17986402C211yesmaximal, characteristic, complemented
4#17986347C211yesmaximal, characteristic, complemented
5#17986346C211yesmaximal, characteristic, complemented
6#17986325C214yesmaximal, complemented
7#17986298C214yesmaximal, complemented
8#17986050C211yesmaximal, characteristic, complemented
9#498×C71C211yesmaximal, characteristic, complemented
10C2×#11765866—21no—
11#14556128C2211yescharacteristic
12#14556125C2211yescharacteristic
13#14556119C2212yes—
14#14556117—21no—
15#14556108—21no—
16#14556104—21no—
17#14556103—21no—
18#14556103C2212yes—
19Q8×Dic71—21no—
20#14556093—21no—
21#14556090—21no—
22#14556090C2212yes—
23#14556086C2212yes—
24C42×D71C2211yescharacteristic
25#14556061C2214yes—
26Dic71⋊C8C2212yes—
27#14556050C2214yes—
28#14556048C2214yes—
29(C2×D71)⋊C8C2212yes—
30C4×C71⋊C8C2211yescharacteristic
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#106×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35#11765866—44no—
36C2×Dic142—21no—
37C2×D284—22no—
38C2×D284—21no—
39C2×D284—21no—
40C2×D284C2312yes—
41C2×C4×D71—22no—
42C2×C4×D71D411yescharacteristic
43C2×C4×D71D411yescharacteristic
44C2×C4×D71C2311yescharacteristic
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—21no—
47C4⋊Dic71—22no—
48C4⋊Dic71C2312yes—
49Dic71⋊C4—22no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53C4×Dic71—22no—
54C4×Dic71D411yescharacteristic
55C4×Dic71D411yescharacteristic
56C4×Dic71C2311yescharacteristic
57C2×Q8×C71—21no—
58C2×Q8×C71—21no—
59C2×C8×C71C2312yes—
60C4⋊C4×C71—21no—
61C4⋊C4×C71—21no—
62C4⋊C4×C71C2312yes—
63C4⋊C4×C71C2312yes—
64C42×C71C2311yescharacteristic
65C22×D71—22no—
66C22×D71C2×D411yescharacteristic
67Q8×C71—42no—
68Q8×C71—42no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2411yescharacteristic
74C2×Dic71—22no—
75C2×Dic71C2×D411yescharacteristic
76C2×Dic71C2×D411yescharacteristic
77C2×Dic71C2×D411yescharacteristic
78D284—24no—
79D284—44no—
80D284—42no—
81D284—42no—
82C4×D71—22no—
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—44no—
86Dic142—42no—
87C8×C71—22no—
88C71⋊C8—22no—
89C22×C71C22×D411yescharacteristic
90C2×D71—21no—
91C2×D71—21no—
92C2×D71—21no—
93C2×D71—42no—
94C2×D71—42no—
95C2×D71—42no—
96C4×C71—42no—
97C4×C71—42no—
98C4×C71#13512yes—
99C4×C71AGL(1,ℤ/8)12yes—
100C4×C712−1+412yes—
101Dic71—21no—
102Dic71—21no—
103Dic71—21no—
104Dic71—42no—
105C2×C71#53611yescharacteristic
106C2×C71C2×#13511yescharacteristic
107C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
108D71—41no—
109D71—82no—
110#3228—711nomaximal, Sylow
111C71#322811yescharacteristic, complemented, Sylow
112#533—711no—
113#532—711no—
114#500—711no—
115#498D7111yescharacteristic, complemented
116#486—711no—
117#486—711no—
118#482—714no—
119#478—714no—
120#431—711no—
121C2×#41—1421no—
122C4⋊Q8—711no—
123C4⋊Q8C2×D7111yescharacteristic
124C4⋊Q8C2×D7111yescharacteristic
125#127—711no—
126#125—712no—
127#124—1421no—
128C22⋊Q8—1421no—
129C22⋊Q8—1421no—
130#121—712no—
131#121—1421no—
132C4×Q8—1421no—
133C4×Q8—1421no—
134C4×D4—712no—
135C4×D4—1421no—
136#117—712no—
137C2×C42—711no—
138#107—714no—
139#106C2×D7114yes—
140C4⋊C8—712no—
141#102—714no—
142#102—714no—
143C22⋊C8—712no—
144C4×C8—711no—
145C4×C8C2×D7111yescharacteristic
146#41—2844no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1421no—
150C2×D4—712no—
151C2×D4—1422no—
152C2×D4—1421no—
153C2×D4—1421no—
154C22×C4—711no—
155C22×C4—711no—
156C22×C4—711no—
157C22×C4—1422no—
158C2×C8—712no—
159C2×C8C22×D7112yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—712no—
163C4⋊C4—1422no—
164C4⋊C4—1422no—
165C4⋊C4—1421no—
166C4⋊C4C22×D7112yes—
167C4⋊C4C22×D7112yes—
168C22⋊C4—1422no—
169C22⋊C4—1422no—
170C22⋊C4—1422no—
171C42—711no—
172C42—711no—
173C42—711no—
174C42—1422no—
175C42C22×D7111yescharacteristic
176C23—711no—
177C23—1422no—
178Q8—42no—
179Q8—42no—
180Q8—2842no—
181D4—2844no—
182D4—2842no—
183D4—2842no—
184D4—1424no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—2844no—
188C2×C4—711no—
189C2×C4—711no—
190C2×C4—711no—
191C2×C4—1422no—
192C2×C4—1422no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4D4×D7111yescharacteristic
196C2×C4D4×D7111yescharacteristic
197C2×C4C23×D7111yescharacteristic
198C8—22no—
199C8—1422no—
200C22—2842no—
201C22—2842no—
202C22—2842no—
203C22—1421no—
204C22—1421no—
205C22—1421no—
206C22C2×D4×D7111yescharacteristic
207C4—42no—
208C4—42no—
209C4—2842no—
210C4—1421no—
211C4—1421no—
212C4—1421no—
213C4#1455614912yes—
214C4#1455615512yes—
215C4#1455617612yes—
216C2—2841no—
217C2—5682no—
218C2#1798657411yescharacteristic
219C2C2×#1455614911yescharacteristic
220C2C2×#1455615511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#498×C71C2
socleC22×C71C22×D4
hypercenter#498D71
hypocenterC71#3228
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986402C2semidirect
4. #17986347C2semidirect
5. #17986346C2semidirect
6. #17986325C2semidirect
7. #17986298C2semidirect
8. #17986050C2semidirect
9. #498×C71C2semidirect
111. C71#3228semidirect
115. #498D71semidirect