Finite Group Databasebeta

← #487×D71 #22598648

Subgroups — #487×D71 #22598648

Order 9088.

424 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D71C211yesmaximal, characteristic
2C4⋊Q8×D71C211yesmaximal, characteristic
3#17986351C211yesmaximal, characteristic, complemented
4#17986350C211yesmaximal, characteristic, complemented
5#17986349C211yesmaximal, characteristic, complemented
6#17986348C214yesmaximal, complemented
7#103×D71C214yesmaximal
8C4×C8×D71C211yesmaximal, characteristic
9#487×C71C211yesmaximal, characteristic, complemented
10C2×Q8×D71—21no—
11C2×Q8×D71C2212yes—
12C2×C8×D71C2212yes—
13#14556128C2211yescharacteristic
14#14556128C2211yescharacteristic
15#14556127—21no—
16#14556127—21no—
17#14556127—21no—
18#14556127C2212yes—
19#14556126—21no—
20#14556126C2212yes—
21C4⋊C4×D71—21no—
22C4⋊C4×D71—21no—
23C4⋊C4×D71—21no—
24C4⋊C4×D71C2212yes—
25C42×D71C2211yescharacteristic
26#14556053C2214yes—
27#14556052C2214yes—
28#14556051C2214yes—
29C4×C71⋊C8C2211yescharacteristic
30C8×Dic71C2212yes—
31C4⋊Q8×C71C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#103×C71C2214yes—
34C4×C8×C71C2211yescharacteristic
35C2×Dic142—21no—
36C2×Dic142—22no—
37C2×Dic142—22no—
38C2×Dic142C2312yes—
39Q8×D71—28no—
40Q8×D71—44no—
41C2×C4×D71—22no—
42C2×C4×D71—22no—
43C2×C4×D71D411yescharacteristic
44C2×C4×D71D411yescharacteristic
45C2×C4×D71C2311yescharacteristic
46C2×C71⋊C8C2312yes—
47C8×D71—24no—
48C4⋊Dic71—21no—
49C4⋊Dic71—21no—
50C4⋊Dic71—21no—
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56C4×Dic71D411yescharacteristic
57C4×Dic71D411yescharacteristic
58C4×Dic71C2311yescharacteristic
59C2×Q8×C71—21no—
60C2×Q8×C71C2312yes—
61C2×C8×C71C2312yes—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—21no—
64C4⋊C4×C71—21no—
65C4⋊C4×C71C2312yes—
66C42×C71C2311yescharacteristic
67C22×D71C2×D411yescharacteristic
68Q8×C71—24no—
69Q8×C71—42no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71C2×D411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2411yescharacteristic
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71C2×D411yescharacteristic
78C2×Dic71C2×D411yescharacteristic
79C2×Dic71C2×D411yescharacteristic
80C4×D71—44no—
81C4×D71—44no—
82C4×D71SD1614yes—
83C4×D71Q1614yes—
84C4×D71#4114yes—
85Dic142—24no—
86Dic142—42no—
87Dic142—44no—
88Dic142—44no—
89C8×C71—22no—
90C71⋊C8—22no—
91C22×C71C22×D411yescharacteristic
92C2×D71#12412yes—
93C2×D71C2×SD1612yes—
94C2×D71C2×Q1612yes—
95C4×C71—42no—
96C4×C71—42no—
97C4×C71C2×SD1612yes—
98C4×C71C2×Q1612yes—
99C4×C71C2×#4112yes—
100Dic71—42no—
101Dic71—42no—
102Dic71#12412yes—
103Dic71C2×SD1612yes—
104Dic71C2×Q1612yes—
105C2×C71C2×#12411yescharacteristic
106C2×C71C22×SD1611yescharacteristic
107C2×C71C22×Q1611yescharacteristic
108D71#48714yesdirect product factor
109C2×#487—711nomaximal, Sylow
110C71C2×#48711yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—711no—
112C2×C4⋊Q8—711no—
113#487—711no—
114#487—711no—
115#487—714no—
116#487—711no—
117#487D7111yescharacteristic, direct product factor
118C2×#103—714no—
119C2×C4×C8—711no—
120C22×Q8—712no—
121C22×Q8—1421no—
122C22×C8—712no—
123C4⋊Q8—712no—
124C4⋊Q8—712no—
125C4⋊Q8—711no—
126C4⋊Q8—711no—
127C4⋊Q8—1421no—
128C4⋊Q8—1421no—
129C4⋊Q8—1421no—
130C4⋊Q8—1421no—
131C4⋊Q8C2×D7111yescharacteristic
132C4⋊Q8C2×D7111yescharacteristic
133C2×C4⋊C4—712no—
134C2×C4⋊C4—1421no—
135C2×C4⋊C4—1421no—
136C2×C4⋊C4—1421no—
137C2×C42—711no—
138#103—714no—
139#103—714no—
140#103—714no—
141#103C2×D7114yes—
142C4×C8—712no—
143C4×C8—711no—
144C4×C8C2×D7111yescharacteristic
145C2×Q8—21no—
146C2×Q8—2844no—
147C2×Q8—712no—
148C2×Q8—1422no—
149C2×Q8—1422no—
150C2×Q8—1428no—
151C2×Q8—1421no—
152C2×Q8C22×D7112yes—
153C22×C4—711no—
154C22×C4—711no—
155C22×C4—711no—
156C22×C4—1422no—
157C22×C4—1422no—
158C2×C8—712no—
159C2×C8—1424no—
160C2×C8C22×D7112yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—712no—
165C4⋊C4—1422no—
166C4⋊C4—1422no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4—1421no—
170C4⋊C4—1421no—
171C4⋊C4—1421no—
172C4⋊C4C22×D7112yes—
173C42—711no—
174C42—711no—
175C42—711no—
176C42C22×D7111yescharacteristic
177C23—711no—
178Q8—24no—
179Q8—42no—
180Q8—2844no—
181Q8—2844no—
182Q8—2842no—
183Q8—1424no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—2844no—
187C2×C4—2844no—
188C2×C4—711no—
189C2×C4—711no—
190C2×C4—711no—
191C2×C4—714no—
192C2×C4—714no—
193C2×C4—714no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4D4×D7111yescharacteristic
197C2×C4D4×D7111yescharacteristic
198C2×C4C23×D7111yescharacteristic
199C8—22no—
200C8—1422no—
201C22—712no—
202C22—712no—
203C22—712no—
204C22C2×D4×D7111yescharacteristic
205C4—42no—
206C4—42no—
207C4—2842no—
208C4—2842no—
209C4—712no—
210C4—712no—
211C4—712no—
212C4SD16×D7112yes—
213C4Q16×D7112yes—
214C4#41×D7112yes—
215C2—714no—
216C2#124×D7111yescharacteristic
217C2C2×SD16×D7111yescharacteristic
218C2C2×Q16×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71C2×#487
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #17986351C2semidirect
4. #17986350C2semidirect
5. #17986349C2semidirect
6. #17986348C2semidirect
9. #487×C71C2semidirect
108. D71#487direct (N × H)
110. C71C2×#487semidirect
117. #487D71direct (N × H)