Finite Group Databasebeta

← #22589299

Subgroups — #22589299

Order 9088.

300 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#14556127C211yesmaximal, characteristic
2C2×C8×Dic71C211yesmaximal, characteristic
3#17985868C211yesmaximal, characteristic
4#17985762C211yesmaximal, characteristic
5#17985761C211yesmaximal, characteristic
6#17985761C211yesmaximal, characteristic
7#340×C71C211yesmaximal, characteristic
8C22×Dic142—21no—
9C22×C71⋊C8C2211yescharacteristic
10#14556127—22no—
11#14556127C414yescomplemented
12C2×C4⋊Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4—21no—
15C2×Dic71⋊C4—21no—
16C2×Dic71⋊C4—21no—
17C2×C4×Dic71C2211yescharacteristic
18C8×Dic71—22no—
19C22×C8×C71C2211yescharacteristic
20C2×C4⋊C4×C71C2211yescharacteristic
21C2×C4⋊C4×C71C2211yescharacteristic
22C2×Dic142—24no—
23C2×Dic142—42no—
24C2×Dic142—42no—
25C22×Dic71—21no—
26C22×Dic71—21no—
27C22×Dic71D411yescharacteristic
28C22×Dic71D411yescharacteristic
29C2×C71⋊C8—21no—
30C2×C71⋊C8—21no—
31C2×C71⋊C8Q812yescomplemented
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35C4⋊Dic71C2×C412yes—
36Dic71⋊C4—24no—
37Dic71⋊C4—42no—
38Dic71⋊C4—42no—
39C4×Dic71C2×C412yes—
40C4×Dic71D412yes—
41C22×C4×C71—21no—
42C22×C4×C71—21no—
43C22×C4×C71C2311yescharacteristic
44C2×C8×C71—21no—
45C2×C8×C71—21no—
46C2×C8×C71Q812yescomplemented
47C4⋊C4×C71—21no—
48C4⋊C4×C71—21no—
49C4⋊C4×C71—21no—
50C4⋊C4×C71C2×C412yes—
51C23×C71C2×D411yescharacteristic
52C2×C4×C71—22no—
53C2×C4×C71—41no—
54C2×C4×C71—41no—
55C2×C4×C71—41no—
56C2×C4×C71—41no—
57C2×C4×C71—41no—
58C2×C4×C71C22×C411yescharacteristic
59C2×C4×C71C2×D411yescharacteristic
60C2×C4×C71C2×Q811yescharacteristic
61C2×C4×C71C2×Q811yescharacteristic
62C2×C4×C71#4111yescharacteristic
63C2×C4×C71#4111yescharacteristic
64C2×Dic71—22no—
65C2×Dic71—41no—
66C2×Dic71—41no—
67C2×Dic71—41no—
68C2×Dic71—41no—
69C2×Dic71—41no—
70C2×Dic71C22⋊C412yes—
71C2×Dic71C22⋊C412yes—
72C2×Dic71D812yes—
73C2×Dic71SD1612yes—
74C2×Dic71SD1612yes—
75C2×Dic71Q1612yes—
76Dic142—48no—
77C8×C71—22no—
78C71⋊C8—22no—
79C22×C71C2×C22⋊C411yescharacteristic
80C22×C71#12411yescharacteristic
81C22×C71C4⋊Q811yescharacteristic
82C22×C71C2×D811yescharacteristic
83C22×C71C2×SD1611yescharacteristic
84C22×C71C2×SD1611yescharacteristic
85C22×C71C2×Q1611yescharacteristic
86C4×C71—42no—
87C4×C71—81no—
88C4×C71C4×Q811yescharacteristic
89C4×C71C4×Q811yescharacteristic
90C4×C71C22⋊Q811yescharacteristic
91C4×C71C22⋊Q811yescharacteristic
92Dic71—42no—
93Dic71—81no—
94Dic71#10214yes—
95Dic71#10314yes—
96C2×C71#39111yescharacteristic
97C2×C71C2×#10211yescharacteristic
98C2×C71C2×#10311yescharacteristic
99C2×C71#48611yescharacteristic
100C2×C71#48711yescharacteristic
101C2×C71#49811yescharacteristic
102C2×C71#50011yescharacteristic
103#1990—711nomaximal, Sylow
104C71#199011yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—711no—
106C2×C4×C8—711no—
107#384—711no—
108#340—711no—
109#340—711no—
110#340—711no—
111#340D7111yescharacteristic
112C22×Q8—1421no—
113C22×C8—711no—
114C22×C8C2×D7111yescharacteristic
115C4⋊Q8—714no—
116C4⋊Q8—1422no—
117C2×C4⋊C4—711no—
118C2×C4⋊C4—711no—
119C2×C4⋊C4—1421no—
120C2×C4⋊C4—1421no—
121C2×C4⋊C4—1421no—
122C2×C4⋊C4C2×D7111yescharacteristic
123C2×C4⋊C4C2×D7111yescharacteristic
124C2×C42—711no—
125C4×C8—1422no—
126C2×Q8—2842no—
127C2×Q8—2842no—
128C2×Q8—1424no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—711no—
132C22×C4—711no—
133C22×C4—1421no—
134C22×C4—1421no—
135C22×C4C22×D7111yescharacteristic
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—712no—
139C2×C8—1421no—
140C2×C8—1421no—
141C2×C8Dic14212yescomplemented
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—2842no—
146C4⋊C4—2842no—
147C4⋊C4—712no—
148C4⋊C4—1424no—
149C4⋊C4—1421no—
150C4⋊C4—1421no—
151C4⋊C4—1421no—
152C4⋊C4C4×D7112yes—
153C42—712no—
154C42—712no—
155C23D4×D7111yescharacteristic
156Q8—2848no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—2841no—
164C2×C4—2841no—
165C2×C4—2841no—
166C2×C4—2841no—
167C2×C4—2841no—
168C2×C4—712no—
169C2×C4—712no—
170C2×C4—712no—
171C2×C4—712no—
172C2×C4—712no—
173C2×C4—712no—
174C2×C4—1422no—
175C2×C4C2×C4×D7111yescharacteristic
176C2×C4D4×D7111yescharacteristic
177C2×C4Q8×D7111yescharacteristic
178C2×C4C2×Dic14211yescharacteristic
179C2×C4#1176586411yescharacteristic
180C2×C4#1176586511yescharacteristic
181C8—22no—
182C8—1422no—
183C22C22⋊C4×D7111yescharacteristic
184C22#1455611411yescharacteristic
185C22#1455612711yescharacteristic
186C22D8×D7111yescharacteristic
187C22SD16×D7111yescharacteristic
188C22SD16×D7111yescharacteristic
189C22Q16×D7111yescharacteristic
190C4—42no—
191C4—81no—
192C4—2842no—
193C4—5681no—
194C4—714no—
195C4—714no—
196C4#1455609311yescharacteristic
197C4C4×Dic14211yescharacteristic
198C4#1455610511yescharacteristic
199C4#1455610611yescharacteristic
200C2#1798590311yescharacteristic
201C2#102×D7111yescharacteristic
202C2#103×D7111yescharacteristic
203C2#1798634411yescharacteristic
204C2#1798634811yescharacteristic
205C2#1798639611yescharacteristic
206C2#1798640311yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#340×C71C2
socleC23×C71C2×D4
hypercenter#340D71
hypocenterC71#1990
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
11. #14556127C4semidirect
31. C2×C71⋊C8Q8semidirect
46. C2×C8×C71Q8semidirect
104. C71#1990semidirect
141. C2×C8Dic142semidirect