Finite Group Databasebeta

← #22586708

Subgroups — #22586708

Order 9088.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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×#14556123C211yesmaximal, characteristic
2#17986053C211yesmaximal, characteristic, complemented
3#17986049C211yesmaximal, characteristic, complemented
4#17985725C211yesmaximal, characteristic, complemented
5#17985722C211yesmaximal, characteristic
6#17985722C211yesmaximal, characteristic
7#324×C71C211yesmaximal, characteristic, complemented
8#14556123—21no—
9#14556123—21no—
10#14556123—21no—
11#14556123—21no—
12C2×C4⋊Dic71C411yescharacteristic
13C2×Dic71⋊C4C411yescharacteristic
14C2×(C2×D71)⋊C4C411yescharacteristic
15C2×(C2×D71)⋊C4C411yescharacteristic
16C2×(C2×D71)⋊C4C2211yescharacteristic
17C2×C4×Dic71C2211yescharacteristic
18Dic71⋊C8—21no—
19Dic71⋊C8—21no—
20C22⋊(C71⋊C8)C2211yescharacteristic
21C22⋊(C71⋊C8)C2211yescharacteristic
22#14556038—21no—
23C8×Dic71—21no—
24C2×C4⋊C4×C71C2211yescharacteristic
25C22⋊C8×C71C2211yescharacteristic
26C22⋊C8×C71C2211yescharacteristic
27C23×D71C2×C411yescharacteristic
28C22×Dic71C2×C411yescharacteristic
29C22×Dic71C2×C411yescharacteristic
30C22×Dic71C2×C411yescharacteristic
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71—21no—
35Dic71⋊C4—21no—
36Dic71⋊C4—21no—
37(C2×D71)⋊C4—21no—
38(C2×D71)⋊C4—21no—
39(C2×D71)⋊C4—21no—
40(C2×D71)⋊C4—21no—
41(C2×D71)⋊C4—21no—
42(C2×D71)⋊C4—21no—
43C4×Dic71D411yescharacteristic
44C4×Dic71D411yescharacteristic
45C4×Dic71D411yescharacteristic
46C4×Dic71D411yescharacteristic
47C22×C4×C71C2×C411yescharacteristic
48C22×C4×C71C2×C411yescharacteristic
49C22×C4×C71C2311yescharacteristic
50C2×C8×C71—21no—
51C2×C8×C71—21no—
52C4⋊C4×C71—21no—
53C4⋊C4×C71—21no—
54C23×C71C22×C411yescharacteristic
55C22×D71—21no—
56C22×D71—21no—
57C22×D71—21no—
58C22×D71—41no—
59C22×D71—41no—
60C2×C4×C71—21no—
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—21no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71C2×D411yescharacteristic
67C2×C4×C71C2×D411yescharacteristic
68C2×Dic71—21no—
69C2×Dic71—21no—
70C2×Dic71—21no—
71C2×Dic71—41no—
72C2×Dic71—41no—
73C2×Dic71C22⋊C411yescharacteristic
74C2×Dic71C22⋊C411yescharacteristic
75C2×Dic71C22⋊C411yescharacteristic
76C2×Dic71C22⋊C411yescharacteristic
77C8×C71—41no—
78C8×C71—41no—
79C71⋊C8—41no—
80C71⋊C8—41no—
81C22×C71—21no—
82C22×C71—21no—
83C22×C71C2×C22⋊C411yescharacteristic
84C22×C71#13111yescharacteristic
85C22×C71#13111yescharacteristic
86C2×D71—41no—
87C2×D71—41no—
88C2×D71—41no—
89C2×D71—81no—
90C2×D71—81no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—41no—
94C4×C71—41no—
95Dic71—21no—
96Dic71—21no—
97Dic71—21no—
98Dic71—21no—
99Dic71—41no—
100C2×C71—21no—
101C2×C71—21no—
102C2×C71#40811yescharacteristic
103C2×C71#41011yescharacteristic
104C2×C71#41311yescharacteristic
105D71—81no—
106#1473—711nomaximal, Sylow
107C71#147311yescharacteristic, complemented, Sylow
108C2×#126—711no—
109#432—711no—
110#431—711no—
111#324—711no—
112#324D7111yescharacteristic, complemented
113#323—711no—
114#323—711no—
115#126—1421no—
116#126—1421no—
117#126—1421no—
118#126—1421no—
119C2×C4⋊C4—711no—
120C2×C4⋊C4—711no—
121C2×C4⋊C4C2×D7111yescharacteristic
122C2×C22⋊C4—711no—
123C2×C22⋊C4—711no—
124C2×C22⋊C4—711no—
125C2×C42—711no—
126C4⋊C8—1421no—
127C4⋊C8—1421no—
128C22⋊C8—711no—
129C22⋊C8—711no—
130C22⋊C8C2×D7111yescharacteristic
131C22⋊C8C2×D7111yescharacteristic
132#97—1421no—
133C4×C8—1421no—
134C24—711no—
135C22×C4—711no—
136C22×C4—711no—
137C22×C4—711no—
138C22×C4C4×D7111yescharacteristic
139C22×C4C4×D7111yescharacteristic
140C22×C4C22×D7111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1421no—
144C2×C8—1421no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—1421no—
148C4⋊C4—1421no—
149C4⋊C4—1421no—
150C4⋊C4—1421no—
151C22⋊C4—1421no—
152C22⋊C4—1421no—
153C22⋊C4—1421no—
154C22⋊C4—1421no—
155C22⋊C4—1421no—
156C22⋊C4—1421no—
157C42—711no—
158C42—711no—
159C42—711no—
160C42—711no—
161C23—2841no—
162C23—2841no—
163C23—1421no—
164C23—1421no—
165C23—1421no—
166C23C2×C4×D7111yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—2841no—
174C2×C4—2841no—
175C2×C4—711no—
176C2×C4—711no—
177C2×C4—711no—
178C2×C4—711no—
179C2×C4—1421no—
180C2×C4—1421no—
181C2×C4—1421no—
182C2×C4D4×D7111yescharacteristic
183C2×C4D4×D7111yescharacteristic
184C8—41no—
185C8—41no—
186C8—2841no—
187C8—2841no—
188C22—21no—
189C22—21no—
190C22—2841no—
191C22—2841no—
192C22—2841no—
193C22—5681no—
194C22—5681no—
195C22C22⋊C4×D7111yescharacteristic
196C22#1455613411yescharacteristic
197C22#1455613611yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—41no—
202C4—2841no—
203C4—1421no—
204C4—1421no—
205C4—1421no—
206C4—1421no—
207C2—21no—
208C2—21no—
209C2—5681no—
210C2#1798596411yescharacteristic
211C2#1798597111yescharacteristic
212C2#1798598211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC23×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#324×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#324D71
hypocenterC71#1473
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #17986053C2semidirect
3. #17986049C2semidirect
4. #17985725C2semidirect
7. #324×C71C2semidirect
107. C71#1473semidirect
112. #324D71semidirect