Finite Group Databasebeta

← #22589779

Subgroups — #22589779

Order 9088.

256 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14556059C211yesmaximal, characteristic, complemented
2C2×(C2×D71)⋊C8C211yesmaximal, characteristic
3#17985856C211yesmaximal, characteristic
4#17985856C211yesmaximal, characteristic
5#17985762C211yesmaximal, characteristic, complemented
6#17985761C211yesmaximal, characteristic, complemented
7#340×C71C211yesmaximal, characteristic, complemented
8C22×C4×D71C2211yescharacteristic
9C22×C71⋊C8C2211yescharacteristic
10C2×C4⋊Dic71C2211yescharacteristic
11C2×C4⋊Dic71C2211yescharacteristic
12C2×(C2×D71)⋊C4—21no—
13C2×(C2×D71)⋊C4—21no—
14#14556059—22no—
15(C2×D71)⋊C8—22no—
16C71⋊#95—21no—
17C71⋊#95—21no—
18C22×C8×C71C2211yescharacteristic
19C2×C4⋊C4×C71C2211yescharacteristic
20C2×C4⋊C4×C71C2211yescharacteristic
21C23×D71D411yescharacteristic
22C22×Dic71—21no—
23C22×Dic71—21no—
24C22×Dic71D411yescharacteristic
25C2×C4×D71—21no—
26C2×C4×D71—21no—
27C2×C4×D71—21no—
28C2×C4×D71—21no—
29C2×C4×D71—21no—
30C2×C4×D71—21no—
31C2×C71⋊C8—21no—
32C2×C71⋊C8—21no—
33C2×C71⋊C8D412yes—
34C4⋊Dic71—21no—
35C4⋊Dic71—21no—
36C4⋊Dic71—21no—
37C4⋊Dic71—21no—
38(C2×D71)⋊C4—42no—
39(C2×D71)⋊C4—42no—
40C22×C4×C71—21no—
41C22×C4×C71—21no—
42C22×C4×C71C2311yescharacteristic
43C2×C8×C71—21no—
44C2×C8×C71—21no—
45C2×C8×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—21no—
48C4⋊C4×C71—21no—
49C4⋊C4×C71—21no—
50C23×C71C2×D411yescharacteristic
51C22×D71—21no—
52C22×D71—21no—
53C22×D71—21no—
54C22×D71—21no—
55C22×D71—21no—
56C22×D71—21no—
57C22×D71SD1612yes—
58C2×C4×C71—41no—
59C2×C4×C71—41no—
60C2×C4×C71—41no—
61C2×C4×C71—41no—
62C2×C4×C71—41no—
63C2×C4×C71—41no—
64C2×C4×C71C2×D411yescharacteristic
65C2×C4×C71#4111yescharacteristic
66C2×C4×C71#4111yescharacteristic
67C2×C4×C71#4111yescharacteristic
68C2×C4×C71#4111yescharacteristic
69C2×C4×C71#4111yescharacteristic
70C2×Dic71—21no—
71C2×Dic71—21no—
72C2×Dic71—21no—
73C2×Dic71—41no—
74C2×Dic71—41no—
75C2×Dic71—41no—
76C2×Dic71—41no—
77C2×Dic71—41no—
78C2×Dic71—41no—
79C4×D71—41no—
80C4×D71—41no—
81C4×D71—41no—
82C4×D71—41no—
83C8×C71—41no—
84C71⋊C8—22no—
85C22×C71#12111yescharacteristic
86C22×C71#12311yescharacteristic
87C22×C71#12311yescharacteristic
88C22×C71C2×SD1611yescharacteristic
89C22×C71#13511yescharacteristic
90C22×C71AGL(1,ℤ/8)11yescharacteristic
91C22×C71#13711yescharacteristic
92C2×D71—22no—
93C2×D71—22no—
94C2×D71—22no—
95C2×D71—41no—
96C2×D71—41no—
97C2×D71—41no—
98C2×D71—41no—
99C4×C71—81no—
100C4×C71—81no—
101C4×C71#12311yescharacteristic
102C4×C71#12411yescharacteristic
103C4×C71#12611yescharacteristic
104C4×C71#12611yescharacteristic
105Dic71—41no—
106Dic71—81no—
107Dic71—81no—
108C2×C71#39711yescharacteristic
109C2×C71#46511yescharacteristic
110C2×C71#46811yescharacteristic
111C2×C71#48111yescharacteristic
112C2×C71#48211yescharacteristic
113C2×C71#48311yescharacteristic
114C2×C71#48511yescharacteristic
115D71—42no—
116#2077—711nomaximal, Sylow
117C71#207711yescharacteristic, complemented, Sylow
118C2×#106—711no—
119C2×C22⋊C8—711no—
120#380—711no—
121#380—711no—
122#340—711no—
123#340—711no—
124#340D7111yescharacteristic, complemented
125C23×C4—711no—
126C22×C8—711no—
127C22×C8C2×D7111yescharacteristic
128C2×C4⋊C4—711no—
129C2×C4⋊C4—711no—
130C2×C4⋊C4C2×D7111yescharacteristic
131C2×C4⋊C4C2×D7111yescharacteristic
132C2×C22⋊C4—1421no—
133C2×C22⋊C4—1421no—
134#106—1422no—
135C22⋊C8—1422no—
136#95—1421no—
137#95—1421no—
138C24—711no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—711no—
142C22×C4—1421no—
143C22×C4—1421no—
144C22×C4—1421no—
145C22×C4—1421no—
146C22×C4—1421no—
147C22×C4—1421no—
148C22×C4—1421no—
149C22×C4—1421no—
150C22×C4C22×D7111yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—712no—
155C2×C8—1421no—
156C2×C8—1421no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1421no—
162C4⋊C4—1421no—
163C4⋊C4—1421no—
164C4⋊C4—1421no—
165C22⋊C4—2842no—
166C22⋊C4—2842no—
167C23—712no—
168C23—1421no—
169C23—1421no—
170C23—1421no—
171C23—1421no—
172C23—1421no—
173C23—1421no—
174C23D4×D7111yescharacteristic
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—2841no—
182C2×C4—2841no—
183C2×C4—2841no—
184C2×C4—2841no—
185C2×C4—2841no—
186C2×C4—2841no—
187C2×C4—2841no—
188C2×C4—2841no—
189C2×C4—2841no—
190C2×C4—2841no—
191C2×C4—1421no—
192C2×C4—1421no—
193C2×C4—1421no—
194C2×C4D4×D7111yescharacteristic
195C2×C4#1176586411yescharacteristic
196C2×C4#1176586511yescharacteristic
197C2×C4#1176586511yescharacteristic
198C2×C4#1176586511yescharacteristic
199C2×C4#1176586611yescharacteristic
200C8—41no—
201C8—1422no—
202C22—2841no—
203C22—2841no—
204C22—2841no—
205C22—2841no—
206C22—1422no—
207C22—1422no—
208C22—1422no—
209C22#1455610111yescharacteristic
210C22#1455611011yescharacteristic
211C22#1455611311yescharacteristic
212C22SD16×D7111yescharacteristic
213C22#1455615111yescharacteristic
214C22#1455615211yescharacteristic
215C22#1455615711yescharacteristic
216C4—81no—
217C4—81no—
218C4—2841no—
219C4—5681no—
220C4—5681no—
221C4#1455611311yescharacteristic
222C4#1455611411yescharacteristic
223C4#1455612211yescharacteristic
224C4#1455612311yescharacteristic
225C2—2842no—
226C2#1798592811yescharacteristic
227C2#1798623111yescharacteristic
228C2#1798624411yescharacteristic
229C2#1798631611yescharacteristic
230C2#1798632611yescharacteristic
231C2#1798633111yescharacteristic
232C2#1798633911yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#340×C71C2
socleC23×C71C2×D4
hypercenter#340D71
hypocenterC71#2077
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#14556059C2semidirect
5. #17985762C2semidirect
6. #17985761C2semidirect
7. #340×C71C2semidirect
117. C71#2077semidirect
124. #340D71semidirect