Finite Group Databasebeta

← #22588540

Subgroups — #22588540

Order 9088.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14556086C211yesmaximal, characteristic
2C2×#14556061C211yesmaximal, characteristic, complemented
3C2×#14556060C211yesmaximal, characteristic, complemented
4C2×(C2×D71)⋊C8C211yesmaximal, characteristic
5#17985856C211yesmaximal, characteristic
6#17985761C211yesmaximal, characteristic, complemented
7#340×C71C211yesmaximal, characteristic, complemented
8C22×C4×D71C2211yescharacteristic
9C22×C71⋊C8C2211yescharacteristic
10#14556086—22no—
11#14556086—22no—
12C2×C4⋊Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×(C2×D71)⋊C4—21no—
15C2×(C2×D71)⋊C4—21no—
16C2×C4×Dic71—21no—
17#14556061—22no—
18#14556060—22no—
19(C2×D71)⋊C8C414yescomplemented
20C71⋊#95—21no—
21C22×C8×C71C2211yescharacteristic
22C2×C4⋊C4×C71C2211yescharacteristic
23C2×C4⋊C4×C71C2211yescharacteristic
24C23×D71D411yescharacteristic
25C22×Dic71—21no—
26C22×Dic71—21no—
27C22×Dic71D411yescharacteristic
28C2×C4×D71—21no—
29C2×C4×D71—21no—
30C2×C4×D71—21no—
31C2×C4×D71—21no—
32C2×C4×D71C2×C412yes—
33C2×C4×D71Q812yes—
34C2×C71⋊C8—21no—
35C2×C71⋊C8—21no—
36C2×C71⋊C8C2×C412yes—
37C4⋊Dic71—21no—
38C4⋊Dic71—21no—
39C4⋊Dic71—21no—
40C4⋊Dic71D412yes—
41(C2×D71)⋊C4—24no—
42(C2×D71)⋊C4—42no—
43C4×Dic71—24no—
44C22×C4×C71—21no—
45C22×C4×C71—21no—
46C22×C4×C71C2311yescharacteristic
47C2×C8×C71—21no—
48C2×C8×C71—21no—
49C2×C8×C71C2×C412yes—
50C4⋊C4×C71—21no—
51C4⋊C4×C71—21no—
52C4⋊C4×C71—21no—
53C4⋊C4×C71D412yes—
54C23×C71C2×D411yescharacteristic
55C22×D71—21no—
56C22×D71—21no—
57C22×D71—21no—
58C22×D71—21no—
59C22×D71—21no—
60C22×D71—21no—
61C22×D71C4⋊C412yes—
62C2×C4×C71—22no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—41no—
67C2×C4×C71—41no—
68C2×C4×C71C22×C411yescharacteristic
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×C4×C71C2×Q811yescharacteristic
72C2×C4×C71#4111yescharacteristic
73C2×C4×C71#4111yescharacteristic
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—22no—
77C2×Dic71—41no—
78C2×Dic71—41no—
79C2×Dic71—41no—
80C2×Dic71—41no—
81C2×Dic71—41no—
82C2×Dic71C4⋊C412yes—
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—22no—
86C4×D71—22no—
87C8×C71—22no—
88C71⋊C8—22no—
89C22×C71C2×C4⋊C411yescharacteristic
90C22×C71#12011yescharacteristic
91C22×C71#12311yescharacteristic
92C22×C71#13511yescharacteristic
93C22×C71#13511yescharacteristic
94C22×C71AGL(1,ℤ/8)11yescharacteristic
95C22×C71#13711yescharacteristic
96C2×D71—22no—
97C2×D71—22no—
98C2×D71—22no—
99C2×D71—41no—
100C2×D71—41no—
101C2×D71—41no—
102C2×D71—41no—
103C4×C71—42no—
104C4×C71—81no—
105C4×C71C4×D411yescharacteristic
106C4×C71C4×D411yescharacteristic
107C4×C71C22⋊Q811yescharacteristic
108C4×C71C22⋊Q811yescharacteristic
109Dic71—22no—
110Dic71—42no—
111Dic71—81no—
112C2×C71#38511yescharacteristic
113C2×C71#42711yescharacteristic
114C2×C71#42811yescharacteristic
115C2×C71#44911yescharacteristic
116C2×C71#45211yescharacteristic
117C2×C71#48211yescharacteristic
118C2×C71#48511yescharacteristic
119D71—42no—
120#1854—711nomaximal, Sylow
121C71#185411yescharacteristic, complemented, Sylow
122C2×#117—711no—
123C2×#107—711no—
124C2×#106—711no—
125C2×C22⋊C8—711no—
126#380—711no—
127#340—711no—
128#340D7111yescharacteristic, complemented
129C23×C4—711no—
130C22×C8—711no—
131C22×C8C2×D7111yescharacteristic
132#117—1422no—
133#117—1422no—
134C2×C4⋊C4—711no—
135C2×C4⋊C4—711no—
136C2×C4⋊C4C2×D7111yescharacteristic
137C2×C4⋊C4C2×D7111yescharacteristic
138C2×C22⋊C4—1421no—
139C2×C22⋊C4—1421no—
140C2×C42—1421no—
141#107—1422no—
142#106—1422no—
143C22⋊C8—714no—
144#95—1421no—
145C24—711no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—711no—
149C22×C4—712no—
150C22×C4—712no—
151C22×C4—1421no—
152C22×C4—1421no—
153C22×C4—1421no—
154C22×C4—1421no—
155C22×C4—1421no—
156C22×C4—1421no—
157C22×C4C22×D7111yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—712no—
161C2×C8—1421no—
162C2×C8—1421no—
163C2×C8C4×D7112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—712no—
168C4⋊C4—1421no—
169C4⋊C4—1421no—
170C4⋊C4—1421no—
171C4⋊C4D28412yes—
172C22⋊C4—2842no—
173C22⋊C4—1424no—
174C42—1424no—
175C23—712no—
176C23—1421no—
177C23—1421no—
178C23—1421no—
179C23—1421no—
180C23—1421no—
181C23—1421no—
182C23D4×D7111yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2841no—
190C2×C4—2841no—
191C2×C4—2841no—
192C2×C4—2841no—
193C2×C4—2841no—
194C2×C4—712no—
195C2×C4—1422no—
196C2×C4—1421no—
197C2×C4—1421no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4C2×C4×D7111yescharacteristic
203C2×C4D4×D7111yescharacteristic
204C2×C4C2×D28411yescharacteristic
205C2×C4Q8×D7111yescharacteristic
206C2×C4#1176586411yescharacteristic
207C2×C4#1176586511yescharacteristic
208C8—22no—
209C8—1422no—
210C22—2841no—
211C22—2841no—
212C22—2841no—
213C22—2841no—
214C22—1422no—
215C22—1422no—
216C22—1422no—
217C22C4⋊C4×D7111yescharacteristic
218C22#1455609811yescharacteristic
219C22#1455611011yescharacteristic
220C22#1455614911yescharacteristic
221C22#1455614911yescharacteristic
222C22#1455615211yescharacteristic
223C22#1455616011yescharacteristic
224C4—42no—
225C4—81no—
226C4—2842no—
227C4—5681no—
228C4—1422no—
229C4(C22⋊D71)⋊C411yescharacteristic
230C4C4×D28411yescharacteristic
231C4#1455610611yescharacteristic
232C4#1455610811yescharacteristic
233C2—2842no—
234C2#1798587311yescharacteristic
235C2#1798603411yescharacteristic
236C2#1798604011yescharacteristic
237C2#1798614511yescharacteristic
238C2#1798616411yescharacteristic
239C2#1798632211yescharacteristic
240C2#1798633811yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#340×C71C2
socleC23×C71C2×D4
hypercenter#340D71
hypocenterC71#1854
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
2. C2×#14556061C2semidirect
3. C2×#14556060C2semidirect
6. #17985761C2semidirect
7. #340×C71C2semidirect
19. (C2×D71)⋊C8C4semidirect
121. C71#1854semidirect
128. #340D71semidirect