Finite Group Databasebeta

← #22589714

Subgroups — #22589714

Order 9088.

284 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 89 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×C4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3C2×#14556049C211yesmaximal, characteristic
4C2×#14556049C211yesmaximal, characteristic
5#17985897C211yesmaximal, characteristic, complemented
6#17985761C211yesmaximal, characteristic, complemented
7#390×C71C211yesmaximal, characteristic
8C22×C71⋊C8C2211yescharacteristic
9C22×C71⋊C8C2211yescharacteristic
10C2×C4⋊Dic71C2211yescharacteristic
11C2×C4⋊Dic71C2211yescharacteristic
12C2×Dic71⋊C4—21no—
13#14556059—22no—
14C4⋊(C71⋊C8)—22no—
15#14556049—22no—
16#14556049—22no—
17C71⋊#95—21no—
18C22×D4×C71C2211yescharacteristic
19C2×C4⋊C4×C71C2211yescharacteristic
20C2×C22⋊C4×C71—21no—
21C2×C22⋊C4×C71—21no—
22C2×C42×C71C2211yescharacteristic
23C22×Dic71—21no—
24C22×Dic71—21no—
25C2×C71⋊C8—21no—
26C2×C71⋊C8—21no—
27C2×C71⋊C8—21no—
28C2×C71⋊C8—21no—
29C2×C71⋊C8—21no—
30C2×C71⋊C8D412yes—
31C4⋊Dic71—21no—
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71D412yes—
35Dic71⋊C4—42no—
36C24×C71—21no—
37C2×D4×C71—21no—
38C2×D4×C71—21no—
39C2×D4×C71—21no—
40C2×D4×C71—21no—
41C2×D4×C71—21no—
42C2×D4×C71Q812yes—
43C22×C4×C71—21no—
44C22×C4×C71D411yescharacteristic
45C22×C4×C71D411yescharacteristic
46C22×C4×C71C2311yescharacteristic
47C4⋊C4×C71—21no—
48C4⋊C4×C71—21no—
49C22⋊C4×C71—42no—
50C22⋊C4×C71—42no—
51C42×C71—21no—
52C42×C71—21no—
53C23×C71—22no—
54C23×C71—41no—
55C23×C71—41no—
56C23×C71—41no—
57C23×C71—41no—
58C23×C71—41no—
59C23×C71—41no—
60C23×C71C2×D411yescharacteristic
61D4×C71—22no—
62D4×C71—22no—
63D4×C71—41no—
64D4×C71—41no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71SD1612yes—
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×Q811yescharacteristic
77C2×C4×C71#4111yescharacteristic
78C2×C4×C71#4111yescharacteristic
79C2×C4×C71#4111yescharacteristic
80C2×Dic71—22no—
81C2×Dic71—41no—
82C2×Dic71—41no—
83C2×Dic71—41no—
84C2×Dic71—41no—
85C2×Dic71—41no—
86C71⋊C8—22no—
87C71⋊C8—41no—
88C22×C71—42no—
89C22×C71—42no—
90C22×C71—42no—
91C22×C71—81no—
92C22×C71—81no—
93C22×C71—81no—
94C22×C71—81no—
95C22×C71#12111yescharacteristic
96C22×C71#12111yescharacteristic
97C22×C71C22⋊Q811yescharacteristic
98C22×C71C2×SD1611yescharacteristic
99C22×C71#13511yescharacteristic
100C22×C71AGL(1,ℤ/8)11yescharacteristic
101C22×C71AGL(1,ℤ/8)11yescharacteristic
102C4×C71—22no—
103C4×C71—41no—
104C4×C71—81no—
105C4×C71#12111yescharacteristic
106C4×C71C22⋊Q811yescharacteristic
107C4×C71C22⋊Q811yescharacteristic
108C4×C71#12611yescharacteristic
109Dic71—42no—
110Dic71—81no—
111C2×C71—82no—
112C2×C71#39611yescharacteristic
113C2×C71#46011yescharacteristic
114C2×C71#46311yescharacteristic
115C2×C71#46511yescharacteristic
116C2×C71#46911yescharacteristic
117C2×C71#47611yescharacteristic
118C2×C71#47811yescharacteristic
119#2065—711nomaximal, Sylow
120C71#206511yescharacteristic, complemented, Sylow
121C2×#106—711no—
122C2×C4⋊C8—711no—
123C2×#102—711no—
124C2×#102—711no—
125#390D7111yescharacteristic
126#389—711no—
127#340—711no—
128C22×D4C2×D7111yescharacteristic
129C22×C8—711no—
130C22×C8—711no—
131C2×C4⋊C4—711no—
132C2×C4⋊C4—711no—
133C2×C4⋊C4—1421no—
134C2×C4⋊C4C2×D7111yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42C2×D7111yescharacteristic
138#106—1422no—
139C4⋊C8—1422no—
140#102—1422no—
141#102—1422no—
142#95—1421no—
143C24—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4Dic14212yes—
150C22×C4—21no—
151C22×C4—1421no—
152C22×C4—1421no—
153C22×C4C22⋊D7111yescharacteristic
154C22×C4C22⋊D7111yescharacteristic
155C22×C4C22×D7111yescharacteristic
156C2×C8—712no—
157C2×C8—1421no—
158C2×C8—1421no—
159C2×C8—1421no—
160C2×C8—1421no—
161C2×C8—1421no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—2842no—
165C4⋊C4—712no—
166C4⋊C4—1421no—
167C4⋊C4—1421no—
168C4⋊C4—1421no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—21no—
172C42—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23C2×C22⋊D7111yescharacteristic
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—2841no—
194C2×C4—2841no—
195C2×C4—2841no—
196C2×C4—2841no—
197C2×C4—2841no—
198C2×C4—1422no—
199C2×C4#1176585312yes—
200C2×C4D4×D7111yescharacteristic
201C2×C4D4×D7111yescharacteristic
202C2×C4C2×Dic14211yescharacteristic
203C2×C4#1176586411yescharacteristic
204C2×C4#1176586411yescharacteristic
205C2×C4#1176586511yescharacteristic
206C8—2841no—
207C8—1422no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#1455609911yescharacteristic
216C22#1455610011yescharacteristic
217C22#1455610711yescharacteristic
218C22C2×#1176585311yescharacteristic
219C22#1455614811yescharacteristic
220C22#1455615311yescharacteristic
221C22#1455615411yescharacteristic
222C4—22no—
223C4—41no—
224C4—81no—
225C4—2842no—
226C4—5681no—
227C4#1455610111yescharacteristic
228C4#1455610511yescharacteristic
229C4#1455610511yescharacteristic
230C4#1455612111yescharacteristic
231C2—82no—
232C2#1798592211yescharacteristic
233C2#1798619611yescharacteristic
234C2#1798621111yescharacteristic
235C2#1798622611yescharacteristic
236C2#1798625011yescharacteristic
237C2#1798628911yescharacteristic
238C2#1798629911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D71
commutatorC22×C4×C71C23
FrattiniC22×C4C22×D71
Fitting#390×C71C2
socleC23×C71C2×D4
hypercenter#390D71
hypocenterC71#2065
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14556059C2semidirect
2. C2×C4⋊(C71⋊C8)C2semidirect
5. #17985897C2semidirect
6. #17985761C2semidirect
120. C71#2065semidirect