Finite Group Databasebeta

← #22586858

Subgroups — #22586858

Order 9088.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C22⋊Q8×D71C211yesmaximal, characteristic
2#17985961C211yesmaximal, characteristic
3#17985846C211yesmaximal, characteristic
4#17985737C211yesmaximal, characteristic, complemented
5#17985736C211yesmaximal, characteristic, complemented
6#17985735C211yesmaximal, characteristic, complemented
7#328×C71C211yesmaximal, characteristic, complemented
8C2×Q8×D71C411yescharacteristic, complemented
9C22×C4×D71C2211yescharacteristic
10C2×#11765848—21no—
11#14556108C411yescharacteristic, complemented
12#14556107C2211yescharacteristic
13#14556106—21no—
14#14556105—21no—
15#14556104C411yescharacteristic, complemented
16C4⋊C4×D71—21no—
17C4⋊C4×D71C411yescharacteristic, complemented
18C2×(C2×D71)⋊C4—21no—
19C22⋊C4×D71—21no—
20C2×C4×Dic71—21no—
21C22⋊(C71⋊C8)C2211yescharacteristic
22(C2×D71)⋊C8—21no—
23C71⋊#95C2211yescharacteristic
24C22⋊Q8×C71C2211yescharacteristic
25C22⋊C8×C71C2211yescharacteristic
26#95×C71C2211yescharacteristic
27C23×D71D411yescharacteristic
28C2×Dic142—21no—
29C2×Dic142C2×C411yescharacteristic
30Q8×D71—42no—
31C22×Dic71—21no—
32C22×Dic71D411yescharacteristic
33C2×C4×D71—21no—
34C2×C4×D71—21no—
35C2×C4×D71—42no—
36C2×C4×D71C2×C411yescharacteristic
37C2×C4×D71C2×C411yescharacteristic
38C2×C4×D71D411yescharacteristic
39C2×C4×D71D411yescharacteristic
40#11765848—42no—
41C2×C71⋊C8—21no—
42C4⋊Dic71—21no—
43C4⋊Dic71C2×C411yescharacteristic
44Dic71⋊C4—21no—
45Dic71⋊C4—21no—
46Dic71⋊C4—21no—
47C22⋊Dic71—21no—
48(C2×D71)⋊C4—24no—
49(C2×D71)⋊C4—21no—
50(C2×D71)⋊C4—21no—
51C4×Dic71—42no—
52C2×Q8×C71C2×C411yescharacteristic
53C22×C4×C71—21no—
54C22×C4×C71C2311yescharacteristic
55C2×C8×C71—21no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71C2×C411yescharacteristic
58C22⋊C4×C71—21no—
59C23×C71C2×D411yescharacteristic
60C22×D71—21no—
61C22×D71—21no—
62C22×D71—22no—
63C22×D71—41no—
64C22×D71C22⋊C411yescharacteristic
65C22×D71C22⋊C411yescharacteristic
66Q8×C71—41no—
67C2×C4×C71—22no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71—41no—
73C2×C4×C71C22×C411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×Dic71—22no—
76C2×Dic71—22no—
77C2×Dic71—21no—
78C2×Dic71—21no—
79C2×Dic71—41no—
80C2×Dic71—41no—
81C2×Dic71C22⋊C411yescharacteristic
82C2×Dic71C22⋊C411yescharacteristic
83C4×D71—22no—
84C4×D71—22no—
85C4×D71—41no—
86C4×D71—41no—
87C4×D71—42no—
88C4×D71—42no—
89Dic142—42no—
90Dic142—41no—
91C8×C71—41no—
92C71⋊C8—41no—
93C22×C71—21no—
94C22×C71—21no—
95C22×C71C2×C22⋊C411yescharacteristic
96C22×C71#13711yescharacteristic
97C22×C71#13711yescharacteristic
98C2×D71—21no—
99C2×D71—21no—
100C2×D71—21no—
101C2×D71—21no—
102C2×D71—21no—
103C2×D71—42no—
104C2×D71—42no—
105C2×D71#10412yes—
106C4×C71—21no—
107C4×C71—21no—
108C4×C71—42no—
109C4×C71—41no—
110C4×C71—41no—
111Dic71—21no—
112Dic71—42no—
113Dic71—41no—
114Dic71—41no—
115Dic71#9912yes—
116C2×C71—21no—
117C2×C71—21no—
118C2×C71C2×#9911yescharacteristic
119C2×C71#41911yescharacteristic
120C2×C71C2×#10411yescharacteristic
121D71—22no—
122D71—41no—
123#1508—711nomaximal, Sylow
124C71#150811yescharacteristic, complemented, Sylow
125C2×C22⋊Q8—711no—
126#407—711no—
127C4×C22⋊C4—711no—
128#328—711no—
129#328—711no—
130#328—711no—
131#328D7111yescharacteristic, complemented
132C22×Q8—711no—
133C23×C4—711no—
134C2×M16—1421no—
135C22⋊Q8—711no—
136C22⋊Q8—711no—
137C22⋊Q8—711no—
138C22⋊Q8—1421no—
139C22⋊Q8—1421no—
140C22⋊Q8C2×D7111yescharacteristic
141C2×C4⋊C4—711no—
142C2×C4⋊C4—1421no—
143C2×C22⋊C4—1421no—
144C2×C22⋊C4—1421no—
145C2×C42—1421no—
146C22⋊C8—711no—
147C22⋊C8—1421no—
148C22⋊C8C2×D7111yescharacteristic
149#95—711no—
150#95C2×D7111yescharacteristic
151C24—711no—
152C2×Q8—2842no—
153C2×Q8—711no—
154C2×Q8—1421no—
155C2×Q8C4×D7111yescharacteristic
156C22×C4—21no—
157C22×C4—2842no—
158C22×C4—711no—
159C22×C4—711no—
160C22×C4—711no—
161C22×C4—711no—
162C22×C4—711no—
163C22×C4—1421no—
164C22×C4—1421no—
165C22×C4—1421no—
166C22×C4C22×D7111yescharacteristic
167M16—2842no—
168C2×C8—21no—
169C2×C8—1421no—
170C4⋊C4—21no—
171C4⋊C4—711no—
172C4⋊C4—1421no—
173C4⋊C4—1421no—
174C4⋊C4—1421no—
175C4⋊C4—1421no—
176C4⋊C4C4×D7111yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—1421no—
179C22⋊C4—1421no—
180C22⋊C4—1424no—
181C22⋊C4—1421no—
182C42—2842no—
183C23—2841no—
184C23—711no—
185C23—711no—
186C23—1422no—
187C23—1421no—
188C23—1421no—
189C23D4×D7111yescharacteristic
190Q8—41no—
191Q8—2841no—
192Q8—2842no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—2841no—
200C2×C4—2841no—
201C2×C4—2842no—
202C2×C4—2842no—
203C2×C4—2841no—
204C2×C4—2841no—
205C2×C4—711no—
206C2×C4—711no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4—1422no—
213C2×C4C2×C4×D7111yescharacteristic
214C2×C4D4×D7111yescharacteristic
215C8—41no—
216C8—2841no—
217C22—21no—
218C22—21no—
219C22—2842no—
220C22—2842no—
221C22—712no—
222C22—1421no—
223C22—1421no—
224C22—1421no—
225C22—1421no—
226C22—1421no—
227C22C22⋊C4×D7111yescharacteristic
228C22#1455615711yescharacteristic
229C22#1455616011yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—2842no—
236C4—2841no—
237C4—2841no—
238C4—712no—
239C4—1421no—
240C2—21no—
241C2—21no—
242C2—2841no—
243C2—1422no—
244C2#99×D7111yescharacteristic
245C2#1798600611yescharacteristic
246C2#104×D7111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#328×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#328D71
hypocenterC71#1508
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #17985737C2semidirect
5. #17985736C2semidirect
6. #17985735C2semidirect
7. #328×C71C2semidirect
8. C2×Q8×D71C4semidirect
11. #14556108C4semidirect
15. #14556104C4semidirect
17. C4⋊C4×D71C4semidirect
124. C71#1508semidirect
131. #328D71semidirect