Finite Group Databasebeta

← #22594714

Subgroups — #22594714

Order 9088.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556106C212yesmaximal, complemented
2C2×#14556104C212yesmaximal, complemented
3#17985923C212yesmaximal, complemented
4#17985921C212yesmaximal, complemented
5#17985919C211yesmaximal, characteristic, complemented
6#17985900C211yesmaximal, characteristic
7#17985896C212yesmaximal, complemented
8#17985877C212yesmaximal
9#395×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12C22×C4×D71C2212yes—
13#14556106—28no—
14#14556104—28no—
15C2×C4⋊Dic71C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×C4×Dic71C2211yescharacteristic
26C71⋊#95C2212yes—
27C71⋊#95C2211yescharacteristic
28C22×Q8×C71C2211yescharacteristic
29C2×C4⋊C4×C71C2212yes—
30C2×C4⋊C4×C71C2211yescharacteristic
31#95×C71C2212yes—
32#95×C71C2211yescharacteristic
33C23×D71C2312yes—
34C2×Dic142—24no—
35C2×Dic142—41no—
36C2×D284—41no—
37C2×D284—41no—
38C2×D284Q814yescomplemented
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C2×C4×D71—24no—
43C2×C4×D71—24no—
44C2×C4×D71—42no—
45C4⋊Dic71—24no—
46Dic71⋊C4—24no—
47Dic71⋊C4—24no—
48Dic71⋊C4—42no—
49Dic71⋊C4D418yescomplemented
50(C2×D71)⋊C4—24no—
51(C2×D71)⋊C4—24no—
52(C2×D71)⋊C4—24no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—42no—
55C4×Dic71—41no—
56C2×Q8×C71—24no—
57C2×Q8×C71—41no—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—24no—
64C4⋊C4×C71—41no—
65C23×C71C2411yescharacteristic
66C22×D71—22no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72C22×D71C2×Q814yes—
73Q8×C71—44no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D414yes—
86C2×C4×C71C2×Q812yes—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71C2×D414yes—
95C2×Dic71C2×D414yes—
96D284—44no—
97C4×D71—48no—
98Dic142—44no—
99C22×C71C22×D412yes—
100C22×C71C22×Q811yescharacteristic
101C22×C71H(5,𝔽2)11yescharacteristic
102C22×C71H(5,𝔽2)11yescharacteristic
103C22×C712−1+412yes—
104C2×D71—24no—
105C2×D71—24no—
106C2×D71—24no—
107C2×D71—42no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—42no—
111C4×C71—22no—
112C4×C71—24no—
113C4×C71—42no—
114C4×C71—41no—
115C4×C71—41no—
116Dic71—24no—
117Dic71—24no—
118Dic71—42no—
119C2×C71#53712yes—
120C2×C71#54311yescharacteristic
121C2×C71D4211yescharacteristic
122C2×C71D4×Q812yes—
123C2×C71#56311yescharacteristic
124D71—44no—
125#2683—711nomaximal, Sylow
126C71#268311yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—712no—
128C2×C22⋊Q8—712no—
129#396—712no—
130#396—712no—
131#395—711no—
132#395D7111yescharacteristic, complemented
133#390—711no—
134#389—712no—
135#386—712no—
136C22×Q8—711no—
137C22×Q8C2×D7111yescharacteristic
138C22×D4—711no—
139C23×C4—712no—
140C22⋊Q8—1428no—
141C22⋊Q8—1428no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4—712no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7112yes—
148C2×C4⋊C4C2×D7111yescharacteristic
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—712no—
152C2×C22⋊C4—712no—
153C2×C22⋊C4—712no—
154C2×C42—711no—
155#95—712no—
156#95—711no—
157#95C2×D7112yes—
158#95C2×D7111yescharacteristic
159C24—712no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—2841no—
163C2×Q8—1424no—
164C2×D4—2841no—
165C2×D4—2841no—
166C2×D4—714no—
167C22×C4—2842no—
168C22×C4—712no—
169C22×C4—712no—
170C22×C4—712no—
171C22×C4—1424no—
172C22×C4—1424no—
173C22×C4C22×D7111yescharacteristic
174C22×C4C22×D7112yes—
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7112yes—
177C22×C4C22×D7111yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—41no—
180C4⋊C4—2842no—
181C4⋊C4—718no—
182C4⋊C4—1424no—
183C4⋊C4—1424no—
184C4⋊C4—1424no—
185C22⋊C4—2842no—
186C22⋊C4—1424no—
187C22⋊C4—1424no—
188C22⋊C4—1424no—
189C22⋊C4—1424no—
190C42—2841no—
191C23—714no—
192C23—1422no—
193C23—1422no—
194C23—1422no—
195C23—1422no—
196C23—1422no—
197C23—1422no—
198C23C23×D7111yescharacteristic
199Q8—44no—
200Q8—2844no—
201D4—2844no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—2848no—
214C2×C4—714no—
215C2×C4—714no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1422no—
223C2×C4D4×D7114yes—
224C2×C4Q8×D7112yes—
225C22—2842no—
226C22—2842no—
227C22—2842no—
228C22—2842no—
229C22—1424no—
230C22—1424no—
231C22—1424no—
232C22C2×D4×D7112yes—
233C22C2×Q8×D7111yescharacteristic
234C22#1455617311yescharacteristic
235C22#1455617411yescharacteristic
236C22#1455617612yes—
237C4—22no—
238C4—24no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—2842no—
243C4—1424no—
244C4—1424no—
245C2—2844no—
246C2#1798657912yes—
247C2#1798662411yescharacteristic
248C2#1798663111yescharacteristic
249C2#1798666412yes—
250C2#1798674811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#395×C71C2
hypercenter#395D71
hypocenterC71#2683
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#14556106C2semidirect
2. C2×#14556104C2semidirect
3. #17985923C2semidirect
4. #17985921C2semidirect
5. #17985919C2semidirect
7. #17985896C2semidirect
9. #395×C71C2semidirect
38. C2×D284Q8semidirect
49. Dic71⋊C4D4semidirect
126. C71#2683semidirect
132. #395D71semidirect