Finite Group Databasebeta

← #22593816

Subgroups — #22593816

Order 9088.

596 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14556103C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#17985937C211yesmaximal, characteristic, complemented
4#17985917C211yesmaximal, characteristic, complemented
5#17985916C211yesmaximal, characteristic, complemented
6#17985891C212yesmaximal, complemented
7#17985888C212yesmaximal, complemented
8#17985877C212yesmaximal, complemented
9#17985877C212yesmaximal, complemented
10#17985846C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C22×D284C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16C22×C4×D71C2211yescharacteristic
17#14556103—24no—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C4⋊C4×D71—24no—
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2312yes—
38C23×D71C2311yescharacteristic
39C2×D284—24no—
40C2×D284—24no—
41C2×D284—24no—
42C2×D284—22no—
43C2×D284—42no—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C22×Dic71C2311yescharacteristic
47C22×Dic71C2311yescharacteristic
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—42no—
53C2×C4×D71—42no—
54C2×C4×D71D414yes—
55C4⋊Dic71—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—42no—
63(C2×D71)⋊C4—42no—
64C4×Dic71—42no—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2312yes—
67C22×C4×C71C2312yes—
68C22×C4×C71C2311yescharacteristic
69C22×C4×C71C2311yescharacteristic
70C4⋊C4×C71—22no—
71C4⋊C4×C71—22no—
72C23×C71C2411yescharacteristic
73C22×D71—21no—
74C22×D71—21no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—22no—
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—22no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71C2×D412yes—
84C22×D71#4114yes—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71C2×D412yes—
97C2×C4×C71#4112yes—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—22no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71C2×D412yes—
107C2×Dic71#4112yes—
108D284—44no—
109D284—48no—
110C4×D71—28no—
111C4×D71—44no—
112C4×D71—44no—
113C22×C71C22×D411yescharacteristic
114C22×C71C2×#4111yescharacteristic
115C22×C71C2×#4112yes—
116C22×C71C2×#4111yescharacteristic
117C22×C71H(5,𝔽2)12yes—
118C2×D71—22no—
119C2×D71—22no—
120C2×D71—24no—
121C2×D71—24no—
122C2×D71—42no—
123C2×D71—42no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—42no—
128C2×D71—42no—
129C2×D71#12414yes—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—42no—
133C4×C71—41no—
134C4×C71—42no—
135Dic71—22no—
136Dic71—22no—
137Dic71—41no—
138Dic71—42no—
139C2×C71#52511yescharacteristic
140C2×C71C2×#12411yescharacteristic
141C2×C71#53411yescharacteristic
142C2×C71#55312yes—
143C2×C71#55312yes—
144D71—24no—
145D71—82no—
146#2591—711nomaximal, Sylow
147C71#259111yescharacteristic, complemented, Sylow
148C2×#121—711no—
149C22×C4⋊C4—711no—
150#400—711no—
151#400D7111yescharacteristic, complemented
152#394—711no—
153#394—711no—
154#388—712no—
155#388—712no—
156#386—712no—
157#386—712no—
158C4×C22⋊C4—711no—
159C22×D4—711no—
160C22×D4—712no—
161C23×C4—711no—
162C23×C4—711no—
163C23×C4—711no—
164#121—1424no—
165C2×C4⋊C4—711no—
166C2×C4⋊C4—711no—
167C2×C4⋊C4—711no—
168C2×C4⋊C4—1424no—
169C2×C4⋊C4C2×D7111yescharacteristic
170C2×C4⋊C4C2×D7111yescharacteristic
171C2×C22⋊C4—712no—
172C2×C22⋊C4—712no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C42—712no—
178#95—711no—
179#95—712no—
180#95—712no—
181#95C2×D7111yescharacteristic
182#95C2×D7112yes—
183#95C2×D7112yes—
184C24—711no—
185C24—712no—
186C2×D4—2842no—
187C2×D4—1424no—
188C2×D4—1424no—
189C2×D4—1424no—
190C2×D4—1422no—
191C22×C4—2842no—
192C22×C4—2842no—
193C22×C4—714no—
194C22×C4—711no—
195C22×C4—711no—
196C22×C4—712no—
197C22×C4—711no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4—1422no—
201C22×C4—1422no—
202C22×C4C22×D7112yes—
203C22×C4C22×D7112yes—
204C22×C4C22×D7111yescharacteristic
205C22×C4C22×D7111yescharacteristic
206C22×C4C22×D7111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—1422no—
210C4⋊C4—1422no—
211C4⋊C4—1422no—
212C22⋊C4—2842no—
213C22⋊C4—2842no—
214C22⋊C4—1424no—
215C22⋊C4—1424no—
216C22⋊C4—1424no—
217C22⋊C4—1424no—
218C42—2842no—
219C23—712no—
220C23—714no—
221C23—1422no—
222C23—1422no—
223C23—1422no—
224C23—1422no—
225C23—1422no—
226C23—1422no—
227C23—1422no—
228C23—1421no—
229C23—1421no—
230C23—1422no—
231C23C23×D7111yescharacteristic
232D4—2844no—
233D4—2848no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2844no—
246C2×C4—2844no—
247C2×C4—712no—
248C2×C4—712no—
249C2×C4—1428no—
250C2×C4—1421no—
251C2×C4—1422no—
252C2×C4—1422no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1421no—
256C2×C4—1421no—
257C2×C4—1422no—
258C2×C4D4×D7112yes—
259C2×C4#1176586612yes—
260C22—2842no—
261C22—2842no—
262C22—2842no—
263C22—2842no—
264C22—2842no—
265C22—2842no—
266C22—2842no—
267C22—714no—
268C22—1422no—
269C22—1424no—
270C22—1424no—
271C22—1422no—
272C22C2×D4×D7111yescharacteristic
273C22#41×D7111yescharacteristic
274C22#41×D7112yes—
275C22C2×#1176586611yescharacteristic
276C22#1455617412yes—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—42no—
282C4—2841no—
283C4—2842no—
284C4—1422no—
285C4—1422no—
286C2—5682no—
287C2—1424no—
288C2#1798651211yescharacteristic
289C2#124×D7111yescharacteristic
290C2#1798656011yescharacteristic
291C2#1798670012yes—
292C2#1798670212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2591
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D71C2semidirect
3. #17985937C2semidirect
4. #17985917C2semidirect
5. #17985916C2semidirect
6. #17985891C2semidirect
7. #17985888C2semidirect
8. #17985877C2semidirect
9. #17985877C2semidirect
10. #17985846C2semidirect
11. #400×C71C2semidirect
147. C71#2591semidirect
151. #400D71semidirect