Finite Group Databasebeta

← #22594227

Subgroups — #22594227

Order 9088.

620 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556102C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D71C211yesmaximal, characteristic, complemented
3#17985939C211yesmaximal, characteristic, complemented
4#17985936C211yesmaximal, characteristic, complemented
5#17985917C211yesmaximal, characteristic, complemented
6#17985915C212yesmaximal, complemented
7#17985909C211yesmaximal, characteristic, complemented
8#17985908C212yesmaximal, complemented
9#17985891C212yesmaximal, complemented
10#17985877C212yesmaximal, complemented
11#381×C71C211yesmaximal, characteristic, complemented
12C22×D284C2212yes—
13C22×D284C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C22⋊D71C2212yes—
16C22×C4×D71C2211yescharacteristic
17#14556102—24no—
18C4×C22⋊D71—24no—
19C2×C4⋊Dic71C2211yescharacteristic
20C2×C4⋊Dic71C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×C4×Dic71C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C23×C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2212yes—
34#95×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C23×D71C2312yes—
38C23×D71C2311yescharacteristic
39C2×D284—24no—
40C2×D284—22no—
41C2×D284—42no—
42C2×D284—42no—
43C2×D284—42no—
44C2×C22⋊D71—22no—
45C2×C22⋊D71—22no—
46C2×C22⋊D71—24no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—42no—
49C2×C22⋊D71—42no—
50C22×Dic71C2311yescharacteristic
51C22×Dic71C2312yes—
52C22×Dic71C2311yescharacteristic
53C22×Dic71C2311yescharacteristic
54C2×C4×D71—22no—
55C2×C4×D71—42no—
56C4⋊Dic71—22no—
57C4⋊Dic71—42no—
58Dic71⋊C4—22no—
59C22⋊Dic71—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—24no—
62(C2×D71)⋊C4—24no—
63(C2×D71)⋊C4—42no—
64(C2×D71)⋊C4—42no—
65(C2×D71)⋊C4—42no—
66C4×Dic71—22no—
67C24×C71C2311yescharacteristic
68C22×C4×C71—42no—
69C22×C4×C71D414yescomplemented
70C22×C4×C71C2312yes—
71C22×C4×C71C2312yes—
72C22×C4×C71C2311yescharacteristic
73C22×C4×C71C2311yescharacteristic
74C22⋊C4×C71—42no—
75C23×C71—21no—
76C23×C71—21no—
77C23×C71—22no—
78C23×C71—22no—
79C23×C71C2×D412yes—
80C23×C71C2411yescharacteristic
81C22×D71—21no—
82C22×D71—21no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—22no—
90C22×D71—22no—
91C22×D71—22no—
92C22×D71#4112yes—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—44no—
102C2×C4×C71C2×D412yes—
103C2×C4×C71C2×D412yes—
104C22⋊D71—28no—
105C22⋊D71—48no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—21no—
109C2×Dic71—22no—
110C2×Dic71—21no—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71#4112yes—
116D284—44no—
117D284—48no—
118C4×D71—44no—
119C22×C71—22no—
120C22×C71—22no—
121C22×C71—22no—
122C22×C71—42no—
123C22×C71—42no—
124C22×C71C22×D411yescharacteristic
125C22×C71C2×#4111yescharacteristic
126C22×C71C2×#4111yescharacteristic
127C22×C71H(5,𝔽2)12yes—
128C22×C71H(5,𝔽2)12yes—
129C2×D71—22no—
130C2×D71—24no—
131C2×D71—41no—
132C2×D71—41no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—42no—
136C2×D71—42no—
137C2×D71—42no—
138C2×D71—42no—
139C2×D71—42no—
140C2×D71—42no—
141C4×C71—22no—
142C4×C71—22no—
143C4×C71—42no—
144C4×C71—42no—
145Dic71—22no—
146Dic71—41no—
147Dic71—41no—
148Dic71—42no—
149C2×C71—42no—
150C2×C71#52511yescharacteristic
151C2×C71#53411yescharacteristic
152C2×C71#53511yescharacteristic
153C2×C71#53812yes—
154C2×C71#54012yes—
155D71—42no—
156D71—82no—
157#2635—711nomaximal, Sylow
158C71#263511yescharacteristic, complemented, Sylow
159C2×#121—711no—
160C2×C4×D4—711no—
161#400—711no—
162#399—711no—
163#394—712no—
164#394—711no—
165#392—712no—
166#392—711no—
167#388—712no—
168#386—712no—
169#381D7111yescharacteristic, complemented
170C22×D4—712no—
171C22×D4—711no—
172C22×D4—711no—
173C22×D4—712no—
174C23×C4—711no—
175C23×C4C2×D7111yescharacteristic
176#121—1424no—
177C4×D4—1424no—
178C2×C4⋊C4—712no—
179C2×C4⋊C4—711no—
180C2×C4⋊C4—711no—
181C2×C22⋊C4—712no—
182C2×C22⋊C4—712no—
183C2×C22⋊C4—712no—
184C2×C22⋊C4—711no—
185C2×C22⋊C4—711no—
186C2×C22⋊C4—712no—
187C2×C22⋊C4—712no—
188C2×C22⋊C4C2×D7112yes—
189C2×C42—711no—
190#95—712no—
191#95—712no—
192#95C2×D7112yes—
193#95C2×D7111yescharacteristic
194#95C2×D7111yescharacteristic
195C24—711no—
196C24—712no—
197C24C22×D7111yescharacteristic
198C2×D4—2842no—
199C2×D4—2842no—
200C2×D4—2842no—
201C2×D4—2842no—
202C2×D4—2842no—
203C2×D4—1424no—
204C2×D4—1422no—
205C2×D4—1424no—
206C2×D4—1422no—
207C2×D4—1422no—
208C2×D4—1422no—
209C22×C4—42no—
210C22×C4—2842no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—712no—
214C22×C4—711no—
215C22×C4—1422no—
216C22×C4D28414yescomplemented
217C22×C4C22×D7112yes—
218C22×C4C22×D7112yes—
219C22×C4C22×D7111yescharacteristic
220C22×C4C22×D7111yescharacteristic
221C4⋊C4—2842no—
222C4⋊C4—1422no—
223C4⋊C4—1422no—
224C22⋊C4—42no—
225C22⋊C4—2842no—
226C22⋊C4—2842no—
227C22⋊C4—2842no—
228C22⋊C4—1424no—
229C22⋊C4—1422no—
230C22⋊C4—1422no—
231C22⋊C4—1424no—
232C42—1422no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—712no—
238C23—1422no—
239C23—1422no—
240C23—1422no—
241C23—1422no—
242C23—1422no—
243C23—1422no—
244C23—1422no—
245C23—1421no—
246C23—1421no—
247C23—1422no—
248C23—1422no—
249C23C2×D28412yes—
250C23C23×D7111yescharacteristic
251D4—2844no—
252D4—2848no—
253D4—2848no—
254D4—1428no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—44no—
264C2×C4—2844no—
265C2×C4—712no—
266C2×C4—1421no—
267C2×C4—1421no—
268C2×C4—1422no—
269C2×C4—1422no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4—1422no—
273C2×C4—1421no—
274C2×C4—1421no—
275C2×C4C2×D28412yes—
276C2×C4C2×D28412yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—2841no—
283C22—2841no—
284C22—2842no—
285C22—2842no—
286C22—2842no—
287C22—2842no—
288C22—2842no—
289C22—2842no—
290C22—2842no—
291C22—2842no—
292C22—1424no—
293C22—1422no—
294C22C22×D28411yescharacteristic
295C22#41×D7111yescharacteristic
296C22#41×D7111yescharacteristic
297C22#1455617312yes—
298C22#1455617412yes—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—42no—
303C4—2841no—
304C4—2842no—
305C4—2841no—
306C4—1422no—
307C2—42no—
308C2—2842no—
309C2—5682no—
310C2#1798651111yescharacteristic
311C2#1798656111yescharacteristic
312C2#1798656811yescharacteristic
313C2#1798658512yes—
314C2#1798660112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2635
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#14556102C2semidirect
2. C2×C4×C22⋊D71C2semidirect
3. #17985939C2semidirect
4. #17985936C2semidirect
5. #17985917C2semidirect
6. #17985915C2semidirect
7. #17985909C2semidirect
8. #17985908C2semidirect
9. #17985891C2semidirect
10. #17985877C2semidirect
11. #381×C71C2semidirect
69. C22×C4×C71D4semidirect
158. C71#2635semidirect
169. #381D71semidirect
216. C22×C4D284semidirect