Finite Group Databasebeta

← #22596366

Subgroups — #22596366

Order 9088.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556104C211yesmaximal, characteristic, complemented
2C2×#14556101C211yesmaximal, characteristic
3#17985936C211yesmaximal, characteristic, complemented
4#17985935C211yesmaximal, characteristic, complemented
5#17985933C212yesmaximal, complemented
6#17985917C212yesmaximal, complemented
7#17985915C212yesmaximal, complemented
8#17985914C212yesmaximal, complemented
9#17985910C211yesmaximal, characteristic, complemented
10#17985861C211yesmaximal, characteristic, complemented
11#393×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16#14556104—24no—
17#14556101—24no—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×C4⋊Dic71C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×C22⋊Dic71C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C22×Q8×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×D284—22no—
40C2×D284—42no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—42no—
44C2×C22⋊D71—42no—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C22×Dic71C2312yes—
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—42no—
50C2×C4×D71D414yes—
51C4⋊Dic71—22no—
52C4⋊Dic71—42no—
53Dic71⋊C4—22no—
54Dic71⋊C4—24no—
55Dic71⋊C4—42no—
56C22⋊Dic71—42no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—42no—
60(C2×D71)⋊C4—42no—
61(C2×D71)⋊C4—42no—
62C24×C71C2311yescharacteristic
63C2×Q8×C71—22no—
64C2×Q8×C71—42no—
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2312yes—
67C22×C4×C71C2312yes—
68C22×C4×C71C2311yescharacteristic
69C22⋊C4×C71—22no—
70C22⋊C4×C71—42no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—21no—
75C23×C71—21no—
76C23×C71C2411yescharacteristic
77C22×D71—22no—
78C22×D71—22no—
79C22×D71—21no—
80C22×D71—21no—
81C22×D71—21no—
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—21no—
85C22×D71—21no—
86C22×D71C2×D412yes—
87Q8×C71—44no—
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×C71—22no—
97C2×C4×C71C2×D412yes—
98C22⋊D71—44no—
99C22⋊D71—44no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71—22no—
108C2×Dic71—22no—
109C2×Dic71C2×D412yes—
110D284—44no—
111C4×D71—44no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71C22×D411yescharacteristic
118C22×C71H(5,𝔽2)12yes—
119C22×C71H(5,𝔽2)12yes—
120C22×C712−1+411yescharacteristic
121C22×C712−1+411yescharacteristic
122C2×D71—22no—
123C2×D71—22no—
124C2×D71—22no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—41no—
128C2×D71—41no—
129C2×D71—41no—
130C2×D71—42no—
131C2×D71—42no—
132C4×C71—22no—
133C4×C71—42no—
134C4×C71—41no—
135C4×C71—42no—
136Dic71—22no—
137Dic71—42no—
138Dic71—41no—
139Dic71—42no—
140C2×C71—81no—
141C2×C71#53411yescharacteristic
142C2×C71#53511yescharacteristic
143C2×C71#53611yescharacteristic
144C2×C71#56212yes—
145C2×C71#56212yes—
146D71—42no—
147D71—81no—
148#2816—711nomaximal, Sylow
149C71#281611yescharacteristic, complemented, Sylow
150C2×C22⋊Q8—711no—
151C2×#121—711no—
152#399—711no—
153#399—711no—
154#398—712no—
155#394—712no—
156#394—712no—
157#394—712no—
158#393—711no—
159#393D7111yescharacteristic, complemented
160#381—711no—
161C22×Q8C2×D7111yescharacteristic
162C22×D4—711no—
163C22×D4—711no—
164C22×D4—711no—
165C23×C4—711no—
166C22⋊Q8—1424no—
167#121—1424no—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—711no—
171C2×C4⋊C4—712no—
172C2×C4⋊C4—711no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—711no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4C2×D7111yescharacteristic
180C2×C22⋊C4C2×D7112yes—
181#95—711no—
182#95—712no—
183#95—712no—
184#95C2×D7111yescharacteristic
185#95C2×D7112yes—
186C24—711no—
187C24—711no—
188C24C22×D7111yescharacteristic
189C2×Q8—22no—
190C2×Q8—42no—
191C2×D4—2842no—
192C2×D4—2842no—
193C2×D4—2842no—
194C2×D4—1422no—
195C2×D4—1422no—
196C2×D4—1422no—
197C22×C4—2842no—
198C22×C4—714no—
199C22×C4—711no—
200C22×C4—712no—
201C22×C4—711no—
202C22×C4—712no—
203C22×C4C22×D7112yes—
204C22×C4C22×D7112yes—
205C22×C4C22×D7111yescharacteristic
206C22×C4C22×D7111yescharacteristic
207C4⋊C4—2842no—
208C4⋊C4—2842no—
209C4⋊C4—1422no—
210C4⋊C4—1424no—
211C4⋊C4—1422no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—2842no—
215C22⋊C4—2842no—
216C22⋊C4—2842no—
217C22⋊C4—2842no—
218C22⋊C4—1424no—
219C22⋊C4—1422no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—712no—
226C23—1421no—
227C23—1422no—
228C23—1422no—
229C23—1421no—
230C23—1421no—
231C23—1422no—
232C23—1422no—
233C23—1421no—
234C23—1421no—
235C23C23×D7111yescharacteristic
236Q8—44no—
237D4—2844no—
238D4—2844no—
239D4—2844no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—2844no—
250C2×C4—712no—
251C2×C4—1422no—
252C2×C4—1421no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4D4×D7112yes—
261C22—41no—
262C22—41no—
263C22—42no—
264C22—42no—
265C22—41no—
266C22—2842no—
267C22—2842no—
268C22—2841no—
269C22—2841no—
270C22—2842no—
271C22—2842no—
272C22—2841no—
273C22—1422no—
274C22—1422no—
275C22—1422no—
276C22C2×D4×D7111yescharacteristic
277C22#1455617312yes—
278C22#1455617412yes—
279C22#1455617511yescharacteristic
280C22#1455617611yescharacteristic
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—2841no—
286C4—2842no—
287C4—2842no—
288C4—1422no—
289C2—81no—
290C2—2842no—
291C2—5681no—
292C2#1798655911yescharacteristic
293C2#1798656711yescharacteristic
294C2#1798657211yescharacteristic
295C2#1798674312yes—
296C2#1798674512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71#2816
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14556104C2semidirect
3. #17985936C2semidirect
4. #17985935C2semidirect
5. #17985933C2semidirect
6. #17985917C2semidirect
7. #17985915C2semidirect
8. #17985914C2semidirect
9. #17985910C2semidirect
10. #17985861C2semidirect
11. #393×C71C2semidirect
149. C71#2816semidirect
159. #393D71semidirect