Finite Group Databasebeta

← #22594133

Subgroups — #22594133

Order 9088.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556101C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic, complemented
4#17985926C211yesmaximal, characteristic, complemented
5#17985924C211yesmaximal, characteristic, complemented
6#17985916C212yesmaximal, complemented
7#17985914C212yesmaximal, complemented
8#17985899C212yesmaximal, complemented
9#17985894C212yesmaximal, complemented
10#17985858C211yesmaximal, characteristic, complemented
11#399×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16#14556101—24no—
17#14556086—24no—
18C2×C4⋊Dic71C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2212yes—
22C2×C22⋊Dic71C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
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×C71C2212yes—
34C2×C22⋊C4×C71C2211yescharacteristic
35C2×C22⋊C4×C71C2212yes—
36#95×C71C2211yescharacteristic
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×D284—22no—
40C2×D284—42no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—24no—
44C2×C22⋊D71—42no—
45C22×Dic71C2311yescharacteristic
46C22×Dic71C2312yes—
47C22×Dic71C2312yes—
48C22×Dic71C2311yescharacteristic
49C2×C4×D71—24no—
50C2×C4×D71D414yes—
51C4⋊Dic71—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—24no—
54Dic71⋊C4—42no—
55C22⋊Dic71—42no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—42no—
59(C2×D71)⋊C4—42no—
60(C2×D71)⋊C4—42no—
61C4×Dic71—24no—
62C24×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2312yes—
65C22×C4×C71C2312yes—
66C22×C4×C71C2311yescharacteristic
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
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—21no—
79C22×D71—22no—
80C22×D71—21no—
81C22×D71—21no—
82C22×D71—22no—
83C22×D71—21no—
84C22×D71—22no—
85C22×D71—21no—
86C22×D71C2×D412yes—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
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—
97C22⋊D71—44no—
98C22⋊D71—44no—
99C2×Dic71—22no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71—22no—
104C2×Dic71—22no—
105C2×Dic71—22no—
106C2×Dic71—22no—
107C2×Dic71C2×D412yes—
108C2×Dic71#4114yes—
109D284—44no—
110C4×D71—28no—
111C22×C71—41no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71C22×D411yescharacteristic
117C22×C71C2×#4112yes—
118C22×C71H(5,𝔽2)12yes—
119C22×C71H(5,𝔽2)11yescharacteristic
120C22×C712−1+411yescharacteristic
121C2×D71—22no—
122C2×D71—22no—
123C2×D71—22no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—41no—
127C2×D71—41no—
128C2×D71—41no—
129C2×D71—42no—
130C2×D71—42no—
131C4×C71—22no—
132C4×C71—41no—
133C4×C71—42no—
134C4×C71—42no—
135Dic71—41no—
136Dic71—42no—
137Dic71—42no—
138Dic71#12314yes—
139C2×C71—81no—
140C2×C71C2×#12311yescharacteristic
141C2×C71#53511yescharacteristic
142C2×C71#53711yescharacteristic
143C2×C71#54012yes—
144C2×C71#54212yes—
145D71—42no—
146D71—81no—
147#2624—711nomaximal, Sylow
148C71#262411yescharacteristic, complemented, Sylow
149C2×#121—711no—
150C2×#117—711no—
151#399—711no—
152#399D7111yescharacteristic, complemented
153#397—711no—
154#397—711no—
155#394—712no—
156#394—712no—
157#390—712no—
158#389—712no—
159#380—711no—
160C22×D4—711no—
161C22×D4—711no—
162C22×D4—711no—
163C23×C4—711no—
164#121—1424no—
165#117—1424no—
166C2×C4⋊C4—712no—
167C2×C4⋊C4—711no—
168C2×C4⋊C4—711no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4C2×D7111yescharacteristic
171C2×C4⋊C4C2×D7112yes—
172C2×C22⋊C4—712no—
173C2×C22⋊C4—712no—
174C2×C22⋊C4—712no—
175C2×C22⋊C4—712no—
176C2×C22⋊C4—712no—
177C2×C22⋊C4—711no—
178C2×C22⋊C4C2×D7111yescharacteristic
179C2×C22⋊C4C2×D7112yes—
180C2×C42—712no—
181#95—711no—
182#95—712no—
183#95—712no—
184#95C2×D7111yescharacteristic
185C24—711no—
186C24—711no—
187C24C22×D7111yescharacteristic
188C2×D4—2842no—
189C2×D4—2842no—
190C2×D4—1422no—
191C2×D4—1422no—
192C2×D4—1424no—
193C2×D4—1422no—
194C22×C4—711no—
195C22×C4—714no—
196C22×C4—712no—
197C22×C4—712no—
198C22×C4—711no—
199C22×C4—1424no—
200C22×C4C22×D7111yescharacteristic
201C22×C4C22×D7112yes—
202C22×C4C22×D7112yes—
203C22×C4C22×D7111yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—2842no—
207C4⋊C4—1424no—
208C4⋊C4—1422no—
209C4⋊C4—1422no—
210C22⋊C4—22no—
211C22⋊C4—42no—
212C22⋊C4—2842no—
213C22⋊C4—2842no—
214C22⋊C4—2842no—
215C22⋊C4—2842no—
216C22⋊C4—1424no—
217C22⋊C4—1422no—
218C42—1424no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—712no—
225C23—1421no—
226C23—1422no—
227C23—1422no—
228C23—1421no—
229C23—1421no—
230C23—1422no—
231C23—1422no—
232C23—1421no—
233C23—1421no—
234C23C23×D7111yescharacteristic
235D4—2844no—
236D4—2844no—
237D4—2844no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—712no—
248C2×C4—714no—
249C2×C4—1428no—
250C2×C4—1422no—
251C2×C4—1421no—
252C2×C4—1422no—
253C2×C4—1422no—
254C2×C4—1422no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4—1422no—
258C2×C4D4×D7112yes—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—2842no—
265C22—2842no—
266C22—2841no—
267C22—2841no—
268C22—2842no—
269C22—2842no—
270C22—2841no—
271C22—1422no—
272C22—1422no—
273C22—1422no—
274C22C2×D4×D7111yescharacteristic
275C22#41×D7112yes—
276C22#1455617312yes—
277C22#1455617411yescharacteristic
278C22#1455617611yescharacteristic
279C4—22no—
280C4—41no—
281C4—42no—
282C4—42no—
283C4—2841no—
284C4—2842no—
285C4—2842no—
286C4—714no—
287C2—81no—
288C2—2842no—
289C2—5681no—
290C2#123×D7111yescharacteristic
291C2#1798656211yescharacteristic
292C2#1798657911yescharacteristic
293C2#1798660112yes—
294C2#1798661912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2624
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#14556086C2semidirect
3. #17985934C2semidirect
4. #17985926C2semidirect
5. #17985924C2semidirect
6. #17985916C2semidirect
7. #17985914C2semidirect
8. #17985899C2semidirect
9. #17985894C2semidirect
10. #17985858C2semidirect
11. #399×C71C2semidirect
148. C71#2624semidirect
152. #399D71semidirect