Finite Group Databasebeta

← #22593257

Subgroups — #22593257

Order 9088.

620 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556106C211yesmaximal, characteristic, complemented
2C2×#14556105C211yesmaximal, characteristic, complemented
3C2×#14556093C211yesmaximal, characteristic, complemented
4C2×(C22⋊D71)⋊C4C211yesmaximal, characteristic
5#17985922C211yesmaximal, characteristic, complemented
6#17985921C211yesmaximal, characteristic, complemented
7#17985921C211yesmaximal, characteristic, complemented
8#17985899C211yesmaximal, characteristic
9#17985899C211yesmaximal, characteristic
10#17985896C211yesmaximal, characteristic, complemented
11#17985894C211yesmaximal, characteristic, complemented
12#17985868C211yesmaximal, characteristic, complemented
13#17985858C211yesmaximal, characteristic, complemented
14#17985857C211yesmaximal, characteristic, complemented
15#396×C71C211yesmaximal, characteristic, complemented
16C22×Dic142C2211yescharacteristic
17C22×C22⋊D71C2211yescharacteristic
18C23×Dic71C2211yescharacteristic
19C22×C4×D71C2211yescharacteristic
20#14556106—24no—
21#14556105—24no—
22#14556093—24no—
23(C22⋊D71)⋊C4—24no—
24C2×C4⋊Dic71C2211yescharacteristic
25C2×C4⋊Dic71C2211yescharacteristic
26C2×Dic71⋊C4C2211yescharacteristic
27C2×Dic71⋊C4C2211yescharacteristic
28C2×Dic71⋊C4C2211yescharacteristic
29C2×Dic71⋊C4C2211yescharacteristic
30C2×Dic71⋊C4C2211yescharacteristic
31C2×Dic71⋊C4C2211yescharacteristic
32C2×Dic71⋊C4C2211yescharacteristic
33C2×Dic71⋊C4C2211yescharacteristic
34C2×Dic71⋊C4C2211yescharacteristic
35C2×C22⋊Dic71C2211yescharacteristic
36C2×C22⋊Dic71C2211yescharacteristic
37C2×(C2×D71)⋊C4C2211yescharacteristic
38C2×(C2×D71)⋊C4C2211yescharacteristic
39C2×(C2×D71)⋊C4C2211yescharacteristic
40C2×(C2×D71)⋊C4C2211yescharacteristic
41C2×(C2×D71)⋊C4C2211yescharacteristic
42C2×C4×Dic71C2211yescharacteristic
43C2×C4×Dic71C2211yescharacteristic
44C2×C4×Dic71C2211yescharacteristic
45C71⋊#95C2211yescharacteristic
46C71⋊#95C2211yescharacteristic
47C71⋊#95C2211yescharacteristic
48C2×C4⋊C4×C71C2211yescharacteristic
49C2×C4⋊C4×C71C2211yescharacteristic
50C2×C4⋊C4×C71C2211yescharacteristic
51C2×C22⋊C4×C71C2211yescharacteristic
52C2×C22⋊C4×C71C2211yescharacteristic
53C2×C22⋊C4×C71C2211yescharacteristic
54#95×C71C2211yescharacteristic
55C23×D71C2311yescharacteristic
56C2×Dic142—22no—
57C2×Dic142—22no—
58C2×Dic142D414yescomplemented
59C2×C22⋊D71—22no—
60C2×C22⋊D71—22no—
61C2×C22⋊D71Q814yescomplemented
62C22×Dic71—22no—
63C22×Dic71—22no—
64C22×Dic71—22no—
65C22×Dic71C2311yescharacteristic
66C22×Dic71C2311yescharacteristic
67C22×Dic71C2311yescharacteristic
68C22×Dic71C2311yescharacteristic
69C22×Dic71C2311yescharacteristic
70C22×Dic71C2311yescharacteristic
71C22×Dic71C2311yescharacteristic
72C2×C4×D71—22no—
73C2×C4×D71—22no—
74C2×C4×D71—22no—
75C4⋊Dic71—22no—
76C4⋊Dic71—41no—
77Dic71⋊C4—22no—
78Dic71⋊C4—22no—
79Dic71⋊C4—22no—
80Dic71⋊C4—22no—
81Dic71⋊C4—22no—
82Dic71⋊C4—22no—
83Dic71⋊C4—22no—
84Dic71⋊C4—22no—
85Dic71⋊C4—22no—
86C22⋊Dic71—22no—
87C22⋊Dic71—41no—
88(C2×D71)⋊C4—22no—
89(C2×D71)⋊C4—22no—
90(C2×D71)⋊C4—22no—
91(C2×D71)⋊C4—41no—
92(C2×D71)⋊C4—41no—
93C4×Dic71—22no—
94C4×Dic71—22no—
95C4×Dic71D414yes—
96C24×C71C2311yescharacteristic
97C22×C4×C71C2311yescharacteristic
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C22×C4×C71C2311yescharacteristic
101C22×C4×C71C2311yescharacteristic
102C22×C4×C71C2311yescharacteristic
103C4⋊C4×C71—22no—
104C4⋊C4×C71—22no—
105C4⋊C4×C71—41no—
106C22⋊C4×C71—22no—
107C22⋊C4×C71—22no—
108C22⋊C4×C71—41no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71C2×Q812yes—
116C23×C71C2411yescharacteristic
117C22×D71—21no—
118C22×D71—21no—
119C22×D71—21no—
120C22×D71—21no—
121C22×D71—21no—
122C22×D71—21no—
123C22×D71C2×Q812yes—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71C2×D412yes—
141C2×C4×C71C2×D412yes—
142C22⋊D71—28no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71—21no—
150C2×Dic71—21no—
151C2×Dic71—21no—
152C2×Dic71—21no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—21no—
157C2×Dic71—44no—
158C2×Dic71C2×D412yes—
159C2×Dic71C2×D412yes—
160C2×Dic71C2×D412yes—
161C2×Dic71C2×D412yes—
162C2×Dic71C2×Q812yes—
163C2×Dic71#4112yes—
164C2×Dic71#4112yes—
165C4×D71—44no—
166Dic142—28no—
167C22×C71—22no—
168C22×C71—22no—
169C22×C71—22no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71—41no—
174C22×C71C22×D411yescharacteristic
175C22×C71C22×D411yescharacteristic
176C22×C71C22×Q811yescharacteristic
177C22×C71C2×#4111yescharacteristic
178C22×C71C2×#4111yescharacteristic
179C22×C71H(5,𝔽2)11yescharacteristic
180C22×C712−1+411yescharacteristic
181C2×D71—22no—
182C2×D71—22no—
183C2×D71—22no—
184C2×D71—41no—
185C2×D71—41no—
186C2×D71—41no—
187C2×D71—41no—
188C4×C71—22no—
189C4×C71—22no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—41no—
193C4×C71—41no—
194Dic71—22no—
195Dic71—41no—
196Dic71—41no—
197Dic71—41no—
198Dic71—41no—
199Dic71#12114yes—
200Dic71C22⋊Q814yescomplemented
201C2×C71—42no—
202C2×C71C2×#12111yescharacteristic
203C2×C71C2×C22⋊Q811yescharacteristic
204C2×C71#53711yescharacteristic
205C2×C71D4×Q811yescharacteristic
206C2×C71#55011yescharacteristic
207C2×C71#55411yescharacteristic
208C2×C71#55611yescharacteristic
209D71—42no—
210#2538—711nomaximal, Sylow
211C71#253811yescharacteristic, complemented, Sylow
212C2×C22⋊Q8—711no—
213C2×C22⋊Q8—711no—
214C2×C4×Q8—711no—
215C2×C4×D4—711no—
216#396—711no—
217#396—711no—
218#396—711no—
219#396D7111yescharacteristic, complemented
220#390—711no—
221#390—711no—
222#389—711no—
223#389—711no—
224#384—711no—
225#380—711no—
226#380—711no—
227C22×Q8—711no—
228C22×D4—711no—
229C23×C4—711no—
230C23×C4—711no—
231C22⋊Q8—1424no—
232C22⋊Q8—1424no—
233C4×Q8—1424no—
234C4×D4—1424no—
235C2×C4⋊C4—711no—
236C2×C4⋊C4—711no—
237C2×C4⋊C4—711no—
238C2×C4⋊C4—711no—
239C2×C4⋊C4—711no—
240C2×C4⋊C4—711no—
241C2×C4⋊C4—711no—
242C2×C4⋊C4—711no—
243C2×C4⋊C4—711no—
244C2×C4⋊C4—711no—
245C2×C4⋊C4—711no—
246C2×C4⋊C4C2×D7111yescharacteristic
247C2×C4⋊C4C2×D7111yescharacteristic
248C2×C4⋊C4C2×D7111yescharacteristic
249C2×C22⋊C4—711no—
250C2×C22⋊C4—711no—
251C2×C22⋊C4—711no—
252C2×C22⋊C4—711no—
253C2×C22⋊C4—711no—
254C2×C22⋊C4—711no—
255C2×C22⋊C4—711no—
256C2×C22⋊C4C2×D7111yescharacteristic
257C2×C22⋊C4C2×D7111yescharacteristic
258C2×C22⋊C4C2×D7111yescharacteristic
259C2×C42—711no—
260C2×C42—711no—
261C2×C42—711no—
262#95—711no—
263#95—711no—
264#95—711no—
265#95C2×D7111yescharacteristic
266C24—711no—
267C24C22×D7111yescharacteristic
268C2×Q8—714no—
269C2×Q8—1422no—
270C2×Q8—1422no—
271C2×D4—714no—
272C2×D4—1422no—
273C2×D4—1422no—
274C22×C4—711no—
275C22×C4—711no—
276C22×C4—711no—
277C22×C4—711no—
278C22×C4—711no—
279C22×C4—711no—
280C22×C4—711no—
281C22×C4—1422no—
282C22×C4—1422no—
283C22×C4—1422no—
284C22×C4—1422no—
285C22×C4—1422no—
286C22×C4—1422no—
287C22×C4C22×D7111yescharacteristic
288C22×C4C22×D7111yescharacteristic
289C22×C4C22×D7111yescharacteristic
290C22×C4C22×D7111yescharacteristic
291C22×C4C22×D7111yescharacteristic
292C22×C4C22×D7111yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—2841no—
297C4⋊C4—1422no—
298C4⋊C4—1422no—
299C4⋊C4—1422no—
300C4⋊C4—1422no—
301C4⋊C4—1422no—
302C4⋊C4—1422no—
303C4⋊C4—1422no—
304C4⋊C4—1422no—
305C4⋊C4—1422no—
306C4⋊C4—1422no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—2841no—
311C22⋊C4—2841no—
312C22⋊C4—2841no—
313C22⋊C4—1422no—
314C22⋊C4—1422no—
315C22⋊C4—1422no—
316C22⋊C4—1422no—
317C42—714no—
318C42—1422no—
319C42—1422no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—712no—
327C23—1421no—
328C23—1421no—
329C23—1421no—
330C23—1421no—
331C23—1421no—
332C23—1421no—
333C23Q8×D7112yes—
334C23C23×D7111yescharacteristic
335Q8—1428no—
336D4—1428no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—2844no—
354C2×C4—2844no—
355C2×C4—712no—
356C2×C4—712no—
357C2×C4—712no—
358C2×C4—712no—
359C2×C4—712no—
360C2×C4—712no—
361C2×C4—712no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4—1421no—
365C2×C4—1421no—
366C2×C4—1421no—
367C2×C4—1421no—
368C2×C4—1421no—
369C2×C4—1421no—
370C2×C4—1421no—
371C2×C4—1421no—
372C2×C4—1421no—
373C2×C4—1421no—
374C2×C4—1421no—
375C2×C4—1421no—
376C2×C4D4×D7112yes—
377C2×C4D4×D7112yes—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—2841no—
386C22—2841no—
387C22—2841no—
388C22—2841no—
389C22—1422no—
390C22—1422no—
391C22—1422no—
392C22C2×D4×D7111yescharacteristic
393C22C2×D4×D7111yescharacteristic
394C22C2×Q8×D7111yescharacteristic
395C22#41×D7111yescharacteristic
396C22#41×D7111yescharacteristic
397C22#1455617311yescharacteristic
398C22#1455617511yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—2841no—
406C4—2841no—
407C4—2841no—
408C4—2841no—
409C4—714no—
410C4—714no—
411C4—1422no—
412C2—42no—
413C2—2842no—
414C2#121×D7111yescharacteristic
415C2C22⋊Q8×D7111yescharacteristic
416C2#1798657611yescharacteristic
417C2#1798666211yescharacteristic
418C2#1798666711yescharacteristic
419C2#1798670311yescharacteristic
420C2#1798671611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#396×C71C2
hypercenter#396D71
hypocenterC71#2538
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#14556106C2semidirect
2. C2×#14556105C2semidirect
3. C2×#14556093C2semidirect
5. #17985922C2semidirect
6. #17985921C2semidirect
7. #17985921C2semidirect
10. #17985896C2semidirect
11. #17985894C2semidirect
12. #17985868C2semidirect
13. #17985858C2semidirect
14. #17985857C2semidirect
15. #396×C71C2semidirect
58. C2×Dic142D4semidirect
61. C2×C22⋊D71Q8semidirect
200. Dic71C22⋊Q8semidirect
211. C71#2538semidirect
219. #396D71semidirect