Finite Group Databasebeta

← #22616178

Subgroups — #22616178

Order 9344.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556580C211yesmaximal, characteristic, complemented
2C2×#14556574C211yesmaximal, characteristic, complemented
3C2×#14556564C211yesmaximal, characteristic
4#17988522C211yesmaximal, characteristic, complemented
5#17988509C211yesmaximal, characteristic, complemented
6#17988506C211yesmaximal, characteristic, complemented
7#17988504C211yesmaximal, characteristic, complemented
8#17988502C211yesmaximal, characteristic, complemented
9#17988502C211yesmaximal, characteristic, complemented
10#17988482C211yesmaximal, characteristic, complemented
11#17988480C211yesmaximal, characteristic, complemented
12#17988466C211yesmaximal, characteristic, complemented
13#17988458C211yesmaximal, characteristic, complemented
14#17988444C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C73C211yesmaximal, characteristic, complemented
16C22×D292C2211yescharacteristic
17C22×C22⋊D73C2211yescharacteristic
18C22×C22⋊D73C2211yescharacteristic
19C22×C4×D73C2211yescharacteristic
20#14556580—24no—
21#14556574—24no—
22#14556564—24no—
23C2×C4⋊Dic73C2211yescharacteristic
24C2×#11924007C2211yescharacteristic
25C2×#11924007C2211yescharacteristic
26C2×#11924007C2211yescharacteristic
27C2×#11924007C2211yescharacteristic
28C2×#11924007C2211yescharacteristic
29C2×C22⋊Dic73C2211yescharacteristic
30C2×C22⋊Dic73C2211yescharacteristic
31C2×#11924005C2211yescharacteristic
32C2×#11924005C2211yescharacteristic
33C2×#11924005C2211yescharacteristic
34C2×#11924005C2211yescharacteristic
35C2×#11924005C2211yescharacteristic
36C2×#11924005C2211yescharacteristic
37C2×#11924005C2211yescharacteristic
38C2×#11924005C2211yescharacteristic
39C2×#11924005C2211yescharacteristic
40C2×C4×Dic73C2211yescharacteristic
41#14556498C2211yescharacteristic
42#14556498C2211yescharacteristic
43#14556498C2211yescharacteristic
44#14556498C2211yescharacteristic
45#14556498C2211yescharacteristic
46#14556498C2211yescharacteristic
47C22×Q8×C73C2211yescharacteristic
48C23×C4×C73C2211yescharacteristic
49C22⋊Q8×C73—24no—
50C2×C4⋊C4×C73C2211yescharacteristic
51C2×C4⋊C4×C73C2211yescharacteristic
52C2×C4⋊C4×C73C2211yescharacteristic
53C2×C22⋊C4×C73C2211yescharacteristic
54C2×C22⋊C4×C73C2211yescharacteristic
55C23×D73C2311yescharacteristic
56C23×D73C2311yescharacteristic
57C2×D292—22no—
58C2×D292—22no—
59C2×D292—41no—
60C2×C22⋊D73—22no—
61C2×C22⋊D73—22no—
62C2×C22⋊D73—22no—
63C2×C22⋊D73—22no—
64C2×C22⋊D73—41no—
65C2×C22⋊D73—41no—
66C22×Dic73C2311yescharacteristic
67C22×Dic73C2311yescharacteristic
68C22×Dic73C2311yescharacteristic
69C22×Dic73C2311yescharacteristic
70C22×Dic73C2311yescharacteristic
71C22×Dic73C2311yescharacteristic
72C2×C4×D73—22no—
73C2×C4×D73—22no—
74C2×C4×D73—41no—
75C4⋊Dic73—41no—
76#11924007—22no—
77#11924007—22no—
78#11924007—22no—
79#11924007—22no—
80#11924007—41no—
81C22⋊Dic73—22no—
82C22⋊Dic73—41no—
83#11924005—22no—
84#11924005—22no—
85#11924005—22no—
86#11924005—22no—
87#11924005—22no—
88#11924005—41no—
89#11924005—41no—
90#11924005—41no—
91#11924005D414yescomplemented
92C4×Dic73—22no—
93C24×C73C2311yescharacteristic
94C2×Q8×C73—22no—
95C2×Q8×C73—22no—
96C2×Q8×C73—41no—
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73—41no—
100C22×C4×C73C2311yescharacteristic
101C22×C4×C73C2311yescharacteristic
102C22×C4×C73C2311yescharacteristic
103C22×C4×C73C2311yescharacteristic
104C22×C4×C73C2311yescharacteristic
105C22×C4×C73C2311yescharacteristic
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—22no—
109C22⋊C4×C73—22no—
110C22⋊C4×C73D414yes—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73C2×D412yes—
118C23×C73C2411yescharacteristic
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—21no—
123C22×D73—21no—
124C22×D73—21no—
125C22×D73—21no—
126C22×D73—21no—
127C22×D73—21no—
128C22×D73—21no—
129C22×D73—21no—
130C22×D73—21no—
131C22×D73—21no—
132C22×D73C2×D412yes—
133Q8×C73—44no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—44no—
150C2×C4×C73C2×D412yes—
151C2×C4×C73C2×D412yes—
152C2×C4×C73C2×D412yes—
153C22⋊D73—44no—
154C22⋊D73—44no—
155C2×Dic73—21no—
156C2×Dic73—21no—
157C2×Dic73—21no—
158C2×Dic73—21no—
159C2×Dic73—21no—
160C2×Dic73—21no—
161C2×Dic73—21no—
162C2×Dic73—21no—
163C2×Dic73—21no—
164C2×Dic73—21no—
165C2×Dic73—21no—
166C2×Dic73—21no—
167C2×Dic73—21no—
168C2×Dic73—21no—
169C2×Dic73—21no—
170C2×Dic73—21no—
171C2×Dic73C2×D412yes—
172C2×Dic73#4112yes—
173D292—44no—
174C4×D73—44no—
175C22×C73—22no—
176C22×C73—22no—
177C22×C73—22no—
178C22×C73—41no—
179C22×C73—41no—
180C22×C73—41no—
181C22×C73—41no—
182C22×C73C22×D411yescharacteristic
183C22×C73C22×D411yescharacteristic
184C22×C73C2×#4111yescharacteristic
185C22×C73H(5,𝔽2)11yescharacteristic
186C22×C73H(5,𝔽2)11yescharacteristic
187C22×C732−1+411yescharacteristic
188C22×C732−1+411yescharacteristic
189C2×D73—22no—
190C2×D73—22no—
191C2×D73—22no—
192C2×D73—41no—
193C2×D73—41no—
194C2×D73—41no—
195C2×D73—41no—
196C2×D73—41no—
197C2×D73—41no—
198C2×D73—41no—
199C2×D73—41no—
200C2×D73—41no—
201C2×D73—41no—
202C2×D73—41no—
203C4×C73—22no—
204C4×C73—22no—
205C4×C73—22no—
206C4×C73—41no—
207C4×C73—41no—
208C4×C73—41no—
209Dic73—22no—
210Dic73—22no—
211Dic73—41no—
212Dic73—41no—
213Dic73—41no—
214Dic73—41no—
215C2×C73—42no—
216C2×C73#53611yescharacteristic
217C2×C73#53711yescharacteristic
218C2×C73#54211yescharacteristic
219C2×C73D4211yescharacteristic
220C2×C73#54611yescharacteristic
221C2×C73#54711yescharacteristic
222C2×C73#56211yescharacteristic
223D73—42no—
224D73—81no—
225#2672—731nomaximal, Sylow
226C73#267211yescharacteristic, complemented, Sylow
227C2×#124—731no—
228C2×#123—731no—
229C2×C22⋊Q8D7311yescharacteristic, complemented
230C2×#121—731no—
231#399—731no—
232#396—731no—
233#395—731no—
234#394—731no—
235#394—731no—
236#394—731no—
237#389—731no—
238#388—731no—
239#386—731no—
240#385—731no—
241#380—731no—
242C22×Q8C2×D7311yescharacteristic
243C22×D4—731no—
244C22×D4—731no—
245C22×D4—731no—
246C23×C4—731no—
247C23×C4C2×D7311yescharacteristic
248#124—1464no—
249#123—1464no—
250C22⋊Q8—24no—
251#121—1464no—
252C2×C4⋊C4—731no—
253C2×C4⋊C4—731no—
254C2×C4⋊C4—731no—
255C2×C4⋊C4—731no—
256C2×C4⋊C4—731no—
257C2×C4⋊C4—731no—
258C2×C4⋊C4C2×D7311yescharacteristic
259C2×C4⋊C4C2×D7311yescharacteristic
260C2×C4⋊C4C2×D7311yescharacteristic
261C2×C22⋊C4—731no—
262C2×C22⋊C4—731no—
263C2×C22⋊C4—731no—
264C2×C22⋊C4—731no—
265C2×C22⋊C4—731no—
266C2×C22⋊C4—731no—
267C2×C22⋊C4—731no—
268C2×C22⋊C4—731no—
269C2×C22⋊C4—731no—
270C2×C22⋊C4—731no—
271C2×C22⋊C4—731no—
272C2×C22⋊C4C2×D7311yescharacteristic
273C2×C22⋊C4C2×D7311yescharacteristic
274C2×C42—731no—
275#95—731no—
276#95—731no—
277#95—731no—
278#95—731no—
279#95—731no—
280#95—731no—
281C24—731no—
282C24—731no—
283C24C22×D7311yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×D4—2921no—
288C2×D4—2921no—
289C2×D4—2921no—
290C2×D4—1462no—
291C2×D4—1462no—
292C2×D4—1462no—
293C2×D4—1462no—
294C2×D4—1462no—
295C2×D4—1462no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—2921no—
300C22×C4—731no—
301C22×C4—731no—
302C22×C4—731no—
303C22×C4—731no—
304C22×C4—731no—
305C22×C4—731no—
306C22×C4—1462no—
307C22×C4—1462no—
308C22×C4C22×D7311yescharacteristic
309C22×C4C22×D7311yescharacteristic
310C22×C4C22×D7311yescharacteristic
311C22×C4C22×D7311yescharacteristic
312C22×C4C22×D7311yescharacteristic
313C22×C4C22×D7311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—2921no—
318C4⋊C4—2921no—
319C4⋊C4—1462no—
320C4⋊C4—1462no—
321C4⋊C4—1462no—
322C4⋊C4—1462no—
323C22⋊C4—22no—
324C22⋊C4—2921no—
325C22⋊C4—2921no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C22⋊C4—734no—
329C22⋊C4—1462no—
330C22⋊C4—1462no—
331C22⋊C4—1462no—
332C22⋊C4—1462no—
333C22⋊C4—1462no—
334C22⋊C4—1462no—
335C22⋊C4C22⋊D7314yes—
336C42—1462no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—732no—
344C23—1461no—
345C23—1461no—
346C23—1461no—
347C23—1461no—
348C23—1461no—
349C23—1461no—
350C23—1461no—
351C23—1461no—
352C23—1461no—
353C23—1461no—
354C23—1461no—
355C23—1461no—
356C23—1461no—
357C23C2×C22⋊D7312yes—
358C23C23×D7311yescharacteristic
359Q8—44no—
360D4—2924no—
361D4—2924no—
362D4—2924no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—44no—
379C2×C4—2924no—
380C2×C4—732no—
381C2×C4—732no—
382C2×C4—1461no—
383C2×C4—1461no—
384C2×C4—1461no—
385C2×C4—1461no—
386C2×C4—1461no—
387C2×C4—1461no—
388C2×C4—1461no—
389C2×C4—1461no—
390C2×C4—1461no—
391C2×C4—1461no—
392C2×C4—1461no—
393C2×C4—1461no—
394C2×C4—1461no—
395C2×C4—1461no—
396C2×C4—1461no—
397C2×C4—1461no—
398C2×C4D4×D7312yes—
399C2×C4C2×C22⋊D7312yes—
400C2×C4C2×C22⋊D7312yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—2921no—
409C22—2921no—
410C22—2921no—
411C22—2921no—
412C22—2921no—
413C22—2921no—
414C22—2921no—
415C22—2921no—
416C22—2921no—
417C22—2921no—
418C22—2921no—
419C22—1462no—
420C22—1462no—
421C22—1462no—
422C22C2×D4×D7311yescharacteristic
423C22C22×C22⋊D7311yescharacteristic
424C22#41×D7311yescharacteristic
425C22#1455663611yescharacteristic
426C22#1455663711yescharacteristic
427C22#1455663911yescharacteristic
428C22#1455663911yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—2921no—
436C4—2921no—
437C4—2921no—
438C4—2921no—
439C4—1462no—
440C4—1462no—
441C2—42no—
442C2—2922no—
443C2—5841no—
444C2#1798916111yescharacteristic
445C2#1798916711yescharacteristic
446C2#1798920711yescharacteristic
447C2D4×C22⋊D7311yescharacteristic
448C2#1798922511yescharacteristic
449C2#1798923911yescharacteristic
450C2#1798933311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×C22⋊Q8×C73C2
hypercenterC2×C22⋊Q8D73
hypocenterC73#2672
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14556580C2semidirect
2. C2×#14556574C2semidirect
4. #17988522C2semidirect
5. #17988509C2semidirect
6. #17988506C2semidirect
7. #17988504C2semidirect
8. #17988502C2semidirect
9. #17988502C2semidirect
10. #17988482C2semidirect
11. #17988480C2semidirect
12. #17988466C2semidirect
13. #17988458C2semidirect
14. #17988444C2semidirect
15. C2×C22⋊Q8×C73C2semidirect
91. #11924005D4semidirect
226. C73#2672semidirect
229. C2×C22⋊Q8D73semidirect