Finite Group Databasebeta

← #22622232

Subgroups — #22622232

Order 9344.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17989238C211yesmaximal, characteristic, complemented
2#17989236C211yesmaximal, characteristic, complemented
3#17988936C211yesmaximal, characteristic, complemented
4#17988922C211yesmaximal, characteristic, complemented
5#17988921C211yesmaximal, characteristic, complemented
6#17988865C211yesmaximal, characteristic, complemented
7#17988836C211yesmaximal, characteristic, complemented
8#17988833C211yesmaximal, characteristic, complemented
9#17988791C211yesmaximal, characteristic, complemented
10#17988733C211yesmaximal, characteristic, complemented
11#17988732C211yesmaximal, characteristic, complemented
12#17988667C211yesmaximal, characteristic, complemented
13#17988663C211yesmaximal, characteristic, complemented
14#17988585C211yesmaximal, characteristic, complemented
15#417×C73C211yesmaximal, characteristic, complemented
16C2×#11924028—21no—
17C2×#11924027C2211yescharacteristic
18C2×#11924017C2211yescharacteristic
19C2×#11924015C2211yescharacteristic
20C2×#11924013C2211yescharacteristic
21C2×#11924011C2211yescharacteristic
22#14556590C2211yescharacteristic
23#14556589C2211yescharacteristic
24#14556576—21no—
25#14556574—21no—
26#14556574—21no—
27#14556572—21no—
28#14556570C2211yescharacteristic
29#14556569—21no—
30#14556567C2211yescharacteristic
31#14556566C2211yescharacteristic
32#14556565C2211yescharacteristic
33#14556563—21no—
34D4×Dic73C2211yescharacteristic
35#14556553C2211yescharacteristic
36C4×C22⋊D73C2211yescharacteristic
37C2×#11924007—21no—
38C4⋊C4×D73C2211yescharacteristic
39#14556525C2211yescharacteristic
40#14556524C2211yescharacteristic
41#14556524C2211yescharacteristic
42#14556520C2211yescharacteristic
43#14556516C2211yescharacteristic
44#14556515C2211yescharacteristic
45#14556514C2211yescharacteristic
46#14556514C2211yescharacteristic
47#14556513C2211yescharacteristic
48#14556512C2211yescharacteristic
49C22⋊#10596831C2211yescharacteristic
50#14556503C2211yescharacteristic
51C8×Dic73C2211yescharacteristic
52C2×#41×C73C2211yescharacteristic
53C2×M16×C73C2211yescharacteristic
54C2×C4⋊C4×C73C2211yescharacteristic
55#103×C73C2211yescharacteristic
56#103×C73C2211yescharacteristic
57#102×C73C2211yescharacteristic
58#102×C73C2211yescharacteristic
59#11924028—44no—
60#11924027—22no—
61#11924027—41no—
62C2×Dic146—21no—
63C2×Dic146C2311yescharacteristic
64C2×D292—21no—
65C2×D292C2311yescharacteristic
66C2×C22⋊D73—21no—
67C2×C22⋊D73—21no—
68C2×C22⋊D73D411yescharacteristic
69C22×Dic73—21no—
70C22×Dic73—21no—
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73C2311yescharacteristic
74#11924017—41no—
75#11924015—41no—
76#11924013—22no—
77#11924011—41no—
78C2×#10596831C2311yescharacteristic
79C2×#10596831C2311yescharacteristic
80C4⋊Dic73C2311yescharacteristic
81C4⋊Dic73C2311yescharacteristic
82#11924007—21no—
83#11924007—21no—
84#11924007—21no—
85#11924007—21no—
86#11924007—21no—
87#11924007—21no—
88#11924007D411yescharacteristic
89C22⋊Dic73—21no—
90C22⋊Dic73—21no—
91C22⋊Dic73D411yescharacteristic
92#11924005—21no—
93#11924005—21no—
94#11924005—21no—
95#11924005—21no—
96#11924005D411yescharacteristic
97C4×Dic73C2311yescharacteristic
98#41×C73—22no—
99#41×C73—41no—
100C2×Q8×C73C2311yescharacteristic
101C2×D4×C73—21no—
102C2×D4×C73C2311yescharacteristic
103C22×C4×C73—21no—
104C22×C4×C73—21no—
105C22×C4×C73C2311yescharacteristic
106M16×C73—41no—
107C2×C8×C73C2311yescharacteristic
108C2×C8×C73C2311yescharacteristic
109C4⋊C4×C73—21no—
110C4⋊C4×C73C2311yescharacteristic
111C4⋊C4×C73C2311yescharacteristic
112C23×C73—21no—
113C23×C73C2×D411yescharacteristic
114C22×D73—21no—
115C22×D73C2×D411yescharacteristic
116Q8×C73—21no—
117Q8×C73—21no—
118D4×C73—21no—
119D4×C73—42no—
120D4×C73#4112yescomplemented
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73C2×D411yescharacteristic
129C2×C4×C73C2411yescharacteristic
130C22⋊D73—22no—
131C22⋊D73—42no—
132C22⋊D73—42no—
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73C2×D411yescharacteristic
139C2×Dic73C2×D411yescharacteristic
140C2×Dic73C2×D411yescharacteristic
141D292—21no—
142D292—21no—
143D292—42no—
144C4×D73—21no—
145C4×D73—21no—
146C4×D73—42no—
147C4×D73—42no—
148Dic146—21no—
149Dic146—21no—
150Dic146—42no—
151C8×C73—21no—
152C8×C73—21no—
153#10596831—21no—
154#10596831—21no—
155C22×C73—22no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—41no—
160C22×C73—41no—
161C22×C73C22×D411yescharacteristic
162C2×D73—21no—
163C2×D73—21no—
164C2×D73—21no—
165C2×D73—41no—
166C2×D73—41no—
167C2×D73—41no—
168C4×C73—21no—
169C4×C73—41no—
170C4×C73—41no—
171C4×C73—41no—
172C4×C73C2×#4111yescharacteristic
173C4×C73H(5,𝔽2)11yescharacteristic
174Dic73—21no—
175Dic73—21no—
176Dic73—41no—
177Dic73—41no—
178Dic73—41no—
179Dic73AGL(1,ℤ/8)12yes—
180C2×C73—41no—
181C2×C73—42no—
182C2×C73#54611yescharacteristic
183C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C73#57811yescharacteristic
185D73—41no—
186D73—81no—
187#3330—731nomaximal, Sylow
188C73#333011yescharacteristic, complemented, Sylow
189#547—731no—
190#547—731no—
191#487—731no—
192#484—731no—
193#484—731no—
194#474—731no—
195#468—731no—
196#468—731no—
197#461—731no—
198#449—731no—
199#449—731no—
200C4×D8—731no—
201#436—731no—
202#417—731no—
203#417D7311yescharacteristic, complemented
204C2×#41—731no—
205C2×#41—1461no—
206C2×#41C2×D7311yescharacteristic
207C2×SD16—731no—
208C2×SD16—731no—
209C2×D8—731no—
210C2×M16—731no—
211C2×M16C2×D7311yescharacteristic
212C4⋊Q8—731no—
213C4⋊Q8—731no—
214#123—1461no—
215#123—1461no—
216#123—1461no—
217#123—1461no—
218C22⋊Q8—731no—
219C22⋊Q8—731no—
220C22⋊Q8—1461no—
221#121—731no—
222#121—731no—
223#121—1461no—
224C4×D4—731no—
225C4×D4—731no—
226C4×D4—731no—
227C2×C4⋊C4—731no—
228C2×C4⋊C4—1461no—
229C2×C4⋊C4C2×D7311yescharacteristic
230#107—731no—
231#107—731no—
232#107—731no—
233C4⋊C8—731no—
234#103—731no—
235#103—731no—
236#103—731no—
237#103—731no—
238#103C2×D7311yescharacteristic
239#103C2×D7311yescharacteristic
240#102—731no—
241#102—731no—
242#102C2×D7311yescharacteristic
243#102C2×D7311yescharacteristic
244C22⋊C8—731no—
245C22⋊C8—731no—
246C4×C8—731no—
247#41—22no—
248#41—41no—
249#41—2924no—
250#41—2921no—
251#41—1462no—
252C2×Q8—731no—
253C2×Q8—1461no—
254C2×Q8C22×D7311yescharacteristic
255C2×D4—21no—
256C2×D4—731no—
257C2×D4—731no—
258C2×D4—1461no—
259C2×D4—1461no—
260C2×D4—1461no—
261C2×D4C22×D7311yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—731no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4C22×D7311yescharacteristic
270SD16—2921no—
271SD16—2921no—
272D8—1462no—
273M16—41no—
274M16—2921no—
275C2×C8—731no—
276C2×C8—731no—
277C2×C8C22×D7311yescharacteristic
278C2×C8C22×D7311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—731no—
281C4⋊C4—731no—
282C4⋊C4—731no—
283C4⋊C4—1461no—
284C4⋊C4—1461no—
285C4⋊C4—1461no—
286C4⋊C4—1461no—
287C4⋊C4—1461no—
288C4⋊C4—1461no—
289C4⋊C4C22×D7311yescharacteristic
290C4⋊C4C22×D7311yescharacteristic
291C22⋊C4—731no—
292C22⋊C4—731no—
293C22⋊C4—1461no—
294C22⋊C4—1461no—
295C22⋊C4—1461no—
296C22⋊C4—1461no—
297C22⋊C4—1461no—
298C22⋊C4—1461no—
299C42—731no—
300C23—21no—
301C23—731no—
302C23—1461no—
303C23D4×D7311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2922no—
307Q8—1461no—
308Q8—1461no—
309D4—21no—
310D4—42no—
311D4—2922no—
312D4—2922no—
313D4—2922no—
314D4—1462no—
315D4—1461no—
316D4—1461no—
317D4#1192402812yescomplemented
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—2922no—
328C2×C4—2922no—
329C2×C4—731no—
330C2×C4—731no—
331C2×C4—731no—
332C2×C4—1461no—
333C2×C4—1461no—
334C2×C4—1461no—
335C2×C4—1461no—
336C2×C4—1461no—
337C2×C4D4×D7311yescharacteristic
338C2×C4C23×D7311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1461no—
342C8—1461no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—2921no—
350C22—2921no—
351C22—2921no—
352C22—1461no—
353C22—1461no—
354C22—1461no—
355C22C2×D4×D7311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2921no—
361C4—2921no—
362C4—2921no—
363C4—732no—
364C4—1461no—
365C4—1461no—
366C4C2×#1192402811yescharacteristic
367C4#1455663611yescharacteristic
368C2—41no—
369C2—42no—
370C2—2921no—
371C2—5841no—
372C2#1798922711yescharacteristic
373C2AGL(1,ℤ/8)×D7311yescharacteristic
374C2#1798940311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#417×C73C2
socleC22×C73C22×D4
hypercenter#417D73
hypocenterC73#3330
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #17989238C2semidirect
2. #17989236C2semidirect
3. #17988936C2semidirect
4. #17988922C2semidirect
5. #17988921C2semidirect
6. #17988865C2semidirect
7. #17988836C2semidirect
8. #17988833C2semidirect
9. #17988791C2semidirect
10. #17988733C2semidirect
11. #17988732C2semidirect
12. #17988667C2semidirect
13. #17988663C2semidirect
14. #17988585C2semidirect
15. #417×C73C2semidirect
120. D4×C73#41semidirect
188. C73#3330semidirect
203. #417D73semidirect
317. D4#11924028semidirect