Finite Group Databasebeta

← #33057773

Subgroups — #33057773

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620501C211yesmaximal, characteristic
2#22618935C212yesmaximal, complemented
3#22608829C211yesmaximal, characteristic, complemented
4#22608817C211yesmaximal, characteristic, complemented
5#22608776C212yesmaximal, complemented
6#22608769C212yesmaximal, complemented
7C8×Dic292C211yesmaximal, characteristic, complemented
8C8×D584C211yesmaximal, characteristic, complemented
9#22608679C211yesmaximal, characteristic, complemented
10#22608340C212yesmaximal, complemented
11C8×M16×C73C211yesmaximal, characteristic, complemented
12#17989121C2212yes—
13#17988989C411yescharacteristic
14#17988971C411yescharacteristic
15#17988942C412yes—
16#17988828C411yescharacteristic
17#17988823C411yescharacteristic
18#17988816C412yes—
19#17988717C2212yes—
20C8×Dic146C2212yes—
21C4×Dic292C2211yescharacteristic
22C4×#11924017C2212yes—
23C4×D584C2211yescharacteristic
24#17988666C2212yes—
25#17988663—22no—
26#17988656—22no—
27C8×D292C2212yes—
28#17988649—22no—
29#17988640C2212yes—
30#17988544—22no—
31#17988340C2211yescharacteristic
32#17988340C2211yescharacteristic
33#17988338C2212yes—
34#17988319C2212yes—
35#17988319C2212yes—
36#17988316C2212yes—
37#17988316C2212yes—
38#431×C73C2211yescharacteristic
39C4×M16×C73C2211yescharacteristic
40C2×C4×C8×C73C2211yescharacteristic
41#322×C73C2211yescharacteristic
42#322×C73C2211yescharacteristic
43C82×C73C2211yescharacteristic
44C82×C73C2211yescharacteristic
45C2×Dic292C2×C411yescharacteristic
46C2×#11924017C2×C412yes—
47C2×D584C2×C411yescharacteristic
48C2×#11924011—22no—
49C2×C8×D73—22no—
50#14556585—22no—
51#14556583C2×C412yes—
52#14556579C2×C412yes—
53#14556574—22no—
54#14556570C2×C412yes—
55#14556565C2×C412yes—
56C4×Dic146C2312yes—
57C4×D292C2312yes—
58C4×C22⋊D73—22no—
59#14556548—22no—
60#14556525C2×C411yescharacteristic
61#14556525C2×C411yescharacteristic
62#14556523C2×C412yes—
63C4⋊#10596831C2312yes—
64C4⋊#10596831C2312yes—
65#14556520—22no—
66#14556520—22no—
67#14556516C2×C412yes—
68#14556516C2×C412yes—
69#14556513C2×C412yes—
70#14556513C2×C412yes—
71C22⋊#10596831—22no—
72#14556503—22no—
73#14556503—22no—
74#14556501—22no—
75C4×#10596831—22no—
76C8×Dic73—22no—
77C2×M16×C73D411yescharacteristic
78C2×M16×C73D411yescharacteristic
79C22×C8×C73C2×C411yescharacteristic
80C22×C8×C73C2×C411yescharacteristic
81C2×C42×C73C2311yescharacteristic
82C4⋊C8×C73—21no—
83C22⋊C8×C73—21no—
84#97×C73D411yescharacteristic
85#97×C73D411yescharacteristic
86C4×C8×C73C2×C411yescharacteristic
87C4×C8×C73C2×C411yescharacteristic
88C4×C8×C73C2311yescharacteristic
89C4×C8×C73C2311yescharacteristic
90C4×C8×C73C2311yescharacteristic
91C4×C8×C73C2311yescharacteristic
92C4×C8×C73C2311yescharacteristic
93C4×C8×C73C2311yescharacteristic
94C2×Dic146C22×C412yes—
95C2×D292C22×C412yes—
96C2×C22⋊D73—22no—
97C2×C4×D73—22no—
98Dic292—22no—
99#11924017—42no—
100D584—22no—
101#11924011—44no—
102C2×#10596831—22no—
103C2×#10596831—22no—
104C8×D73—44no—
105C4⋊Dic73C22×C412yes—
106C4⋊Dic73C22×C412yes—
107#11924007—22no—
108#11924007—22no—
109#11924007—22no—
110#11924007—22no—
111C22⋊Dic73—22no—
112#11924005—22no—
113#11924005—22no—
114#11924005—22no—
115#11924005—22no—
116C4×Dic73—22no—
117C22×C4×C73C22×C411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73#4111yescharacteristic
120M16×C73—22no—
121M16×C73—22no—
122C2×C8×C73—22no—
123C2×C8×C73—21no—
124C2×C8×C73—21no—
125C2×C8×C73—41no—
126C2×C8×C73C22×C411yescharacteristic
127C2×C8×C73C22×C411yescharacteristic
128C2×C8×C73C22×C411yescharacteristic
129C2×C8×C73C22×C411yescharacteristic
130C2×C8×C73C2×D411yescharacteristic
131C2×C8×C73C2×D411yescharacteristic
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73C2×D411yescharacteristic
134C42×C73C22×C411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73#4111yescharacteristic
137C42×C73C2411yescharacteristic
138C23×C73C4×D411yescharacteristic
139C22×D73—22no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—41no—
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73C23×C411yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C2×C4×C73C2×#4111yescharacteristic
151C22⋊D73—82no—
152C2×Dic73—22no—
153C2×Dic73—22no—
154C2×Dic73—22no—
155D292—22no—
156D292—22no—
157C4×D73—42no—
158C4×D73—42no—
159Dic146—22no—
160Dic146—22no—
161C8×C73—21no—
162C8×C73—21no—
163C8×C73—21no—
164C8×C73—21no—
165C8×C73—21no—
166C8×C73—41no—
167C8×C73—41no—
168C8×C73#13112yes—
169C8×C73#13512yes—
170C8×C73#13512yes—
171#10596831—42no—
172#10596831—42no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73C2×C4×D411yescharacteristic
177C2×D73—42no—
178C2×D73—42no—
179C2×D73—42no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73C2×#13111yescharacteristic
185C4×C73#56811yescharacteristic
186C4×C73C2×#13511yescharacteristic
187C4×C73C2×#13511yescharacteristic
188Dic73—42no—
189Dic73—42no—
190Dic73—42no—
191#18069—731nomaximal, Sylow
192C2×C73—41no—
193C2×C73#293511yescharacteristic
194C2×C73C4×#13511yescharacteristic
195C2×C73#295711yescharacteristic
196D73—82no—
197#3168—731no—
198#2969—732no—
199#1591—731no—
200#1589—731no—
201#1583—732no—
202#1582—732no—
203C8×Q16—731no—
204C8×D8—731no—
205#1564—731no—
206#1482—732no—
207C8×M16D7311yescharacteristic, complemented
208C73#1806911yescharacteristic, complemented, Sylow
209#529—732no—
210#499—731no—
211#495—731no—
212#488—732no—
213#467—731no—
214#466—731no—
215#465—732no—
216#446—732no—
217C8×Q8—732no—
218C4×Q16—731no—
219C4×SD16—732no—
220C4×D8—731no—
221#436—732no—
222#436—1462no—
223#435—1462no—
224C8×D4—732no—
225#433—1462no—
226#431—732no—
227#431C2×D7311yescharacteristic
228#405—1462no—
229C4×M16C2×D7311yescharacteristic
230C2×C4×C8C2×D7311yescharacteristic
231#335—731no—
232#335—731no—
233#334—732no—
234Q8⋊C8—732no—
235Q8⋊C8—732no—
236D4⋊C8—732no—
237D4⋊C8—732no—
238#322C2×D7311yescharacteristic
239#322C2×D7311yescharacteristic
240C82C2×D7311yescharacteristic
241C82C2×D7311yescharacteristic
242C2×Q16—731no—
243C2×SD16—732no—
244C2×D8—731no—
245C2×M16—1462no—
246C2×M16D29211yescharacteristic
247C2×M16D29211yescharacteristic
248C22×C8—1462no—
249C22×C8C4×D7311yescharacteristic
250C22×C8C4×D7311yescharacteristic
251#126—1462no—
252#125—732no—
253#124—732no—
254#123—1462no—
255C22⋊Q8—732no—
256#121—732no—
257C4×Q8—732no—
258C4×D4—732no—
259C4×D4—1462no—
260#117—1462no—
261C2×C42C22×D7311yescharacteristic
262#107—731no—
263#107—731no—
264#106—732no—
265C4⋊C8—21no—
266C4⋊C8—732no—
267C4⋊C8—732no—
268C4⋊C8—1462no—
269C4⋊C8—1462no—
270#103—732no—
271#103—732no—
272#102—732no—
273#102—732no—
274C22⋊C8—21no—
275C22⋊C8—1462no—
276C22⋊C8—1462no—
277C22⋊C8—1462no—
278#97—1462no—
279#97D29211yescharacteristic
280#97D29211yescharacteristic
281C4×C8—1462no—
282C4×C8—1462no—
283C4×C8C4×D7311yescharacteristic
284C4×C8C4×D7311yescharacteristic
285C4×C8C22×D7311yescharacteristic
286C4×C8C22×D7311yescharacteristic
287C4×C8C22×D7311yescharacteristic
288C4×C8C22×D7311yescharacteristic
289C4×C8C22×D7311yescharacteristic
290C4×C8C22×D7311yescharacteristic
291C2×Q8—732no—
292C2×D4—732no—
293C2×D4—1462no—
294C22×C4—1462no—
295C22×C4C2×C4×D7311yescharacteristic
296C22×C4C2×D29211yescharacteristic
297C22×C4#1192402811yescharacteristic
298Q16—1462no—
299SD16—2922no—
300D8—1462no—
301M16—22no—
302M16—22no—
303M16—2924no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—2924no—
309C2×C8—1462no—
310C2×C8—1462no—
311C2×C8C2×C4×D7311yescharacteristic
312C2×C8C2×C4×D7311yescharacteristic
313C2×C8C2×C4×D7311yescharacteristic
314C2×C8C2×C4×D7311yescharacteristic
315C2×C8C2×D29211yescharacteristic
316C2×C8C2×D29211yescharacteristic
317C2×C8C2×D29211yescharacteristic
318C2×C8C2×D29211yescharacteristic
319C4⋊C4—732no—
320C4⋊C4—732no—
321C4⋊C4—1462no—
322C4⋊C4—1462no—
323C4⋊C4—1462no—
324C4⋊C4—1462no—
325C22⋊C4—1462no—
326C22⋊C4—1462no—
327C22⋊C4—1462no—
328C22⋊C4—1462no—
329C22⋊C4—1462no—
330C42—1462no—
331C42C2×C4×D7311yescharacteristic
332C42C2×D29211yescharacteristic
333C42#1192402811yescharacteristic
334C42C23×D7311yescharacteristic
335C23—1462no—
336C23C4×D29211yescharacteristic
337Q8—1462no—
338Q8—1462no—
339D4—5842no—
340D4—1462no—
341D4—1462no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—2922no—
348C2×C4—2922no—
349C2×C4—1462no—
350C2×C4—1462no—
351C2×C4—1462no—
352C2×C4C4×D29211yescharacteristic
353C2×C4C4×D29211yescharacteristic
354C2×C4C4×D29211yescharacteristic
355C2×C4C22×C4×D7311yescharacteristic
356C2×C4C22×D29211yescharacteristic
357C2×C4C2×#1192402811yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—41no—
364C8—41no—
365C8—2922no—
366C8—2922no—
367C8#1455659912yes—
368C8#1455661312yes—
369C8#1455661312yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—2922no—
374C22—2922no—
375C22—2922no—
376C22C2×C4×D29211yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—2922no—
382C4—2922no—
383C4—2922no—
384C4C2×#1455659911yescharacteristic
385C4#1798935311yescharacteristic
386C4C2×#1455661311yescharacteristic
387C4C2×#1455661311yescharacteristic
388C2—41no—
389C2—5842no—
390C2#2261867711yescharacteristic
391C2C4×#1455661311yescharacteristic
392C2#2261884911yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×M16×C73C2
socleC22×C73C2×C4×D4
prime radical#17988823C4
cosocleC42×C73C24
hypercenterC8×M16D73
hypocenterC73#18069
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #22618935C2semidirect
3. #22608829C2semidirect
4. #22608817C2semidirect
5. #22608776C2semidirect
6. #22608769C2semidirect
7. C8×Dic292C2semidirect
8. C8×D584C2semidirect
9. #22608679C2semidirect
10. #22608340C2semidirect
11. C8×M16×C73C2semidirect
207. C8×M16D73semidirect
208. C73#18069semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226205012
#226189352
#226088292
#226088172
#226087762
#226087692
C8×Dic2922
C8×D5842
#226086792
#226083402
C8×M16×C732
#1806973