Finite Group Databasebeta

← #33115871

Subgroups — #33115871

Order 18688.

988 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618494C212yesmaximal, complemented
2C2×#17988603C212yesmaximal
3#22618375C211yesmaximal, characteristic
4#22611483C218yesmaximal, complemented
5C2×#17988344C211yesmaximal, characteristic, complemented
6C2×#430×C73C211yesmaximal, characteristic, complemented
7C22×#11924028C2211yescharacteristic
8C22×#11924012C2211yescharacteristic
9C2×#14556550C2212yes—
10#17988615C2218yescomplemented
11#17988613—24no—
12#17988610—24no—
13#17988603—24no—
14#17988603C2218yes—
15C2×#14556517—24no—
16#17988555C2214yes—
17#17988553—22no—
18#17988550—22no—
19#17988344C2214yescomplemented
20C22×M16×C73C2212yes—
21C22×M16×C73C2211yescharacteristic
22#430×C73—22no—
23#430×C73C2214yescomplemented
24C2×#108×C73—22no—
25C2×#11924028—21no—
26C2×#11924028—22no—
27C2×#11924028—22no—
28C2×#11924028—21no—
29C2×#11924028—22no—
30C2×#11924028—22no—
31C2×#11924028Q812yes—
32C2×#11924028C2312yes—
33C22×Dic146Q811yescharacteristic
34C22×D292Q811yescharacteristic
35C22×C22⋊D73—21no—
36C22×C4×D73—21no—
37C2×#11924012—22no—
38C2×#11924012—22no—
39C2×#11924012—21no—
40C2×#11924012—21no—
41C22×#10596831C2312yes—
42#14556550—22no—
43#14556550—22no—
44#14556550D414yes—
45#14556550C2314yes—
46C2×C4⋊Dic73—22no—
47C2×C22⋊Dic73—22no—
48C2×C4×Dic73D414yes—
49#14556520—216no—
50#14556517—216no—
51#14556503—28no—
52C23×C4×C73C2311yescharacteristic
53C2×M16×C73—22no—
54C2×M16×C73—22no—
55C2×M16×C73—22no—
56C2×M16×C73—22no—
57C2×M16×C73—22no—
58C2×M16×C73—22no—
59C2×M16×C73—22no—
60C2×M16×C73D412yes—
61C2×M16×C73C2314yes—
62C2×M16×C73C2312yes—
63C22×C8×C73—22no—
64C22×C8×C73D412yes—
65#108×C73—28no—
66C23×D73—21no—
67#11924028—24no—
68#11924028—24no—
69#11924028—24no—
70#11924028—24no—
71#11924028—41no—
72#11924028—41no—
73#11924028—42no—
74#11924028—42no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—22no—
78C2×Dic146—22no—
79C2×Dic146—21no—
80C2×Dic146C2×Q812yes—
81C2×D292—22no—
82C2×D292—22no—
83C2×D292—21no—
84C2×D292C2×Q812yes—
85C2×C22⋊D73—22no—
86C2×C22⋊D73—22no—
87C2×C22⋊D73—42no—
88C2×C22⋊D73—42no—
89C22×Dic73—21no—
90C22×Dic73—24no—
91C2×C4×D73—22no—
92C2×C4×D73—22no—
93C2×C4×D73—42no—
94C2×C4×D73—42no—
95#11924012—41no—
96#11924012—42no—
97#11924012—41no—
98C2×#10596831—22no—
99C2×#10596831—24no—
100C4⋊Dic73—24no—
101C4⋊Dic73—24no—
102C22⋊Dic73—24no—
103C22⋊Dic73—24no—
104C4×Dic73—24no—
105C4×Dic73C2×D418yes—
106C24×C73#4111yescharacteristic
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2×D412yes—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×Q811yescharacteristic
112C22×C4×C73C2×Q811yescharacteristic
113C22×C4×C73#4112yes—
114C22×C4×C73#4112yes—
115C22×C4×C73#4111yescharacteristic
116C22×C4×C73C2411yescharacteristic
117M16×C73—24no—
118M16×C73—24no—
119M16×C73—24no—
120M16×C73—24no—
121M16×C73—42no—
122M16×C73—42no—
123C2×C8×C73—22no—
124C2×C8×C73—22no—
125C2×C8×C73—24no—
126C2×C8×C73—42no—
127C2×C8×C73—42no—
128C2×C8×C73C2×D414yes—
129C23×C73—81no—
130C23×C73C22⋊Q811yescharacteristic
131C23×C73C22⋊Q811yescharacteristic
132C23×C73#12312yes—
133C23×C73#12312yes—
134C23×C73C2×#4111yescharacteristic
135C22×D73—22no—
136C22×D73—42no—
137C22×D73—82no—
138C2×C4×C73C22⋊Q811yescharacteristic
139C2×C4×C73C22⋊Q811yescharacteristic
140C2×C4×C73C22⋊Q812yes—
141C2×C4×C73C22⋊Q812yes—
142C2×C4×C73C22⋊Q812yes—
143C2×C4×C73C22⋊Q812yes—
144C2×C4×C73#12311yescharacteristic
145C2×C4×C73#12311yescharacteristic
146C2×C4×C73#12312yes—
147C2×C4×C73#12312yes—
148C2×C4×C73#12312yes—
149C2×C4×C73#12711yescharacteristic
150C2×C4×C73#12712yes—
151C2×C4×C73#12711yescharacteristic
152C2×C4×C73C22×D412yes—
153C2×C4×C73C22×D411yescharacteristic
154C2×C4×C73C22×Q811yescharacteristic
155C2×C4×C73C2×#4112yes—
156C22⋊D73—44no—
157C22⋊D73—42no—
158C22⋊D73—42no—
159C2×Dic73—22no—
160C2×Dic73—28no—
161C2×Dic73—42no—
162C2×Dic73—84no—
163D292—24no—
164D292—41no—
165D292—41no—
166C4×D73—42no—
167C4×D73—42no—
168C4×D73—44no—
169Dic146—24no—
170Dic146—41no—
171Dic146—41no—
172C8×C73—24no—
173C8×C73—44no—
174#10596831—42no—
175C22×C73—42no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—42no—
179C22×C73—41no—
180C22×C73#39912yes—
181C22×C73#39911yescharacteristic
182C22×C73#39911yescharacteristic
183C22×C73C2×C22⋊Q811yescharacteristic
184C22×C73C2×#12312yes—
185C2×D73—44no—
186C2×D73—42no—
187C2×D73—84no—
188C4×C73#39911yescharacteristic
189C4×C73#39911yescharacteristic
190C4×C73#39912yes—
191C4×C73C2×C22⋊Q812yes—
192C4×C73C2×#12311yescharacteristic
193C4×C73C2×#12711yescharacteristic
194Dic73—48no—
195Dic73—42no—
196#23740—731nomaximal, Sylow
197C2×C73—22no—
198C2×C73—21no—
199C2×C73—22no—
200C2×C73—21no—
201C2×C73—21no—
202C2×C73C2×#39911yescharacteristic
203D73—82no—
204C2×#430D7311yescharacteristic, complemented
205#2905—732no—
206C2×#421—732no—
207#2879—731no—
208#2098—738no—
209C2×H(3,ℤ/4)—731no—
210C73#2374011yescharacteristic, complemented, Sylow
211C22×#41—731no—
212C22×M16—731no—
213C22×M16C2×D7312yes—
214C22×M16C2×D7311yescharacteristic
215C2×#117—732no—
216#430—22no—
217#430C2×D7314yescomplemented
218C2×#108—22no—
219#424—738no—
220#424—1464no—
221#423—1464no—
222#421—738no—
223#421—1464no—
224C2×#104—1464no—
225#408—734no—
226#408—1462no—
227#407—1462no—
228H(3,ℤ/4)—734no—
229C2×#41—732no—
230C2×#41—732no—
231C2×#41—1462no—
232C2×#41—1462no—
233C2×#41—1462no—
234C2×#41—1462no—
235C2×#41—1461no—
236C2×#41—1461no—
237C22×Q8—731no—
238C22×D4—731no—
239C22×D4—1461no—
240C23×C4—1461no—
241C23×C4C22×D7311yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—1462no—
250C2×M16—1462no—
251C2×M16—1461no—
252C2×M16—1461no—
253C2×M16C22⋊D7312yes—
254C2×M16C22×D7314yes—
255C2×M16C22×D7312yes—
256C22×C8—22no—
257C22×C8—732no—
258C22×C8C22⋊D7312yes—
259#117—734no—
260#117—734no—
261#117—1462no—
262#117—1462no—
263C2×C4⋊C4—1462no—
264C2×C22⋊C4—1462no—
265C2×C42—734no—
266#108—28no—
267C4⋊C8—14616no—
268#104—14616no—
269C22⋊C8—1468no—
270C24—1461no—
271C24#1192402911yescharacteristic
272#41—2921no—
273#41—2921no—
274#41—2922no—
275#41—2922no—
276#41—2921no—
277#41—2921no—
278#41—1464no—
279#41—1464no—
280#41—1464no—
281#41—1464no—
282C2×Q8—732no—
283C2×Q8—1462no—
284C2×Q8—1462no—
285C2×Q8—1461no—
286C2×D4—2922no—
287C2×D4—2922no—
288C2×D4—732no—
289C2×D4—1462no—
290C2×D4—1462no—
291C2×D4—1461no—
292C2×D4—1462no—
293C2×D4—1462no—
294C22×C4—2922no—
295C22×C4—2922no—
296C22×C4—1461no—
297C22×C4—1462no—
298C22×C4—1462no—
299C22×C4—1464no—
300C22×C4D4×D7312yes—
301C22×C4D4×D7312yes—
302C22×C4C2×C22⋊D7311yescharacteristic
303C22×C4C2×C22⋊D7311yescharacteristic
304C22×C4Q8×D7311yescharacteristic
305C22×C4Q8×D7311yescharacteristic
306C22×C4#1192402812yes—
307C22×C4#1192402812yes—
308C22×C4#1192402911yescharacteristic
309C22×C4C23×D7311yescharacteristic
310M16—24no—
311M16—24no—
312M16—24no—
313M16—24no—
314M16—42no—
315M16—42no—
316M16—2921no—
317M16—2922no—
318M16—2921no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—1464no—
325C2×C8—1462no—
326C2×C8C2×C22⋊D7314yes—
327C4⋊C4—1464no—
328C4⋊C4—1464no—
329C22⋊C4—1464no—
330C22⋊C4—1464no—
331C42—738no—
332C42—1464no—
333C23—81no—
334C23—2922no—
335C23—5842no—
336C23—1462no—
337C23#1455656711yescharacteristic
338C23#1455656711yescharacteristic
339C23#1455657612yes—
340C23#1455657612yes—
341C23C2×#1192402911yescharacteristic
342Q8—2921no—
343Q8—2921no—
344Q8—1464no—
345D4—2922no—
346D4—2922no—
347D4—2924no—
348D4—2921no—
349D4—2921no—
350D4—1464no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4—2924no—
354C2×C4—2922no—
355C2×C4—5844no—
356C2×C4—1462no—
357C2×C4—1468no—
358C2×C4#1455656711yescharacteristic
359C2×C4#1455656711yescharacteristic
360C2×C4#1455656912yes—
361C2×C4#1455656912yes—
362C2×C4#1455656912yes—
363C2×C4#1455656912yes—
364C2×C4#1455657411yescharacteristic
365C2×C4#1455657411yescharacteristic
366C2×C4#1455657412yes—
367C2×C4#1455657612yes—
368C2×C4#1455657612yes—
369C2×C4#1455658711yescharacteristic
370C2×C4#1455658712yes—
371C2×C4#1455658711yescharacteristic
372C2×C4C2×D4×D7312yes—
373C2×C4C22×C22⋊D7311yescharacteristic
374C2×C4C2×Q8×D7311yescharacteristic
375C2×C4C2×#1192402812yes—
376C8—24no—
377C8—44no—
378C8—2922no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—2924no—
385C22—2922no—
386C22—5844no—
387C22#1798852212yes—
388C22#1798852211yescharacteristic
389C22#1798852211yescharacteristic
390C22C2×#1455656711yescharacteristic
391C22C2×#1455657612yes—
392C4—2928no—
393C4—2922no—
394C4#1798852212yes—
395C4#1798852211yescharacteristic
396C4#1798852211yescharacteristic
397C4C2×#1455656912yes—
398C4C2×#1455657411yescharacteristic
399C4C2×#1455658711yescharacteristic
400C2—22no—
401C2—21no—
402C2—22no—
403C2—21no—
404C2—21no—
405C2—5842no—
406C2C2×#1798852211yescharacteristic
subgroupG/H
centerC4C2×#14556587
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#430×C73C2
socleC2×C73C2×#399
prime radicalC22×#11924028C22
hypercenterC2×#430D73
hypocenterC73#23740
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
1. #22618494C2semidirect
4. #22611483C2semidirect
5. C2×#17988344C2semidirect
6. C2×#430×C73C2semidirect
10. #17988615C22semidirect
19. #17988344C22semidirect
23. #430×C73C22semidirect
204. C2×#430D73semidirect
210. C73#23740semidirect
217. #430C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226184942
C2×#179886032
#226183752
#226114832
C2×#179883442
C2×#430×C732
#2374073