Finite Group Databasebeta

← #33240581

Subgroups — #33240581

Order 18688.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989027C211yesmaximal, characteristic, complemented
2#22623034C211yesmaximal, characteristic, complemented
3C2×#17988862C211yesmaximal, characteristic, complemented
4#22612059C214yesmaximal, complemented
5#22612054C214yesmaximal, complemented
6#22611725C211yesmaximal, characteristic, complemented
7#22611680C211yesmaximal, characteristic, complemented
8C2×#17988402C211yesmaximal, characteristic, complemented
9C73×#3067C211yesmaximal, characteristic
10C2×#14556621—21no—
11C22×Dic292C2211yescharacteristic
12#17989027—22no—
13#17989027—22no—
14C2×#14556537C2212yescomplemented
15C2×#14556535C2212yescomplemented
16#17989004—22no—
17#17989004C2214yes—
18C2×#14556531—22no—
19C2×#14556531C2211yescharacteristic
20C2×#14556531C2211yescharacteristic
21#17988862—22no—
22#17988862C2214yes—
23C2×#14556527C2211yescharacteristic
24C2×#14556509—21no—
25#17988416C2214yescomplemented
26#17988403C2212yescomplemented
27#17988402C2214yescomplemented
28#17988400C2212yescomplemented
29#575×C73—21no—
30C2×#135×C73C2211yescharacteristic
31C2×#131×C73C2211yescharacteristic
32#473×C73C2212yes—
33#472×C73C2212yes—
34#471×C73—22no—
35C2×#108×C73C2211yescharacteristic
36#413×C73—21no—
37C22×Dic146—21no—
38#14556621—24no—
39#14556621—42no—
40C2×Dic292—21no—
41C2×Dic292—21no—
42C2×Dic292—21no—
43C2×Dic292—21no—
44C2×Dic292—21no—
45C2×Dic292C2312yes—
46C2×#11924018—41no—
47C2×#11924016—41no—
48C2×#11924012—21no—
49#14556537—24no—
50#14556535—24no—
51#14556531—24no—
52#14556531—21no—
53#14556531D412yes—
54#14556531D412yes—
55#14556531C2312yes—
56C2×#11924003—24no—
57C2×#11924003—21no—
58C2×#11924003C2312yes—
59#14556527—21no—
60#14556527C2312yes—
61#14556509—24no—
62C2×#41×C73—21no—
63C2×#41×C73D411yescharacteristic
64#137×C73—22no—
65#137×C73—41no—
66AGL(1,ℤ/8)×C73—22no—
67AGL(1,ℤ/8)×C73—41no—
68#135×C73—21no—
69#135×C73—21no—
70#135×C73—21no—
71#135×C73—21no—
72#135×C73—42no—
73C2×Q16×C73C2311yescharacteristic
74C2×SD16×C73—21no—
75C2×D8×C73C2311yescharacteristic
76#131×C73—21no—
77#131×C73—21no—
78#131×C73D412yes—
79#131×C73C2312yes—
80C2×M16×C73—21no—
81C2×M16×C73—21no—
82C2×M16×C73D411yescharacteristic
83C22×C8×C73—21no—
84C22×C8×C73C2311yescharacteristic
85#108×C73—21no—
86#108×C73C2312yes—
87#101×C73—22no—
88#100×C73—22no—
89C2×Dic146—22no—
90C2×Dic146—41no—
91C2×Dic146—41no—
92C2×Dic146—41no—
93C2×Dic146—41no—
94C2×Dic146—41no—
95C22×Dic73—41no—
96Dic292—22no—
97Dic292—22no—
98Dic292—41no—
99Dic292—41no—
100#11924018—44no—
101#11924016—44no—
102#11924012—22no—
103#11924012—41no—
104C2×#10596831—41no—
105#11924003—22no—
106#11924003C2×D414yes—
107#41×C73—21no—
108#41×C73—21no—
109#41×C73—41no—
110#41×C73—41no—
111#41×C73—41no—
112#41×C73—41no—
113#41×C73D812yes—
114#41×C73C2×D412yes—
115C2×Q8×C73—21no—
116C2×Q8×C73D811yescharacteristic
117C2×D4×C73—21no—
118C2×D4×C73—21no—
119C2×D4×C73—41no—
120C2×D4×C73D811yescharacteristic
121C22×C4×C73—21no—
122C22×C4×C73—41no—
123C22×C4×C73C2×D411yescharacteristic
124Q16×C73—21no—
125Q16×C73—21no—
126Q16×C73—42no—
127SD16×C73—42no—
128SD16×C73—42no—
129SD16×C73—41no—
130SD16×C73—41no—
131D8×C73—21no—
132D8×C73—21no—
133D8×C73—42no—
134M16×C73—21no—
135M16×C73—22no—
136M16×C73—22no—
137M16×C73—22no—
138M16×C73—41no—
139M16×C73C2×D412yes—
140C2×C8×C73—21no—
141C2×C8×C73—22no—
142C2×C8×C73—22no—
143C2×C8×C73—41no—
144C2×C8×C73C2×D411yescharacteristic
145C2×C8×C73C2×D411yescharacteristic
146C2×C8×C73C2×D411yescharacteristic
147C2×C8×C73#4111yescharacteristic
148C2×C8×C73#4111yescharacteristic
149C2×C8×C73C2411yescharacteristic
150C23×C73—21no—
151C23×C73—41no—
152C23×C73#13511yescharacteristic
153Q8×C73—21no—
154Q8×C73—41no—
155Q8×C73—41no—
156Q8×C73C2×D812yes—
157D4×C73—22no—
158D4×C73—22no—
159D4×C73—21no—
160D4×C73—41no—
161D4×C73—41no—
162D4×C73—81no—
163D4×C73—81no—
164D4×C73C2×D812yes—
165C2×C4×C73—21no—
166C2×C4×C73—22no—
167C2×C4×C73—22no—
168C2×C4×C73—41no—
169C2×C4×C73—81no—
170C2×C4×C73—81no—
171C2×C4×C73#12111yescharacteristic
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73C2×D811yescharacteristic
174C2×C4×C73C2×D811yescharacteristic
175C2×C4×C73#13511yescharacteristic
176C2×C4×C73C22×D411yescharacteristic
177C2×Dic73—42no—
178C2×Dic73—81no—
179C2×Dic73—81no—
180Dic146—42no—
181Dic146—42no—
182Dic146—81no—
183Dic146—81no—
184C8×C73—22no—
185C8×C73—42no—
186C8×C73#12111yescharacteristic
187C8×C73#12111yescharacteristic
188C8×C73C22×D411yescharacteristic
189C8×C73C2×#4111yescharacteristic
190#10596831—42no—
191C22×C73—22no—
192C22×C73—41no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—161no—
196C22×C73#46611yescharacteristic
197C22×C73#46611yescharacteristic
198C22×C73C2×#13511yescharacteristic
199C4×C73—22no—
200C4×C73—81no—
201C4×C73#46611yescharacteristic
202C4×C73#46611yescharacteristic
203C4×C73C2×#12111yescharacteristic
204C4×C73C22×D811yescharacteristic
205Dic73—82no—
206#33619—731nomaximal, Sylow
207C2×C73—21no—
208C2×C73—21no—
209C2×C73—21no—
210C2×C73—42no—
211C2×C73—161no—
212C2×C73C2×#46611yescharacteristic
213C2×#510—731no—
214#3407—731no—
215#3067D7311yescharacteristic
216C2×#473—731no—
217#2223—734no—
218#2222—734no—
219#2161—731no—
220#2148—731no—
221C2×#362—731no—
222C73#3361911yescharacteristic, complemented, Sylow
223#575—21no—
224C2×#137—1461no—
225C2×#135C2×D7311yescharacteristic
226C22×Q16—731no—
227C2×#131C2×D7311yescharacteristic
228#510—1462no—
229#510—1462no—
230C2×Q32—732no—
231C2×SD32—732no—
232#504—734no—
233#504—1462no—
234C2×M32—731no—
235C2×M32—731no—
236C2×M32—1462no—
237#473—734no—
238#473—1462no—
239#473C2×D7312yes—
240#472C2×D7312yes—
241#471—22no—
242C2×#108—731no—
243C2×#108C2×D7311yescharacteristic
244#413—21no—
245C2×#101—1461no—
246#368—734no—
247#362—732no—
248#362—734no—
249#361—732no—
250C2×#41—21no—
251C2×#41C22⋊D7311yescharacteristic
252C22×Q8—1461no—
253#137—22no—
254#137—41no—
255#137—2922no—
256#137—1464no—
257AGL(1,ℤ/8)—22no—
258AGL(1,ℤ/8)—41no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—42no—
264C2×Q16—2921no—
265C2×Q16—732no—
266C2×Q16—1461no—
267C2×Q16—1461no—
268C2×Q16—1461no—
269C2×Q16—1461no—
270C2×Q16—1461no—
271C2×Q16C22×D7311yescharacteristic
272C2×SD16—21no—
273C2×SD16—2921no—
274C2×D8C22×D7311yescharacteristic
275#131—21no—
276#131—21no—
277#131C22⋊D7312yes—
278#131C22×D7312yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—1461no—
282C2×M16C22⋊D7311yescharacteristic
283C22×C8—21no—
284C22×C8C22×D7311yescharacteristic
285Q32—1464no—
286SD32—1464no—
287M32—732no—
288M32—732no—
289M32—732no—
290M32—1464no—
291M32—1461no—
292C2×C16—732no—
293C2×C16—1464no—
294C2×C16—1461no—
295#108—21no—
296#108—732no—
297#108—1461no—
298#108C22×D7312yes—
299#101—22no—
300#101—1464no—
301#100—22no—
302#41—21no—
303#41—21no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41#1192401312yes—
309#41C2×C22⋊D7312yes—
310C2×Q8—21no—
311C2×Q8—2921no—
312C2×Q8—2921no—
313C2×Q8—2921no—
314C2×Q8—2921no—
315C2×Q8—2921no—
316C2×Q8—1462no—
317C2×Q8#1192401311yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4#1192401311yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—2921no—
325C22×C4C2×C22⋊D7311yescharacteristic
326Q16—21no—
327Q16—21no—
328Q16—42no—
329Q16—2924no—
330Q16—2921no—
331Q16—2921no—
332Q16—1462no—
333Q16—1462no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—2924no—
339D8—21no—
340D8—21no—
341D8—42no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—2921no—
348M16—1462no—
349M16C2×C22⋊D7312yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—2921no—
355C2×C8D4×D7311yescharacteristic
356C2×C8D4×D7311yescharacteristic
357C2×C8C2×C22⋊D7311yescharacteristic
358C2×C8#1192402711yescharacteristic
359C2×C8#1192402711yescharacteristic
360C2×C8C23×D7311yescharacteristic
361C16—734no—
362C16—1462no—
363C23—21no—
364C23—41no—
365C23#1455661111yescharacteristic
366Q8—21no—
367Q8—41no—
368Q8—41no—
369Q8—2922no—
370Q8—2922no—
371Q8—5841no—
372Q8—5841no—
373Q8C2×#1192401312yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4C2×#1192401312yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—2922no—
389C2×C4—5841no—
390C2×C4—5841no—
391C2×C4#1455656311yescharacteristic
392C2×C4#1455656311yescharacteristic
393C2×C4C2×#1192401311yescharacteristic
394C2×C4C2×#1192401311yescharacteristic
395C2×C4#1455661111yescharacteristic
396C2×C4C22×C22⋊D7311yescharacteristic
397C8—22no—
398C8—42no—
399C8—2922no—
400C8#1455656311yescharacteristic
401C8#1455656311yescharacteristic
402C8C2×D4×D7311yescharacteristic
403C8C2×#1192402711yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—161no—
409C22#1798882111yescharacteristic
410C22#1798882111yescharacteristic
411C22C2×#1455661111yescharacteristic
412C4—22no—
413C4—81no—
414C4—5842no—
415C4#1798882111yescharacteristic
416C4#1798882111yescharacteristic
417C4C2×#1455656311yescharacteristic
418C4C22×#1192401311yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#1798882111yescharacteristic
subgroupG/H
centerC2C2×#17988821
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3067C2
socleC2×C73C2×#466
hypercenter#3067D73
hypocenterC73#33619
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#17989027C2semidirect
2. #22623034C2semidirect
3. C2×#17988862C2semidirect
4. #22612059C2semidirect
5. #22612054C2semidirect
6. #22611725C2semidirect
7. #22611680C2semidirect
8. C2×#17988402C2semidirect
14. C2×#14556537C22semidirect
15. C2×#14556535C22semidirect
25. #17988416C22semidirect
26. #17988403C22semidirect
27. #17988402C22semidirect
28. #17988400C22semidirect
222. C73#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#179890272
#226230342
C2×#179888622
#226120592
#226120542
#226117252
#226116802
C2×#179884022
C73×#30672
#3361973