Finite Group Databasebeta

← #33144900

Subgroups — #33144900

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619778C212yesmaximal
2#22618528C211yesmaximal, characteristic
3#22611439C212yesmaximal, complemented
4#22611273C212yesmaximal, complemented
5#22610716C211yesmaximal, characteristic, complemented
6#22610680C211yesmaximal, characteristic, complemented
7#22610154C212yesmaximal
8#22610089C211yesmaximal, characteristic, complemented
9#22609996C211yesmaximal, characteristic
10#22609770C211yesmaximal, characteristic
11C73×#3164C211yesmaximal, characteristic, complemented
12C2×#14556597C2211yescharacteristic
13#17989067C2212yes—
14#17988891—22no—
15#17988884—22no—
16#17988879—22no—
17#17988864—22no—
18C2×#14556527C2211yescharacteristic
19#17988630C2212yes—
20#17988630C2212yes—
21#17988626—22no—
22#17988624—21no—
23C2×#14556525C2211yescharacteristic
24C2×#14556524C2212yes—
25#107×D73—21no—
26#17988614C2212yes—
27C2×#14556517C2211yescharacteristic
28#17988586C2212yes—
29#17988581C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988456C2212yes—
32#17988356C2212yes—
33#17988355C2212yes—
34#17988351C2211yescharacteristic
35#17988346C2211yescharacteristic
36#17988346C2212yes—
37#544×C73C2212yes—
38#501×C73—22no—
39#500×C73—21no—
40#499×C73—21no—
41#428×C73C2212yes—
42#427×C73C2211yescharacteristic
43C2×#107×C73C2211yescharacteristic
44#405×C73C2211yescharacteristic
45C2×#11924028C2311yescharacteristic
46#14556597—22no—
47#14556597Q814yescomplemented
48C2×#11924012C2312yes—
49C2×#11924012C2311yescharacteristic
50C2×#11924011—21no—
51C22×#10596831C2311yescharacteristic
52C2×C8×D73—21no—
53C4×Dic146—22no—
54C4×D292—22no—
55#14556551—42no—
56#14556549—22no—
57C2×C4⋊Dic73C2312yes—
58C2×#11924007—42no—
59C4⋊C4×D73—22no—
60C2×C4×Dic73C2311yescharacteristic
61#14556527—22no—
62#14556525—21no—
63#14556525—21no—
64#14556524—22no—
65#14556524—22no—
66#14556524—22no—
67#14556524—22no—
68#14556522—22no—
69#14556522—22no—
70C4⋊#10596831—22no—
71C4⋊#10596831—22no—
72#14556517Q814yescomplemented
73#14556514—22no—
74#14556514D411yescharacteristic
75#14556514D411yescharacteristic
76#14556511—22no—
77#14556511D411yescharacteristic
78#14556511D411yescharacteristic
79C8×Dic73—22no—
80C2×M16×C73C2311yescharacteristic
81C22×C8×C73C2311yescharacteristic
82C4⋊Q8×C73—22no—
83C4⋊Q8×C73—22no—
84#125×C73—22no—
85#125×C73—22no—
86C22⋊Q8×C73—42no—
87#117×C73C2312yes—
88#117×C73C2311yescharacteristic
89C2×C4⋊C4×C73C2312yes—
90#107×C73—21no—
91#107×C73—21no—
92#107×C73—22no—
93#107×C73D411yescharacteristic
94#107×C73D411yescharacteristic
95#106×C73—22no—
96#106×C73D412yes—
97#97×C73—21no—
98C4×C8×C73—21no—
99#11924028—21no—
100#11924028—21no—
101#11924028C2×Q812yes—
102#11924028C2×Q812yes—
103C2×Dic146C2×D411yescharacteristic
104C2×D292C2×D411yescharacteristic
105C2×C22⋊D73—21no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C2×C4×D73—21no—
109C2×C4×D73—42no—
110#11924012—21no—
111#11924012—22no—
112#11924012—22no—
113#11924012C2×Q812yes—
114#11924011—24no—
115C2×#10596831—21no—
116C2×#10596831—21no—
117C2×#10596831—21no—
118C2×#10596831—22no—
119C2×#10596831C2×D411yescharacteristic
120C2×#10596831C2×D411yescharacteristic
121C8×D73—24no—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124#11924007—42no—
125#11924007—84no—
126#11924005—42no—
127C4×Dic73—21no—
128C4×Dic73—42no—
129C4×Dic73C2×Q812yes—
130C2×Q8×C73—42no—
131C22×C4×C73—22no—
132C22×C4×C73C2411yescharacteristic
133M16×C73—21no—
134M16×C73C2×Q812yes—
135C2×C8×C73—21no—
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73C2×D411yescharacteristic
138C2×C8×C73C2×Q812yes—
139C2×C8×C73#4112yes—
140C4⋊C4×C73—22no—
141C4⋊C4×C73—22no—
142C4⋊C4×C73—42no—
143C4⋊C4×C73—42no—
144C4⋊C4×C73—42no—
145C4⋊C4×C73C2×D411yescharacteristic
146C4⋊C4×C73C2×D411yescharacteristic
147C4⋊C4×C73C2×D412yes—
148C4⋊C4×C73C2×D412yes—
149C22⋊C4×C73—21no—
150C22⋊C4×C73—22no—
151C42×C73—21no—
152C42×C73—22no—
153C23×C73H(5,𝔽2)11yescharacteristic
154C22×D73—21no—
155Q8×C73—162no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—22no—
159C2×C4×C73—22no—
160C2×C4×C73—42no—
161C2×C4×C73—82no—
162C2×C4×C73C22×D411yescharacteristic
163C2×C4×C73C22×D411yescharacteristic
164C2×C4×C73C22×Q811yescharacteristic
165C2×C4×C73C22×Q811yescharacteristic
166C2×C4×C73C2×#4111yescharacteristic
167C2×C4×C732−1+411yescharacteristic
168C22⋊D73—22no—
169C22⋊D73—22no—
170C2×Dic73—22no—
171C2×Dic73—22no—
172C2×Dic73—21no—
173C2×Dic73—22no—
174C2×Dic73—42no—
175C2×Dic73—82no—
176D292—21no—
177D292C4⋊Q812yes—
178C4×D73—22no—
179C4×D73—22no—
180C4×D73—162no—
181Dic146—21no—
182Dic146C4⋊Q812yes—
183C8×C73—22no—
184C8×C73C22⋊Q812yes—
185C8×C73C22⋊Q812yes—
186#10596831—22no—
187#10596831—21no—
188#10596831—21no—
189#10596831—42no—
190C22×C73—21no—
191C22×C73—21no—
192C22×C73#54411yescharacteristic
193C22×C73D4211yescharacteristic
194C22×C73#55411yescharacteristic
195C2×D73—22no—
196C2×D73—41no—
197C2×D73—41no—
198C4×C73—42no—
199C4×C73—41no—
200C4×C73—41no—
201C4×C73—42no—
202C4×C73—82no—
203C4×C73C2×C22⋊Q811yescharacteristic
204C4×C73C2×C4⋊Q811yescharacteristic
205C4×C73#54711yescharacteristic
206C4×C73D4×Q811yescharacteristic
207Dic73—24no—
208Dic73—22no—
209Dic73—82no—
210#26083—731nomaximal, Sylow
211C2×C73—21no—
212C2×C73—21no—
213C2×C73#244011yescharacteristic
214C2×C73#329211yescharacteristic
215C2×C73#329311yescharacteristic
216D73—42no—
217#3164D7311yescharacteristic, complemented
218#3083—732no—
219#2913—731no—
220#2089—732no—
221#2060—732no—
222#1952—731no—
223#1945—731no—
224#1859—732no—
225#1845—731no—
226#1816—731no—
227#1764—731no—
228C73#2608311yescharacteristic, complemented, Sylow
229C2×#131—731no—
230#544C2×D7312yes—
231#520—732no—
232#501—22no—
233#500—21no—
234#499—21no—
235#479—1462no—
236#478—1462no—
237#477—1462no—
238#474—1462no—
239C2×#108—731no—
240#428—732no—
241#428—732no—
242#428—1462no—
243#428C2×D7312yes—
244#427—1461no—
245#427C2×D7311yescharacteristic
246C2×#107—731no—
247C2×#107—732no—
248C2×#107—1461no—
249C2×#107C2×D7311yescharacteristic
250#424—732no—
251C2×#104—731no—
252#417—732no—
253#416—731no—
254#405C2×D7311yescharacteristic
255C2×C4×C8—731no—
256#384—732no—
257#343—732no—
258#343—732no—
259#341—731no—
260#339—731no—
261#339—732no—
262C2×#41—731no—
263#131—734no—
264#131—1462no—
265C2×M16—731no—
266C2×M16—732no—
267C2×M16—1461no—
268C2×M16C22×D7311yescharacteristic
269C22×C8—731no—
270C22×C8—1461no—
271C22×C8C22×D7311yescharacteristic
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#125—22no—
275#125—22no—
276C22⋊Q8—42no—
277C4×Q8—1462no—
278C4×D4—2922no—
279C4×D4—1462no—
280#117—1462no—
281#117C22×D7311yescharacteristic
282#117C22×D7312yes—
283C2×C4⋊C4—2922no—
284C2×C4⋊C4—732no—
285C2×C4⋊C4—1462no—
286C2×C4⋊C4C22×D7312yes—
287C2×C42—731no—
288#108—1462no—
289#107—21no—
290#107—21no—
291#107—22no—
292#107—1462no—
293#107—1462no—
294#107—1461no—
295#107—1461no—
296#107—1462no—
297#107—1462no—
298#107C22⋊D7311yescharacteristic
299#107C22⋊D7311yescharacteristic
300#106—22no—
301#106—1462no—
302#106—1462no—
303#106C22⋊D7312yes—
304C4⋊C8—1462no—
305C4⋊C8—1462no—
306#104—734no—
307#103—731no—
308#103—731no—
309#103—1462no—
310#102—731no—
311#102—731no—
312#102—1462no—
313#97—21no—
314C4×C8—21no—
315C4×C8—1462no—
316#41—732no—
317#41—732no—
318#41—1461no—
319#41—1461no—
320C2×Q8—42no—
321C2×Q8—731no—
322C2×D4—731no—
323C2×D4—1461no—
324C22×C4—22no—
325C22×C4—2922no—
326C22×C4—1462no—
327C22×C4—1461no—
328C22×C4—1461no—
329C22×C4C23×D7311yescharacteristic
330M16—21no—
331M16—732no—
332M16—1462no—
333M16—1462no—
334M16—1461no—
335M16—1464no—
336M16Q8×D7312yes—
337C2×C8—21no—
338C2×C8—731no—
339C2×C8—731no—
340C2×C8—1462no—
341C2×C8—1461no—
342C2×C8—1461no—
343C2×C8—1461no—
344C2×C8—1464no—
345C2×C8C2×C22⋊D7311yescharacteristic
346C2×C8C2×C22⋊D7311yescharacteristic
347C2×C8Q8×D7312yes—
348C2×C8#1192402912yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—2922no—
355C4⋊C4—5844no—
356C4⋊C4—1462no—
357C4⋊C4—1462no—
358C4⋊C4D4×D7311yescharacteristic
359C4⋊C4D4×D7311yescharacteristic
360C4⋊C4C2×C22⋊D7312yes—
361C4⋊C4C2×C22⋊D7312yes—
362C22⋊C4—22no—
363C22⋊C4—21no—
364C22⋊C4—2922no—
365C42—22no—
366C42—21no—
367C42—2922no—
368C42—732no—
369C42—1461no—
370C23—1461no—
371C23#1455663611yescharacteristic
372Q8—162no—
373Q8—732no—
374Q8—1461no—
375D4—732no—
376D4—1462no—
377D4—1462no—
378D4—1461no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—2922no—
386C2×C4—5842no—
387C2×C4—11682no—
388C2×C4—1462no—
389C2×C4—1462no—
390C2×C4—1461no—
391C2×C4—1462no—
392C2×C4—1462no—
393C2×C4—1462no—
394C2×C4C2×D4×D7311yescharacteristic
395C2×C4C22×C22⋊D7311yescharacteristic
396C2×C4C2×Q8×D7311yescharacteristic
397C2×C4C2×Q8×D7311yescharacteristic
398C2×C4C2×#1192402911yescharacteristic
399C2×C4#1455663911yescharacteristic
400C8—22no—
401C8—2922no—
402C8—1461no—
403C8—1461no—
404C8—1462no—
405C8#1455656712yes—
406C8#1455656712yes—
407C22—21no—
408C22—21no—
409C22—2921no—
410C22—2921no—
411C22—1462no—
412C22#1798921411yescharacteristic
413C22D4×C22⋊D7311yescharacteristic
414C22#1798929811yescharacteristic
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—82no—
420C4—5842no—
421C4—1464no—
422C4—1462no—
423C4C2×#1455656711yescharacteristic
424C4C4⋊Q8×D7311yescharacteristic
425C4#1798924011yescharacteristic
426C4Q8×C22⋊D7311yescharacteristic
427C2—21no—
428C2—21no—
429C2—2922no—
430C2#2261362911yescharacteristic
431C2#2262172811yescharacteristic
432C2#2262173311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#3164C2
socleC22×C73D42
prime radicalC2×#11924028C23
hypercenter#3164D73
hypocenterC73#26083
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3164

Complements

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

NH ≅ G/Ntype
3. #22611439C2semidirect
4. #22611273C2semidirect
5. #22610716C2semidirect
6. #22610680C2semidirect
8. #22610089C2semidirect
11. C73×#3164C2semidirect
47. #14556597Q8semidirect
72. #14556517Q8semidirect
217. #3164D73semidirect
228. C73#26083semidirect

Maximal subgroups

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

subgroupindex
#226197782
#226185282
#226114392
#226112732
#226107162
#226106802
#226101542
#226100892
#226099962
#226097702
C73×#31642
#2608373