Finite Group Databasebeta

← #32980791

Subgroups — #32980791

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988864C211yesmaximal, characteristic
2#22612882C211yesmaximal, characteristic
3#22610855C211yesmaximal, characteristic, complemented
4C2×#17988316C211yesmaximal, characteristic
5#22607814C211yesmaximal, characteristic, complemented
6#22607771C211yesmaximal, characteristic, complemented
7C73×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9C2×#14556549—21no—
10#17988864—22no—
11#17988864C414yescomplemented
12C2×#14556524—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×C4⋊#10596831C2211yescharacteristic
15C2×#14556515—21no—
16C2×#14556515—21no—
17C2×#14556511—21no—
18#17988488—21no—
19#17988477—22no—
20#17988444—21no—
21C4⋊C4×Dic73—21no—
22C4×C4⋊Dic73C2211yescharacteristic
23#17988316—22no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26#384×C73C2211yescharacteristic
27#340×C73—22no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31C22×#10596831—21no—
32C4×D292—21no—
33C4×D292—21no—
34C4×D292C2×C412yes—
35C4×D292D412yes—
36#14556549—42no—
37#14556549—42no—
38C2×C4⋊Dic73D411yescharacteristic
39C2×C4⋊Dic73D411yescharacteristic
40C2×C4⋊Dic73D411yescharacteristic
41C2×#11924005—22no—
42C2×#11924005—22no—
43C2×#11924005—21no—
44C2×C4×Dic73—22no—
45C2×C4×Dic73—21no—
46#14556524—24no—
47C4⋊#10596831—21no—
48C4⋊#10596831—21no—
49C4⋊#10596831—21no—
50C4⋊#10596831C2×C412yes—
51#14556515—42no—
52#14556515—42no—
53#14556511—24no—
54#14556498—22no—
55#14556498—21no—
56C22×Q8×C73—21no—
57C22×C8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73C2×C412yes—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73D412yes—
67C2×C42×C73C2311yescharacteristic
68C4×C8×C73—21no—
69C4×C8×C73—21no—
70C23×D73—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292C22⋊C412yes—
77C22×Dic73—21no—
78C22×Dic73—21no—
79C22×Dic73—21no—
80C2×C4×D73—22no—
81C2×C4×D73—22no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×#10596831—22no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C4⋊Dic73—21no—
93C4⋊Dic73—21no—
94C4⋊Dic73—21no—
95C4⋊Dic73—21no—
96C4⋊Dic73C22⋊C412yes—
97C4⋊Dic73D812yes—
98#11924005—22no—
99#11924005—22no—
100#11924005—44no—
101#11924005—82no—
102C4×Dic73—44no—
103C4×Dic73—41no—
104C4×Dic73—41no—
105C2×Q8×C73—22no—
106C2×Q8×C73—41no—
107C2×Q8×C73—41no—
108C2×Q8×C73—41no—
109C2×Q8×C73—41no—
110C2×Q8×C73—41no—
111C22×C4×C73—22no—
112C22×C4×C73—41no—
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73C2×D411yescharacteristic
115C22×C4×C73C2×D411yescharacteristic
116C2×C8×C73—22no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—42no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C4⋊C4×C73—41no—
128C42×C73C22×C411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73#4111yescharacteristic
132C23×C73#12011yescharacteristic
133C22×D73—22no—
134C22×D73—41no—
135C22×D73—41no—
136C22×D73—41no—
137C22×D73—41no—
138C22×D73—41no—
139C22×D73—41no—
140Q8×C73—42no—
141Q8×C73—42no—
142Q8×C73—81no—
143Q8×C73—81no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73—42no—
149C2×C4×C73—42no—
150C2×C4×C73—81no—
151C2×C4×C73—81no—
152C2×C4×C73C2×C22⋊C411yescharacteristic
153C2×C4×C73C4×D411yescharacteristic
154C2×C4×C73C4×D411yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#12311yescharacteristic
158C2×C4×C73C2×D811yescharacteristic
159C2×C4×C73#13511yescharacteristic
160C2×C4×C73#13511yescharacteristic
161C2×C4×C73#13511yescharacteristic
162C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C73#13711yescharacteristic
165C2×C4×C73#13711yescharacteristic
166C2×Dic73—22no—
167C2×Dic73—22no—
168C2×Dic73—41no—
169C2×Dic73—41no—
170C2×Dic73—41no—
171C2×Dic73—41no—
172C2×Dic73—41no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175D292—22no—
176D292—22no—
177D292—41no—
178D292—41no—
179C4×D73—42no—
180C4×D73—42no—
181C4×D73—42no—
182C4×D73—42no—
183C8×C73—42no—
184#10596831—42no—
185#10596831—81no—
186C22×C73#38611yescharacteristic
187C22×C73#44711yescharacteristic
188C22×C73#44911yescharacteristic
189C22×C73#45211yescharacteristic
190C22×C73#45211yescharacteristic
191C22×C73#45511yescharacteristic
192C22×C73#45611yescharacteristic
193C2×D73—42no—
194C2×D73—42no—
195C2×D73—42no—
196C2×D73—81no—
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—81no—
200C4×C73—21no—
201C4×C73—21no—
202C4×C73—82no—
203C4×C73—82no—
204C4×C73#41611yescharacteristic
205C4×C73#41911yescharacteristic
206C4×C73C4×D811yescharacteristic
207C4×C73#44211yescharacteristic
208C4×C73#45911yescharacteristic
209C4×C73#46311yescharacteristic
210C4×C73#48511yescharacteristic
211C4×C73#48511yescharacteristic
212Dic73—42no—
213Dic73—42no—
214Dic73—81no—
215#8984—731nomaximal, Sylow
216C2×C73#164111yescharacteristic
217C2×C73#164611yescharacteristic
218C2×C73#164711yescharacteristic
219C2×C73#165211yescharacteristic
220C2×C73#187311yescharacteristic
221C2×C73#187911yescharacteristic
222C2×C73#188211yescharacteristic
223D73—82no—
224C2×#474—731no—
225#2354—731no—
226#1990D7311yescharacteristic, complemented
227#1981—731no—
228C2×D4⋊C8—731no—
229#1296—731no—
230#1279—731no—
231C73#898411yescharacteristic, complemented, Sylow
232C2×C4⋊Q8C2×D7311yescharacteristic
233C2×C4×D4—731no—
234C2×#117—1461no—
235#474—734no—
236#474—1462no—
237C2×#107—1461no—
238C2×C4⋊C8—731no—
239C2×C4⋊C8—731no—
240C2×#103—1461no—
241C2×#103—1461no—
242C2×#102—1461no—
243C2×C4×C8C2×D7311yescharacteristic
244#390—1461no—
245#388—1462no—
246#384C2×D7311yescharacteristic
247#380—1461no—
248C4×C4⋊C4—731no—
249C4×C4⋊C4—1461no—
250#340—22no—
251D4⋊C8—1462no—
252C22×Q8—21no—
253C22×D4—731no—
254C23×C4—1461no—
255C22×C8—21no—
256C22×C8—1461no—
257C22×C8—1461no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8C4×D7312yes—
262C4×D4—732no—
263C4×D4—732no—
264C4×D4—1461no—
265C4×D4—1461no—
266#117—2922no—
267#117—2922no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—731no—
273C2×C4⋊C4—731no—
274C2×C4⋊C4—731no—
275C2×C4⋊C4C22⋊D7312yes—
276C2×C22⋊C4—1462no—
277C2×C22⋊C4—1462no—
278C2×C22⋊C4—1461no—
279C2×C42—1462no—
280C2×C42—1461no—
281C2×C42C22×D7311yescharacteristic
282#107—1464no—
283C4⋊C8—732no—
284C4⋊C8—1461no—
285C4⋊C8—1461no—
286C4⋊C8—1461no—
287#103—2922no—
288#103—2922no—
289#102—1464no—
290C4×C8—21no—
291C4×C8—21no—
292#95—1462no—
293#95—1461no—
294C24—1461no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—732no—
302C2×D4—1461no—
303C2×D4—1461no—
304C2×D4—1461no—
305C2×D4—1461no—
306C2×D4—1461no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—2921no—
310C22×C4—2921no—
311C22×C4—2921no—
312C22×C4—2921no—
313C22×C4—1461no—
314C22×C4—1461no—
315C22×C4—1461no—
316C22×C4—1462no—
317C22×C4—1462no—
318C22×C4D4×D7311yescharacteristic
319C22×C4D4×D7311yescharacteristic
320C22×C4C2×C22⋊D7311yescharacteristic
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—2921no—
325C2×C8—2921no—
326C2×C8—2921no—
327C2×C8—2921no—
328C2×C8—2921no—
329C2×C8—1462no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—732no—
340C4⋊C4—732no—
341C4⋊C4—1461no—
342C4⋊C4—1461no—
343C4⋊C4—1461no—
344C4⋊C4—1461no—
345C22⋊C4—2924no—
346C22⋊C4—5842no—
347C22⋊C4—1462no—
348C22⋊C4—1462no—
349C42—2924no—
350C42—2921no—
351C42—2921no—
352C42C2×C4×D7311yescharacteristic
353C42D4×D7311yescharacteristic
354C42#1192402711yescharacteristic
355C42#1192402811yescharacteristic
356C23—2921no—
357C23—2921no—
358C23—2921no—
359C23—2921no—
360C23—2921no—
361C23—2921no—
362C23—1462no—
363C23#1455655911yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—81no—
367Q8—81no—
368D4—2921no—
369D4—2921no—
370D4—1462no—
371D4—1462no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—2921no—
381C2×C4—2921no—
382C2×C4—2921no—
383C2×C4—2921no—
384C2×C4—2922no—
385C2×C4—2922no—
386C2×C4—2922no—
387C2×C4—2922no—
388C2×C4—2921no—
389C2×C4—2921no—
390C2×C4—2921no—
391C2×C4—1462no—
392C2×C4—1462no—
393C2×C4C22⋊C4×D7311yescharacteristic
394C2×C4#1455655111yescharacteristic
395C2×C4C4×C22⋊D7311yescharacteristic
396C2×C4#1455656211yescharacteristic
397C2×C4#1455656411yescharacteristic
398C2×C4#1455657311yescharacteristic
399C2×C4D8×D7311yescharacteristic
400C2×C4#1455661211yescharacteristic
401C2×C4#1455661211yescharacteristic
402C2×C4#1455661411yescharacteristic
403C2×C4#1455661511yescharacteristic
404C2×C4#1455661811yescharacteristic
405C2×C4#1455662011yescharacteristic
406C2×C4#1455662311yescharacteristic
407C8—42no—
408C8—2922no—
409C8—5841no—
410C22—2922no—
411C22—2922no—
412C22—2922no—
413C22—5841no—
414C22—5841no—
415C22—5841no—
416C22—5841no—
417C22#1798846311yescharacteristic
418C22#1798871911yescharacteristic
419C22#1798873111yescharacteristic
420C22#1798874611yescharacteristic
421C22#1798874811yescharacteristic
422C22#1798876411yescharacteristic
423C22#1798876911yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—82no—
428C4—2922no—
429C4—2922no—
430C4—5841no—
431C4#1798858211yescharacteristic
432C4#1798859411yescharacteristic
433C4#1798866711yescharacteristic
434C4#1798869611yescharacteristic
435C4#1798878111yescharacteristic
436C4#1798880111yescharacteristic
437C4#1798892611yescharacteristic
438C4#1798892711yescharacteristic
439C2—5842no—
440C2#2260912111yescharacteristic
441C2#2260915611yescharacteristic
442C2#2260916311yescharacteristic
443C2#2260919811yescharacteristic
444C2#2261024011yescharacteristic
445C2#2261027611yescharacteristic
446C2#2261030311yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1990C2
socleC23×C73#120
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1990D73
hypocenterC73#8984
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #22610855C2semidirect
5. #22607814C2semidirect
6. #22607771C2semidirect
7. C73×#1990C2semidirect
11. #17988864C4semidirect
226. #1990D73semidirect
231. C73#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
#226128822
#226108552
C2×#179883162
#226078142
#226077712
C73×#19902
#898473