Finite Group Databasebeta

← #32982154

Subgroups — #32982154

Order 18688.

594 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×#17988864C211yesmaximal, characteristic
2C2×C8×D292C211yesmaximal, characteristic
3#22609928C211yesmaximal, characteristic
4C2×#17988333C211yesmaximal, characteristic, complemented
5#22607814C211yesmaximal, characteristic, complemented
6#22607766C211yesmaximal, characteristic, complemented
7C73×#1956C211yesmaximal, characteristic, complemented
8C22×C8×D73—21no—
9C2×C4×D292C2211yescharacteristic
10#17988864—22no—
11#17988864C414yescomplemented
12C8×D292—22no—
13C8×D292—22no—
14C2×#14556524—21no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×C4⋊#10596831C2211yescharacteristic
17C2×C4⋊#10596831C2211yescharacteristic
18C2×#14556511—21no—
19C2×#14556511—21no—
20C2×#14556503—21no—
21#17988349—21no—
22#17988333—22no—
23C2×C4⋊Q8×C73C2211yescharacteristic
24C2×#103×C73—21no—
25C2×#103×C73—21no—
26C2×C4×C8×C73C2211yescharacteristic
27#384×C73C2211yescharacteristic
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31C22×#10596831—21no—
32C22×#10596831—21no—
33C2×C8×D73—24no—
34C2×C8×D73—42no—
35C2×C8×D73—42no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292C2×C412yes—
39C4×D292D412yes—
40C2×C4⋊Dic73D411yescharacteristic
41C2×#11924005—21no—
42#14556524—24no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831—21no—
46C4⋊#10596831—21no—
47C4⋊#10596831C2×C412yes—
48C4⋊#10596831D412yes—
49#14556511—24no—
50#14556511—42no—
51#14556503—24no—
52C22×Q8×C73—21no—
53C22×C8×C73D411yescharacteristic
54C22×C8×C73D411yescharacteristic
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73C2×C412yes—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65#103×C73—42no—
66#103×C73—42no—
67C4×C8×C73—21no—
68C4×C8×C73—21no—
69C23×D73—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292C22⋊C412yes—
74C2×D292SD1612yes—
75C2×D292Q1612yes—
76C22×Dic73—21no—
77C2×C4×D73—22no—
78C2×C4×D73—22no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×#10596831—22no—
84C2×#10596831—22no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C8×D73—48no—
93C4⋊Dic73—21no—
94C4⋊Dic73C22⋊C412yes—
95#11924005—22no—
96#11924005—22no—
97C2×Q8×C73—22no—
98C2×Q8×C73—41no—
99C2×Q8×C73—41no—
100C2×Q8×C73—41no—
101C2×Q8×C73—41no—
102C2×Q8×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73#4111yescharacteristic
108C2×C8×C73—21no—
109C2×C8×C73—21no—
110C2×C8×C73—21no—
111C2×C8×C73—21no—
112C2×C8×C73—21no—
113C2×C8×C73D812yes—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73—81no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73#12111yescharacteristic
128C22×D73—22no—
129C22×D73—22no—
130C22×D73—22no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135Q8×C73—42no—
136Q8×C73—42no—
137Q8×C73—81no—
138Q8×C73—81no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—42no—
142C2×C4×C73—81no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73—81no—
146C2×C4×C73—81no—
147C2×C4×C73C2×C22⋊C411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73C4×D411yescharacteristic
150C2×C4×C73#12011yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12311yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73C2×SD1611yescharacteristic
155C2×C4×C73C2×Q1611yescharacteristic
156C2×C4×C73#13511yescharacteristic
157C2×C4×C73#13511yescharacteristic
158C2×C4×C73#13511yescharacteristic
159C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
161C2×Dic73—22no—
162C2×Dic73—41no—
163C2×Dic73—41no—
164D292—22no—
165D292—22no—
166D292—22no—
167D292#10314yescomplemented
168C4×D73—42no—
169C4×D73—42no—
170C4×D73—42no—
171C4×D73—42no—
172C8×C73—22no—
173C8×C73—41no—
174#10596831—42no—
175#10596831—42no—
176#10596831—81no—
177C22×C73#38611yescharacteristic
178C22×C73#45911yescharacteristic
179C22×C73#46311yescharacteristic
180C22×C73#46511yescharacteristic
181C22×C73#46711yescharacteristic
182C22×C73#47111yescharacteristic
183C22×C73#47311yescharacteristic
184C2×D73—24no—
185C2×D73—42no—
186C2×D73—42no—
187C2×D73—42no—
188C2×D73—42no—
189C2×D73—42no—
190C2×D73—42no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—82no—
194C4×C73—161no—
195C4×C73C2×#10311yescharacteristic
196C4×C73#41611yescharacteristic
197C4×C73C4×D811yescharacteristic
198C4×C73#44211yescharacteristic
199C4×C73#44711yescharacteristic
200C4×C73#44911yescharacteristic
201C4×C73#46511yescharacteristic
202C4×C73#46711yescharacteristic
203Dic73—42no—
204#9219—731nomaximal, Sylow
205C2×C73#166511yescharacteristic
206C2×C73#166911yescharacteristic
207C2×C73#167911yescharacteristic
208C2×C73#168311yescharacteristic
209C2×C73#187911yescharacteristic
210C2×C73#189211yescharacteristic
211C2×C73#190811yescharacteristic
212D73—44no—
213C2×#474—731no—
214C2×C8×D4—731no—
215#1956D7311yescharacteristic, complemented
216#1802—731no—
217C2×#332—731no—
218#1296—731no—
219#1277—731no—
220C73#921911yescharacteristic, complemented, Sylow
221C23×C8—1461no—
222C2×C4⋊Q8C2×D7311yescharacteristic
223C2×C4×D4—731no—
224#474—734no—
225#474—1462no—
226C8×D4—1462no—
227C8×D4—1462no—
228C2×#107—1461no—
229C2×C4⋊C8—731no—
230C2×C4⋊C8—731no—
231C2×C4⋊C8—731no—
232C2×#103—21no—
233C2×#103—21no—
234C2×#102—1461no—
235C2×#102—1461no—
236C2×C22⋊C8—1461no—
237C2×C4×C8C2×D7311yescharacteristic
238#384C2×D7311yescharacteristic
239#340—1461no—
240#332—1462no—
241C22×Q8—21no—
242C22×D4—731no—
243C23×C4—1461no—
244C22×C8—2922no—
245C22×C8—2922no—
246C22×C8—1464no—
247C22×C8—1461no—
248C22×C8—1461no—
249C22×C8—1461no—
250C22×C8C22⋊D7311yescharacteristic
251C22×C8C22⋊D7311yescharacteristic
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C4×D7312yes—
256C4×D4—732no—
257C4×D4—732no—
258C4×D4—1461no—
259C4×D4—1461no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—731no—
266C2×C22⋊C4—1461no—
267C2×C42C22×D7311yescharacteristic
268#107—1464no—
269C4⋊C8—732no—
270C4⋊C8—732no—
271C4⋊C8—1461no—
272C4⋊C8—1461no—
273C4⋊C8—1461no—
274C4⋊C8—1461no—
275#103—42no—
276#103—42no—
277#102—2922no—
278#102—1464no—
279C22⋊C8—1464no—
280C4×C8—21no—
281C4×C8—21no—
282C24—1461no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—732no—
290C2×D4—732no—
291C2×D4—732no—
292C2×D4—1461no—
293C2×D4—1461no—
294C2×D4—1461no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—2921no—
298C22×C4—2921no—
299C22×C4—2921no—
300C22×C4—2921no—
301C22×C4—1461no—
302C22×C4—1462no—
303C22×C4—1462no—
304C22×C4D4×D7311yescharacteristic
305C22×C4C2×C22⋊D7311yescharacteristic
306C22×C4#1192402711yescharacteristic
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—2928no—
313C2×C8—2921no—
314C2×C8—2921no—
315C2×C8—2921no—
316C2×C8—2921no—
317C2×C8—2921no—
318C2×C8—2921no—
319C2×C8—2921no—
320C2×C8—1462no—
321C2×C8—1462no—
322C2×C8#1192401312yes—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—81no—
332C4⋊C4—732no—
333C4⋊C4—1461no—
334C22⋊C4—1462no—
335C22⋊C4—1462no—
336C42C2×C4×D7311yescharacteristic
337C42D4×D7311yescharacteristic
338C42D4×D7311yescharacteristic
339C42#1192402811yescharacteristic
340C23—2921no—
341C23—2921no—
342C23—2921no—
343C23—2921no—
344C23—1462no—
345C23—1462no—
346C23—1462no—
347C23#1455656211yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—734no—
353D4—1462no—
354D4—1462no—
355D4—1462no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—2921no—
365C2×C4—2921no—
366C2×C4—2922no—
367C2×C4—2922no—
368C2×C4—2922no—
369C2×C4—2922no—
370C2×C4—1462no—
371C2×C4C22⋊C4×D7311yescharacteristic
372C2×C4#1455655111yescharacteristic
373C2×C4C4×C22⋊D7311yescharacteristic
374C2×C4#1455655911yescharacteristic
375C2×C4#1455656411yescharacteristic
376C2×C4#1455657311yescharacteristic
377C2×C4C2×#1192401311yescharacteristic
378C2×C4SD16×D7311yescharacteristic
379C2×C4Q16×D7311yescharacteristic
380C2×C4#1455661111yescharacteristic
381C2×C4#1455661211yescharacteristic
382C2×C4#1455661411yescharacteristic
383C2×C4#1455661611yescharacteristic
384C2×C4#1455661711yescharacteristic
385C8—22no—
386C8—41no—
387C8—2922no—
388C8—2922no—
389C8—5841no—
390C22—2922no—
391C22—2922no—
392C22—2922no—
393C22—2922no—
394C22—2922no—
395C22—2922no—
396C22—1464no—
397C22#1798846311yescharacteristic
398C22#1798877911yescharacteristic
399C22#1798879911yescharacteristic
400C22#1798881311yescharacteristic
401C22#1798882511yescharacteristic
402C22#1798884711yescharacteristic
403C22#1798885911yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—161no—
408C4—2922no—
409C4#103×D7311yescharacteristic
410C4#1798858211yescharacteristic
411C4C4×#1192401311yescharacteristic
412C4#1798869711yescharacteristic
413C4#1798872011yescharacteristic
414C4#1798873011yescharacteristic
415C4#1798881911yescharacteristic
416C4#1798882711yescharacteristic
417C2—2924no—
418C2#2260926911yescharacteristic
419C2#2260929311yescharacteristic
420C2#2260935111yescharacteristic
421C2#2260937911yescharacteristic
422C2#2261028211yescharacteristic
423C2#2261035111yescharacteristic
424C2#2261044911yescharacteristic
subgroupG/H
centerC23#14556562
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1956C2
socleC23×C73#121
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1956D73
hypocenterC73#9219
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
4. C2×#17988333C2semidirect
5. #22607814C2semidirect
6. #22607766C2semidirect
7. C73×#1956C2semidirect
11. #17988864C4semidirect
167. D292#103semidirect
215. #1956D73semidirect
220. C73#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
C2×C8×D2922
#226099282
C2×#179883332
#226078142
#226077662
C73×#19562
#921973