Finite Group Databasebeta

← #33002931

Subgroups — #33002931

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988829C211yesmaximal, characteristic
2C2×#107×D73C211yesmaximal, characteristic
3C2×#17988392C211yesmaximal, characteristic, complemented
4C2×#17988390C211yesmaximal, characteristic, complemented
5C2×#17988366C211yesmaximal, characteristic
6#22608022C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×C8×D73C2211yescharacteristic
10C2×#14556571—21no—
11C2×C4⋊C4×D73—21no—
12C22×#11924003C2211yescharacteristic
13#17988829—22no—
14#17988829C414yescomplemented
15C2×#14556525C2211yescharacteristic
16C2×#14556524—21no—
17#107×D73—22no—
18#107×D73—22no—
19C2×#14556514—21no—
20#17988392—22no—
21#17988390—22no—
22#17988366—22no—
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#107×C73C2211yescharacteristic
26C22×Dic146—21no—
27C22×C4×D73—41no—
28C22×C4×D73D411yescharacteristic
29C2×Dic292—21no—
30C2×Dic292—21no—
31C2×Dic292—21no—
32C2×Dic292—21no—
33C2×Dic292—21no—
34C2×Dic292C2×C412yescomplemented
35C22×#10596831D411yescharacteristic
36C2×C8×D73—21no—
37C2×C8×D73—21no—
38C2×C8×D73—21no—
39C2×C8×D73—21no—
40C2×C8×D73C2×C412yes—
41C2×C8×D73D412yes—
42#14556571—24no—
43#14556571—42no—
44C2×C4⋊Dic73—21no—
45C2×C4⋊Dic73—41no—
46C2×#11924007—41no—
47C4⋊C4×D73—24no—
48C4⋊C4×D73—42no—
49C2×#11924005—41no—
50C2×#11924003—21no—
51C2×#11924003—21no—
52C2×#11924003—21no—
53#14556525—21no—
54#14556525D412yes—
55#14556524—24no—
56#14556514—24no—
57C22×C8×C73C2311yescharacteristic
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C16×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63#107×C73—21no—
64#107×C73—21no—
65#107×C73C2×C412yescomplemented
66#107×C73D412yes—
67C23×D73D811yescharacteristic
68C2×Dic146—22no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×Dic146—41no—
74C22×Dic73—41no—
75C22×Dic73—41no—
76C22×Dic73D811yescharacteristic
77C2×C4×D73—21no—
78C2×C4×D73—21no—
79C2×C4×D73—21no—
80C2×C4×D73—21no—
81C2×C4×D73—44no—
82C2×C4×D73—82no—
83C2×C4×D73—82no—
84C2×C4×D73C22⋊C412yes—
85C2×C4×D73SD1612yes—
86Dic292—22no—
87Dic292—22no—
88Dic292—41no—
89Dic292—41no—
90C2×#10596831—21no—
91C2×#10596831—21no—
92C2×#10596831C22⋊C412yes—
93C8×D73—22no—
94C8×D73—22no—
95C8×D73—22no—
96C8×D73—22no—
97C4⋊Dic73—22no—
98C4⋊Dic73—41no—
99C4⋊Dic73—44no—
100#11924007—44no—
101#11924005—44no—
102#11924003—41no—
103C22×C4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73C2×D411yescharacteristic
106C2×C8×C73C22×C411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73#4111yescharacteristic
111C2×C8×C73#4111yescharacteristic
112C4⋊C4×C73—22no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—41no—
116C16×C73—41no—
117C23×C73C2×D811yescharacteristic
118C22×D73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73#10212yes—
123C22×D73SD3212yes—
124C22×D73Q3212yes—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—81no—
128C2×C4×C73—81no—
129C2×C4×C73—81no—
130C2×C4×C73—81no—
131C2×C4×C73C2×C22⋊C411yescharacteristic
132C2×C4×C73#12011yescharacteristic
133C2×C4×C73#12311yescharacteristic
134C2×C4×C73C2×SD1611yescharacteristic
135C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C73#13711yescharacteristic
137C2×Dic73—21no—
138C2×Dic73—21no—
139C2×Dic73—42no—
140C2×Dic73—42no—
141C2×Dic73—81no—
142C2×Dic73—81no—
143C2×Dic73—81no—
144C2×Dic73—81no—
145C2×Dic73#10212yes—
146C4×D73—22no—
147C4×D73—22no—
148C4×D73—22no—
149C4×D73—22no—
150C4×D73—88no—
151Dic146—42no—
152Dic146—42no—
153Dic146—81no—
154Dic146—81no—
155C8×C73C4×D411yescharacteristic
156C8×C73C4×D411yescharacteristic
157C8×C73#12111yescharacteristic
158C8×C73#12111yescharacteristic
159#10596831—22no—
160C22×C73C2×#10211yescharacteristic
161C22×C73#44711yescharacteristic
162C22×C73#48011yescharacteristic
163C22×C73C2×SD3211yescharacteristic
164C22×C73C2×Q3211yescharacteristic
165C22×C73#50911yescharacteristic
166C22×C73#51011yescharacteristic
167C2×D73—22no—
168C2×D73—22no—
169C2×D73—22no—
170C2×D73—22no—
171C2×D73—22no—
172C2×D73—22no—
173C2×D73#35814yescomplemented
174C4×C73—82no—
175C4×C73—82no—
176C4×C73#38611yescharacteristic
177C4×C73#41711yescharacteristic
178C4×C73#44811yescharacteristic
179C4×C73#48111yescharacteristic
180Dic73—22no—
181Dic73—82no—
182Dic73—82no—
183#12993—731nomaximal, Sylow
184C2×C73#187411yescharacteristic
185C2×C73C2×#35811yescharacteristic
186C2×C73#213911yescharacteristic
187C2×C73#218711yescharacteristic
188C2×C73#218811yescharacteristic
189C2×C73#223511yescharacteristic
190C2×C73#223611yescharacteristic
191D73—24no—
192C2×#467—731no—
193C22×#107—731no—
194C2×#358—731no—
195C2×#358—731no—
196C2×C22⋊C16—731no—
197#1380—731no—
198#1380D7311yescharacteristic, complemented
199C73#1299311yescharacteristic, complemented, Sylow
200C22×Q16—731no—
201C23×C8—731no—
202C2×C22⋊Q8—1461no—
203C22×C4⋊C4—1461no—
204C22×C16—731no—
205C22×C16C2×D7311yescharacteristic
206#467—734no—
207#467—1462no—
208C2×#107—731no—
209C2×#107—1462no—
210C2×#107—1461no—
211C2×#107—1462no—
212C2×#107C2×D7311yescharacteristic
213C2×#107C2×D7311yescharacteristic
214C2×#103—1461no—
215#358—1462no—
216#358—1462no—
217C22⋊C16—1462no—
218C22×Q8—1461no—
219C23×C4—2921no—
220C23×C4—731no—
221C2×Q16—732no—
222C2×Q16—1461no—
223C2×Q16—1461no—
224C2×Q16—1461no—
225C2×Q16—1461no—
226C2×Q16—1461no—
227C22×C8—731no—
228C22×C8—732no—
229C22×C8—732no—
230C22×C8—1461no—
231C22×C8—1461no—
232C22×C8—1461no—
233C22×C8—1461no—
234C22×C8C22×D7311yescharacteristic
235C22⋊Q8—2922no—
236C22⋊Q8—1464no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—2922no—
240C2×C4⋊C4—2921no—
241C2×C4⋊C4—2921no—
242C2×C4⋊C4—1464no—
243C2×C4⋊C4—1461no—
244C2×C22⋊C4—2921no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1461no—
249C2×C16—1461no—
250C2×C16—1461no—
251#107—21no—
252#107—21no—
253#107—732no—
254#107—1464no—
255#107—1461no—
256#107C4×D7312yescomplemented
257#107D29212yes—
258#103—1464no—
259C24—731no—
260C2×Q8—2921no—
261C2×Q8—2921no—
262C2×Q8—2921no—
263C2×Q8—2921no—
264C2×Q8—2921no—
265C2×Q8—1462no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—2924no—
269C22×C4—2921no—
270C22×C4—2921no—
271C22×C4—5842no—
272C22×C4—5842no—
273C22×C4—731no—
274C22×C4—732no—
275C22×C4—732no—
276C22×C4—1461no—
277C22×C4—1461no—
278C22×C4—1461no—
279C22×C4—1461no—
280C22×C4D4×D7311yescharacteristic
281Q16—2921no—
282Q16—2921no—
283Q16—1462no—
284Q16—1462no—
285C2×C8—732no—
286C2×C8—1461no—
287C2×C8—1461no—
288C2×C8—1462no—
289C2×C8—1462no—
290C2×C8—1462no—
291C2×C8—1462no—
292C2×C8C2×C4×D7311yescharacteristic
293C2×C8D4×D7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8C2×D29211yescharacteristic
296C2×C8#1192402811yescharacteristic
297C2×C8#1192402911yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—2924no—
303C4⋊C4—2924no—
304C4⋊C4—2921no—
305C4⋊C4—1462no—
306C22⋊C4—2924no—
307C16—41no—
308C16—2921no—
309C23—732no—
310C23—732no—
311C23—732no—
312C23—1461no—
313C23—1461no—
314C23—1461no—
315C23—1461no—
316C23D8×D7311yescharacteristic
317Q8—2922no—
318Q8—2922no—
319Q8—5841no—
320Q8—5841no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—2922no—
328C2×C4—2922no—
329C2×C4—5848no—
330C2×C4—5841no—
331C2×C4—5841no—
332C2×C4—5841no—
333C2×C4—5841no—
334C2×C4—732no—
335C2×C4—1461no—
336C2×C4—1461no—
337C2×C4—1462no—
338C2×C4—1462no—
339C2×C4—1462no—
340C2×C4—1462no—
341C2×C4C22⋊C4×D7311yescharacteristic
342C2×C4#1455656111yescharacteristic
343C2×C4#1455657611yescharacteristic
344C2×C4SD16×D7311yescharacteristic
345C2×C4#1455661811yescharacteristic
346C2×C4#1455662311yescharacteristic
347C8—1462no—
348C8#1455655311yescharacteristic
349C8C4×D29211yescharacteristic
350C8#1455656411yescharacteristic
351C8#1455656611yescharacteristic
352C22—734no—
353C22—1462no—
354C22—1462no—
355C22—1462no—
356C22—1462no—
357C22—1462no—
358C22—1462no—
359C22#102×D7311yescharacteristic
360C22#1798872211yescharacteristic
361C22#1798890211yescharacteristic
362C22SD32×D7311yescharacteristic
363C22Q32×D7311yescharacteristic
364C22#1798902111yescharacteristic
365C22#1798902611yescharacteristic
366C4—82no—
367C4—82no—
368C4—5842no—
369C4—5842no—
370C4—1462no—
371C4#1798846511yescharacteristic
372C4#1798858711yescharacteristic
373C4#1798872711yescharacteristic
374C4#1798890711yescharacteristic
375C2—1464no—
376C2#2261024711yescharacteristic
377C2#358×D7311yescharacteristic
378C2#2261164011yescharacteristic
379C2#2261186111yescharacteristic
380C2#2261186611yescharacteristic
381C2#2261213011yescharacteristic
382C2#2261213511yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC23×D73D8
cosocleC22×C8×C73C23
hypercenter#1380D73
hypocenterC73#12993
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#17988392C2semidirect
4. C2×#17988390C2semidirect
6. #22608022C2semidirect
7. C73×#1380C2semidirect
14. #17988829C4semidirect
34. C2×Dic292C2×C4semidirect
65. #107×C73C2×C4semidirect
173. C2×D73#358semidirect
198. #1380D73semidirect
199. C73#12993semidirect
256. #107C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888292
C2×#107×D732
C2×#179883922
C2×#179883902
C2×#179883662
#226080222
C73×#13802
#1299373