Finite Group Databasebeta

← #33002924

Subgroups — #33002924

Order 18688.

582 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988820C211yesmaximal, characteristic
2C2×#17988624C211yesmaximal, characteristic, complemented
3C2×#17988388C211yesmaximal, characteristic, complemented
4C2×#17988366C211yesmaximal, characteristic, complemented
5#22608023C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#357×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556562—21no—
10C2×#14556549—21no—
11C22×#11924003C2211yescharacteristic
12#17988820—22no—
13#17988820C414yescomplemented
14#17988624—22no—
15#17988624—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556522—21no—
19C2×#14556512—21no—
20#17988388—22no—
21#17988366—22no—
22C22×D8×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#357×C73—22no—
26C22×C22⋊D73—41no—
27C22×C4×D73D411yescharacteristic
28C22×#10596831D411yescharacteristic
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73—21no—
33C2×C8×D73C2×C412yes—
34C2×C8×D73D412yes—
35#14556562—24no—
36#14556562—42no—
37#14556549—24no—
38#14556549—42no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—21no—
41C2×C22⋊Dic73—41no—
42C2×#11924005—41no—
43C2×C4×Dic73—41no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47#14556525—21no—
48#14556525—21no—
49#14556525C2×C412yes—
50#14556525D412yescomplemented
51#14556522—24no—
52#14556512—24no—
53C22×D4×C73—21no—
54C2×D8×C73—21no—
55C2×D8×C73—21no—
56C2×D8×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73—21no—
59C2×D8×C73C2×C412yes—
60C22×C8×C73C2311yescharacteristic
61C2×C4⋊C4×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64C2×C16×C73—21no—
65#107×C73—21no—
66#107×C73D412yescomplemented
67C23×D73D811yescharacteristic
68C2×C22⋊D73—44no—
69C2×C22⋊D73—82no—
70C2×C22⋊D73—82no—
71C22×Dic73—41no—
72C22×Dic73—41no—
73C22×Dic73D811yescharacteristic
74C2×C4×D73—21no—
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77C2×C4×D73—21no—
78C2×C4×D73C22⋊C412yes—
79C2×C4×D73SD1612yes—
80C2×#10596831—21no—
81C2×#10596831—21no—
82C2×#10596831C22⋊C412yes—
83C8×D73—22no—
84C8×D73—22no—
85C8×D73—22no—
86C8×D73—22no—
87C4⋊Dic73—22no—
88C4⋊Dic73—22no—
89C4⋊Dic73—41no—
90C4⋊Dic73—41no—
91C22⋊Dic73—44no—
92#11924005—44no—
93C4×Dic73—44no—
94#11924003—41no—
95C24×C73—41no—
96C2×D4×C73—22no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73C2×D411yescharacteristic
104D8×C73—22no—
105D8×C73—22no—
106D8×C73—41no—
107D8×C73—41no—
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C4⋊C4×C73—22no—
115C4⋊C4×C73—41no—
116C16×C73—41no—
117C23×C73—42no—
118C23×C73—81no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73—81no—
124C23×C73C2×D811yescharacteristic
125C22×D73—21no—
126C22×D73—21no—
127C22×D73—21no—
128C22×D73—21no—
129C22×D73—21no—
130C22×D73—21no—
131C22×D73#10212yes—
132D4×C73—42no—
133D4×C73—42no—
134D4×C73—81no—
135D4×C73—81no—
136C2×C4×C73—42no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73C2×C22⋊C411yescharacteristic
140C2×C4×C73#12011yescharacteristic
141C2×C4×C73#12311yescharacteristic
142C2×C4×C73C2×SD1611yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C73#13711yescharacteristic
145C22⋊D73—88no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—42no—
149C2×Dic73—42no—
150C2×Dic73—81no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154C2×Dic73#10212yes—
155C4×D73—22no—
156C4×D73—22no—
157C4×D73—22no—
158C4×D73—22no—
159C8×C73C4×D411yescharacteristic
160C8×C73C4×D411yescharacteristic
161C8×C73#12111yescharacteristic
162C8×C73#12111yescharacteristic
163#10596831—22no—
164C22×C73—82no—
165C22×C73—82no—
166C22×C73—82no—
167C22×C73—161no—
168C22×C73—161no—
169C22×C73—161no—
170C22×C73—161no—
171C22×C73C2×#10211yescharacteristic
172C22×C73#44711yescharacteristic
173C22×C73#48011yescharacteristic
174C22×C73#50811yescharacteristic
175C22×C73#50811yescharacteristic
176C22×C73#50911yescharacteristic
177C22×C73#50911yescharacteristic
178C2×D73—22no—
179C2×D73—22no—
180C2×D73—22no—
181C2×D73—41no—
182C2×D73—41no—
183C2×D73—41no—
184C2×D73—41no—
185C4×C73—82no—
186C4×C73#38611yescharacteristic
187C4×C73#41711yescharacteristic
188C4×C73#44811yescharacteristic
189C4×C73#48111yescharacteristic
190Dic73—22no—
191Dic73—82no—
192Dic73—82no—
193#12992—731nomaximal, Sylow
194C2×C73—162no—
195C2×C73#187411yescharacteristic
196C2×C73#213811yescharacteristic
197C2×C73#214011yescharacteristic
198C2×C73#218611yescharacteristic
199C2×C73#218611yescharacteristic
200C2×C73#223411yescharacteristic
201C2×C73#223411yescharacteristic
202D73—42no—
203C2×#466—731no—
204C2×#427—731no—
205C2×#357—731no—
206C2×#357D7311yescharacteristic, complemented
207C2×C22⋊C16—731no—
208#1380—731no—
209#1380—731no—
210C73#1299211yescharacteristic, complemented, Sylow
211C22×D8C2×D7311yescharacteristic
212C23×C8—731no—
213C2×#121—1461no—
214C2×#117—1461no—
215C22×C16—731no—
216C22×C16C2×D7311yescharacteristic
217#466—734no—
218#466—1462no—
219#427—1462no—
220#427—1462no—
221C2×#107—731no—
222C2×#107—731no—
223C2×#107C2×D7311yescharacteristic
224C2×#106—1461no—
225C2×#102—1461no—
226#357—22no—
227#357—1462no—
228C22⋊C16—1462no—
229C22×D4—21no—
230C22×D4—2921no—
231C23×C4—731no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8C4×D7312yes—
238C22×C8—731no—
239C22×C8—732no—
240C22×C8—732no—
241C22×C8—1461no—
242C22×C8—1461no—
243C22×C8—1461no—
244C22×C8—1461no—
245C22×C8C22×D7311yescharacteristic
246#121—2922no—
247#121—1464no—
248#117—2922no—
249#117—1464no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1461no—
252C2×C4⋊C4—1461no—
253C2×C22⋊C4—2921no—
254C2×C22⋊C4—2921no—
255C2×C42—2921no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1461no—
260C2×C16—1461no—
261C2×C16—1461no—
262#107—21no—
263#107—732no—
264#107—732no—
265#107—1461no—
266#107—1461no—
267#107C22⋊D7312yescomplemented
268#106—1464no—
269#102—1464no—
270C24—41no—
271C24—731no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—2924no—
279C2×D4—5842no—
280C2×D4—5842no—
281C22×C4—41no—
282C22×C4—2921no—
283C22×C4—2921no—
284C22×C4—731no—
285C22×C4—732no—
286C22×C4—732no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4D4×D7311yescharacteristic
292D8—22no—
293D8—22no—
294D8—41no—
295D8—41no—
296C2×C8—732no—
297C2×C8—1461no—
298C2×C8—1461no—
299C2×C8—1462no—
300C2×C8—1462no—
301C2×C8—1462no—
302C2×C8—1462no—
303C2×C8C2×C4×D7311yescharacteristic
304C2×C8D4×D7311yescharacteristic
305C2×C8D4×D7311yescharacteristic
306C2×C8C2×C22⋊D7311yescharacteristic
307C2×C8#1192402711yescharacteristic
308C2×C8#1192402811yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—2921no—
312C4⋊C4—2921no—
313C4⋊C4—1462no—
314C4⋊C4—1462no—
315C22⋊C4—2924no—
316C22⋊C4—2924no—
317C42—2924no—
318C16—41no—
319C16—2921no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—732no—
328C23—1461no—
329C23—1461no—
330C23—1461no—
331C23—1461no—
332C23—1461no—
333C23—1461no—
334C23D8×D7311yescharacteristic
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—5848no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—5841no—
346C2×C4—5841no—
347C2×C4—5841no—
348C2×C4—5841no—
349C2×C4—732no—
350C2×C4—1461no—
351C2×C4—1461no—
352C2×C4—1462no—
353C2×C4—1462no—
354C2×C4—1462no—
355C2×C4—1462no—
356C2×C4C22⋊C4×D7311yescharacteristic
357C2×C4#1455655911yescharacteristic
358C2×C4#1455657311yescharacteristic
359C2×C4SD16×D7311yescharacteristic
360C2×C4#1455661511yescharacteristic
361C2×C4#1455662011yescharacteristic
362C8—1462no—
363C8#1455655111yescharacteristic
364C8C4×C22⋊D7311yescharacteristic
365C8#1455656211yescharacteristic
366C8#1455656411yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—2921no—
375C22—2921no—
376C22—2921no—
377C22—2921no—
378C22—1462no—
379C22—1462no—
380C22—1462no—
381C22#102×D7311yescharacteristic
382C22#1798871911yescharacteristic
383C22#1798889911yescharacteristic
384C22#1798901611yescharacteristic
385C22#1798901811yescharacteristic
386C22#1798902111yescharacteristic
387C22#1798902411yescharacteristic
388C4—82no—
389C4—5842no—
390C4—5842no—
391C4—1462no—
392C4#1798846311yescharacteristic
393C4#1798858511yescharacteristic
394C4#1798872411yescharacteristic
395C4#1798890411yescharacteristic
396C2—162no—
397C2—2922no—
398C2#2261024511yescharacteristic
399C2#2261163511yescharacteristic
400C2#2261164511yescharacteristic
401C2#2261185111yescharacteristic
402C2#2261185311yescharacteristic
403C2#2261212011yescharacteristic
404C2#2261212111yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#357×C73C2
socleC23×C73C2×D8
prime radicalC2×#17988820C2
cosocleC22×C8×C73C23
hypercenterC2×#357D73
hypocenterC73#12992
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#17988624C2semidirect
3. C2×#17988388C2semidirect
4. C2×#17988366C2semidirect
5. #22608023C2semidirect
6. #22608022C2semidirect
7. C2×#357×C73C2semidirect
13. #17988820C4semidirect
50. #14556525D4semidirect
66. #107×C73D4semidirect
206. C2×#357D73semidirect
210. C73#12992semidirect
267. #107C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888202
C2×#179886242
C2×#179883882
C2×#179883662
#226080232
#226080222
C2×#357×C732
#1299273