Finite Group Databasebeta

← #33002934

Subgroups — #33002934

Order 18688.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988825C211yesmaximal, characteristic
2C2×#17988624C211yesmaximal, characteristic
3C2×#17988391C211yesmaximal, characteristic, complemented
4C2×#17988366C211yesmaximal, characteristic
5#22608023C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#358×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556567—21no—
10C2×#14556549—21no—
11C22×#11924003C2211yescharacteristic
12#17988825—22no—
13#17988825C414yescomplemented
14#17988624—22no—
15#17988624—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556522—21no—
19C2×#14556515—21no—
20#17988391—22no—
21#17988366—22no—
22C22×Q16×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#358×C73—22no—
26C22×C4×D73D411yescharacteristic
27C22×#10596831D411yescharacteristic
28C2×C8×D73—21no—
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73C2×C412yes—
33C2×C8×D73D412yes—
34#14556567—24no—
35#14556567—42no—
36#14556549—24no—
37#14556549—42no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73—21no—
40C2×#11924007—41no—
41C2×#11924005—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#14556525D412yes—
51#14556522—24no—
52#14556515—24no—
53C22×Q8×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73—21no—
58C2×Q16×C73—21no—
59C2×Q16×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×C73D412yes—
67C23×D73D811yescharacteristic
68C22×Dic73—41no—
69C22×Dic73—41no—
70C22×Dic73D811yescharacteristic
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73—21no—
74C2×C4×D73—21no—
75C2×C4×D73C22⋊C412yes—
76C2×C4×D73SD1612yes—
77C2×#10596831—21no—
78C2×#10596831—21no—
79C2×#10596831C22⋊C412yes—
80C8×D73—22no—
81C8×D73—22no—
82C8×D73—22no—
83C8×D73—22no—
84C4⋊Dic73—22no—
85C4⋊Dic73—22no—
86C4⋊Dic73—41no—
87C4⋊Dic73—41no—
88#11924007—44no—
89#11924005—44no—
90#11924005—44no—
91C4×Dic73—44no—
92#11924003—41no—
93C2×Q8×C73—22no—
94C2×Q8×C73—41no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×Q8×C73—41no—
98C2×Q8×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73C2×D411yescharacteristic
102Q16×C73—22no—
103Q16×C73—22no—
104Q16×C73—41no—
105Q16×C73—41no—
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—41no—
114C16×C73—41no—
115C23×C73C2×D811yescharacteristic
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73#10212yes—
123Q8×C73—42no—
124Q8×C73—42no—
125Q8×C73—81no—
126Q8×C73—81no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73—81no—
130C2×C4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73—81no—
133C2×C4×C73C2×C22⋊C411yescharacteristic
134C2×C4×C73#12011yescharacteristic
135C2×C4×C73#12311yescharacteristic
136C2×C4×C73C2×SD1611yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C73#13711yescharacteristic
139C2×Dic73—21no—
140C2×Dic73—21no—
141C2×Dic73—42no—
142C2×Dic73—42no—
143C2×Dic73—81no—
144C2×Dic73—81no—
145C2×Dic73—81no—
146C2×Dic73—81no—
147C2×Dic73#10212yes—
148C4×D73—22no—
149C4×D73—22no—
150C4×D73—22no—
151C4×D73—22no—
152C8×C73C4×D411yescharacteristic
153C8×C73C4×D411yescharacteristic
154C8×C73#12111yescharacteristic
155C8×C73#12111yescharacteristic
156#10596831—22no—
157C22×C73C2×#10211yescharacteristic
158C22×C73#44711yescharacteristic
159C22×C73#48011yescharacteristic
160C22×C73#50811yescharacteristic
161C22×C73#50811yescharacteristic
162C22×C73#51011yescharacteristic
163C22×C73#51011yescharacteristic
164C2×D73—22no—
165C2×D73—22no—
166C2×D73—22no—
167C2×D73—41no—
168C2×D73—41no—
169C2×D73—41no—
170C2×D73—41no—
171C4×C73—82no—
172C4×C73—82no—
173C4×C73#38611yescharacteristic
174C4×C73#41711yescharacteristic
175C4×C73#44811yescharacteristic
176C4×C73#48111yescharacteristic
177Dic73—22no—
178Dic73—82no—
179Dic73—82no—
180#12994—731nomaximal, Sylow
181C2×C73#187411yescharacteristic
182C2×C73#213811yescharacteristic
183C2×C73#214111yescharacteristic
184C2×C73#218911yescharacteristic
185C2×C73#218911yescharacteristic
186C2×C73#223711yescharacteristic
187C2×C73#223711yescharacteristic
188D73—42no—
189C2×#467—731no—
190C2×#427—731no—
191C2×#358—731no—
192C2×#358D7311yescharacteristic, complemented
193C2×C22⋊C16—731no—
194#1380—731no—
195#1380—731no—
196C73#1299411yescharacteristic, complemented, Sylow
197C22×Q16C2×D7311yescharacteristic
198C23×C8—731no—
199C2×C22⋊Q8—1461no—
200C2×#117—1461no—
201C22×C16—731no—
202C22×C16C2×D7311yescharacteristic
203#467—734no—
204#467—1462no—
205#427—1462no—
206#427—1462no—
207C2×#107—731no—
208C2×#107—731no—
209C2×#107C2×D7311yescharacteristic
210C2×#106—1461no—
211C2×#103—1461no—
212#358—22no—
213#358—1462no—
214C22⋊C16—1462no—
215C22×Q8—21no—
216C23×C4—731no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16C4×D7312yes—
223C22×C8—731no—
224C22×C8—732no—
225C22×C8—732no—
226C22×C8—1461no—
227C22×C8—1461no—
228C22×C8—1461no—
229C22×C8—1461no—
230C22×C8C22×D7311yescharacteristic
231C22⋊Q8—2922no—
232C22⋊Q8—1464no—
233#117—2922no—
234#117—1464no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—2921no—
237C2×C4⋊C4—1461no—
238C2×C4⋊C4—1461no—
239C2×C22⋊C4—2921no—
240C2×C22⋊C4—2921no—
241C2×C42—2921no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1461no—
246C2×C16—1461no—
247C2×C16—1461no—
248#107—21no—
249#107—732no—
250#107—732no—
251#107—1461no—
252#107—1461no—
253#107C22⋊D7312yes—
254#106—1464no—
255#103—1464no—
256C24—731no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—2921no—
266C22×C4—2921no—
267C22×C4—731no—
268C22×C4—732no—
269C22×C4—732no—
270C22×C4—1461no—
271C22×C4—1461no—
272C22×C4—1461no—
273C22×C4—1461no—
274C22×C4D4×D7311yescharacteristic
275Q16—22no—
276Q16—22no—
277Q16—41no—
278Q16—41no—
279C2×C8—732no—
280C2×C8—1461no—
281C2×C8—1461no—
282C2×C8—1462no—
283C2×C8—1462no—
284C2×C8—1462no—
285C2×C8—1462no—
286C2×C8C2×C4×D7311yescharacteristic
287C2×C8D4×D7311yescharacteristic
288C2×C8D4×D7311yescharacteristic
289C2×C8C2×C22⋊D7311yescharacteristic
290C2×C8#1192402711yescharacteristic
291C2×C8#1192402811yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—2924no—
295C4⋊C4—2921no—
296C4⋊C4—2921no—
297C4⋊C4—1462no—
298C4⋊C4—1462no—
299C22⋊C4—2924no—
300C22⋊C4—2924no—
301C42—2924no—
302C16—41no—
303C16—2921no—
304C23—732no—
305C23—1461no—
306C23—1461no—
307C23—1461no—
308C23—1461no—
309C23—1461no—
310C23—1461no—
311C23D8×D7311yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—81no—
315Q8—81no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—2922no—
323C2×C4—2922no—
324C2×C4—5841no—
325C2×C4—5841no—
326C2×C4—5841no—
327C2×C4—5841no—
328C2×C4—732no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1462no—
332C2×C4—1462no—
333C2×C4—1462no—
334C2×C4—1462no—
335C2×C4C22⋊C4×D7311yescharacteristic
336C2×C4#1455655911yescharacteristic
337C2×C4#1455657311yescharacteristic
338C2×C4SD16×D7311yescharacteristic
339C2×C4#1455661511yescharacteristic
340C2×C4#1455662011yescharacteristic
341C8—1462no—
342C8#1455655111yescharacteristic
343C8C4×C22⋊D7311yescharacteristic
344C8#1455656211yescharacteristic
345C8#1455656411yescharacteristic
346C22—2921no—
347C22—2921no—
348C22—2921no—
349C22—2921no—
350C22—1462no—
351C22—1462no—
352C22—1462no—
353C22#102×D7311yescharacteristic
354C22#1798871911yescharacteristic
355C22#1798889911yescharacteristic
356C22#1798901811yescharacteristic
357C22#1798902011yescharacteristic
358C22#1798902611yescharacteristic
359C22#1798902911yescharacteristic
360C4—82no—
361C4—82no—
362C4—5842no—
363C4—5842no—
364C4—1462no—
365C4#1798846311yescharacteristic
366C4#1798858511yescharacteristic
367C4#1798872411yescharacteristic
368C4#1798890411yescharacteristic
369C2—2922no—
370C2#2261024511yescharacteristic
371C2#2261163711yescharacteristic
372C2#2261164911yescharacteristic
373C2#2261186811yescharacteristic
374C2#2261187011yescharacteristic
375C2#2261213711yescharacteristic
376C2#2261213811yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socleC23×C73C2×D8
prime radicalC23×D73D8
cosocleC22×C8×C73C23
hypercenterC2×#358D73
hypocenterC73#12994
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#17988391C2semidirect
5. #22608023C2semidirect
6. #22608022C2semidirect
7. C2×#358×C73C2semidirect
13. #17988825C4semidirect
192. C2×#358D73semidirect
196. C73#12994semidirect

Maximal subgroups

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

subgroupindex
C2×#179888252
C2×#179886242
C2×#179883912
C2×#179883662
#226080232
#226080222
C2×#358×C732
#1299473