Finite Group Databasebeta

← #33239020

Subgroups — #33239020

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D73C211yesmaximal, characteristic
2C2×#17988624C211yesmaximal, characteristic
3#427×D73C211yesmaximal, characteristic
4#22611720C211yesmaximal, characteristic, complemented
5#22611719C211yesmaximal, characteristic, complemented
6#22611718C212yesmaximal, complemented
7#22611717C211yesmaximal, characteristic, complemented
8#22611716C212yesmaximal, complemented
9#22611714C212yesmaximal, complemented
10#22611712C212yesmaximal, complemented
11C73×#2159C211yesmaximal, characteristic, complemented
12C22×C8×D73C2211yescharacteristic
13C2×#14556549—21no—
14#117×D73—21no—
15C2×#14556531C2211yescharacteristic
16C2×#14556530C2212yes—
17M32×D73C418yescomplemented
18C2×C16×D73C2212yes—
19#17988625C2211yescharacteristic
20#17988624—21no—
21#17988624—21no—
22#17988624C2211yescharacteristic
23#17988624C2212yes—
24#17988624C2211yescharacteristic
25#17988624C2212yes—
26#17988623—21no—
27#17988622—21no—
28C2×#14556525C2211yescharacteristic
29#107×D73C2211yescharacteristic
30#107×D73C2211yescharacteristic
31C2×#14556522—21no—
32#106×D73—21no—
33#17988415C2212yescomplemented
34#17988414C2212yescomplemented
35#17988414C2212yescomplemented
36#17988413C2212yescomplemented
37#17988412C2212yescomplemented
38#17988412C2212yescomplemented
39C2×M32×C73C2211yescharacteristic
40#427×C73C2211yescharacteristic
41C2×#107×C73C2211yescharacteristic
42#367×C73C2212yescomplemented
43#366×C73C2212yescomplemented
44C22×C4×D73D411yescharacteristic
45C22×#10596831D411yescharacteristic
46C2×C8×D73C2×C414yes—
47C2×C8×D73D411yescharacteristic
48C2×C8×D73D411yescharacteristic
49C2×C8×D73Q814yes—
50C2×C8×D73C2311yescharacteristic
51C2×C8×D73C2311yescharacteristic
52#14556550—21no—
53#14556549—21no—
54#14556549—21no—
55#14556549—22no—
56#14556549—22no—
57#14556549—41no—
58#14556549—41no—
59#14556548—41no—
60#14556547—41no—
61C2×C4⋊Dic73—21no—
62C4⋊C4×D73—21no—
63C4⋊C4×D73—21no—
64C2×#11924005—41no—
65C22⋊C4×D73—41no—
66C2×C4×Dic73—41no—
67C42×D73—41no—
68#14556531C2×C414yescomplemented
69#14556530C2×C418yescomplemented
70C2×#11924003C2312yes—
71C16×D73C2×C418yescomplemented
72#14556525—21no—
73#14556525C2311yescharacteristic
74#14556525C2311yescharacteristic
75#14556525C2312yes—
76#14556524—21no—
77#14556524—21no—
78#14556523—21no—
79#14556522—21no—
80#14556522—21no—
81#14556522—22no—
82#14556522—22no—
83C22×C8×C73C2311yescharacteristic
84#117×C73—21no—
85C2×C4⋊C4×C73—21no—
86M32×C73C2×C414yescomplemented
87C2×C16×C73C2312yes—
88#107×C73—21no—
89#107×C73C2311yescharacteristic
90#107×C73C2311yescharacteristic
91#107×C73C2312yes—
92#106×C73—21no—
93C23×D73D811yescharacteristic
94C22×Dic73—41no—
95C22×Dic73D811yescharacteristic
96C2×C4×D73—41no—
97C2×C4×D73—41no—
98C2×C4×D73C4⋊C414yes—
99C2×C4×D73D811yescharacteristic
100C2×C4×D73D811yescharacteristic
101C2×C4×D73Q1614yes—
102C2×C4×D73C2×D411yescharacteristic
103C2×C4×D73C2×D411yescharacteristic
104C2×#10596831C4⋊C414yes—
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C8×D73C4⋊C414yes—
108C8×D73C4⋊C414yes—
109C8×D73C22×C412yes—
110C8×D73C22×C412yes—
111C8×D73C2×Q812yes—
112C8×D73C2×Q812yes—
113C4⋊Dic73—22no—
114C4⋊Dic73—21no—
115C4⋊Dic73—21no—
116C4⋊Dic73—41no—
117#11924007—41no—
118#11924007—41no—
119C22⋊Dic73—41no—
120#11924005—41no—
121#11924005—41no—
122#11924005—42no—
123#11924005—42no—
124C4×Dic73—41no—
125C4×Dic73—41no—
126C4×Dic73—41no—
127C4×Dic73—42no—
128C4×Dic73—42no—
129#11924003C22×C414yes—
130C22×C4×C73—41no—
131C22×C4×C73C2×D411yescharacteristic
132C2×C8×C73C22×C412yes—
133C2×C8×C73C2×D411yescharacteristic
134C2×C8×C73C2×Q812yes—
135C2×C8×C73C2411yescharacteristic
136C4⋊C4×C73—21no—
137C4⋊C4×C73—21no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—41no—
140C22⋊C4×C73—41no—
141C42×C73—41no—
142C16×C73C22×C414yes—
143C23×C73C2×D811yescharacteristic
144C22×D73—41no—
145C22×D73—41no—
146C22×D73#10714yes—
147C22×D73C2×D811yescharacteristic
148C22×D73C2×D811yescharacteristic
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—42no—
152C2×C4×C73—82no—
153C2×C4×C73C2×C4⋊C412yes—
154C2×C4×C73C2×D811yescharacteristic
155C2×C4×C73C2×Q1612yes—
156C2×C4×C73C22×D411yescharacteristic
157C2×Dic73—42no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—82no—
161C2×Dic73#10714yes—
162C2×Dic73C2×D811yescharacteristic
163C2×Dic73C2×D811yescharacteristic
164C4×D73—82no—
165C4×D73—82no—
166C4×D73#10714yes—
167C4×D73#10714yes—
168C4×D73C2×C4⋊C412yes—
169C4×D73C2×C4⋊C412yes—
170C4×D73C2×Q1612yes—
171C4×D73C2×Q1612yes—
172C8×C73C2×C4⋊C412yes—
173C8×C73C23×C411yescharacteristic
174C8×C73C22×Q811yescharacteristic
175#10596831C2×C4⋊C412yes—
176#10596831C2×C4⋊C412yes—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73C2×#10712yes—
180C22×C73C22×D811yescharacteristic
181C2×D73—21no—
182C2×D73—21no—
183C2×D73—21no—
184C2×D73—21no—
185C2×D73—44no—
186C2×D73C2×#10712yes—
187C2×D73C2×#10712yes—
188C4×C73—81no—
189C4×C73—81no—
190C4×C73—82no—
191C4×C73C2×#10712yes—
192C4×C73C22×C4⋊C411yescharacteristic
193C4×C73C22×Q1611yescharacteristic
194Dic73—81no—
195Dic73—81no—
196Dic73—82no—
197Dic73C2×#10712yes—
198Dic73C2×#10712yes—
199#33409—731nomaximal, Sylow
200C2×C73—22no—
201C2×C73C22×#10711yescharacteristic
202C2×C73#341411yescharacteristic
203C2×C73#341411yescharacteristic
204D73—22no—
205D73—22no—
206C22×M32—731no—
207C2×#427—731no—
208C2×#427—731no—
209#2159—731no—
210#2159—731no—
211#2159—731no—
212#2159—732no—
213#2159—732no—
214#2159D7311yescharacteristic, complemented
215#2158—732no—
216#2158—732no—
217C73#3340911yescharacteristic, complemented, Sylow
218C23×C8—731no—
219C2×#117—1461no—
220C2×#117—1461no—
221C2×M32—731no—
222C2×M32—738no—
223C2×M32—732no—
224C2×M32C2×D7311yescharacteristic
225C22×C16—732no—
226#427—731no—
227#427—731no—
228#427—732no—
229#427—731no—
230#427—732no—
231#427—1461no—
232#427—1461no—
233#427—1461no—
234#427—1461no—
235#427C2×D7311yescharacteristic
236C2×#107—731no—
237C2×#107—731no—
238C2×#107—731no—
239C2×#107C2×D7311yescharacteristic
240C2×#106—1461no—
241C2×#106—1461no—
242#367—732no—
243#367—732no—
244#367—732no—
245#367C2×D7312yescomplemented
246#366—732no—
247#366—732no—
248#366—732no—
249#366C2×D7312yescomplemented
250C23×C4—731no—
251C22×C8—731no—
252C22×C8—734no—
253C22×C8—731no—
254C22×C8—731no—
255C22×C8—734no—
256C22×C8—731no—
257C22×C8—731no—
258C22×C8C22×D7311yescharacteristic
259#117—21no—
260#117—2921no—
261#117—2921no—
262#117—2921no—
263#117—2921no—
264#117—1461no—
265#117—1461no—
266#117—1462no—
267#117—1462no—
268#117—1461no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—1461no—
271C2×C4⋊C4—1461no—
272C2×C4⋊C4—1461no—
273C2×C22⋊C4—2921no—
274C2×C22⋊C4—2921no—
275C2×C42—2921no—
276C2×C42—2921no—
277M32—734no—
278M32—738no—
279M32C4×D7314yescomplemented
280C2×C16—732no—
281C2×C16—738no—
282C2×C16C22×D7312yes—
283#107—21no—
284#107—731no—
285#107—731no—
286#107—732no—
287#107—1461no—
288#107—1461no—
289#107—1461no—
290#107C22×D7311yescharacteristic
291#107C22×D7312yes—
292#107C22×D7311yescharacteristic
293#106—21no—
294#106—1461no—
295#106—1461no—
296#106—1462no—
297#106—1462no—
298#106—1461no—
299C24—731no—
300C22×C4—41no—
301C22×C4—2921no—
302C22×C4—2921no—
303C22×C4—2921no—
304C22×C4—731no—
305C22×C4—734no—
306C22×C4—731no—
307C22×C4—731no—
308C22×C4—734no—
309C22×C4—731no—
310C22×C4—731no—
311C22×C4D4×D7311yescharacteristic
312C2×C8—734no—
313C2×C8—731no—
314C2×C8—731no—
315C2×C8—734no—
316C2×C8—734no—
317C2×C8—732no—
318C2×C8—732no—
319C2×C8—732no—
320C2×C8—732no—
321C2×C8C2×C4×D7312yes—
322C2×C8D4×D7311yescharacteristic
323C2×C8Q8×D7312yes—
324C2×C8C23×D7311yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—2921no—
330C4⋊C4—2921no—
331C4⋊C4—2921no—
332C4⋊C4—1462no—
333C4⋊C4—1461no—
334C4⋊C4—1461no—
335C22⋊C4—41no—
336C22⋊C4—2921no—
337C22⋊C4—2921no—
338C22⋊C4—2922no—
339C22⋊C4—2922no—
340C22⋊C4—2921no—
341C42—41no—
342C42—2921no—
343C42—2921no—
344C42—2922no—
345C42—2922no—
346C42—2921no—
347C16—734no—
348C16C2×C4×D7314yes—
349C23—2921no—
350C23—2921no—
351C23—734no—
352C23—731no—
353C23—731no—
354C23D8×D7311yescharacteristic
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—2921no—
360C2×C4—2922no—
361C2×C4—2921no—
362C2×C4—5842no—
363C2×C4—5842no—
364C2×C4—5842no—
365C2×C4—734no—
366C2×C4—731no—
367C2×C4—731no—
368C2×C4—734no—
369C2×C4—734no—
370C2×C4—732no—
371C2×C4—732no—
372C2×C4—732no—
373C2×C4—732no—
374C2×C4C4⋊C4×D7312yes—
375C2×C4D8×D7311yescharacteristic
376C2×C4Q16×D7312yes—
377C2×C4C2×D4×D7311yescharacteristic
378C8—732no—
379C8—732no—
380C8C4⋊C4×D7312yes—
381C8C22×C4×D7311yescharacteristic
382C8C2×Q8×D7311yescharacteristic
383C22—21no—
384C22—21no—
385C22—2924no—
386C22—732no—
387C22—732no—
388C22—1461no—
389C22—1461no—
390C22—1461no—
391C22—1461no—
392C22#107×D7312yes—
393C22C2×D8×D7311yescharacteristic
394C4—82no—
395C4—81no—
396C4—81no—
397C4—5841no—
398C4—5841no—
399C4—5842no—
400C4—732no—
401C4—732no—
402C4#107×D7312yes—
403C4C2×C4⋊C4×D7311yescharacteristic
404C4C2×Q16×D7311yescharacteristic
405C2—22no—
406C2—1462no—
407C2—1462no—
408C2C2×#107×D7311yescharacteristic
409C2#2262306311yescharacteristic
410C2#2262306611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC23×D73D8
cosocleC2×C8×C73C24
hypercenter#2159D73
hypocenterC73#33409
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #22611720C2semidirect
5. #22611719C2semidirect
6. #22611718C2semidirect
7. #22611717C2semidirect
8. #22611716C2semidirect
9. #22611714C2semidirect
10. #22611712C2semidirect
11. C73×#2159C2semidirect
17. M32×D73C4semidirect
33. #17988415C22semidirect
34. #17988414C22semidirect
35. #17988414C22semidirect
36. #17988413C22semidirect
37. #17988412C22semidirect
38. #17988412C22semidirect
42. #367×C73C22semidirect
43. #366×C73C22semidirect
68. #14556531C2×C4semidirect
69. #14556530C2×C4semidirect
71. C16×D73C2×C4semidirect
86. M32×C73C2×C4semidirect
214. #2159D73semidirect
217. C73#33409semidirect
245. #367C2×D73semidirect
249. #366C2×D73semidirect
279. M32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D732
C2×#179886242
#427×D732
#226117202
#226117192
#226117182
#226117172
#226117162
#226117142
#226117122
C73×#21592
#3340973