Finite Group Databasebeta

← #33086348

Subgroups — #33086348

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556597C211yesmaximal, characteristic
2#22610077C212yesmaximal, complemented
3#22610056C211yesmaximal, characteristic, complemented
4#22610054C211yesmaximal, characteristic, complemented
5#22610034C212yesmaximal
6#22609978C211yesmaximal, characteristic
7#22609974C211yesmaximal, characteristic
8#22609810C211yesmaximal, characteristic, complemented
9#22609732C212yesmaximal
10#22609686C212yesmaximal, complemented
11C73×#1772C211yesmaximal, characteristic, complemented
12C2×#14556597C2211yescharacteristic
13C2×#14556597C2211yescharacteristic
14C4×#11924028C2211yescharacteristic
15C2×#14556527C2211yescharacteristic
16#17988625C2211yescharacteristic
17#17988569C2212yes—
18#17988560C2212yes—
19#17988542C2212yes—
20#17988542C2211yescharacteristic
21#17988542C2211yescharacteristic
22#17988541C412yescomplemented
23#17988541C412yescomplemented
24C4×#11924012C2211yescharacteristic
25C4×#11924011C412yescomplemented
26C4×C8×D73C412yescomplemented
27#17988359C2212yes—
28#17988357C2212yes—
29#17988347C2212yes—
30#17988345C2212yes—
31#17988344C2212yes—
32#17988343C2212yes—
33#17988343C2212yes—
34C2×#108×C73C2211yescharacteristic
35#427×C73C2211yescharacteristic
36#405×C73C2212yes—
37C2×C4×C8×C73C2211yescharacteristic
38#339×C73C2212yes—
39C2×#11924028C2311yescharacteristic
40#14556597—21no—
41#14556597—21no—
42#14556597—21no—
43#14556597—21no—
44#14556597—21no—
45#14556597—21no—
46#14556597—21no—
47#14556597—21no—
48C2×#11924012C2312yes—
49C2×#11924012C2311yescharacteristic
50C2×#11924012C2311yescharacteristic
51C2×#11924011C2×C412yes—
52C2×#11924011C2×C412yes—
53C2×C8×D73C2×C412yes—
54C2×C8×D73C2×C412yes—
55C4×Dic146—21no—
56C4×D292—21no—
57C4×C22⋊D73—22no—
58#14556550C2312yes—
59#14556550C2311yescharacteristic
60#14556548C2×C412yes—
61C42×D73C2×C412yes—
62#14556527—21no—
63#14556527—21no—
64#14556525—21no—
65#14556523—21no—
66#14556509—22no—
67#14556507—22no—
68#14556505—24no—
69#14556502C2×C412yes—
70#14556501—22no—
71#14556501C2×C412yes—
72#14556501C2×C412yes—
73C4×#10596831C2×C412yes—
74C8×Dic73—22no—
75C8×Dic73C2×C412yes—
76C8×Dic73C2×C412yes—
77C2×M16×C73C2312yes—
78C22×C8×C73C2311yescharacteristic
79C22×C8×C73C2311yescharacteristic
80#117×C73C2312yes—
81C2×C42×C73C2311yescharacteristic
82#108×C73—21no—
83#108×C73—21no—
84#107×C73—21no—
85#106×C73—21no—
86#97×C73—22no—
87C4×C8×C73—22no—
88C4×C8×C73C2×C411yescharacteristic
89C4×C8×C73C2×C411yescharacteristic
90C4×C8×C73C2×C411yescharacteristic
91C4×C8×C73C2×C411yescharacteristic
92#11924028—21no—
93#11924028—21no—
94#11924028—21no—
95#11924028—21no—
96C2×Dic146#4111yescharacteristic
97C2×D292#4111yescharacteristic
98C2×C22⋊D73#4112yes—
99C2×C4×D73—22no—
100C2×C4×D73C22×C412yes—
101#11924012—21no—
102#11924012—21no—
103#11924012—21no—
104#11924012—21no—
105#11924012—22no—
106#11924012—22no—
107#11924011—22no—
108#11924011—22no—
109#11924011—22no—
110#11924011—22no—
111C2×#10596831—22no—
112C2×#10596831C22×C412yes—
113C2×#10596831C22×C412yes—
114C8×D73—22no—
115C8×D73—22no—
116C8×D73—22no—
117C8×D73—22no—
118C4⋊Dic73—21no—
119C4⋊Dic73—22no—
120#11924007—22no—
121C22⋊Dic73—21no—
122C22⋊Dic73—22no—
123#11924005—22no—
124C4×Dic73—22no—
125C4×Dic73—22no—
126C4×Dic73C22×C412yes—
127C22×C4×C73—21no—
128C22×C4×C73C2411yescharacteristic
129M16×C73—22no—
130M16×C73—22no—
131C2×C8×C73—22no—
132C2×C8×C73C22×C411yescharacteristic
133C2×C8×C73C22×C411yescharacteristic
134C2×C8×C73C22×C411yescharacteristic
135C2×C8×C73C22×C411yescharacteristic
136C2×C8×C73#4111yescharacteristic
137C2×C8×C73#4111yescharacteristic
138C2×C8×C73#4111yescharacteristic
139C2×C8×C73#4111yescharacteristic
140C2×C8×C73#4111yescharacteristic
141C2×C8×C73#4111yescharacteristic
142C2×C8×C73#4111yescharacteristic
143C2×C8×C73#4111yescharacteristic
144C4⋊C4×C73—22no—
145C22⋊C4×C73—22no—
146C42×C73—21no—
147C42×C73—22no—
148C42×C73C22×C411yescharacteristic
149C42×C73C22×C411yescharacteristic
150C23×C73C2×#4111yescharacteristic
151C22×D73#11712yes—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—42no—
155C2×C4×C73—42no—
156C2×C4×C73C23×C411yescharacteristic
157C2×C4×C73C2×#4111yescharacteristic
158C2×C4×C73C2×#4111yescharacteristic
159C2×C4×C73C2×#4111yescharacteristic
160C2×C4×C73C2×#4111yescharacteristic
161C2×C4×C73C2×#4111yescharacteristic
162C22⋊D73—22no—
163C22⋊D73—22no—
164C2×Dic73—22no—
165C2×Dic73—42no—
166C2×Dic73#11712yes—
167D292—21no—
168D292—21no—
169C4×D73—22no—
170C4×D73—22no—
171C4×D73—44no—
172Dic146—21no—
173Dic146—21no—
174C8×C73—42no—
175C8×C73#11711yescharacteristic
176C8×C73#11711yescharacteristic
177C8×C73#11711yescharacteristic
178C8×C73#11711yescharacteristic
179C8×C73#11711yescharacteristic
180C8×C73#11711yescharacteristic
181C8×C73#11711yescharacteristic
182C8×C73#11711yescharacteristic
183#10596831—22no—
184#10596831—22no—
185#10596831—42no—
186C22×C73—41no—
187C22×C73C2×#11711yescharacteristic
188C22×C73#52911yescharacteristic
189C22×C73#52911yescharacteristic
190C2×D73—22no—
191C2×D73—42no—
192C4×C73—22no—
193C4×C73—41no—
194C4×C73—82no—
195C4×C73C2×#11711yescharacteristic
196C4×C73C2×#11711yescharacteristic
197C4×C73#52911yescharacteristic
198C4×C73#52911yescharacteristic
199Dic73—22no—
200Dic73—42no—
201Dic73—82no—
202#20787—731nomaximal, Sylow
203C2×C73—21no—
204C2×C73—21no—
205C2×C73—21no—
206C2×C73#228311yescharacteristic
207D73—42no—
208C4×#131—731no—
209#1841—732no—
210#1833—731no—
211#1833—731no—
212#1827—732no—
213#1813—731no—
214#1812—731no—
215#1772—731no—
216#1772D7311yescharacteristic, complemented
217#1757—732no—
218#1747—732no—
219C73#2078711yescharacteristic, complemented, Sylow
220C2×#131—731no—
221C2×#131—731no—
222C4×#41—731no—
223C2×#108—731no—
224C2×#108C2×D7311yescharacteristic
225#427—731no—
226#427C2×D7311yescharacteristic
227#413—732no—
228#410—732no—
229#405—731no—
230#405—731no—
231#405—732no—
232#405—732no—
233#405—732no—
234#405C2×D7312yes—
235C4×M16—731no—
236C4×M16—732no—
237C2×C4×C8—732no—
238C2×C4×C8C2×D7311yescharacteristic
239#344—732no—
240#344—732no—
241#339—732no—
242#339C2×D7312yes—
243#338—732no—
244H(3,ℤ/4)—732no—
245H(3,ℤ/4)—732no—
246H(3,ℤ/4)—732no—
247C2×#41—731no—
248#131—1461no—
249#131—1461no—
250#131—1461no—
251#131—1461no—
252#131—1461no—
253#131—1461no—
254#131—1461no—
255#131—1461no—
256C2×M16—731no—
257C2×M16—731no—
258C2×M16—732no—
259C2×M16—732no—
260C2×M16—732no—
261C2×M16C22×D7312yes—
262C22×C8—732no—
263C22×C8—732no—
264C22×C8C22×D7311yescharacteristic
265C22×C8C22×D7311yescharacteristic
266C4×Q8—1461no—
267C4×D4—1461no—
268C4×D4—1462no—
269#117—731no—
270#117—732no—
271#117—732no—
272#117C22×D7312yes—
273C2×C42—732no—
274C2×C42C22×D7311yescharacteristic
275#108—21no—
276#108—21no—
277#108—1461no—
278#108—1461no—
279#107—21no—
280#107—1461no—
281#106—21no—
282#106—1461no—
283#101—1462no—
284#100—1462no—
285#99—1464no—
286#97—22no—
287#97—732no—
288#97—732no—
289#97—732no—
290#97—1462no—
291C4×C8—22no—
292C4×C8—732no—
293C4×C8—732no—
294C4×C8—732no—
295C4×C8—1462no—
296C4×C8C4×D7311yescharacteristic
297C4×C8C4×D7311yescharacteristic
298C4×C8C4×D7311yescharacteristic
299C4×C8C4×D7311yescharacteristic
300#41—1461no—
301#41—1461no—
302#41—1461no—
303#41—1461no—
304C2×Q8—731no—
305C2×D4—732no—
306C2×D4—731no—
307C22×C4—21no—
308C22×C4—732no—
309C22×C4—1462no—
310C22×C4C23×D7311yescharacteristic
311M16—22no—
312M16—22no—
313M16—1462no—
314M16—1462no—
315M16—1461no—
316M16—1461no—
317M16—1461no—
318M16—1461no—
319M16—1462no—
320M16—1462no—
321M16—1462no—
322M16—1462no—
323C2×C8—22no—
324C2×C8—732no—
325C2×C8—732no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8—1462no—
329C2×C8—1462no—
330C2×C8—1462no—
331C2×C8C2×C4×D7311yescharacteristic
332C2×C8C2×C4×D7311yescharacteristic
333C2×C8C2×C4×D7311yescharacteristic
334C2×C8C2×C4×D7311yescharacteristic
335C2×C8#1192402711yescharacteristic
336C2×C8#1192402711yescharacteristic
337C2×C8#1192402811yescharacteristic
338C2×C8#1192402811yescharacteristic
339C2×C8#1192402811yescharacteristic
340C2×C8#1192402811yescharacteristic
341C2×C8#1192402911yescharacteristic
342C2×C8#1192402911yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—1462no—
345C4⋊C4—1462no—
346C4⋊C4—1461no—
347C22⋊C4—22no—
348C22⋊C4—1462no—
349C22⋊C4—1462no—
350C22⋊C4—1461no—
351C42—21no—
352C42—22no—
353C42—732no—
354C42—1462no—
355C42—1462no—
356C42C2×C4×D7311yescharacteristic
357C42C2×C4×D7311yescharacteristic
358C23—732no—
359C23#41×D7311yescharacteristic
360Q8—1461no—
361Q8—1461no—
362D4—1462no—
363D4—1462no—
364D4—1461no—
365D4—1461no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—2924no—
371C2×C4—2922no—
372C2×C4—732no—
373C2×C4—1462no—
374C2×C4—1462no—
375C2×C4—1462no—
376C2×C4C22×C4×D7311yescharacteristic
377C2×C4#41×D7311yescharacteristic
378C2×C4C2×#1192402711yescharacteristic
379C2×C4C2×#1192402811yescharacteristic
380C2×C4C2×#1192402811yescharacteristic
381C2×C4C2×#1192402911yescharacteristic
382C8—42no—
383C8—2922no—
384C8—1462no—
385C8—1462no—
386C8#1455654811yescharacteristic
387C8#1455654811yescharacteristic
388C8#1455654811yescharacteristic
389C8#1455654811yescharacteristic
390C8#1455654911yescharacteristic
391C8#1455654911yescharacteristic
392C8#1455654911yescharacteristic
393C8#1455654911yescharacteristic
394C22—41no—
395C22—2922no—
396C22—1462no—
397C22#117×D7311yescharacteristic
398C22#1798911911yescharacteristic
399C22#1798912411yescharacteristic
400C4—22no—
401C4—41no—
402C4—82no—
403C4—2922no—
404C4—5842no—
405C4—1462no—
406C4C2×#1455654811yescharacteristic
407C4C2×#1455654911yescharacteristic
408C4#1798911911yescharacteristic
409C4#1798912411yescharacteristic
410C2—21no—
411C2—21no—
412C2—21no—
413C2—2922no—
414C2#2261232711yescharacteristic
subgroupG/H
centerC8#14556549
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1772C2
socleC2×C73#2283
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1772D73
hypocenterC73#20787
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. #22610077C2semidirect
3. #22610056C2semidirect
4. #22610054C2semidirect
8. #22609810C2semidirect
10. #22609686C2semidirect
11. C73×#1772C2semidirect
22. #17988541C4semidirect
23. #17988541C4semidirect
25. C4×#11924011C4semidirect
26. C4×C8×D73C4semidirect
216. #1772D73semidirect
219. C73#20787semidirect

Maximal subgroups

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

subgroupindex
C4×#145565972
#226100772
#226100562
#226100542
#226100342
#226099782
#226099742
#226098102
#226097322
#226096862
C73×#17722
#2078773