Finite Group Databasebeta

← #29725147

Subgroups — #29725147

Order 15616.

620 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×#16840034C211yesmaximal, characteristic
2#21454687C211yesmaximal, characteristic, complemented
3#21447283C212yesmaximal, complemented
4#21447098C212yesmaximal
5#21446610C211yesmaximal, characteristic, complemented
6#21446311C212yesmaximal
7#21445943C211yesmaximal, characteristic, complemented
8#21445931C212yesmaximal, complemented
9#21445874C211yesmaximal, characteristic
10#21445620C211yesmaximal, characteristic
11C61×#1979C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2211yescharacteristic
14#16840240C2212yes—
15#16840034—22no—
16#16840034—22no—
17#16839869—22no—
18Q16×Dic61—22no—
19#16839811C2212yes—
20C2×#14519713C2211yescharacteristic
21#16839800C2211yescharacteristic
22#16839778C2212yes—
23#16839771C2212yes—
24#16839744C2211yescharacteristic
25C2×#14519696C2212yes—
26#16839735C2211yescharacteristic
27#16839717C2211yescharacteristic
28#16839534C2211yescharacteristic
29#16839534C2212yes—
30#16839533C2212yes—
31#16839533C2212yes—
32#16839532C2211yescharacteristic
33#16839518C2212yes—
34C22×Q16×C61C2211yescharacteristic
35C2×#108×C61C2211yescharacteristic
36C61×#419C2212yes—
37C2×#101×C61C2212yes—
38C61×#405C2211yescharacteristic
39C2×#11758222C2311yescharacteristic
40#14519808—24no—
41#14519808—42no—
42C2×#11758202—22no—
43C2×#11758201—22no—
44#14519783—21no—
45#14519783—21no—
46#14519783—21no—
47#14519783—21no—
48C2×#11758194C2311yescharacteristic
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2312yes—
51C2×#11758189—21no—
52C2×C8×D61—21no—
53Q8×Dic61—24no—
54Q8×Dic61—42no—
55#14519736—24no—
56#14519736C2312yes—
57#14519736C2311yescharacteristic
58#14519713—21no—
59#14519713—21no—
60#14519713—42no—
61#14519711—21no—
62#14519709—21no—
63#14519703—42no—
64#14519701—24no—
65#14519696—24no—
66#14519695—21no—
67#14519693—21no—
68#14519691—22no—
69#14519687—21no—
70C8×Dic61—21no—
71C22×Q8×C61C2312yes—
72C2×Q16×C61—21no—
73C2×Q16×C61—21no—
74C2×Q16×C61—21no—
75C2×Q16×C61—21no—
76C2×Q16×C61D412yes—
77C2×Q16×C61D412yes—
78C2×M16×C61C2311yescharacteristic
79C2×M16×C61C2312yes—
80C22×C8×C61C2311yescharacteristic
81#117×C61C2311yescharacteristic
82#108×C61—21no—
83#108×C61—21no—
84#103×C61—42no—
85#101×C61—42no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61C2×D414yes—
94C22×C4×C61—22no—
95C22×C4×C61C2411yescharacteristic
96C2×C22⋊D61—21no—
97#11758222—21no—
98#11758222—21no—
99#11758222—21no—
100#11758222—21no—
101C2×D244#4111yescharacteristic
102C2×C4×D61—21no—
103C2×Dic122#4111yescharacteristic
104Q16×C61—22no—
105Q16×C61—22no—
106Q16×C61—22no—
107Q16×C61—22no—
108M16×C61—22no—
109M16×C61—21no—
110M16×C61—21no—
111M16×C61—22no—
112C2×C8×C61—21no—
113C2×C8×C61—22no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C2×C8×C61#4111yescharacteristic
119C2×C8×C61#4111yescharacteristic
120C4⋊C4×C61—21no—
121C22⋊C4×C61—21no—
122C42×C61—21no—
123C22⋊Dic61—22no—
124C22⋊Dic61#4112yes—
125#11758202—44no—
126#11758201—44no—
127C4⋊Dic61—21no—
128C4⋊Dic61—22no—
129C4⋊Dic61—44no—
130C4×Dic61—21no—
131C4×Dic61—22no—
132C4×Dic61—44no—
133#11758194—22no—
134#11758194—21no—
135#11758194—21no—
136#11758194—21no—
137#11758194—22no—
138#11758194—21no—
139C2×#92418—21no—
140C2×#92418—21no—
141C2×#92418—22no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—41no—
145C8×D61—41no—
146C23×C61C2×#4111yescharacteristic
147C22×D61—21no—
148Q8×C61—24no—
149Q8×C61—24no—
150Q8×C61—42no—
151Q8×C61—42no—
152C2×C4×C61—24no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×C4×C61H(5,𝔽2)11yescharacteristic
161C2×C4×C61H(5,𝔽2)11yescharacteristic
162C22⋊D61—41no—
163C22⋊D61—41no—
164C2×Dic61—21no—
165C2×Dic61—22no—
166C2×Dic61—42no—
167D244—21no—
168D244—21no—
169C4×D61—41no—
170C4×D61—41no—
171Dic122—21no—
172Dic122—21no—
173C8×C61—42no—
174C8×C61—41no—
175C8×C61#12311yescharacteristic
176C8×C61#12311yescharacteristic
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179#92418—41no—
180#92418—41no—
181#92418—42no—
182#24989—611nomaximal, Sylow
183C22×C61—41no—
184C22×C61#52911yescharacteristic
185C22×C61#54611yescharacteristic
186C22×C61#55511yescharacteristic
187C2×D61—41no—
188C2×D61—81no—
189C4×C61—44no—
190C4×C61—81no—
191C4×C61C2×#12311yescharacteristic
192C4×C61#54011yescharacteristic
193C4×C61#54611yescharacteristic
194C4×C61#55311yescharacteristic
195Dic61—42no—
196Dic61—41no—
197Dic61—82no—
198C2×#473—611no—
199#2950—611no—
200#2082—612no—
201#2049—612no—
202#1979D6111yescharacteristic, complemented
203#1953—611no—
204#1905—612no—
205#1833—611no—
206#1830—612no—
207#1814—611no—
208#1757—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#242011yescharacteristic
213D61—81no—
214C2×#137—612no—
215C22×Q16C2×D6111yescharacteristic
216C2×#131—611no—
217#519—612no—
218#473—1222no—
219#473—1222no—
220#441—1222no—
221C4×Q16—1222no—
222#430—612no—
223C2×#108—611no—
224C2×#108C2×D6111yescharacteristic
225#427—611no—
226#421—612no—
227#419—612no—
228#419C2×D6112yes—
229#413—611no—
230C2×#101—612no—
231C2×#101C2×D6112yes—
232#410—611no—
233#405—611no—
234#405C2×D6111yescharacteristic
235#344—612no—
236#344—612no—
237#344—611no—
238#344—612no—
239#344—611no—
240H(3,ℤ/4)—612no—
241C61#2498911yescharacteristic, complemented, Sylow
242C2×#41—611no—
243C22×Q8C22×D6112yes—
244#137—1224no—
245#137—2442no—
246C2×Q16—21no—
247C2×Q16—21no—
248C2×Q16—21no—
249C2×Q16—21no—
250C2×Q16—1222no—
251C2×Q16D24412yes—
252C2×Q16D24412yes—
253C2×SD16—1222no—
254#131—1221no—
255#131—1221no—
256#131—1221no—
257#131—1221no—
258C2×M16—611no—
259C2×M16—612no—
260C2×M16—611no—
261C2×M16—1221no—
262C2×M16C22×D6111yescharacteristic
263C2×M16C22×D6112yes—
264C22×C8—1221no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—1224no—
267C4×Q8—2442no—
268#117—611no—
269#117—612no—
270#117—1224no—
271#117C22×D6111yescharacteristic
272#108—21no—
273#108—21no—
274#108—1221no—
275#108—1221no—
276#108—2442no—
277#107—1221no—
278#106—1221no—
279#104—2442no—
280#103—42no—
281#103—1224no—
282#101—42no—
283#101—1221no—
284#101—1224no—
285#100—1221no—
286#99—1222no—
287#97—21no—
288#97—1221no—
289C4×C8—21no—
290C4×C8—1221no—
291#41—1221no—
292#41—1221no—
293#41—1221no—
294#41—1221no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—611no—
301C2×Q8C2×D24414yes—
302C2×D4—611no—
303C2×D4—1221no—
304C22×C4—22no—
305C22×C4—1221no—
306C22×C4C23×D6111yescharacteristic
307Q16—22no—
308Q16—22no—
309Q16—22no—
310Q16—22no—
311Q16—2444no—
312SD16—2444no—
313M16—21no—
314M16—22no—
315M16—21no—
316M16—22no—
317M16—1221no—
318M16—1222no—
319M16—1221no—
320M16—1221no—
321M16—1221no—
322M16—1222no—
323M16—2441no—
324M16—2441no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—1221no—
328C2×C8—1222no—
329C2×C8—1221no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8C2×D24411yescharacteristic
333C2×C8C2×D24411yescharacteristic
334C2×C8#1175822411yescharacteristic
335C2×C8#1175822411yescharacteristic
336C2×C8#1175822411yescharacteristic
337C2×C8#1175822411yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—1221no—
340C4⋊C4—1222no—
341C4⋊C4—2444no—
342C22⋊C4—21no—
343C22⋊C4—612no—
344C22⋊C4—1222no—
345C42—21no—
346C42—1221no—
347C42—1222no—
348C42—2444no—
349C23—1221no—
350C23#41×D6111yescharacteristic
351Q8—24no—
352Q8—24no—
353Q8—42no—
354Q8—42no—
355Q8—1221no—
356Q8—1221no—
357D4—1221no—
358D4—1221no—
359D4—2441no—
360D4—2441no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—1221no—
366C2×C4—1222no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4C22×D24411yescharacteristic
371C2×C4#41×D6111yescharacteristic
372C2×C4C2×#1175822411yescharacteristic
373C2×C4C2×#1175822411yescharacteristic
374C2×C4#1451982211yescharacteristic
375C2×C4#1451982211yescharacteristic
376C8—41no—
377C8—42no—
378C8—2441no—
379C8—2441no—
380C8—2442no—
381C8#1451976111yescharacteristic
382C8#1451976111yescharacteristic
383C8#1451976111yescharacteristic
384C8#1451976111yescharacteristic
385C22—41no—
386C22—4881no—
387C22—2441no—
388C22#1684029111yescharacteristic
389C22#1684040711yescharacteristic
390C22#1684047611yescharacteristic
391C4—44no—
392C4—81no—
393C4—4882no—
394C4—2442no—
395C4—2441no—
396C4C2×#1451976111yescharacteristic
397C4#1684036611yescharacteristic
398C4#1684040711yescharacteristic
399C4#1684045911yescharacteristic
400C2—21no—
401C2—21no—
402C2—21no—
403C2—4881no—
404C2#2144930911yescharacteristic
subgroupG/H
centerC2#21449309
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1979C2
socleC2×C61#2420
prime radicalC2×#11758222C23
hypercenter#1979D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1979

Complements

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

NH ≅ G/Ntype
2. #21454687C2semidirect
3. #21447283C2semidirect
5. #21446610C2semidirect
7. #21445943C2semidirect
8. #21445931C2semidirect
11. C61×#1979C2semidirect
202. #1979D61semidirect
241. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214546872
#214472832
#214470982
#214466102
#214463112
#214459432
#214459312
#214458742
#214456202
C61×#19792
#2498961