Finite Group Databasebeta

← #29725150

Subgroups — #29725150

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×#16840038C211yesmaximal, characteristic
2#21454688C211yesmaximal, characteristic, complemented
3#21447284C212yesmaximal, complemented
4#21447101C212yesmaximal
5#21446732C211yesmaximal, characteristic, complemented
6#21446312C212yesmaximal
7#21445944C211yesmaximal, characteristic, complemented
8#21445932C212yesmaximal, complemented
9#21445874C211yesmaximal, characteristic
10#21445620C211yesmaximal, characteristic
11C61×#1953C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840239C2212yes—
16#16840038—22no—
17#16840038—22no—
18#16839870—22no—
19C4×Dic244—22no—
20#16839800C2211yescharacteristic
21#16839778C2212yes—
22#16839772C2212yes—
23#16839772C2212yes—
24#16839744C2211yescharacteristic
25C2×#14519695C2212yes—
26C2×#14519695C2212yes—
27#16839735C2211yescharacteristic
28#16839717C2211yescharacteristic
29#16839534C2212yes—
30#16839534C2211yescharacteristic
31#16839533C2212yes—
32#16839532C2211yescharacteristic
33#16839519C2212yes—
34C61×#430C2212yes—
35C2×#108×C61C2211yescharacteristic
36C2×#108×C61C2211yescharacteristic
37C61×#405C2211yescharacteristic
38C61×#344C2212yes—
39C2×#11758222C2311yescharacteristic
40C22×Dic122C2312yes—
41#14519810—24no—
42#14519810—42no—
43C2×Dic244—21no—
44C2×Dic244—21no—
45C2×Dic244—21no—
46C2×Dic244—21no—
47C2×Dic244—22no—
48C2×Dic244D412yes—
49C2×Dic244D412yes—
50C2×#11758190—22no—
51#14519783—21no—
52#14519783—21no—
53#14519783—21no—
54#14519783—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758189—21no—
58C2×C8×D61—21no—
59C4×Dic122—24no—
60#14519742—42no—
61#14519736C2311yescharacteristic
62#14519736C2312yes—
63#14519733—24no—
64#14519711—21no—
65#14519709—21no—
66#14519703—42no—
67#14519702—24no—
68#14519702—42no—
69#14519695—24no—
70#14519695—21no—
71#14519695—42no—
72#14519693—21no—
73#14519691—22no—
74#14519687—21no—
75C8×Dic61—21no—
76C2×M16×C61C2312yes—
77C2×M16×C61C2312yes—
78C2×M16×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80#117×C61C2311yescharacteristic
81#108×C61—21no—
82#108×C61—21no—
83#108×C61—21no—
84#108×C61—21no—
85#108×C61—42no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C22×C4×C61C2411yescharacteristic
89C2×C22⋊D61—21no—
90C22×Dic61—22no—
91#11758222—21no—
92#11758222—21no—
93#11758222—21no—
94#11758222—21no—
95C2×D244#4111yescharacteristic
96C2×C4×D61—21no—
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—22no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122C2×D414yes—
103C2×Dic122#4111yescharacteristic
104M16×C61—22no—
105M16×C61—22no—
106M16×C61—21no—
107M16×C61—22no—
108M16×C61—21no—
109M16×C61—22no—
110C2×C8×C61—21no—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C2×C8×C61#4111yescharacteristic
119C4⋊C4×C61—21no—
120C22⋊C4×C61#4112yes—
121C42×C61—21no—
122C22⋊Dic61—21no—
123C22⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126#11758196—44no—
127C4×Dic61—21no—
128C4×Dic61—22no—
129C4×Dic61—44no—
130#11758194—21no—
131#11758194—21no—
132#11758194—21no—
133#11758194—21no—
134C2×#92418—21no—
135C2×#92418—21no—
136Dic244—22no—
137Dic244—22no—
138Dic244—22no—
139Dic244—22no—
140Dic244—44no—
141#11758190—44no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—41no—
145C8×D61—41no—
146C23×C61C2×#4111yescharacteristic
147C22×D61—21no—
148C2×C4×C61—22no—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C2×#4111yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×C4×C61H(5,𝔽2)11yescharacteristic
154C2×C4×C61H(5,𝔽2)11yescharacteristic
155C22⋊D61—41no—
156C22⋊D61—41no—
157C2×Dic61—21no—
158C2×Dic61—24no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—41no—
163D244—21no—
164D244—21no—
165C4×D61—41no—
166C4×D61—41no—
167Dic122—24no—
168Dic122—24no—
169Dic122—21no—
170Dic122—21no—
171Dic122—42no—
172Dic122—42no—
173C8×C61—42no—
174C8×C61—42no—
175C8×C61—41no—
176C8×C61#12311yescharacteristic
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180#92418—41no—
181#92418—41no—
182#24989—611nomaximal, Sylow
183C22×C61—41no—
184C22×C61#52911yescharacteristic
185C22×C61#54611yescharacteristic
186C22×C61#55511yescharacteristic
187C2×D61—41no—
188C2×D61—81no—
189C4×C61—42no—
190C4×C61C2×#12311yescharacteristic
191C4×C61#54011yescharacteristic
192C4×C61#54611yescharacteristic
193C4×C61#55311yescharacteristic
194Dic61—44no—
195Dic61—41no—
196Dic61—82no—
197Dic61—81no—
198C2×#473—611no—
199#2950—611no—
200#2082—612no—
201#2049—612no—
202#1979—611no—
203#1953D6111yescharacteristic, complemented
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×Q16—611no—
216C2×#131—611no—
217#519—612no—
218#473—1222no—
219#473—1222no—
220#441—1222no—
221C4×Q16—1222no—
222#430C2×D6112yes—
223C2×#108C2×D6111yescharacteristic
224C2×#108C2×D6111yescharacteristic
225#427—611no—
226#421—612no—
227#419—612no—
228#419—612no—
229#413—611no—
230C2×#101—612no—
231C2×#101—612no—
232#410—611no—
233#405—611no—
234#405C2×D6111yescharacteristic
235#344—612no—
236#344—612no—
237#344—611no—
238#344—611no—
239#344C2×D6112yes—
240H(3,ℤ/4)—612no—
241C61#2498911yescharacteristic, complemented, Sylow
242C2×#41—611no—
243C22×Q8—612no—
244#137—1224no—
245#137—2442no—
246C2×Q16—612no—
247C2×Q16—612no—
248C2×Q16—1222no—
249C2×Q16—1221no—
250C2×Q16—1221no—
251C2×Q16—1221no—
252C2×Q16—1221no—
253C2×SD16—1222no—
254#131—1221no—
255#131—1221no—
256#131—1221no—
257#131—1221no—
258C2×M16—611no—
259C2×M16—611no—
260C2×M16—1221no—
261C2×M16C22×D6112yes—
262C2×M16C22×D6112yes—
263C2×M16C22×D6111yescharacteristic
264C22×C8—1221no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—1224no—
267C4×Q8—2442no—
268#117—612no—
269#117—611no—
270#117—1224no—
271#117C22×D6111yescharacteristic
272#108—21no—
273#108—21no—
274#108—21no—
275#108—21no—
276#108—42no—
277#107—1221no—
278#106—1221no—
279#104—2442no—
280#103—1224no—
281#103—2442no—
282#101—1224no—
283#101—1221no—
284#101—2442no—
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—614no—
296C2×Q8—611no—
297C2×Q8—1222no—
298C2×Q8—1222no—
299C2×Q8—1222no—
300C2×Q8—1222no—
301C2×Q8—1222no—
302C2×D4—611no—
303C2×D4—1221no—
304C22×C4—1222no—
305C22×C4—1221no—
306C22×C4C23×D6111yescharacteristic
307Q16—1222no—
308Q16—1222no—
309Q16—1222no—
310Q16—1222no—
311Q16—2444no—
312SD16—2444no—
313M16—22no—
314M16—21no—
315M16—22no—
316M16—21no—
317M16—22no—
318M16—22no—
319M16—1221no—
320M16—1221no—
321M16—1221no—
322M16—1221no—
323M16—2441no—
324M16—2441no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8#1175822211yescharacteristic
333C2×C8#1175822211yescharacteristic
334C2×C8D4×D6111yescharacteristic
335C2×C8D4×D6111yescharacteristic
336C2×C8#1175822611yescharacteristic
337C2×C8#1175822611yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—1222no—
340C4⋊C4—1221no—
341C4⋊C4—2444no—
342C22⋊C4—1221no—
343C22⋊C4—1222no—
344C22⋊C4#1175822612yes—
345C42—21no—
346C42—1221no—
347C42—1222no—
348C42—2444no—
349C23—1221no—
350C23C2×#1175822611yescharacteristic
351Q8—1221no—
352Q8—1224no—
353Q8—1224no—
354Q8—1221no—
355Q8—2442no—
356Q8—2442no—
357D4—1221no—
358D4—1221no—
359D4—2441no—
360D4—2441no—
361C2×C4—22no—
362C2×C4—1221no—
363C2×C4—1224no—
364C2×C4—2441no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4C2×D4×D6111yescharacteristic
371C2×C4#41×D6111yescharacteristic
372C2×C4C2×#1175822211yescharacteristic
373C2×C4C2×#1175822611yescharacteristic
374C2×C4#1451982211yescharacteristic
375C2×C4#1451982311yescharacteristic
376C8—42no—
377C8—41no—
378C8—42no—
379C8—2441no—
380C8—2441no—
381C8#1451976211yescharacteristic
382C8#1451976211yescharacteristic
383C8#1451976211yescharacteristic
384C8#1451976211yescharacteristic
385C22—41no—
386C22—4881no—
387C22—2441no—
388C22#1684029911yescharacteristic
389C22#1684040311yescharacteristic
390C22#1684047811yescharacteristic
391C4—42no—
392C4—4882no—
393C4—4881no—
394C4—2441no—
395C4—2444no—
396C4C2×#1451976211yescharacteristic
397C4#1684036311yescharacteristic
398C4#1684040911yescharacteristic
399C4#1684046511yescharacteristic
400C2—21no—
401C2—21no—
402C2—21no—
403C2—4881no—
404C2#2144931211yescharacteristic
subgroupG/H
centerC2#21449312
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1953C2
socleC2×C61#2420
prime radicalC2×#11758222C23
hypercenter#1953D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #21454688C2semidirect
3. #21447284C2semidirect
5. #21446732C2semidirect
7. #21445944C2semidirect
8. #21445932C2semidirect
11. C61×#1953C2semidirect
203. #1953D61semidirect
241. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400382
#214546882
#214472842
#214471012
#214467322
#214463122
#214459442
#214459322
#214458742
#214456202
C61×#19532
#2498961