Finite Group Databasebeta

← #29710908

Subgroups — #29710908

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454756C211yesmaximal, characteristic, complemented
2#21454722C211yesmaximal, characteristic
3#21447400C211yesmaximal, characteristic, complemented
4#21447394C211yesmaximal, characteristic, complemented
5#21447272C212yesmaximal, complemented
6#21447122C211yesmaximal, characteristic
7#21446991C212yesmaximal, complemented
8#21445664C212yesmaximal, complemented
9#21445622C212yesmaximal, complemented
10#21445575C211yesmaximal, characteristic, complemented
11C61×#2050C211yesmaximal, characteristic
12C2×#14519784C2212yes—
13#16840319C2211yescharacteristic
14#16839891—22no—
15#16839891—22no—
16#16839884—22no—
17#16839879—22no—
18#16839811C2211yescharacteristic
19#16839790C2211yescharacteristic
20#16839790C2211yescharacteristic
21#16839779C2212yes—
22#16839758C2212yes—
23#16839736C2212yes—
24#16839720C2212yes—
25#16839720C2211yescharacteristic
26#16839720C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839534C2212yes—
29#16839534C2212yes—
30#16839526C2212yes—
31#16839522C2212yes—
32#16839519C2212yes—
33#16839519C2211yescharacteristic
34C61×#575C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C2×#104×C61C2212yes—
37C61×#413C2212yes—
38C61×#338C2211yescharacteristic
39#14519784—22no—
40#14519784—22no—
41#14519784—22no—
42#14519784—22no—
43C2×#11758194—22no—
44C2×#11758194C2312yes—
45C22×#92418—22no—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48#14519777—21no—
49#14519769—21no—
50#14519756—22no—
51C4×Dic122—22no—
52#14519736C2312yes—
53#14519736C2311yescharacteristic
54#14519736C2311yescharacteristic
55#14519713—22no—
56#14519706—22no—
57#14519706—22no—
58#14519704—24no—
59#14519701—22no—
60#14519698—22no—
61#14519692—24no—
62#14519688—21no—
63#14519688—21no—
64#14519687—21no—
65#14519687—22no—
66#14519687D412yes—
67C8×Dic61—21no—
68C8×Dic61—22no—
69C8×Dic61D412yes—
70C2×#41×C61C2312yes—
71#137×C61—22no—
72AGL(1,ℤ/8)×C61—22no—
73#135×C61—22no—
74C2×M16×C61C2312yes—
75C2×M16×C61C2311yescharacteristic
76C2×M16×C61C2311yescharacteristic
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—22no—
79#104×C61—22no—
80#104×C61—22no—
81#101×C61—22no—
82#100×C61—22no—
83#41×C61—22no—
84#41×C61—22no—
85#41×C61—22no—
86#41×C61—22no—
87C2×Q8×C61#4112yes—
88C2×D4×C61—22no—
89C2×D4×C61#4112yes—
90C22×C4×C61—21no—
91C22×C4×C61—22no—
92C22×C4×C61C2411yescharacteristic
93C2×Dic122—22no—
94Q16×C61—22no—
95SD16×C61—24no—
96D8×C61—22no—
97M16×C61—22no—
98M16×C61—22no—
99M16×C61—22no—
100M16×C61—22no—
101C2×C8×C61—21no—
102C2×C8×C61—22no—
103C2×C8×C61C2×D412yes—
104C42×C61—21no—
105C42×C61#4111yescharacteristic
106C42×C61#4111yescharacteristic
107C22⋊Dic61—21no—
108C22⋊Dic61—22no—
109C22⋊Dic61#4112yes—
110C4⋊Dic61—22no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—22no—
114#11758196—24no—
115C4×Dic61—21no—
116C4×Dic61—22no—
117C4×Dic61C2×D412yes—
118#11758194—22no—
119#11758194—22no—
120#11758194—42no—
121#11758194—42no—
122C2×#92418—21no—
123C2×#92418—22no—
124C2×#92418—22no—
125C2×#92418—42no—
126C2×#92418—42no—
127C2×#92418C2×D411yescharacteristic
128C2×#92418C2×D411yescharacteristic
129C2×#92418#4112yes—
130C2×#92418#4112yes—
131C23×C61—22no—
132C23×C61C2×#4111yescharacteristic
133Q8×C61—22no—
134Q8×C61—22no—
135D4×C61—22no—
136D4×C61—22no—
137D4×C61—42no—
138D4×C61—42no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—81no—
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C2×#4112yes—
147C2×C4×C61C2×#4112yes—
148C2×C4×C61C2×#4111yescharacteristic
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—42no—
152Dic122—82no—
153C8×C61—22no—
154C8×C61—41no—
155C8×C61—42no—
156#92418—41no—
157#92418—42no—
158#92418#12411yescharacteristic
159#92418#12412yes—
160#92418#12411yescharacteristic
161#23834—611nomaximal, Sylow
162C22×C61—41no—
163C22×C61—42no—
164C22×C61—82no—
165C22×C61#52511yescharacteristic
166C22×C61#52912yes—
167C4×C61—41no—
168C4×C61—41no—
169C4×C61—42no—
170C4×C61#52511yescharacteristic
171C4×C61C2×#12411yescharacteristic
172C4×C61#52912yes—
173Dic61—42no—
174Dic61—42no—
175Dic61—82no—
176#2961—611no—
177#2955—611no—
178#2103—611no—
179#2102—611no—
180#2080—612no—
181#2053—611no—
182#2050D6111yescharacteristic
183#2031—612no—
184#1765—612no—
185#1757—612no—
186#1747—611no—
187C2×C61—21no—
188C2×C61—22no—
189C2×C61—82no—
190C2×C61#239811yescharacteristic
191#575C2×D6111yescharacteristic
192C2×#131—612no—
193#533—611no—
194#446—1222no—
195#446—1222no—
196#444—1222no—
197#443—1222no—
198#430—611no—
199#424—611no—
200#424—611no—
201#423C2×D6111yescharacteristic
202#421—612no—
203C2×#104C2×D6112yes—
204#416—612no—
205#413C2×D6112yes—
206#410—612no—
207#405—611no—
208#405—611no—
209#405—612no—
210C2×#97—611no—
211#344—612no—
212#344—612no—
213#341—612no—
214#339—612no—
215#338C2×D6111yescharacteristic
216H(3,ℤ/4)—611no—
217H(3,ℤ/4)—612no—
218C61#2383411yescharacteristic, complemented, Sylow
219C2×#41C22×D6112yes—
220#137—22no—
221AGL(1,ℤ/8)—22no—
222#135—22no—
223#131—1222no—
224#131—1222no—
225#131—1222no—
226#131—1222no—
227C2×M16—612no—
228C2×M16—1222no—
229C2×M16C22×D6112yes—
230C2×M16C22×D6111yescharacteristic
231C2×M16C22×D6111yescharacteristic
232C22×C8—611no—
233C22×C8—611no—
234C22×C8—1222no—
235C4⋊Q8—1221no—
236#125—1221no—
237C22⋊Q8—1222no—
238C4×Q8—1222no—
239#117—611no—
240#117—611no—
241#117—612no—
242C2×C42C22×D6111yescharacteristic
243#108—1222no—
244C4⋊C8—22no—
245C4⋊C8—1222no—
246C4⋊C8—1222no—
247#104—22no—
248#104—22no—
249#104—1224no—
250#103—1222no—
251#102—1222no—
252#101—22no—
253#100—22no—
254#99—1224no—
255#97—612no—
256#97—1221no—
257#97—1221no—
258#97—1222no—
259#97—1221no—
260C4×C8—612no—
261C4×C8—1222no—
262C4×C8—1221no—
263#41—22no—
264#41—22no—
265#41—22no—
266#41—22no—
267C2×Q8—1222no—
268C2×Q8#1175822412yes—
269C2×D4—22no—
270C2×D4#1175822412yes—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4C23×D6111yescharacteristic
274Q16—22no—
275SD16—24no—
276D8—22no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—1222no—
282M16—1222no—
283M16—2442no—
284M16—2442no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—611no—
288C2×C8—611no—
289C2×C8—612no—
290C2×C8—612no—
291C2×C8—1221no—
292C2×C8—1222no—
293C2×C8—1222no—
294C2×C8—2442no—
295C2×C8—2442no—
296C2×C8D4×D6112yes—
297C4⋊C4—1224no—
298C4⋊C4—1222no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1222no—
302C22⋊C4—612no—
303C22⋊C4—1221no—
304C22⋊C4—1222no—
305C42—21no—
306C42—612no—
307C42—1221no—
308C42—1222no—
309C42#1175822411yescharacteristic
310C42#1175822411yescharacteristic
311C23—22no—
312C23#41×D6111yescharacteristic
313Q8—22no—
314Q8—22no—
315Q8—4882no—
316D4—22no—
317D4—22no—
318D4—42no—
319D4—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—1222no—
327C2×C4—1222no—
328C2×C4—2442no—
329C2×C4C2×D4×D6111yescharacteristic
330C2×C4#41×D6112yes—
331C2×C4C2×#1175822412yes—
332C2×C4C2×#1175822411yescharacteristic
333C8—22no—
334C8—41no—
335C8—42no—
336C8—611no—
337C8—612no—
338C8—611no—
339C8—2441no—
340C8—2442no—
341C22—41no—
342C22—42no—
343C22—82no—
344C22#1684027211yescharacteristic
345C22#1684028912yes—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—4882no—
350C4—2442no—
351C4—2442no—
352C4#1684026811yescharacteristic
353C4#124×D6111yescharacteristic
354C4#1684029312yes—
355C2—21no—
356C2—22no—
357C2—82no—
358C2#2144910711yescharacteristic
subgroupG/H
centerC4#124×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2050C2
socleC2×C61#2398
prime radicalC61×#575C22
hypercenter#2050D61
hypocenterC61#23834
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2050

Complements

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

NH ≅ G/Ntype
1. #21454756C2semidirect
3. #21447400C2semidirect
4. #21447394C2semidirect
5. #21447272C2semidirect
7. #21446991C2semidirect
8. #21445664C2semidirect
9. #21445622C2semidirect
10. #21445575C2semidirect
218. C61#23834semidirect

Maximal subgroups

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

subgroupindex
#214547562
#214547222
#214474002
#214473942
#214472722
#214471222
#214469912
#214456642
#214456222
#214455752
C61×#20502
#2383461