Finite Group Databasebeta

← #29592386

Subgroups — #29592386

Order 15616.

354 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447048C211yesmaximal, characteristic
2#21444855C211yesmaximal, characteristic
3#21444821C211yesmaximal, characteristic
4#21443846C211yesmaximal, characteristic, complemented
5#21443825C211yesmaximal, characteristic, complemented
6#21443763C211yesmaximal, characteristic, complemented
7C61×#2769C211yesmaximal, characteristic, complemented
8#16840343C2211yescharacteristic
9#16840080—21no—
10#16840075—21no—
11C2×#14519705—21no—
12#16839761—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#400—21no—
19C61×#400C2211yescharacteristic
20C61×#398—21no—
21C61×#398C2211yescharacteristic
22C61×#389—21no—
23C61×#387—21no—
24C61×#387—21no—
25C61×#383—21no—
26C61×#382C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758194—21no—
29#14519757—41no—
30#14519752—41no—
31#14519751—21no—
32#14519751—21no—
33#14519710—41no—
34#14519708—41no—
35#14519705—42no—
36#14519700—41no—
37#14519697—41no—
38#14519697—41no—
39#14519697—41no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C4⋊C4×C61C2311yescharacteristic
50C2×C42×C61—21no—
51C2×C42×C61D411yescharacteristic
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—41no—
73#11758222—41no—
74#11758222—41no—
75#11758222—41no—
76#11758222—41no—
77C2×D244—21no—
78C2×D244—21no—
79C2×D244—21no—
80C2×C4×D61—41no—
81C2×Dic122—21no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—42no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C42×C61—21no—
92C42×C61—21no—
93C42×C61—42no—
94#11758198—41no—
95#11758198—41no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98#11758194—41no—
99#11758194—41no—
100C2×#92418—41no—
101C2×#92418—41no—
102C2×#92418—41no—
103C23×C61#12011yescharacteristic
104C22×D61—41no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61#13511yescharacteristic
130C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
131C22⋊D61—81no—
132C22⋊D61—81no—
133C2×Dic61—41no—
134D244—41no—
135D244—41no—
136D244—81no—
137D244—81no—
138C4×D61—81no—
139C4×D61—81no—
140Dic122—41no—
141Dic122—41no—
142#92418—81no—
143#92418—81no—
144#92418—81no—
145#12147—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61#44911yescharacteristic
149C22×C61#45411yescharacteristic
150C22×C61B(4,𝔽2)11yescharacteristic
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—42no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161C4×C61—81no—
162C4×C61—81no—
163C4×C61—81no—
164Dic61—81no—
165#2769D6111yescharacteristic, complemented
166#2040—611no—
167#1616—611no—
168#1610—611no—
169#1353—611no—
170#1346—611no—
171#1324—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#159711yescharacteristic
175C2×C61#220111yescharacteristic
176C2×C61#220411yescharacteristic
177D61—161no—
178#537—611no—
179#481—1221no—
180#480—1221no—
181C2×#104—1221no—
182#417—1221no—
183#400—21no—
184#400C2×D6111yescharacteristic
185#398—21no—
186#398C2×D6111yescharacteristic
187#389—21no—
188#387—21no—
189#387—21no—
190#383—21no—
191#382C2×D6111yescharacteristic
192#343—611no—
193#327—1221no—
194#327—1221no—
195#324—611no—
196#324—611no—
197C61#1214711yescharacteristic, complemented, Sylow
198C2×#41—1221no—
199C2×M16—1221no—
200C22⋊Q8—2441no—
201#121—1221no—
202#121—1221no—
203#121—2441no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22⋊D6111yescharacteristic
209C2×C4⋊C4C22⋊D6111yescharacteristic
210C2×C4⋊C4C22⋊D6111yescharacteristic
211C2×C4⋊C4C22×D6111yescharacteristic
212C2×C42—21no—
213C2×C42C22⋊D6111yescharacteristic
214#107—2441no—
215#106—2441no—
216#104—2442no—
217#103—2441no—
218#102—2441no—
219#102—2441no—
220#102—2441no—
221C22⋊C8—1221no—
222C22⋊C8—1221no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C22⋊D6111yescharacteristic
233#95C22⋊D6111yescharacteristic
234#41—2441no—
235#41—2441no—
236#41—2441no—
237#41—2441no—
238C2×Q8—1221no—
239C2×D4—1221no—
240C2×D4—1221no—
241C2×D4—1221no—
242C2×D4—2441no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—2441no—
250C22×C4C2×C22⋊D6111yescharacteristic
251C22×C4C2×C22⋊D6111yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253M16—2441no—
254M16—2441no—
255C2×C8—2441no—
256C2×C8—2441no—
257C2×C8—2441no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C42—21no—
272C42—21no—
273C42—42no—
274C23—2441no—
275C23#1451974611yescharacteristic
276Q8—2441no—
277Q8—2441no—
278D4—4881no—
279D4—4881no—
280D4—4881no—
281D4—4881no—
282D4—2441no—
283D4—2441no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—4881no—
309C2×C4—4881no—
310C2×C4—2441no—
311C2×C4#1451979711yescharacteristic
312C2×C4#1451980211yescharacteristic
313C8—4881no—
314C8—4881no—
315C8—4881no—
316C22—21no—
317C22—21no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22#1683990711yescharacteristic
322C22#1683993511yescharacteristic
323C22#1683994911yescharacteristic
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—4881no—
335C2—21no—
336C2—21no—
337C2—9761no—
338C2#2144474811yescharacteristic
339C2#2144783211yescharacteristic
340C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2769C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2769D61
hypocenterC61#12147
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2769

Complements

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

NH ≅ G/Ntype
4. #21443846C2semidirect
5. #21443825C2semidirect
6. #21443763C2semidirect
7. C61×#2769C2semidirect
165. #2769D61semidirect
197. C61#12147semidirect

Maximal subgroups

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

subgroupindex
#214470482
#214448552
#214448212
#214438462
#214438252
#214437632
C61×#27692
#1214761