Finite Group Databasebeta

← #29593152

Subgroups — #29593152

Order 15616.

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

normal subgroups only

This group has 372 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#21446986C211yesmaximal, characteristic
2#21444809C211yesmaximal, characteristic
3#21444775C211yesmaximal, characteristic
4#21443840C211yesmaximal, characteristic
5#21443819C211yesmaximal, characteristic
6#21443749C211yesmaximal, characteristic
7C61×#2764C211yesmaximal, characteristic
8#16840388C2211yescharacteristic
9#16840092—21no—
10#16839918—21no—
11#16839625—21no—
12#16839529C2211yescharacteristic
13#16839528—21no—
14#16839498—21no—
15#16839498—21no—
16#16839486C2211yescharacteristic
17#16839486C2211yescharacteristic
18C61×#400—21no—
19C61×#398—21no—
20C61×#397C2211yescharacteristic
21C61×#393C2211yescharacteristic
22C61×#391—21no—
23C61×#388—21no—
24C61×#385—21no—
25C61×#383—21no—
26C61×#381C2211yescharacteristic
27C2×#11758202—41no—
28#14519756—21no—
29#14519756—21no—
30#14519754—41no—
31#14519754—41no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34#14519708—41no—
35#14519701—41no—
36#14519700—41no—
37C22⋊#92418—21no—
38C22⋊#92418—21no—
39#14519684—41no—
40#14519684—41no—
41C22×Q8×C61D411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61C2311yescharacteristic
51C2×C42×C61—21no—
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×C61D411yescharacteristic
61#95×C61D411yescharacteristic
62C24×C61C2×D411yescharacteristic
63C2×Q8×C61—42no—
64C2×Q8×C61—41no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—41no—
80C2×Dic122—21no—
81C2×Dic122—21no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—81no—
84C4⋊C4×C61—81no—
85C4⋊C4×C61—81no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—42no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C42×C61—42no—
92C22⋊Dic61—41no—
93C22⋊Dic61—41no—
94C22⋊Dic61—41no—
95C22⋊Dic61—41no—
96#11758202—82no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99#11758196—41no—
100#11758196—41no—
101C2×#92418—41no—
102C2×#92418—41no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61#12011yescharacteristic
108C23×C61#13511yescharacteristic
109C23×C61#13711yescharacteristic
110Q8×C61—42no—
111Q8×C61—81no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
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—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—81no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—81no—
139C2×Dic61—81no—
140C2×Dic61—81no—
141C2×Dic61—81no—
142C2×Dic61—81no—
143Dic122—81no—
144Dic122—81no—
145#92418—81no—
146#92418—81no—
147#12269—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—81no—
157C22×C61#45211yescharacteristic
158C22×C61#45511yescharacteristic
159C22×C61B(4,𝔽2)11yescharacteristic
160C4×C61—41no—
161C4×C61—42no—
162C4×C61—41no—
163C4×C61—81no—
164C4×C61—81no—
165C4×C61—81no—
166C4×C61—81no—
167C4×C61—81no—
168Dic61—81no—
169Dic61—81no—
170Dic61—161no—
171#2764D6111yescharacteristic
172#2030—611no—
173#1608—611no—
174#1602—611no—
175#1351—611no—
176#1344—611no—
177#1317—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61—41no—
181C2×C61—41no—
182C2×C61#160611yescharacteristic
183C2×C61#219711yescharacteristic
184C2×C61#220111yescharacteristic
185#543—611no—
186#483—1221no—
187#451—1221no—
188#400—21no—
189#398—21no—
190#397C2×D6111yescharacteristic
191#393C2×D6111yescharacteristic
192#391—21no—
193#388—21no—
194#385—21no—
195#383—21no—
196#381—1221no—
197#381C2×D6111yescharacteristic
198#342—611no—
199#342—1221no—
200#328—1221no—
201#328—1221no—
202#323—611no—
203#323—611no—
204C61#1226911yescharacteristic, complemented, Sylow
205C22×Q8C22⋊D6111yescharacteristic
206C23×C4C22⋊D6111yescharacteristic
207C2×Q16—2441no—
208C22⋊Q8—1221no—
209C22⋊Q8—1221no—
210C22⋊Q8—2441no—
211C22⋊Q8—2441no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22⋊D6111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4C22⋊D6111yescharacteristic
221C2×C22⋊C4C22×D6111yescharacteristic
222C2×C42—21no—
223#106—2441no—
224#103—2441no—
225#103—2441no—
226C22⋊C8—1221no—
227C22⋊C8—1221no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—2441no—
237#95—2441no—
238#95C22⋊D6111yescharacteristic
239#95C22⋊D6111yescharacteristic
240C24C2×C22⋊D6111yescharacteristic
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—1221no—
244C2×Q8—1221no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—2441no—
258C22×C4C2×C22⋊D6111yescharacteristic
259C22×C4C2×C22⋊D6111yescharacteristic
260Q16—4882no—
261C2×C8—2441no—
262C2×C8—2441no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C4⋊C4—2441no—
270C4⋊C4—2441no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2441no—
279C22⋊C4—2441no—
280C42—42no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23#1451974611yescharacteristic
286C23#1451979711yescharacteristic
287C23#1451980711yescharacteristic
288Q8—42no—
289Q8—81no—
290Q8—4881no—
291Q8—4881no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
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—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—4881no—
317C2×C4—4881no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4881no—
321C2×C4—2441no—
322C2×C4—2441no—
323C8—4881no—
324C8—4881no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22#1683992411yescharacteristic
335C22#1683994011yescharacteristic
336C22#1683994911yescharacteristic
337C4—41no—
338C4—42no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—9761no—
346C4—4881no—
347C4—4881no—
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2#2144479911yescharacteristic
353C2#2144780911yescharacteristic
354C2#2144783211yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2764C2
socleC22×C61B(4,𝔽2)
prime radicalC24×C61C2×D4
hypercenter#2764D61
hypocenterC61#12269
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2764

Complements

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

NH ≅ G/Ntype
204. C61#12269semidirect

Maximal subgroups

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

subgroupindex
#214469862
#214448092
#214447752
#214438402
#214438192
#214437492
C61×#27642
#1226961