Finite Group Databasebeta

← #29581443

Subgroups — #29581443

Order 15616.

474 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 85 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#21457021C211yesmaximal, characteristic
2#21451939C211yesmaximal, characteristic
3#21444536C211yesmaximal, characteristic
4#21444420C211yesmaximal, characteristic
5#21444331C211yesmaximal, characteristic
6#21443713C211yesmaximal, characteristic
7C61×#3165C211yesmaximal, characteristic
8C2×#14519769C2211yescharacteristic
9#16840168—21no—
10#16840130—21no—
11#16840095—21no—
12#16840090—21no—
13#16840069—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16#16839693—22no—
17#16839681—22no—
18#16839631—21no—
19#16839627—21no—
20#16839608C2211yescharacteristic
21#16839508—22no—
22#16839503—22no—
23#16839499—22no—
24C61×#533C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#501—22no—
27C61×#500—21no—
28C61×#498—21no—
29C61×#428—22no—
30C4×M16×C61C2211yescharacteristic
31#14519769—21no—
32#14519769—21no—
33#14519769—21no—
34#14519769—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×#11758196—24no—
40C2×C4×Dic61—21no—
41#14519710—22no—
42#14519708—22no—
43C4⋊#92418—21no—
44C4⋊#92418D412yes—
45#14519701—22no—
46#14519701—22no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49#14519688—21no—
50#14519688—21no—
51#14519684—24no—
52#14519684—21no—
53#14519684D412yes—
54C22×Q8×C61—21no—
55C2×M16×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61D411yescharacteristic
61C4⋊Q8×C61D411yescharacteristic
62#125×C61—21no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67#117×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D412yes—
70C2×C42×C61C2311yescharacteristic
71#107×C61—24no—
72#106×C61—24no—
73#97×C61D412yes—
74C4×C8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—82no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—22no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87M16×C61—24no—
88C2×C8×C61—22no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61#4111yescharacteristic
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109#11758196—41no—
110#11758196—41no—
111#11758196—41no—
112#11758196—41no—
113#11758196—84no—
114C4×Dic61—24no—
115C2×#92418—22no—
116C2×#92418—41no—
117C23×C61#12011yescharacteristic
118Q8×C61—42no—
119Q8×C61—82no—
120Q8×C61—82no—
121Q8×C61—81no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—44no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12711yescharacteristic
135C2×Dic61—24no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—42no—
142C8×C61—24no—
143#92418—42no—
144#92418—81no—
145#10101—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61#39211yescharacteristic
149C22×C61#45612yes—
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—81no—
156C4×C61—82no—
157C4×C61—81no—
158C4×C61#45612yes—
159Dic61—44no—
160Dic61—81no—
161Dic61—81no—
162#3236—611no—
163#3165D6111yescharacteristic
164#2660—611no—
165#1556—611no—
166#1528—611no—
167#1506—611no—
168#1300—611no—
169C2×C61—22no—
170C2×C61#200511yescharacteristic
171C2×C61#200511yescharacteristic
172C2×C61#200811yescharacteristic
173#533C2×D6111yescharacteristic
174C2×C4⋊Q8C2×D6111yescharacteristic
175C2×#125—611no—
176#501—22no—
177#501—1221no—
178#500—21no—
179#498—21no—
180#491—1221no—
181#484—1221no—
182#483—1221no—
183#479—1222no—
184#432—611no—
185#432—611no—
186#428—22no—
187C4×M16C2×D6111yescharacteristic
188#398—1222no—
189#395—1222no—
190#384—1221no—
191#382—1221no—
192#376—611no—
193#332—1222no—
194#330—1222no—
195#328—1222no—
196C61#1010111yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C2×M16—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C22⋊D6111yescharacteristic
204C4⋊Q8C22⋊D6111yescharacteristic
205#125—21no—
206#125—1221no—
207#125—1221no—
208#125—1221no—
209#125—1221no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214#117—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—611no—
218C2×C4⋊C4—1224no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4C22⋊D6112yes—
222C2×C42—1221no—
223C2×C42C22×D6111yescharacteristic
224#107—24no—
225#107—1222no—
226#106—24no—
227#106—1222no—
228C4⋊C8—612no—
229C4⋊C8—1221no—
230#103—1222no—
231#103—1222no—
232C22⋊C8—1221no—
233C22⋊C8—1221no—
234#97—1221no—
235#97—1221no—
236#97C22⋊D6112yes—
237C4×C8—21no—
238#95—612no—
239#95—1224no—
240#95—1221no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—1222no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4D4×D6111yescharacteristic
251C22×C4D4×D6111yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253M16—24no—
254C2×C8—22no—
255C2×C8—1222no—
256C2×C8—2441no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—4884no—
272C4⋊C4—2441no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C22⋊C4—41no—
277C42—41no—
278C42—1224no—
279C42D4×D6111yescharacteristic
280C42#1175822411yescharacteristic
281C42C2×C22⋊D6111yescharacteristic
282C42C2×C22⋊D6111yescharacteristic
283C23#1451974511yescharacteristic
284Q8—42no—
285Q8—81no—
286Q8—82no—
287Q8—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—1224no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4#1451974511yescharacteristic
303C2×C4#1451974611yescharacteristic
304C2×C4#1451974811yescharacteristic
305C2×C4#1451974911yescharacteristic
306C2×C4#1451974911yescharacteristic
307C2×C4#1451977311yescharacteristic
308C8—24no—
309C8—4881no—
310C8—2442no—
311C22—21no—
312C22—21no—
313C22#1683967011yescharacteristic
314C22#1683994412yes—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—81no—
321C4—82no—
322C4—81no—
323C4—4881no—
324C4—4881no—
325C4—2444no—
326C4#1683994312yes—
327C2—22no—
328C2#2144685211yescharacteristic
329C2#2144685511yescharacteristic
330C2#2144687011yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3165C2
socleC22×C61#392
prime radicalC23×C61#120
hypercenter#3165D61
hypocenterC61#10101
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
196. C61#10101semidirect

Maximal subgroups

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

subgroupindex
#214570212
#214519392
#214445362
#214444202
#214443312
#214437132
C61×#31652
#1010161