Finite Group Databasebeta

← #29592599

Subgroups — #29592599

Order 15616.

362 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 57 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#21454085C211yesmaximal, characteristic, complemented
2#21447040C211yesmaximal, characteristic
3#21444856C211yesmaximal, characteristic
4#21443849C211yesmaximal, characteristic, complemented
5#21443847C211yesmaximal, characteristic, complemented
6#21443764C211yesmaximal, characteristic, complemented
7C61×#1604C211yesmaximal, characteristic
8#16840078—21no—
9C2×#14519704—21no—
10#16839763—21no—
11#16839703—21no—
12#16839701C2211yescharacteristic
13#16839696—21no—
14#16839695—21no—
15#16839693—21no—
16#16839681—21no—
17#16839631C2211yescharacteristic
18#16839629—21no—
19#16839531C2211yescharacteristic
20#16839496—21no—
21#16839488C2211yescharacteristic
22C61×#537C2211yescharacteristic
23C61×#448—21no—
24C61×#395C2211yescharacteristic
25C61×#327—21no—
26C61×#324C2211yescharacteristic
27C2×#11758194—21no—
28C2×C4⋊Dic61D411yescharacteristic
29C2×C4⋊Dic61D411yescharacteristic
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×C4×Dic61D411yescharacteristic
36#14519708—41no—
37#14519704—42no—
38#14519701—41no—
39#14519698—41no—
40#14519698—41no—
41C22⋊#92418—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684D411yescharacteristic
50C2×#41×C61—21no—
51C22×Q8×C61D411yescharacteristic
52C2×SD16×C61—41no—
53C22⋊Q8×C61—41no—
54#121×C61—21no—
55#121×C61—21no—
56#121×C61—41no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61C2311yescharacteristic
59#103×C61—41no—
60C22⋊C8×C61—21no—
61#95×C61—21no—
62#95×C61D411yescharacteristic
63#41×C61—41no—
64#41×C61—41no—
65#41×C61—41no—
66#41×C61—41no—
67C2×Q8×C61—21no—
68C2×Q8×C61—41no—
69C2×Q8×C61—81no—
70C2×D4×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—21no—
73C2×D4×C61—41no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—41no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84SD16×C61—82no—
85C2×C8×C61—41no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—81no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94#11758196—81no—
95#11758196—81no—
96#11758196—81no—
97#11758196—81no—
98#11758196—81no—
99C4×Dic61—21no—
100C4×Dic61SD1612yescomplemented
101#11758194—41no—
102#11758194—41no—
103C2×#92418—41no—
104C2×#92418—41no—
105C23×C61—41no—
106C23×C61#12011yescharacteristic
107Q8×C61—41no—
108Q8×C61—41no—
109Q8×C61—42no—
110Q8×C61—81no—
111D4×C61—41no—
112D4×C61—41no—
113D4×C61—81no—
114D4×C61—81no—
115D4×C61—81no—
116D4×C61—81no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
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—41no—
130C2×C4×C61—81no—
131C2×C4×C61—81no—
132C2×C4×C61C2×SD1611yescharacteristic
133C2×C4×C61#13711yescharacteristic
134C2×Dic61—22no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C8×C61—81no—
146#92418—81no—
147#92418—81no—
148#12183—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61#44811yescharacteristic
155C22×C61#45411yescharacteristic
156C22×C61B(4,𝔽2)11yescharacteristic
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—81no—
162C4×C61—81no—
163C4×C61—81no—
164Dic61—42no—
165Dic61—81no—
166Dic61—81no—
167Dic61—81no—
168#2843—611no—
169#2039—611no—
170#1616—611no—
171#1604D6111yescharacteristic
172#1354—611no—
173#1353—611no—
174#1325—611no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—161no—
178C2×C61#159911yescharacteristic
179C2×C61#220011yescharacteristic
180C2×C61#220311yescharacteristic
181#537C2×D6111yescharacteristic
182#481—1221no—
183#448—21no—
184C2×#104—1221no—
185#417—1221no—
186#400—611no—
187#400—1221no—
188#398—1221no—
189#398—1221no—
190#398—1221no—
191#395—1221no—
192#395C2×D6111yescharacteristic
193#384—611no—
194#383—1221no—
195#343—611no—
196#327—21no—
197#327—1221no—
198#324—611no—
199#324C2×D6111yescharacteristic
200C61#1218311yescharacteristic, complemented, Sylow
201C2×#41—21no—
202C22×Q8C22⋊D6111yescharacteristic
203C2×SD16—41no—
204C2×M16—1221no—
205C22⋊Q8—41no—
206#121—21no—
207#121—21no—
208#121—41no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4C22×D6111yescharacteristic
218C2×C42—611no—
219#106—2441no—
220#104—2442no—
221#103—41no—
222#103—2441no—
223#102—2441no—
224#102—2441no—
225C22⋊C8—21no—
226C22⋊C8—1221no—
227#95—21no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95—1221no—
236#95C22⋊D6111yescharacteristic
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×Q8—81no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4D4×D6111yescharacteristic
256C22×C4D4×D6111yescharacteristic
257C22×C4C2×C22⋊D6111yescharacteristic
258SD16—82no—
259M16—2441no—
260M16—2441no—
261C2×C8—41no—
262C2×C8—2441no—
263C2×C8—2441no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—4881no—
269C4⋊C4—4881no—
270C4⋊C4—4881no—
271C4⋊C4—4881no—
272C4⋊C4—4881no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—612no—
278C42—1221no—
279C23—41no—
280C23#1451974511yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—42no—
284Q8—81no—
285D4—41no—
286D4—41no—
287D4—81no—
288D4—81no—
289D4—81no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
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—81no—
305C2×C4—81no—
306C2×C4—1222no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4SD16×D6111yescharacteristic
318C2×C4#1451980611yescharacteristic
319C8—81no—
320C8—4881no—
321C8—4881no—
322C22—21no—
323C22—21no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#1683989911yescharacteristic
328C22#1683993311yescharacteristic
329C22#1683994811yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—4881no—
338C4—4881no—
339C4—4881no—
340C4—2442no—
341C2—21no—
342C2—21no—
343C2—161no—
344C2#2144475711yescharacteristic
345C2#2144782411yescharacteristic
346C2#2144783811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1604C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#1604D61
hypocenterC61#12183
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1604

Complements

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

NH ≅ G/Ntype
1. #21454085C2semidirect
4. #21443849C2semidirect
5. #21443847C2semidirect
6. #21443764C2semidirect
100. C4×Dic61SD16semidirect
200. C61#12183semidirect

Maximal subgroups

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

subgroupindex
#214540852
#214470402
#214448562
#214438492
#214438472
#214437642
C61×#16042
#1218361