Finite Group Databasebeta

← #29577665

Subgroups — #29577665

Order 15616.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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#21454457C211yesmaximal, characteristic, complemented
2#21450277C211yesmaximal, characteristic
3#21444347C212yesmaximal
4#21443671C212yesmaximal, complemented
5C61×#1820C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7#16839823—24no—
8#16839786—21no—
9#16839726C2212yes—
10C4×#11758194—21no—
11#16839677—21no—
12#16839677—21no—
13#16839663C411yescharacteristic, complemented
14#16839663C411yescharacteristic, complemented
15#16839653—21no—
16#16839653—21no—
17#16839631C411yescharacteristic, complemented
18#16839627C411yescharacteristic, complemented
19C4×C22⋊Dic61C2211yescharacteristic
20#16839495—24no—
21C2×#117×C61C2211yescharacteristic
22C61×#381C2212yes—
23C61×#342—21no—
24C61×#342—21no—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C2×#11758194—22no—
28#14519751—22no—
29#14519751—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61C2×C411yescharacteristic
41C4⋊#92418—24no—
42C22⋊#92418—22no—
43C22⋊#92418D414yes—
44#14519688—22no—
45C4×#92418—22no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C23×C4×C61C2311yescharacteristic
50#117×C61—22no—
51#117×C61—22no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59#95×C61—22no—
60#95×C61—22no—
61C24×C61C2×D411yescharacteristic
62C23×D61—21no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D412yes—
73C22×C4×C61#4112yes—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—81no—
76C2×C22⋊D61—81no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C2×D244—22no—
81C2×D244—41no—
82C2×D244—41no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—22no—
86C22⋊C4×C61—22no—
87C22⋊C4×C61—42no—
88C22⋊C4×C61—42no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—21no—
92C42×C61—21no—
93C22⋊Dic61—42no—
94#11758198—42no—
95#11758198—42no—
96#11758198—41no—
97#11758198—41no—
98#11758198—81no—
99#11758198—81no—
100C4⋊Dic61—22no—
101#11758196—81no—
102#11758196—81no—
103C4×Dic61—41no—
104C4×Dic61—41no—
105#11758194—42no—
106#11758194—42no—
107C2×#92418—24no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61C2×C22⋊C411yescharacteristic
113C23×C61#12112yes—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—41no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—82no—
136C2×C4×C61—82no—
137C2×C4×C61—82no—
138C2×C4×C61C4×D412yes—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61#12311yescharacteristic
141C22⋊D61—84no—
142C2×Dic61—22no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151D244—82no—
152#92418—44no—
153#9446—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61#38611yescharacteristic
162C22×C61#41011yescharacteristic
163C22×C61#41011yescharacteristic
164C2×D61—81no—
165C2×D61—81no—
166C2×D61—81no—
167C2×D61—81no—
168C2×D61—81no—
169C2×D61—81no—
170C2×D61—81no—
171C4×C61—21no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—41no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61—41no—
178C4×C61—162no—
179Dic61—42no—
180Dic61—81no—
181Dic61—81no—
182#2921—611no—
183#2500—611no—
184#1820D6111yescharacteristic, complemented
185#1510—612no—
186#1284—612no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—42no—
190C2×C61#188411yescharacteristic
191C2×C61#189911yescharacteristic
192C2×C61#191011yescharacteristic
193D61—161no—
194C2×#121—611no—
195C2×#117C2×D6111yescharacteristic
196#433—1224no—
197#423—1221no—
198#407—612no—
199C4×M16—1221no—
200#394—1221no—
201#394—1221no—
202#390—611no—
203#390—611no—
204#388—1221no—
205#388—1221no—
206#384—611no—
207#382—611no—
208#381C2×D6112yes—
209C4×C22⋊C4—611no—
210#342—21no—
211#342—21no—
212#327—1224no—
213C61#944611yescharacteristic, complemented, Sylow
214C22×D4—611no—
215C22×D4—1221no—
216C23×C4C22×D6111yescharacteristic
217C2×M16—1222no—
218#121—1222no—
219#121—1222no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4C4×D6111yescharacteristic
226C2×C4⋊C4C4×D6111yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4C22⋊D6111yescharacteristic
235C2×C22⋊C4C22⋊D6111yescharacteristic
236C2×C42—611no—
237C2×C42—611no—
238C2×C42D24411yescharacteristic
239C2×C42D24411yescharacteristic
240C4⋊C8—1224no—
241C22⋊C8—614no—
242C22⋊C8—1222no—
243#97—1222no—
244C4×C8—1222no—
245#95—22no—
246#95—22no—
247#95—1221no—
248#95—1221no—
249#95—1221no—
250C24—1221no—
251C24C2×C22⋊D6111yescharacteristic
252C2×D4—1222no—
253C2×D4—4881no—
254C2×D4—4881no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2442no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4#1175819811yescharacteristic
265C22×C4#1175819811yescharacteristic
266C22×C4#1175819811yescharacteristic
267C22×C4#1175819811yescharacteristic
268C22×C4C2×C4×D6111yescharacteristic
269C22×C4C2×D24411yescharacteristic
270C22×C4D4×D6112yes—
271C22×C4#1175822412yes—
272M16—2442no—
273M16—2442no—
274C2×C8—1224no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—1222no—
278C4⋊C4—4881no—
279C4⋊C4—4881no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—4881no—
285C22⋊C4—4881no—
286C22⋊C4—2442no—
287C22⋊C4—2442no—
288C22⋊C4—2442no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—2441no—
296C42—2441no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—2441no—
302C23—2441no—
303C23—2441no—
304C23—2441no—
305C23—2441no—
306C23—2441no—
307C23—2441no—
308C23C2×#1175819811yescharacteristic
309C23#1451974912yes—
310D4—4882no—
311D4—4884no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—1222no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4#1451973712yes—
339C2×C4#1451974711yescharacteristic
340C2×C4#1451976111yescharacteristic
341C8—2444no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—4881no—
350C22—4881no—
351C22—4881no—
352C22—4881no—
353C22—4881no—
354C22—4881no—
355C22—4881no—
356C22#1683963911yescharacteristic
357C22#1683973511yescharacteristic
358C22#1683973511yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—162no—
367C4—4881no—
368C4—4881no—
369C4—2442no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—9761no—
374C2#2144620011yescharacteristic
375C2#2144627511yescharacteristic
376C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1820C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1820D61
hypocenterC61#9446
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
1. #21454457C2semidirect
4. #21443671C2semidirect
5. C61×#1820C2semidirect
13. #16839663C4semidirect
14. #16839663C4semidirect
17. #16839631C4semidirect
18. #16839627C4semidirect
184. #1820D61semidirect
213. C61#9446semidirect

Maximal subgroups

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

subgroupindex
#214544572
#214502772
#214443472
#214436712
C61×#18202
#944661