Finite Group Databasebeta

← #29689182

Subgroups — #29689182

Order 15616.

1270 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 231 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#21459464C211yesmaximal, characteristic
2#21454295C212yesmaximal, complemented
3#21446370C214yesmaximal, complemented
4#21446236C214yesmaximal, complemented
5#21445930C211yesmaximal, characteristic, complemented
6#21445744C212yesmaximal, complemented
7C61×#1829C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9C22×#11758224C2211yescharacteristic
10#16840528C2214yes—
11#16840416—24no—
12#16840407—24no—
13#16840338C2212yes—
14#16840324C414yes—
15#16840274C2212yes—
16C2×#14519761C414yes—
17#16839744C2214yes—
18#16839743—24no—
19C2×#14519695C2214yescomplemented
20C2×#14519694C2212yes—
21#100×D61—24no—
22#16839725C2214yescomplemented
23#16839532C2214yescomplemented
24C61×#518C2211yescharacteristic
25C2×#100×C61C2212yes—
26C61×#344C2214yescomplemented
27#14519822—44no—
28C2×#11758222C2312yes—
29C2×#11758224—24no—
30C2×#11758224—44no—
31C2×#11758224D414yes—
32C22×Dic122C2311yescharacteristic
33C22×C22⋊D61C2×C414yes—
34C22×C22⋊D61C2×C412yes—
35C2×D4×D61D414yes—
36C23×Dic61C2×C412yes—
37C22×C4×D61C2311yescharacteristic
38#14519785—44no—
39#14519783—44no—
40C2×#11758194C2314yes—
41M16×D61—28no—
42#14519777—22no—
43#14519765—22no—
44#14519761—28no—
45#14519761—24no—
46#14519761C2×C414yes—
47#14519761C2×C418yes—
48#14519757—24no—
49#14519757—22no—
50#14519752—22no—
51#14519751—28no—
52#14519747—24no—
53#14519747C2×C414yes—
54C4×D244—24no—
55C2×C4⋊Dic61—24no—
56C2×#11758198—24no—
57#14519695—24no—
58#14519695—44no—
59#14519694—24no—
60#14519693—24no—
61#14519689—44no—
62C22×D4×C61C2311yescharacteristic
63C2×M16×C61C2314yes—
64C4×D4×C61—24no—
65#117×C61C2312yes—
66C2×C22⋊C4×C61C2×C414yes—
67#100×C61—24no—
68C24×C61C22×C412yes—
69C23×D61C22⋊C414yes—
70C23×D61C22×C411yescharacteristic
71C2×D4×C61—24no—
72C2×D4×C61C2×D414yes—
73C22×C4×C61—24no—
74C22×C4×C61C2411yescharacteristic
75C2×C22⋊D61—24no—
76C2×C22⋊D61—44no—
77C2×C22⋊D61—48no—
78C2×C22⋊D61C22⋊C418yes—
79C2×C22⋊D61C22×C414yes—
80C22×Dic61—24no—
81C22×Dic61—44no—
82C22×Dic61C22⋊C414yes—
83C22×Dic61C22×C411yescharacteristic
84C22×Dic61C22×C412yes—
85#11758224—48no—
86#11758224—416no—
87D4×D61—48no—
88#11758222—24no—
89#11758222—24no—
90C2×D244—24no—
91C2×D244C2×D412yes—
92C2×C4×D61—22no—
93C2×C4×D61—24no—
94C2×C4×D61C2×D414yes—
95C2×Dic122—22no—
96C2×Dic122—24no—
97C2×Dic122C2×D412yes—
98M16×C61—24no—
99M16×C61—24no—
100C2×C8×C61—24no—
101C4⋊C4×C61—22no—
102C22⋊C4×C61—24no—
103C22⋊C4×C61C22×C414yes—
104C42×C61—22no—
105#11758198—24no—
106#11758198—28no—
107#11758198—24no—
108C4⋊Dic61—24no—
109C4⋊Dic61—28no—
110#11758194—24no—
111#11758194—24no—
112C2×#92418—24no—
113#11758189—48no—
114C8×D61—48no—
115C23×C61—24no—
116C23×C61—82no—
117C23×C61C2×C22⋊C414yes—
118C23×C61C23×C411yescharacteristic
119C22×D61—21no—
120C22×D61—22no—
121C22×D61—44no—
122C22×D61—81no—
123C22×D61—84no—
124C22×D61C2×C22⋊C414yes—
125D4×C61—28no—
126C2×C4×C61—24no—
127C2×C4×C61—84no—
128C2×C4×C61C22×D412yes—
129C2×C4×C61H(5,𝔽2)14yes—
130C22⋊D61—24no—
131C22⋊D61—24no—
132C22⋊D61—48no—
133C22⋊D61—48no—
134C2×Dic61—22no—
135C2×Dic61—21no—
136C2×Dic61—24no—
137C2×Dic61—22no—
138C2×Dic61—44no—
139C2×Dic61—48no—
140C2×Dic61C2×C22⋊C414yes—
141D244—24no—
142D244—84no—
143C4×D61—28no—
144C4×D61—44no—
145Dic122—24no—
146Dic122—44no—
147C8×C61—44no—
148#92418—44no—
149#21779—611nomaximal, Sylow
150C22×C61—24no—
151C22×C61—41no—
152C22×C61—44no—
153C22×C61—84no—
154C22×C61C22×C22⋊C411yescharacteristic
155C22×C61#51812yes—
156C2×D61—24no—
157C2×D61—44no—
158C2×D61—42no—
159C2×D61—44no—
160C2×D61—41no—
161C2×D61—164no—
162C4×C61—44no—
163C4×C61#53414yes—
164Dic61—24no—
165Dic61—42no—
166Dic61—41no—
167#3485—611no—
168#2888—612no—
169#1915—614no—
170#1891—614no—
171#1829—611no—
172#1829D6111yescharacteristic, complemented
173#1785—612no—
174C2×C61—21no—
175C2×C61—22no—
176C2×C61—44no—
177C2×C61#230811yescharacteristic
178D61—44no—
179D61—81no—
180C2×H(5,𝔽2)—611no—
181C22×#41—611no—
182#568—614no—
183#547—1224no—
184#546—1224no—
185#536—612no—
186#534—614no—
187#525—612no—
188C2×#123—614no—
189#518C2×D6111yescharacteristic
190#413—614no—
191#413—1224no—
192C2×#101—614no—
193C2×#100—612no—
194C2×#100—1224no—
195C2×#100C2×D6112yes—
196#407—614no—
197#344—614no—
198#344C2×D6114yescomplemented
199C61#2177911yescharacteristic, complemented, Sylow
200H(5,𝔽2)—2444no—
201C2×#41—614no—
202C2×#41—612no—
203C2×#41—1224no—
204C2×#41—2444no—
205C22×Q8—611no—
206C22×D4—614no—
207C22×D4—614no—
208C22×D4—612no—
209C22×D4C22×D6111yescharacteristic
210C23×C4—612no—
211C23×C4—611no—
212#131—2444no—
213#131—2444no—
214C2×M16—614no—
215C2×M16—1228no—
216C2×M16C22×D6114yes—
217C4⋊Q8—1222no—
218#124—1222no—
219#123—614no—
220#123—618no—
221#123—1224no—
222#123—1228no—
223C22⋊Q8—1224no—
224C22⋊Q8—1222no—
225#121—1222no—
226#121—1228no—
227#120—614no—
228#120—1224no—
229C4×D4—24no—
230C4×D4—1224no—
231#117C22×D6112yes—
232C2×C4⋊C4—1224no—
233C2×C22⋊C4—1224no—
234C2×C22⋊C4C4×D6114yes—
235#101—1224no—
236#101—2444no—
237#100—24no—
238#100—1224no—
239#100—1224no—
240C22⋊C8—2444no—
241C24—614no—
242C24—611no—
243C24C2×C4×D6112yes—
244#41—1224no—
245#41—1224no—
246#41—2448no—
247#41—24416no—
248C2×Q8—612no—
249C2×Q8—1222no—
250C2×Q8—1224no—
251C2×D4—24no—
252C2×D4—614no—
253C2×D4—618no—
254C2×D4—612no—
255C2×D4—1224no—
256C2×D4—1224no—
257C2×D4—2448no—
258C2×D4—2444no—
259C2×D4—2448no—
260C2×D4D4×D6114yes—
261C22×C4—24no—
262C22×C4—614no—
263C22×C4—614no—
264C22×C4—611no—
265C22×C4—612no—
266C22×C4—1224no—
267C22×C4—1222no—
268C22×C4—1224no—
269C22×C4—2444no—
270C22×C4C23×D6111yescharacteristic
271M16—24no—
272M16—24no—
273M16—1224no—
274M16—1224no—
275M16—2448no—
276C2×C8—24no—
277C2×C8—1224no—
278C2×C8—2448no—
279C4⋊C4—22no—
280C4⋊C4—1224no—
281C4⋊C4—1228no—
282C22⋊C4—24no—
283C22⋊C4—1224no—
284C22⋊C4—1224no—
285C22⋊C4—1228no—
286C22⋊C4C2×C4×D6114yes—
287C42—22no—
288C23—24no—
289C23—82no—
290C23—614no—
291C23—1221no—
292C23—1222no—
293C23—4884no—
294C23—4881no—
295C23—2444no—
296C23C22⋊C4×D6114yes—
297C23C22×C4×D6111yescharacteristic
298Q8—1224no—
299Q8—2444no—
300D4—28no—
301D4—1224no—
302D4—1224no—
303D4—1224no—
304D4—4884no—
305D4—2448no—
306D4—2448no—
307C2×C4—24no—
308C2×C4—84no—
309C2×C4—614no—
310C2×C4—1228no—
311C2×C4—1221no—
312C2×C4—1222no—
313C2×C4—1222no—
314C2×C4—1224no—
315C2×C4—2444no—
316C2×C4—2448no—
317C2×C4—2444no—
318C2×C4C2×D4×D6112yes—
319C2×C4#1451982314yes—
320C8—44no—
321C8—2444no—
322C22—24no—
323C22—41no—
324C22—44no—
325C22—84no—
326C22—1224no—
327C22—9764no—
328C22—2442no—
329C22—2444no—
330C22—2444no—
331C22—2441no—
332C22C2×C22⋊C4×D6111yescharacteristic
333C22#1684023612yes—
334C4—44no—
335C4—1224no—
336C4—2441no—
337C4—2442no—
338C4#1684032314yes—
339C2—21no—
340C2—22no—
341C2—44no—
342C2—4881no—
343C2—2444no—
344C2#2144839611yescharacteristic
subgroupG/H
centerC2#21448396
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1829C2
socleC2×C61#2308
prime radical#21459464C2
cosocleC22×C4×C61C24
hypercenter#1829D61
hypocenterC61#21779
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1829

Complements

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

NH ≅ G/Ntype
2. #21454295C2semidirect
3. #21446370C2semidirect
4. #21446236C2semidirect
5. #21445930C2semidirect
6. #21445744C2semidirect
7. C61×#1829C2semidirect
19. C2×#14519695C22semidirect
22. #16839725C22semidirect
23. #16839532C22semidirect
26. C61×#344C22semidirect
172. #1829D61semidirect
198. #344C2×D61semidirect
199. C61#21779semidirect

Maximal subgroups

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

subgroupindex
#214594642
#214542952
#214463702
#214462362
#214459302
#214457442
C61×#18292
#2177961