Finite Group Databasebeta

← #29600410

Subgroups — #29600410

Order 15616.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456891C211yesmaximal, characteristic
2#21447488C211yesmaximal, characteristic
3#21443995C212yesmaximal, complemented
4#21443980C212yesmaximal, complemented
5C61×#2134C211yesmaximal, characteristic
6#16840374C411yescharacteristic, complemented
7#16840157C2211yescharacteristic
8#16840134C411yescharacteristic, complemented
9#16840126C411yescharacteristic, complemented
10#16840116C2211yescharacteristic
11#16839981—21no—
12#16839959—21no—
13#16839924—21no—
14#16839913—21no—
15#16839823C411yescharacteristic, complemented
16C2×#14519692—21no—
17#16839557C2212yes—
18#16839555—21no—
19C61×#535C2211yescharacteristic
20C61×#413—21no—
21C61×#355—21no—
22C61×#355C2212yes—
23C2×#11758202—21no—
24C2×#11758200—21no—
25C2×#11758200—21no—
26C2×#11758200—21no—
27#14519777C2×C411yescharacteristic, complemented
28#14519770—41no—
29#14519769C2×C411yescharacteristic, complemented
30#14519767—41no—
31#14519756—21no—
32C4×Dic122—21no—
33C2×C22⋊Dic61—41no—
34C4⋊#92418—21no—
35#14519701—21no—
36#14519698—21no—
37#14519698—21no—
38#14519698—21no—
39#14519692—22no—
40#14519692—41no—
41C22⋊#92418—21no—
42#14519688C2×C411yescharacteristic, complemented
43C4×#92418C2×C411yescharacteristic, complemented
44C2×#41×C61D411yescharacteristic
45C22×D4×C61D411yescharacteristic
46C2×M16×C61—41no—
47#127×C61C2×C411yescharacteristic
48#127×C61D411yescharacteristic
49#124×C61D411yescharacteristic
50#124×C61C2311yescharacteristic
51#121×C61—41no—
52#120×C61—41no—
53#117×C61C2×C411yescharacteristic
54#101×C61—22no—
55#100×C61—41no—
56C24×C61—21no—
57#41×C61—81no—
58C2×Q8×C61C2×D411yescharacteristic
59C2×D4×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61—81no—
64C2×D4×C61C22⋊C411yescharacteristic
65C2×D4×C61C22⋊C411yescharacteristic
66C2×D4×C61C2×D411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×Dic61—81no—
70C2×Dic122—21no—
71M16×C61—42no—
72M16×C61—81no—
73C2×C8×C61—81no—
74C4⋊C4×C61—21no—
75C22⋊C4×C61—21no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C42×C61C22⋊C411yescharacteristic
79C42×C61C22×C411yescharacteristic
80C22⋊Dic61—41no—
81C22⋊Dic61—42no—
82C22⋊Dic61—81no—
83#11758202—81no—
84#11758200—81no—
85#11758200—81no—
86#11758200—81no—
87C4⋊Dic61—21no—
88C4⋊Dic61—41no—
89#11758196—41no—
90#11758196—41no—
91#11758196—41no—
92C4×Dic61—41no—
93C2×#92418—21no—
94C2×#92418—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—161no—
101C23×C61#9911yescharacteristic
102Q8×C61—41no—
103D4×C61—41no—
104D4×C61—41no—
105D4×C61—41no—
106D4×C61—81no—
107D4×C61—81no—
108D4×C61—161no—
109D4×C61—161no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121C2×Dic61—162no—
122Dic122—41no—
123Dic122—41no—
124C8×C61—82no—
125#92418—41no—
126#92418—41no—
127#13607—611nomaximal, Sylow
128C22×C61—21no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—41no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61—82no—
137C22×C61—161no—
138C22×C61C2×#9911yescharacteristic
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—41no—
142C4×C61—41no—
143C4×C61—41no—
144Dic61—81no—
145Dic61—81no—
146Dic61—162no—
147#3222—611no—
148#2134D6111yescharacteristic
149#2127—611no—
150#1414—612no—
151#1407—612no—
152C2×C61—21no—
153C2×C61—41no—
154C2×C61—81no—
155C2×C61—81no—
156C2×C61—81no—
157C2×C61#212711yescharacteristic
158#541—611no—
159#535C2×D6111yescharacteristic
160#497—611no—
161#492—611no—
162#490—611no—
163#488—611no—
164#464—1221no—
165#460—1221no—
166#452—1221no—
167#450—1221no—
168#433—611no—
169#413—21no—
170C2×#99—1221no—
171#355—21no—
172#355C2×D6112yes—
173#354—612no—
174#353—1221no—
175C61#1360711yescharacteristic, complemented, Sylow
176C2×#41C22⋊D6111yescharacteristic
177C22×D4D24411yescharacteristic
178C2×Q16—1221no—
179C2×SD16—1221no—
180C2×SD16—1221no—
181C2×SD16—1221no—
182C2×M16—41no—
183C4⋊Q8—611no—
184#127C4×D6111yescharacteristic
185#127D24411yescharacteristic
186#126—2441no—
187#125—611no—
188#125—2441no—
189#124C22⋊D6111yescharacteristic
190#124C22×D6111yescharacteristic
191C22⋊Q8—1221no—
192#121—41no—
193#120—41no—
194C4×Q8—1221no—
195#117C4×D6111yescharacteristic
196C2×C22⋊C4—2441no—
197C4⋊C8—1221no—
198#103—1221no—
199#102—1221no—
200#102—1221no—
201#102—1221no—
202#101—22no—
203#100—41no—
204#99—1222no—
205#99—2441no—
206C22⋊C8—1221no—
207#97—611no—
208C4×C8—611no—
209C24—21no—
210#41—81no—
211C2×Q8—1221no—
212C2×Q8C2×C22⋊D6111yescharacteristic
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—81no—
218C2×D4#1175819811yescharacteristic
219C2×D4#1175819811yescharacteristic
220C2×D4C2×D24411yescharacteristic
221C22×C4—21no—
222C22×C4—4881no—
223C22×C4#1175819811yescharacteristic
224Q16—4881no—
225SD16—4881no—
226SD16—4881no—
227SD16—4881no—
228M16—42no—
229M16—81no—
230C2×C8—81no—
231C2×C8—1221no—
232C2×C8—1221no—
233C4⋊C4—21no—
234C4⋊C4—1221no—
235C4⋊C4—2441no—
236C4⋊C4—2441no—
237C4⋊C4—2441no—
238C4⋊C4—2441no—
239C22⋊C4—21no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—4881no—
243C22⋊C4—2442no—
244C22⋊C4—2441no—
245C42—2441no—
246C42#1175819811yescharacteristic
247C42C2×C4×D6111yescharacteristic
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—161no—
254C23#1451969111yescharacteristic
255Q8—41no—
256Q8—2441no—
257Q8—2441no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—81no—
262D4—81no—
263D4—161no—
264D4—161no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—9762no—
272C2×C4—4882no—
273C2×C4—2441no—
274C2×C4—2441no—
275C2×C4#1451969111yescharacteristic
276C2×C4C2×#1175819811yescharacteristic
277C8—82no—
278C8—2441no—
279C8—2441no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—82no—
289C22—161no—
290C22C2×#1451969111yescharacteristic
291C4—21no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—9762no—
297C4—4881no—
298C4—4881no—
299C2—21no—
300C2—41no—
301C2—81no—
302C2—81no—
303C2—81no—
304C2#2144748611yescharacteristic
subgroupG/H
centerC2#21447486
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#2134C2
socleC2×C61#2127
prime radicalC61×#535C22
cosocle#124×C61C23
hypercenter#2134D61
hypocenterC61#13607
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ #2134

Complements

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

NH ≅ G/Ntype
3. #21443995C2semidirect
4. #21443980C2semidirect
6. #16840374C4semidirect
8. #16840134C4semidirect
9. #16840126C4semidirect
15. #16839823C4semidirect
27. #14519777C2×C4semidirect
29. #14519769C2×C4semidirect
42. #14519688C2×C4semidirect
43. C4×#92418C2×C4semidirect
175. C61#13607semidirect

Maximal subgroups

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

subgroupindex
#214568912
#214474882
#214439952
#214439802
C61×#21342
#1360761