Finite Group Databasebeta

← #29599391

Subgroups — #29599391

Order 15616.

540 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16840137C211yesmaximal, characteristic
2C2×#16839574C211yesmaximal, characteristic, complemented
3#21447536C211yesmaximal, characteristic, complemented
4#21447535C211yesmaximal, characteristic, complemented
5#21446565C211yesmaximal, characteristic, complemented
6#21443900C211yesmaximal, characteristic, complemented
7C2×C61×#361C211yesmaximal, characteristic, complemented
8C22×D488C2211yescharacteristic
9C22×#11758199—21no—
10C2×#14519773—21no—
11C2×#14519717C2211yescharacteristic
12#16840137—22no—
13#16840137—22no—
14#16839800C2211yescharacteristic
15#16839790—21no—
16C2×C8×Dic61C2211yescharacteristic
17#16839574—22no—
18#16839564—22no—
19#16839563—22no—
20#16839523—21no—
21C22×D8×C61C2211yescharacteristic
22C2×M32×C61C2211yescharacteristic
23C2×#108×C61C2211yescharacteristic
24C61×#361—22no—
25C22×D244—21no—
26C22×C22⋊D61—42no—
27C2×D488—21no—
28C2×D488—21no—
29C2×D488—21no—
30C2×D488—21no—
31C2×D488—21no—
32C2×D488D412yes—
33C2×#11758199—24no—
34C2×#11758199—42no—
35C2×#11758199—42no—
36C22×#92418D411yescharacteristic
37#14519773—24no—
38#14519773—42no—
39#14519736—21no—
40C2×C4×Dic61D411yescharacteristic
41#14519717—21no—
42#14519717—21no—
43C2×#11758185—21no—
44#14519711D412yescomplemented
45#14519709—21no—
46#14519706—42no—
47C8×Dic61—21no—
48C8×Dic61—21no—
49C22×D4×C61—21no—
50C2×D8×C61—21no—
51C2×D8×C61—21no—
52C2×D8×C61—21no—
53C2×D8×C61—21no—
54C2×D8×C61—21no—
55C2×D8×C61D412yes—
56C2×M16×C61—21no—
57C22×C8×C61C2311yescharacteristic
58M32×C61—21no—
59M32×C61—21no—
60C2×C16×C61—21no—
61#108×C61—21no—
62#108×C61—21no—
63C24×C61—41no—
64C23×D61—41no—
65C2×D4×C61—22no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C22×C4×C61C2×D411yescharacteristic
72C2×C22⋊D61—48no—
73C2×C22⋊D61—88no—
74C22×Dic61—21no—
75C2×D244—22no—
76C2×D244—41no—
77C2×D244—41no—
78C2×D244—41no—
79C2×D244—41no—
80C2×D244—41no—
81D8×C61—22no—
82D8×C61—22no—
83D8×C61—41no—
84D8×C61—41no—
85M16×C61—41no—
86M16×C61—41no—
87C2×C8×C61—41no—
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61C2×D411yescharacteristic
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61#4111yescharacteristic
93C2×C8×C61#4111yescharacteristic
94C22⋊Dic61—41no—
95#11758199—48no—
96C4⋊Dic61—22no—
97C4×Dic61—21no—
98C4×Dic61—41no—
99C4×Dic61SD1612yes—
100C2×#92418—21no—
101C2×#92418—21no—
102C2×#92418D812yes—
103D488—22no—
104D488—22no—
105D488—41no—
106D488—41no—
107C16×C61—41no—
108#11758185—41no—
109C23×C61—42no—
110C23×C61—84no—
111C23×C61—81no—
112C23×C61—81no—
113C23×C61AGL(1,ℤ/8)11yescharacteristic
114C22×D61—42no—
115C22×D61—84no—
116C22×D61—81no—
117C22×D61—81no—
118D4×C61—42no—
119D4×C61—42no—
120D4×C61—81no—
121D4×C61—81no—
122C2×C4×C61#12011yescharacteristic
123C2×C4×C61#12111yescharacteristic
124C2×C4×C61#12411yescharacteristic
125C2×C4×C61C2×D811yescharacteristic
126C2×C4×C61C2×SD1611yescharacteristic
127C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
128C22⋊D61—816no—
129C2×Dic61—24no—
130C2×Dic61—22no—
131C2×Dic61—42no—
132D244—42no—
133D244—42no—
134D244—81no—
135D244—81no—
136C8×C61—81no—
137C8×C61#12111yescharacteristic
138C8×C61#12111yescharacteristic
139C8×C61#12311yescharacteristic
140C8×C61#12311yescharacteristic
141#92418—22no—
142#13413—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61—84no—
145C22×C61—82no—
146C22×C61—162no—
147C22×C61—162no—
148C22×C61#45011yescharacteristic
149C22×C61#45911yescharacteristic
150C22×C61#49011yescharacteristic
151C2×D61—84no—
152C2×D61—82no—
153C2×D61—162no—
154C2×D61—162no—
155C4×C61#39411yescharacteristic
156C4×C61#44711yescharacteristic
157C4×C61#46011yescharacteristic
158C4×C61#48611yescharacteristic
159Dic61—24no—
160Dic61—82no—
161C2×#493—611no—
162C2×#361—611no—
163C2×#361D6111yescharacteristic, complemented
164#2140—611no—
165#2140—611no—
166#1945—611no—
167#1376—611no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61—162no—
171C2×C61#204111yescharacteristic
172C2×C61#221312yes—
173D61—162no—
174C22×D8—611no—
175C22×D8—1221no—
176C22×D8C2×D6111yescharacteristic
177C2×#127—1221no—
178C2×M32—611no—
179C2×M32C2×D6111yescharacteristic
180#493—1222no—
181#493—1222no—
182C2×#108C2×D6111yescharacteristic
183#427—611no—
184#424—1221no—
185C2×C4×C8—611no—
186#361—22no—
187#361—1222no—
188#357—1222no—
189#357—1222no—
190#339—1221no—
191C61#1341311yescharacteristic, complemented, Sylow
192C22×D4—21no—
193C22×D4—1221no—
194C22×D4—2442no—
195C2×D8—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—612no—
201C2×D8—1224no—
202C2×D8—1221no—
203C2×D8—1221no—
204C2×D8—1221no—
205C2×D8—1221no—
206C2×D8—1221no—
207C2×D8—2442no—
208C2×D8—2442no—
209C2×D8C22⋊D6112yes—
210C2×M16—21no—
211C22×C8—611no—
212C22×C8C22×D6111yescharacteristic
213#127—1224no—
214#127—2442no—
215#117—1221no—
216C2×C42—611no—
217M32—21no—
218M32—21no—
219M32—1221no—
220M32—1221no—
221C2×C16—21no—
222C2×C16—1221no—
223#108—21no—
224#108—21no—
225#107—612no—
226#106—1221no—
227C4⋊C8—2442no—
228C4×C8—1221no—
229C4×C8—1221no—
230C24—41no—
231C24—2441no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—1222no—
239C2×D4—4888no—
240C2×D4—2448no—
241C2×D4—2441no—
242C2×D4—2441no—
243C2×D4—2441no—
244C2×D4—2441no—
245C2×D4—2441no—
246C22×C4—1221no—
247C22×C4D4×D6111yescharacteristic
248D8—22no—
249D8—22no—
250D8—41no—
251D8—41no—
252D8—1222no—
253D8—1222no—
254D8—2448no—
255D8—2441no—
256D8—2441no—
257M16—41no—
258M16—41no—
259C2×C8—41no—
260C2×C8—612no—
261C2×C8—1221no—
262C2×C8—1221no—
263C2×C8#1175822211yescharacteristic
264C2×C8#1175822211yescharacteristic
265C2×C8D4×D6111yescharacteristic
266C2×C8D4×D6111yescharacteristic
267C2×C8#1175822411yescharacteristic
268C2×C8C2×C22⋊D6111yescharacteristic
269C4⋊C4—1222no—
270C22⋊C4—2441no—
271C42—612no—
272C42—1221no—
273C42—2441no—
274C16—41no—
275C16—2441no—
276C23—42no—
277C23—84no—
278C23—81no—
279C23—81no—
280C23—4884no—
281C23—4881no—
282C23—4881no—
283C23—2442no—
284C23#1451980411yescharacteristic
285D4—42no—
286D4—42no—
287D4—81no—
288D4—81no—
289D4—48816no—
290D4—4881no—
291D4—4881no—
292D4—2442no—
293D4—2442no—
294C2×C4—1224no—
295C2×C4—1222no—
296C2×C4—2442no—
297C2×C4#1451974511yescharacteristic
298C2×C4#1451975011yescharacteristic
299C2×C4#1451976311yescharacteristic
300C2×C4D8×D6111yescharacteristic
301C2×C4SD16×D6111yescharacteristic
302C2×C4#1451980111yescharacteristic
303C8—81no—
304C8—1222no—
305C8#1451974911yescharacteristic
306C8#1451975011yescharacteristic
307C8#1451975911yescharacteristic
308C8#1451976011yescharacteristic
309C22—22no—
310C22—84no—
311C22—82no—
312C22—162no—
313C22—162no—
314C22—9762no—
315C22—9762no—
316C22—4884no—
317C22—4882no—
318C22#1683991111yescharacteristic
319C22#1683995611yescharacteristic
320C22#1684012511yescharacteristic
321C4—1224no—
322C4—4882no—
323C4#1683967711yescharacteristic
324C4#1683989411yescharacteristic
325C4#1683996111yescharacteristic
326C4#1684010711yescharacteristic
327C2—21no—
328C2—21no—
329C2—162no—
330C2—9762no—
331C2#2144705111yescharacteristic
332C2#2144789412yes—
subgroupG/H
centerC22#16839956
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC2×C61×#361C2
socleC22×C61#459
prime radicalC2×#16840137C2
hypercenterC2×#361D61
hypocenterC61#13413
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
2. C2×#16839574C2semidirect
3. #21447536C2semidirect
4. #21447535C2semidirect
5. #21446565C2semidirect
6. #21443900C2semidirect
7. C2×C61×#361C2semidirect
44. #14519711D4semidirect
163. C2×#361D61semidirect
191. C61#13413semidirect

Maximal subgroups

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

subgroupindex
C2×#168401372
C2×#168395742
#214475362
#214475352
#214465652
#214439002
C2×C61×#3612
#1341361