Finite Group Databasebeta

← #29592504

Subgroups — #29592504

Order 15616.

354 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 59 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#21457258C211yesmaximal, characteristic
2#21447736C212yesmaximal, complemented
3#21443833C212yesmaximal, complemented
4#21443761C211yesmaximal, characteristic, complemented
5C61×#2047C211yesmaximal, characteristic, complemented
6#16840367—21no—
7C2×#14519720—22no—
8#16840063—22no—
9#16840015—21no—
10#16839998C2212yes—
11#16839958—22no—
12#16839566—22no—
13C22⋊#11758185C2212yes—
14C22×Q16×C61C2211yescharacteristic
15C61×#424C2211yescharacteristic
16C61×#419—21no—
17C2×#101×C61—21no—
18C61×#341C2211yescharacteristic
19C2×D488—21no—
20C2×D488D412yes—
21#14519774—21no—
22#14519769—21no—
23#14519762—42no—
24#14519751—22no—
25#14519750—42no—
26C4×D244—22no—
27#14519720—44no—
28C2×#11758185—22no—
29#14519711D412yes—
30#14519709—21no—
31#14519699—22no—
32#14519699—22no—
33C22×Q8×C61—21no—
34C2×Q16×C61—22no—
35C2×Q16×C61—42no—
36C2×M16×C61—21no—
37C2×M16×C61D411yescharacteristic
38C22×C8×C61C2311yescharacteristic
39#117×C61D411yescharacteristic
40C4⋊C8×C61—22no—
41#103×C61—42no—
42#101×C61—42no—
43C2×Q8×C61—22no—
44C2×Q8×C61—41no—
45C2×Q8×C61—41no—
46C2×Q8×C61—41no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C22×C4×C61—41no—
50C22×C4×C61C2×D411yescharacteristic
51C2×C22⋊D61—42no—
52C2×D244—22no—
53C2×D244—42no—
54C2×C4×D61—42no—
55Q16×C61—24no—
56Q16×C61—42no—
57M16×C61—41no—
58M16×C61—41no—
59M16×C61—41no—
60C2×C8×C61—21no—
61C2×C8×C61—41no—
62C2×C8×C61—41no—
63C2×C8×C61C2×D412yes—
64C4⋊C4×C61D812yes—
65C22⋊C4×C61—21no—
66C42×C61—21no—
67#11758198—42no—
68#11758198—42no—
69C4⋊Dic61—22no—
70#11758196—42no—
71D488—42no—
72D488—81no—
73#11758185—42no—
74C23×C61AGL(1,ℤ/8)11yescharacteristic
75C22×D61—42no—
76Q8×C61—42no—
77Q8×C61—42no—
78Q8×C61—81no—
79Q8×C61—81no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—42no—
84C2×C4×C61—81no—
85C2×C4×C61—81no—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61C2×D811yescharacteristic
88C22⋊D61—162no—
89C2×Dic61—42no—
90D244—42no—
91D244—42no—
92D244—162no—
93C4×D61—162no—
94C8×C61—22no—
95C8×C61—41no—
96C8×C61—81no—
97#12167—611nomaximal, Sylow
98C22×C61—21no—
99C22×C61—22no—
100C22×C61#44711yescharacteristic
101C2×D61—82no—
102C2×D61—82no—
103C2×D61—82no—
104C4×C61—21no—
105C4×C61—42no—
106C4×C61—82no—
107C4×C61#45411yescharacteristic
108C4×C61B(4,𝔽2)11yescharacteristic
109Dic61—82no—
110#3259—611no—
111#2185—612no—
112#2047D6111yescharacteristic, complemented
113#1349—612no—
114#1323—611no—
115C2×C61—41no—
116C2×C61#159811yescharacteristic
117C2×C61#219412yes—
118D61—162no—
119C22×Q16C2×D6111yescharacteristic
120#540—1221no—
121C2×SD32—1222no—
122#478—1222no—
123#469—1221no—
124#466—612no—
125#459—1222no—
126#424C2×D6111yescharacteristic
127#419—21no—
128C2×#101—21no—
129#358—1222no—
130C22⋊C16—612no—
131#341C2×D6111yescharacteristic
132C61#1216711yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C2×Q16—22no—
135C2×Q16—42no—
136C2×D8—612no—
137C2×D8—1221no—
138C2×M16—21no—
139C2×M16C22⋊D6111yescharacteristic
140C22×C8C22×D6111yescharacteristic
141#127—1221no—
142#125—1221no—
143#123—2442no—
144#121—1222no—
145#121—2442no—
146C4×D4—1222no—
147#117C22⋊D6111yescharacteristic
148SD32—2444no—
149C2×C16—1222no—
150#107—612no—
151#106—1221no—
152C4⋊C8—22no—
153#103—42no—
154#102—1222no—
155#102—1222no—
156#101—42no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—1222no—
164C2×D4—2442no—
165C2×D4—2442no—
166C22×C4—41no—
167C22×C4—2442no—
168C22×C4C2×C22⋊D6111yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—4881no—
172D8—2442no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8D4×D6112yes—
180C4⋊C4—1222no—
181C4⋊C4—2442no—
182C4⋊C4#1175819912yes—
183C22⋊C4—21no—
184C22⋊C4—2442no—
185C22⋊C4—2442no—
186C42—21no—
187C16—2442no—
188C23—2442no—
189C23#1451980211yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—9762no—
195D4—9762no—
196D4—2442no—
197D4—2442no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—9762no—
205C2×C4—2442no—
206C2×C4#1451974511yescharacteristic
207C2×C4C2×#1175819911yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—22no—
212C22—21no—
213C22—4882no—
214C22—4882no—
215C22—4882no—
216C22#1683989511yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—4882no—
221C4#1683993411yescharacteristic
222C4#1683994811yescharacteristic
223C2—41no—
224C2—9762no—
225C2#2144475311yescharacteristic
226C2#2144778812yes—
subgroupG/H
centerC22#16839895
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#2047C2
socleC22×C61#447
prime radical#21457258C2
hypercenter#2047D61
hypocenterC61#12167
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #2047

Complements

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

NH ≅ G/Ntype
2. #21447736C2semidirect
3. #21443833C2semidirect
4. #21443761C2semidirect
5. C61×#2047C2semidirect
112. #2047D61semidirect
132. C61#12167semidirect

Maximal subgroups

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

subgroupindex
#214572582
#214477362
#214438332
#214437612
C61×#20472
#1216761