Finite Group Databasebeta

← #29592759

Subgroups — #29592759

Order 15616.

388 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446986C211yesmaximal, characteristic
2#21444809C212yesmaximal
3#21443819C212yesmaximal
4#21443749C211yesmaximal, characteristic
5C61×#2817C211yesmaximal, characteristic
6#16840388C2211yescharacteristic
7#16840092—22no—
8#16839625—21no—
9#16839529C2211yescharacteristic
10#16839528—21no—
11#16839498—22no—
12#16839486C2212yes—
13C2×C22⋊Q8×C61—21no—
14C61×#400—22no—
15C61×#397—21no—
16C61×#397C2212yes—
17C61×#395—22no—
18C61×#381C2211yescharacteristic
19#14519756—22no—
20#14519754—42no—
21C2×C22⋊Dic61—21no—
22C2×C22⋊Dic61—21no—
23#14519708—42no—
24#14519700—42no—
25C22⋊#92418—22no—
26#14519684—42no—
27C22×Q8×C61—21no—
28C23×C4×C61D411yescharacteristic
29C22⋊Q8×C61—42no—
30C22⋊Q8×C61—42no—
31C2×C4⋊C4×C61—22no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61D412yes—
35C2×C22⋊C4×C61—22no—
36C2×C22⋊C4×C61D411yescharacteristic
37C2×C22⋊C4×C61C2311yescharacteristic
38#95×C61—22no—
39#95×C61—21no—
40#95×C61—22no—
41#95×C61—22no—
42#95×C61D412yes—
43C24×C61C2×D411yescharacteristic
44C2×Q8×C61—42no—
45C2×Q8×C61—81no—
46C2×Q8×C61—81no—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61SD1612yes—
55C22×C4×C61C2×D412yes—
56C22×Dic61—22no—
57C22×Dic61—41no—
58C2×Dic122—22no—
59C4⋊C4×C61—42no—
60C4⋊C4×C61—44no—
61C4⋊C4×C61—42no—
62C4⋊C4×C61—81no—
63C22⋊C4×C61—41no—
64C22⋊C4×C61—42no—
65C22⋊C4×C61—42no—
66C22⋊C4×C61—41no—
67C22⋊Dic61—41no—
68C22⋊Dic61—41no—
69C22⋊Dic61—41no—
70C22⋊Dic61—41no—
71C4⋊Dic61—22no—
72#11758196—42no—
73C2×#92418—42no—
74C23×C61—21no—
75C23×C61—21no—
76C23×C61—42no—
77C23×C61#12011yescharacteristic
78C23×C61C2×SD1611yescharacteristic
79C23×C61AGL(1,ℤ/8)11yescharacteristic
80Q8×C61—84no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—44no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—41no—
92C2×C4×C61—42no—
93C2×C4×C61—42no—
94C2×C4×C61—81no—
95C2×Dic61—42no—
96C2×Dic61—82no—
97C2×Dic61—81no—
98C2×Dic61—82no—
99Dic122—82no—
100#92418—82no—
101#12207—611nomaximal, Sylow
102C22×C61—22no—
103C22×C61—21no—
104C22×C61—21no—
105C22×C61—41no—
106C22×C61—41no—
107C22×C61—42no—
108C22×C61—81no—
109C22×C61#45011yescharacteristic
110C22×C61#45511yescharacteristic
111C22×C61B(4,𝔽2)11yescharacteristic
112C4×C61—42no—
113C4×C61—42no—
114C4×C61—82no—
115C4×C61—81no—
116C4×C61—82no—
117Dic61—82no—
118Dic61—161no—
119#2817D6111yescharacteristic
120#2030—611no—
121#1608—612no—
122#1344—612no—
123#1317—611no—
124C2×C61—21no—
125C2×C61—21no—
126C2×C61—41no—
127C2×C61—41no—
128C2×C61#160111yescharacteristic
129C2×C61#220112yes—
130#543—611no—
131C2×C22⋊Q8—21no—
132#483—1222no—
133#400—22no—
134#397—21no—
135#397C2×D6112yes—
136#395—22no—
137#381—1221no—
138#381C2×D6111yescharacteristic
139#342—611no—
140#342—1221no—
141#328—1222no—
142#323—612no—
143C61#1220711yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C23×C4C22⋊D6111yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—1222no—
149C22⋊Q8—2442no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊D6112yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—1221no—
156C2×C22⋊C4—1221no—
157C2×C22⋊C4C22⋊D6111yescharacteristic
158C2×C22⋊C4C22×D6111yescharacteristic
159#106—2442no—
160#103—2442no—
161C22⋊C8—1222no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—2442no—
167#95C22⋊D6112yes—
168C24C2×C22⋊D6111yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—1222no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—1222no—
181C22×C4—2441no—
182C22×C4#1175820012yes—
183C22×C4C2×C22⋊D6112yes—
184C2×C8—2442no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—1222no—
190C4⋊C4—2442no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—2441no—
196C22⋊C4—2441no—
197C22⋊C4—2441no—
198C22⋊C4—2441no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1451974611yescharacteristic
203C23C2×#1175820011yescharacteristic
204C23#1451980311yescharacteristic
205Q8—84no—
206Q8—4882no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—4881no—
222C2×C4—4882no—
223C2×C4—4882no—
224C2×C4—2442no—
225C8—4882no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1683991311yescharacteristic
234C22#1683994011yescharacteristic
235C22#1683994911yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—9761no—
242C4—4882no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2144477111yescharacteristic
248C2#2144783212yes—
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2817C2
socleC22×C61B(4,𝔽2)
prime radicalC24×C61C2×D4
hypercenter#2817D61
hypocenterC61#12207
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
143. C61#12207semidirect

Maximal subgroups

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

subgroupindex
#214469862
#214448092
#214438192
#214437492
C61×#28172
#1220761