Finite Group Databasebeta

← #29584138

Subgroups — #29584138

Order 15616.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#21446797C212yesmaximal
2#21445970C211yesmaximal, characteristic
3#21444509C212yesmaximal
4#21443694C211yesmaximal, characteristic
5C61×#3249C211yesmaximal, characteristic
6#16839786C2212yes—
7C2×C4⋊#92418C2211yescharacteristic
8C2×#14519706—22no—
9C42×Dic61C2211yescharacteristic
10#16839521—22no—
11#16839521—22no—
12#16839511—22no—
13#16839508—22no—
14C61×#533C2212yes—
15C61×#477—22no—
16C61×#467—22no—
17C61×#462—22no—
18C2×C4⋊C8×C61C2211yescharacteristic
19C2×#11758194—22no—
20C22×#92418—21no—
21C2×C4×Dic61—21no—
22C2×C4×Dic61—21no—
23C2×C4×Dic61D411yescharacteristic
24C2×C4×Dic61D411yescharacteristic
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27C4⋊#92418—22no—
28#14519706—44no—
29C2×Q16×C61—22no—
30C22×C8×C61D412yes—
31C4⋊Q8×C61D412yes—
32C4⋊Q8×C61D412yes—
33#125×C61—22no—
34C22⋊Q8×C61—22no—
35C22⋊Q8×C61—22no—
36C4×Q8×C61—22no—
37C4×Q8×C61—22no—
38#117×C61—22no—
39C2×C42×C61C2311yescharacteristic
40#107×C61—22no—
41C4⋊C8×C61—22no—
42#103×C61—24no—
43C2×Q8×C61—22no—
44C2×Q8×C61—22no—
45C22×C4×C61C2×D411yescharacteristic
46C22×C4×C61C2×D411yescharacteristic
47C22×C4×C61#4111yescharacteristic
48C22×Dic61—21no—
49C22×Dic61—21no—
50Q16×C61—82no—
51C2×C8×C61—22no—
52C2×C8×C61—42no—
53C4⋊C4×C61—22no—
54C4⋊C4×C61—22no—
55C4⋊C4×C61—22no—
56C4⋊C4×C61—22no—
57C4⋊C4×C61—42no—
58C4⋊C4×C61—42no—
59C22⋊C4×C61—42no—
60C42×C61—42no—
61C42×C61C2×D412yes—
62C42×C61#4112yes—
63C4×Dic61—22no—
64C4×Dic61—22no—
65C4×Dic61—21no—
66C4×Dic61—22no—
67C4×Dic61—42no—
68C4×Dic61Q1612yescomplemented
69#11758194—42no—
70#11758194—42no—
71C2×#92418—42no—
72C2×#92418—41no—
73C2×#92418—42no—
74C23×C61#12111yescharacteristic
75Q8×C61—42no—
76Q8×C61—42no—
77Q8×C61—82no—
78C2×C4×C61—21no—
79C2×C4×C61—21no—
80C2×C4×C61—41no—
81C2×C4×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61#12011yescharacteristic
84C2×C4×C61#12112yes—
85C2×C4×C61#12312yes—
86C2×C4×C61#12411yescharacteristic
87C2×C4×C61C2×Q1611yescharacteristic
88C2×C4×C61#13511yescharacteristic
89C2×C4×C61#13712yes—
90C2×Dic61—22no—
91C2×Dic61—22no—
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—41no—
95C2×Dic61—41no—
96C8×C61—42no—
97#92418—81no—
98#92418—82no—
99#10522—611nomaximal, Sylow
100C22×C61—21no—
101C22×C61—21no—
102C22×C61#39411yescharacteristic
103C22×C61#46711yescharacteristic
104C22×C61#47011yescharacteristic
105C4×C61—22no—
106C4×C61—21no—
107C4×C61—82no—
108C4×C61—82no—
109C4×C61#45311yescharacteristic
110C4×C61#45611yescharacteristic
111C4×C61#46212yes—
112C4×C61#46412yes—
113Dic61—24no—
114Dic61—42no—
115#3249D6111yescharacteristic
116#1994—612no—
117#1842—611no—
118#1550—612no—
119#1292—611no—
120C2×C61—21no—
121C2×C61—21no—
122C2×C61#203911yescharacteristic
123C2×C61#205011yescharacteristic
124C2×C61#205611yescharacteristic
125#533C2×D6112yes—
126#477—22no—
127#467—22no—
128#462—22no—
129#423—612no—
130C2×C4⋊C8—611no—
131C2×C4⋊C8—1222no—
132C2×C4⋊C8C2×D6111yescharacteristic
133C43—611no—
134#339—1222no—
135#339—1222no—
136#333—1222no—
137#332—1222no—
138C61#1052211yescharacteristic, complemented, Sylow
139C2×Q16—22no—
140C2×M16—1222no—
141C22×C8—1221no—
142C22×C8C22⋊D6112yes—
143C4⋊Q8C22⋊D6112yes—
144C4⋊Q8C22⋊D6112yes—
145#125—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—22no—
151C2×C42—611no—
152C2×C42—611no—
153C2×C42—1221no—
154C2×C42—1221no—
155C2×C42C22×D6111yescharacteristic
156#107—22no—
157C4⋊C8—22no—
158C4⋊C8—1222no—
159C4⋊C8—1222no—
160C4⋊C8—1222no—
161C4⋊C8—2444no—
162#103—24no—
163C2×Q8—22no—
164C2×Q8—22no—
165C22×C4—1221no—
166C22×C4—1221no—
167C22×C4D4×D6111yescharacteristic
168C22×C4#1175822411yescharacteristic
169C22×C4C2×C22⋊D6111yescharacteristic
170Q16—82no—
171M16—2442no—
172M16—2442no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—2442no—
176C2×C8—2441no—
177C2×C8—2442no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C42—42no—
186C42—612no—
187C42—1222no—
188C42—1222no—
189C42—1221no—
190C42—1222no—
191C42—2442no—
192C42#1175822412yes—
193C42C2×C22⋊D6112yes—
194C23#1451974811yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1222no—
204C2×C4—1222no—
205C2×C4—1222no—
206C2×C4—1222no—
207C2×C4—2441no—
208C2×C4—2441no—
209C2×C4#1451974611yescharacteristic
210C2×C4#1451974912yes—
211C2×C4#1451975812yes—
212C2×C4#1451976411yescharacteristic
213C2×C4Q16×D6111yescharacteristic
214C2×C4#1451980011yescharacteristic
215C2×C4#1451980912yes—
216C8—42no—
217C8—4882no—
218C8—4881no—
219C22—21no—
220C22—21no—
221C22#1683967611yescharacteristic
222C22#1684000011yescharacteristic
223C22#1684001711yescharacteristic
224C4—21no—
225C4—22no—
226C4—82no—
227C4—82no—
228C4—1224no—
229C4—2442no—
230C4#1683993011yescharacteristic
231C4#1683994511yescharacteristic
232C4#1683997012yes—
233C4#1683998612yes—
234C2—21no—
235C2—21no—
236C2#2144704011yescharacteristic
237C2#2144710311yescharacteristic
238C2#2144713911yescharacteristic
subgroupG/H
centerC22#16839676
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3249C2
socleC22×C61#394
prime radicalC23×C61#121
hypercenter#3249D61
hypocenterC61#10522
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
68. C4×Dic61Q16semidirect
138. C61#10522semidirect

Maximal subgroups

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

subgroupindex
#214467972
#214459702
#214445092
#214436942
C61×#32492
#1052261