Finite Group Databasebeta

← #29886436

Subgroups — #29886436

Order 15616.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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#21454083C213yesmaximal
2#21454079C2112yesmaximal, complemented
3#21449086C2112yesmaximal, complemented
4#21448197C213yesmaximal
5C61×#2840C211yesmaximal, characteristic
6C24×Dic61C2211yescharacteristic
7C2×Q8×Dic61C2214yescomplemented
8C2×#14519733C2216yes—
9#16839702C22112yescomplemented
10#16839698C22112yes—
11#16839696C22124yescomplemented
12#16839685C22124yes—
13#16839681C22112yescomplemented
14#16839623C2216yes—
15#16839620C22112yes—
16#16839618C2213yes—
17#16839613C22112yescomplemented
18C22⋊C4×Dic61C22112yes—
19C61×#393C22112yes—
20C61×#379C2213yes—
21C23×Dic61—44no—
22C23×Dic61Q818yescomplemented
23C23×Dic61C2313yes—
24C23×Dic61C2313yes—
25Q8×Dic61—48no—
26#14519733—412no—
27C2×C4⋊Dic61C23112yes—
28C2×#11758196C23124yes—
29C2×#11758196C23112yes—
30C2×C22⋊Dic61C2316yes—
31C2×C22⋊Dic61C23112yes—
32C2×C4×Dic61C23112yes—
33#14519684C23112yes—
34#14519684C23124yes—
35C25×C61C2311yescharacteristic
36C22×Q8×C61C2314yes—
37C2×C22⋊C4×C61C23112yes—
38C2×C22⋊C4×C61C2316yes—
39#95×C61C23112yes—
40C24×C61—42no—
41C24×C61—44no—
42C24×C61C2×Q814yes—
43C24×C61C2413yes—
44C2×Q8×C61—48no—
45C2×Q8×C61—44no—
46C22×C4×C61C24112yes—
47C22×Dic61—212no—
48C22×Dic61—416no—
49C22×Dic61—46no—
50C22×Dic61C2×Q8112yes—
51C22×Dic61C2×Q8112yes—
52C22×Dic61C24112yes—
53C22×Dic61C2413yes—
54C22×Dic61C2411yescharacteristic
55C22⋊C4×C61—412no—
56C22⋊C4×C61—46no—
57C22⋊Dic61—412no—
58C22⋊Dic61—46no—
59C4⋊Dic61—412no—
60#11758196—424no—
61#11758196—412no—
62C4×Dic61—224no—
63C23×C61—26no—
64C23×C61—212no—
65C23×C61—44no—
66C23×C61—48no—
67C23×C61—416no—
68C23×C61C22×Q816yes—
69C23×C61C2511yescharacteristic
70Q8×C61—416no—
71C2×C4×C61—212no—
72C2×C4×C61—224no—
73C2×Dic61—26no—
74C2×Dic61—224no—
75C2×Dic61—212no—
76C2×Dic61—224no—
77C2×Dic61—412no—
78C2×Dic61C22×Q816yes—
79C2×Dic61C22×Q812yes—
80C2×Dic61H(5,𝔽2)14yes—
81#36674—611nomaximal, Sylow
82C22×C61—26no—
83C22×C61—212no—
84C22×C61—416no—
85C22×C61—412no—
86C22×C61C23×Q811yescharacteristic
87C22×C61C2×H(5,𝔽2)12yes—
88C22×C61#58514yes—
89C4×C61—412no—
90Dic61—26no—
91Dic61—412no—
92Dic61#54314yes—
93#2842—613no—
94#2841—6112no—
95#2840D6111yescharacteristic
96#2396—6112no—
97#2280—613no—
98C2×C61—46no—
99C2×C61C2×#54311yescharacteristic
100C2×C61#345812yes—
101C2×C61#352914yes—
102C24×C4—611no—
103C2×C4×Q8—614no—
104C2×#117—616no—
105#400—6112no—
106#399—6112no—
107#398—6124no—
108#396—6124no—
109#395—6112no—
110#393C2×D61112yes—
111#381—616no—
112#380—6112no—
113#379—613no—
114#379C2×D6113yes—
115C4×C4⋊C4—6112no—
116C4×C22⋊C4—6112no—
117C61#3667411yescharacteristic, complemented, Sylow
118C25C22×D6111yescharacteristic
119C22×Q8C22×D6114yes—
120C23×C4—613no—
121C23×C4—618no—
122C23×C4—613no—
123C23×C4—2444no—
124C4×Q8—2448no—
125#117—24412no—
126C2×C4⋊C4—6112no—
127C2×C4⋊C4—6112no—
128C2×C4⋊C4—6124no—
129C2×C22⋊C4—616no—
130C2×C22⋊C4—6112no—
131C2×C22⋊C4C22×D6116yes—
132C2×C22⋊C4C22×D61112yes—
133C2×C42—6112no—
134#95—6124no—
135#95—6112no—
136#95C22×D61112yes—
137C24—42no—
138C24—44no—
139C24Q8×D6114yes—
140C24C23×D6113yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—611no—
144C22×C4—6112no—
145C22×C4—613no—
146C22×C4—6112no—
147C22×C4—6112no—
148C22×C4—12212no—
149C22×C4—24416no—
150C22×C4—2446no—
151C22×C4C23×D61112yes—
152C4⋊C4—24424no—
153C4⋊C4—24412no—
154C4⋊C4—24412no—
155C22⋊C4—46no—
156C22⋊C4—412no—
157C22⋊C4—2446no—
158C22⋊C4—24412no—
159C42—12224no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D6116yes—
166C23C24×D6111yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—612no—
171C2×C4—614no—
172C2×C4—616no—
173C2×C4—12224no—
174C2×C4—1226no—
175C2×C4—12212no—
176C2×C4—12224no—
177C2×C4—24412no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D6111yescharacteristic
183C22H(5,𝔽2)×D6112yes—
184C22#1684059714yes—
185C4—412no—
186C4—614no—
187C4—1226no—
188C4—24412no—
189C2—46no—
190C2#543×D6111yescharacteristic
191C2#2145922112yes—
192C2#2145995914yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2840C2
prime radicalC25×C61C23
hypercenter#2840D61
hypocenterC61#36674
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #21454079C2semidirect
3. #21449086C2semidirect
7. C2×Q8×Dic61C22semidirect
9. #16839702C22semidirect
11. #16839696C22semidirect
13. #16839681C22semidirect
17. #16839613C22semidirect
22. C23×Dic61Q8semidirect
117. C61#36674semidirect

Maximal subgroups

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

subgroupindex
#214540832
#214540792
#214490862
#214481972
C61×#28402
#3667461