Finite Group Databasebeta

← #29593355

Subgroups — #29593355

Order 15616.

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

normal subgroups only

This group has 342 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#21448031C212yesmaximal, complemented
2#21447372C211yesmaximal, characteristic, complemented
3#21443804C212yesmaximal, complemented
4#21443757C211yesmaximal, characteristic, complemented
5C61×#3199C211yesmaximal, characteristic
6#16839811C2211yescharacteristic, complemented
7#16839788—21no—
8#16839779—21no—
9#16839584—22no—
10#16839575—22no—
11#16839544C2212yescomplemented
12#16839519C2211yescharacteristic, complemented
13C61×#535—21no—
14C61×#482—22no—
15C61×#469C2212yes—
16C61×#466—21no—
17C61×#447—22no—
18C61×#408C2211yescharacteristic
19C2×#11758194—21no—
20#14519736—21no—
21#14519717—22no—
22#14519713—22no—
23#14519706—42no—
24#14519704—42no—
25C2×#41×C61D411yescharacteristic
26C22×D4×C61—21no—
27C2×D8×C61—21no—
28C2×D8×C61D412yes—
29C2×M16×C61C2311yescharacteristic
30C22×C8×C61D411yescharacteristic
31#127×C61—41no—
32#124×C61—41no—
33#121×C61—22no—
34#121×C61—22no—
35#120×C61—41no—
36#120×C61—41no—
37#117×C61—21no—
38#107×C61—21no—
39#106×C61D412yes—
40#102×C61—22no—
41#102×C61—22no—
42C22⋊C8×C61—22no—
43C24×C61—41no—
44#41×C61—41no—
45#41×C61—41no—
46C2×Q8×C61SD1611yescharacteristic
47C2×D4×C61—21no—
48C2×D4×C61—22no—
49C2×D4×C61—41no—
50C2×D4×C61—42no—
51C2×D4×C61—42no—
52C2×D4×C61—81no—
53C2×D4×C61—81no—
54C2×D4×C61SD1611yescharacteristic
55C22×C4×C61—21no—
56C22×C4×C61C2×D411yescharacteristic
57D8×C61—42no—
58D8×C61—81no—
59M16×C61—41no—
60C2×C8×C61—21no—
61C2×C8×C61—41no—
62C2×C8×C61C2×D412yes—
63C4⋊C4×C61—22no—
64C22⋊C4×C61—42no—
65C22⋊C4×C61—41no—
66C22⋊C4×C61—42no—
67C42×C61—41no—
68C22⋊Dic61—41no—
69C4⋊Dic61—41no—
70C4×Dic61—22no—
71#11758194—41no—
72#11758194—41no—
73C2×#92418—41no—
74#11758185—42no—
75C23×C61—21no—
76C23×C61—42no—
77C23×C61—82no—
78C23×C61—162no—
79C23×C61AGL(1,ℤ/8)11yescharacteristic
80Q8×C61—41no—
81D4×C61—41no—
82D4×C61—42no—
83D4×C61—42no—
84D4×C61—81no—
85D4×C61—82no—
86D4×C61—162no—
87D4×C61—162no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—42no—
94C2×C4×C61#12011yescharacteristic
95C2×C4×C61C2×SD1611yescharacteristic
96C2×Dic61—42no—
97C8×C61—22no—
98C8×C61—41no—
99#92418—81no—
100#12301—611nomaximal, Sylow
101C22×C61—21no—
102C22×C61—41no—
103C22×C61—42no—
104C22×C61—41no—
105C22×C61—82no—
106C22×C61—82no—
107C22×C61—82no—
108C22×C61—164no—
109C22×C61#45011yescharacteristic
110C4×C61—21no—
111C4×C61—41no—
112C4×C61—82no—
113C4×C61#45311yescharacteristic
114C4×C61#45811yescharacteristic
115Dic61—82no—
116#3199D6111yescharacteristic
117#2238—612no—
118#2098—611no—
119#1339—612no—
120#1321—611no—
121C2×C61—21no—
122C2×C61—41no—
123C2×C61—81no—
124C2×C61—162no—
125C2×C61#161311yescharacteristic
126#535—21no—
127#482—22no—
128#469C2×D6112yes—
129#466—21no—
130#447—22no—
131#430—611no—
132#424—1221no—
133#421—1221no—
134#408C2×D6111yescharacteristic
135#365—1222no—
136#361—1222no—
137#349—612no—
138H(3,ℤ/4)—611no—
139C61#1230111yescharacteristic, complemented, Sylow
140C2×#41C22⋊D6111yescharacteristic
141C22×D4—21no—
142C2×D8—21no—
143C2×D8C22⋊D6112yes—
144C2×M16—1221no—
145C2×M16C22×D6111yescharacteristic
146C22×C8C22⋊D6111yescharacteristic
147#127—41no—
148#124—41no—
149#121—22no—
150#121—22no—
151#120—41no—
152#120—41no—
153#117—21no—
154#117—1221no—
155M32—1222no—
156#108—1222no—
157#107—21no—
158#106C22⋊D6112yes—
159C4⋊C8—2442no—
160#104—2442no—
161#102—22no—
162#102—22no—
163C22⋊C8—22no—
164C24—41no—
165#41—41no—
166#41—41no—
167C2×Q8#1175820011yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4#1175820011yescharacteristic
176C22×C4—21no—
177C22×C4C2×C22⋊D6111yescharacteristic
178D8—42no—
179D8—81no—
180M16—41no—
181M16—2441no—
182M16—2441no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—2441no—
186C2×C8C2×C22⋊D6112yes—
187C4⋊C4—22no—
188C4⋊C4—2441no—
189C22⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—2441no—
193C42—41no—
194C42—1222no—
195C16—2442no—
196C23—21no—
197C23—42no—
198C23—82no—
199C23—162no—
200C23#1451980311yescharacteristic
201Q8—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—82no—
207D4—162no—
208D4—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—2442no—
216C2×C4#1451974611yescharacteristic
217C2×C4C2×#1175820011yescharacteristic
218C8—22no—
219C8—41no—
220C8—4881no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—164no—
229C22#1683991311yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—4882no—
234C4#1683993011yescharacteristic
235C4#1683995211yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—162no—
240C2#2144483911yescharacteristic
subgroupG/H
centerC2#21444839
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
Fitting, prime radicalC61×#3199C2
socleC2×C61#1613
hypercenter#3199D61
hypocenterC61#12301
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3199

Complements

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

NH ≅ G/Ntype
1. #21448031C2semidirect
2. #21447372C2semidirect
3. #21443804C2semidirect
4. #21443757C2semidirect
6. #16839811C22semidirect
11. #16839544C22semidirect
12. #16839519C22semidirect
139. C61#12301semidirect

Maximal subgroups

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

subgroupindex
#214480312
#214473722
#214438042
#214437572
C61×#31992
#1230161