Finite Group Databasebeta

← #29567540

Subgroups — #29567540

Order 15616.

268 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 41 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#21446865C211yesmaximal, characteristic
2#21443645C211yesmaximal, characteristic, complemented
3C61×#1394C211yesmaximal, characteristic, complemented
4C2×#14519808—21no—
5#16840343C411yescharacteristic
6C2×#14519705—21no—
7#16839531C411yescharacteristic
8#16839517—21no—
9C2×#101×C61—21no—
10C61×#391C2211yescharacteristic
11C2×#11758222—21no—
12#14519808—42no—
13#14519808—41no—
14#14519808—41no—
15C2×#11758202—41no—
16C2×#11758201—41no—
17C2×#11758194—21no—
18C22×#92418—41no—
19#14519757—41no—
20#14519752—41no—
21#14519751—41no—
22#14519705—42no—
23C22×Q8×C61D411yescharacteristic
24C2×M16×C61—41no—
25C2×C4⋊C4×C61C2×C411yescharacteristic
26C2×C42×C61D411yescharacteristic
27#101×C61—22no—
28#101×C61—41no—
29#95×C61—41no—
30C2×Q8×C61—21no—
31C2×Q8×C61—41no—
32C2×Q8×C61—41no—
33C2×Q8×C61D811yescharacteristic
34C2×Q8×C61SD1611yescharacteristic
35C22×C4×C61—21no—
36C22×C4×C61—21no—
37C22×C4×C61—21no—
38C22×C4×C61C22⋊C411yescharacteristic
39C2×C22⋊D61—41no—
40#11758222—41no—
41#11758222—41no—
42#11758222—41no—
43#11758222—41no—
44C2×D244—21no—
45C2×D244—41no—
46C2×C4×D61—41no—
47C2×Dic122—21no—
48M16×C61—42no—
49M16×C61—81no—
50C2×C8×C61—81no—
51C4⋊C4×C61—41no—
52C42×C61—41no—
53#11758202—42no—
54#11758202—42no—
55#11758201—42no—
56#11758201—42no—
57#11758198—42no—
58C4⋊Dic61—41no—
59#11758194—22no—
60#11758194—41no—
61C2×#92418—41no—
62C2×#92418—42no—
63C2×#92418—81no—
64C2×#92418—81no—
65C23×C61#9911yescharacteristic
66C22×D61—41no—
67Q8×C61—41no—
68Q8×C61—41no—
69Q8×C61—81no—
70C2×C4×C61—21no—
71C2×C4×C61—21no—
72C2×C4×C61—21no—
73C2×C4×C61—21no—
74C2×C4×C61—21no—
75C2×C4×C61—21no—
76C2×C4×C61—41no—
77C2×C4×C61—41no—
78C2×C4×C61—41no—
79C2×C4×C61—41no—
80C2×C4×C61—41no—
81C2×C4×C61—81no—
82C2×C4×C61#10211yescharacteristic
83C2×C4×C61#10411yescharacteristic
84C22⋊D61—81no—
85C22⋊D61—81no—
86C2×Dic61—41no—
87D244—41no—
88D244—41no—
89D244—161no—
90C4×D61—81no—
91C4×D61—81no—
92Dic122—41no—
93Dic122—41no—
94C8×C61—82no—
95#92418—42no—
96#92418—82no—
97#7409—611nomaximal, Sylow
98C22×C61—21no—
99C22×C61—21no—
100C22×C61#32711yescharacteristic
101C22×C61AΣL(1,𝔽16)11yescharacteristic
102C22×C61#35311yescharacteristic
103C2×D61—81no—
104C2×D61—81no—
105C2×D61—81no—
106C4×C61—21no—
107C4×C61—21no—
108C4×C61—41no—
109C4×C61—41no—
110C4×C61—41no—
111C4×C61—41no—
112C4×C61—81no—
113Dic61—81no—
114#2007—611no—
115#1394D6111yescharacteristic, complemented
116#1272—611no—
117C2×C61—21no—
118C2×C61—21no—
119C2×C61#134311yescharacteristic
120C2×C61#140311yescharacteristic
121C2×C61#140511yescharacteristic
122D61—161no—
123C2×#137—1221no—
124#537—611no—
125C2×#104—1221no—
126C2×#101—21no—
127#391C2×D6111yescharacteristic
128#343—611no—
129#336—1221no—
130C61#740911yescharacteristic, complemented, Sylow
131C2×#41—1221no—
132C22×Q8C22⋊D6111yescharacteristic
133#137—2442no—
134#137—2441no—
135#137—2441no—
136C2×Q16—2441no—
137C2×SD16—2441no—
138C2×M16—41no—
139C2×M16—1221no—
140C22×C8—2441no—
141C22⋊Q8—2441no—
142#121—2441no—
143#121—2441no—
144C2×C4⋊C4C4×D6111yescharacteristic
145C2×C42D24411yescharacteristic
146#104—2442no—
147#101—22no—
148#101—41no—
149#95—41no—
150#41—2441no—
151#41—2441no—
152#41—2441no—
153#41—2441no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—1221no—
158C2×Q8#1175819911yescharacteristic
159C2×Q8#1175820111yescharacteristic
160C2×D4—1221no—
161C2×D4—2441no—
162C2×D4—2441no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2441no—
167C22×C4#1175819811yescharacteristic
168Q16—2442no—
169Q16—2442no—
170SD16—2442no—
171SD16—2442no—
172M16—42no—
173M16—81no—
174M16—1222no—
175M16—2441no—
176C2×C8—81no—
177C2×C8—4881no—
178C2×C8—4881no—
179C2×C8—2442no—
180C2×C8—2441no—
181C4⋊C4—41no—
182C4⋊C4—2441no—
183C22⋊C4—2442no—
184C42—41no—
185C23—2441no—
186C23#1451969111yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—81no—
190Q8—2441no—
191Q8—2441no—
192D4—9761no—
193D4—4881no—
194D4—4881no—
195D4—2441no—
196D4—2441no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—4881no—
210C2×C4—4881no—
211C2×C4—2441no—
212C2×C4#1451969711yescharacteristic
213C2×C4#1451970511yescharacteristic
214C8—82no—
215C8—4882no—
216C8—2442no—
217C22—21no—
218C22—21no—
219C22—4881no—
220C22—4881no—
221C22—4881no—
222C22#1683949511yescharacteristic
223C22#1683954811yescharacteristic
224C22#1683955411yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—4881no—
233C2—21no—
234C2—21no—
235C2—9761no—
236C2#2144381511yescharacteristic
237C2#2144396611yescharacteristic
238C2#2144397211yescharacteristic
subgroupG/H
centerC22#16839548
commutatorC2×C4⋊C4×C61C2×C4
Frattini#391C2×D61
FittingC61×#1394C2
socleC22×C61AΣL(1,𝔽16)
prime radical#16840343C4
cosocleC61×#391C22
hypercenter#1394D61
hypocenterC61#7409
LCSG ▷ C2×C4⋊C4×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1394

Complements

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

NH ≅ G/Ntype
2. #21443645C2semidirect
3. C61×#1394C2semidirect
115. #1394D61semidirect
130. C61#7409semidirect

Maximal subgroups

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

subgroupindex
#214468652
#214436452
C61×#13942
#740961