Finite Group Databasebeta

← #29567787

Subgroups — #29567787

Order 15616.

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

normal subgroups only

This group has 290 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#21447504C211yesmaximal, characteristic
2#21446648C211yesmaximal, characteristic
3C61×#2126C211yesmaximal, characteristic
4#16839786C411yescharacteristic, complemented
5C2×#14519704—21no—
6C2×#14519704—21no—
7#16839746—21no—
8C42×Dic61C411yescharacteristic, complemented
9#16839561C412yescomplemented
10#16839559—21no—
11C61×#532C2211yescharacteristic
12C61×#410—21no—
13C61×#354—21no—
14C61×#353C412yescomplemented
15C2×#11758194—21no—
16C2×#11758194—41no—
17C2×C4×Dic61—21no—
18C2×C4×Dic61—41no—
19C4⋊#92418—41no—
20#14519704—42no—
21#14519704—42no—
22#14519696—22no—
23#14519694—41no—
24C2×#41×C61D411yescharacteristic
25C2×#41×C61D411yescharacteristic
26C4⋊Q8×C61C2×C411yescharacteristic
27#127×C61C2×C411yescharacteristic
28#124×C61D411yescharacteristic
29#124×C61Q811yescharacteristic
30#121×C61—41no—
31C4×D4×C61—41no—
32#117×C61—41no—
33C2×C42×C61C2×C411yescharacteristic
34#99×C61—22no—
35#99×C61—41no—
36#41×C61—41no—
37#41×C61—41no—
38#41×C61—41no—
39#41×C61—41no—
40C2×Q8×C61C22⋊C411yescharacteristic
41C2×Q8×C61C4⋊C411yescharacteristic
42C2×D4×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61C22⋊C411yescharacteristic
46C2×D4×C61C4⋊C411yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C411yescharacteristic
51C22×Dic61—41no—
52C4⋊C4×C61—41no—
53C4⋊C4×C61—81no—
54C22⋊C4×C61—41no—
55C22⋊C4×C61—41no—
56C22⋊C4×C61—42no—
57C42×C61—21no—
58C42×C61—81no—
59C42×C61C4211yescharacteristic
60C42×C61C4⋊C411yescharacteristic
61C4×Dic61—22no—
62C4×Dic61—41no—
63C4×Dic61—44no—
64#11758194—42no—
65#11758194—41no—
66#11758194—41no—
67#11758194—81no—
68C2×#92418—41no—
69C2×#92418—81no—
70C23×C61—21no—
71C23×C61—21no—
72C23×C61#9911yescharacteristic
73Q8×C61—41no—
74Q8×C61—41no—
75D4×C61—41no—
76D4×C61—41no—
77D4×C61—82no—
78D4×C61—81no—
79D4×C61—81no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—42no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61—41no—
91C2×C4×C61—82no—
92C2×C4×C61#9511yescharacteristic
93C2×C4×C61#9911yescharacteristic
94C2×Dic61—42no—
95C2×Dic61—44no—
96#92418—82no—
97#92418—81no—
98#7497—611nomaximal, Sylow
99C22×C61—21no—
100C22×C61—41no—
101C22×C61—41no—
102C22×C61—41no—
103C22×C61—81no—
104C22×C61—81no—
105C22×C61#34211yescharacteristic
106C4×C61—22no—
107C4×C61—21no—
108C4×C61—41no—
109C4×C61—41no—
110C4×C61—41no—
111C4×C61—162no—
112C4×C61#35311yescharacteristic
113C4×C61#35411yescharacteristic
114Dic61—44no—
115#2132—611no—
116#2126D6111yescharacteristic
117#1962—611no—
118C2×C61—21no—
119C2×C61—41no—
120C2×C61—81no—
121C2×C61—81no—
122C2×C61#139311yescharacteristic
123#532C2×D6111yescharacteristic
124#423—611no—
125C2×#104—1221no—
126C2×#104—1221no—
127#413—1221no—
128#410—21no—
129C43—611no—
130#356—612no—
131#355—1221no—
132#354—21no—
133#353Dic6112yescomplemented
134C61#749711yescharacteristic, complemented, Sylow
135C2×#41D24411yescharacteristic
136C2×#41C22⋊D6111yescharacteristic
137C2×M16—1221no—
138C2×M16—2441no—
139C4⋊Q8C4×D6111yescharacteristic
140#127C2×Dic6111yescharacteristic
141#124Dic12211yescharacteristic
142#124C22⋊D6111yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—41no—
146C2×C42—1221no—
147C2×C42—2441no—
148C2×C42C4×D6111yescharacteristic
149C4⋊C8—2441no—
150#104—2442no—
151#104—2442no—
152#101—1222no—
153#100—2441no—
154#99—22no—
155#99—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8#1175819611yescharacteristic
161C2×Q8#1175819811yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C4⋊Dic6111yescharacteristic
166C2×D4C22⋊Dic6111yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2441no—
171C22×C4#1175819811yescharacteristic
172M16—4881no—
173M16—2442no—
174M16—2441no—
175M16—2441no—
176C2×C8—4881no—
177C2×C8—2441no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—21no—
184C42—81no—
185C42—1222no—
186C42—2441no—
187C42—2444no—
188C42C4×Dic6111yescharacteristic
189C42#1175819611yescharacteristic
190C23—21no—
191C23—21no—
192C23#1451969111yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—2442no—
213C2×C4—2444no—
214C2×C4#1451968411yescharacteristic
215C2×C4#1451969111yescharacteristic
216C8—4882no—
217C8—4881no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1683952811yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—162no—
231C4—2444no—
232C4#1683955411yescharacteristic
233C4#1683955611yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2144394311yescharacteristic
subgroupG/H
centerC4#16839554
commutatorC42×C61C42
Frattini#532C2×D61
FittingC61×#2126C2
socleC2×C61#1393
prime radical, cosocleC61×#532C22
hypercenter#2126D61
hypocenterC61#7497
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #16839786C4semidirect
8. C42×Dic61C4semidirect
9. #16839561C4semidirect
14. C61×#353C4semidirect
133. #353Dic61semidirect
134. C61#7497semidirect

Maximal subgroups

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

subgroupindex
#214475042
#214466482
C61×#21262
#749761