Subgroups — #335×Dic₆₁ #29571200

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₆₁	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839515	C₂	1	1	yes	maximal, characteristic
4	#21443653	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
11	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
12	#16839516	—	2	2	no	—
13	#16839515	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#335	C₄	1	4	yes	complemented
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	C₂×C₄	1	4	yes	—
25	C₄×#92418	—	2	1	no	—
26	C₄×#92418	—	2	1	no	—
27	C₈×Dic₆₁	—	2	4	no	—
28	C₈×Dic₆₁	—	4	4	no	—
29	C₈×Dic₆₁	C₈	1	8	yes	complemented
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	—	4	2	no	—
46	C₂×C₈×C₆₁	—	4	2	no	—
47	C₂×C₈×C₆₁	C₄²	1	2	yes	—
48	C₂×C₈×C₆₁	C₄²	1	2	yes	—
49	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	#41	1	1	yes	characteristic
53	C₄²×C₆₁	#41	1	1	yes	characteristic
54	C₄×Dic₆₁	—	2	1	no	—
55	C₄×Dic₆₁	—	2	1	no	—
56	C₄×Dic₆₁	—	2	4	no	—
57	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
58	C₄×Dic₆₁	M₁₆	1	4	yes	—
59	C₄×Dic₆₁	D₈	1	2	yes	—
60	C₄×Dic₆₁	Q₁₆	1	2	yes	—
61	C₂×#92418	—	2	2	no	—
62	C₂×#92418	—	2	2	no	—
63	C₂×#92418	—	2	4	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂×#92418	C₂×C₈	1	4	yes	—
67	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
81	C₂×Dic₆₁	—	2	2	no	—
82	C₂×Dic₆₁	—	2	2	no	—
83	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
84	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
85	C₂×Dic₆₁	#107	1	4	yes	—
86	C₂×Dic₆₁	#108	1	4	yes	—
87	C₈×C₆₁	—	2	2	no	—
88	C₈×C₆₁	—	4	4	no	—
89	C₈×C₆₁	C₄×C₈	1	4	yes	complemented
90	#92418	—	2	4	no	—
91	#92418	—	4	4	no	—
92	C₄×#335	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
96	C₂²×C₆₁	#427	1	1	yes	characteristic
97	C₂²×C₆₁	C₂×#108	1	2	yes	—
98	C₄×C₆₁	—	2	1	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
101	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₆₁	#431	1	2	yes	—
103	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
104	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
105	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
106	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
107	Dic₆₁	—	2	4	no	—
108	Dic₆₁	#335	1	8	yes	direct product factor
109	C₄×C₄⋊C₈	—	61	2	no	—
110	C₄²×C₈	—	61	1	no	—
111	C₂×#335	—	61	1	no	—
112	C₂×#335	D₆₁	1	1	yes	characteristic
113	#1277	—	61	2	no	—
114	C₂×C₆₁	C₂×#335	1	2	yes	—
115	C₂×C₆₁	#1564	1	2	yes	—
116	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
118	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	61	2	no	—
120	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
121	C₂×C₄×C₈	—	61	2	no	—
122	C₂×C₄×C₈	—	61	2	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₂×C₄×C₈	—	122	2	no	—
125	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
126	C₄³	—	61	1	no	—
127	#336	—	122	2	no	—
128	#335	—	61	4	no	—
129	#335	Dic₆₁	1	4	yes	direct product factor
130	C₆₁	C₄×#335	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	61	2	no	—
133	C₂²×C₈	—	122	2	no	—
134	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
136	C₂×C₄²	—	61	2	no	—
137	C₂×C₄²	—	61	1	no	—
138	C₂×C₄²	—	61	1	no	—
139	C₂×C₄²	—	61	2	no	—
140	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	61	4	no	—
143	C₄⋊C₈	—	122	2	no	—
144	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
145	C₄×C₈	—	61	8	no	—
146	C₄×C₈	—	122	4	no	—
147	C₄×C₈	—	122	1	no	—
148	C₄×C₈	—	122	1	no	—
149	C₄×C₈	—	244	4	no	—
150	C₄×C₈	C₄×D₆₁	1	2	yes	—
151	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
152	C₂²×C₄	—	61	2	no	—
153	C₂²×C₄	—	61	2	no	—
154	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
156	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	61	4	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	2	no	—
164	C₂×C₈	—	122	4	no	—
165	C₂×C₈	—	244	2	no	—
166	C₂×C₈	—	244	2	no	—
167	C₂×C₈	C₈×D₆₁	1	4	yes	—
168	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
169	C₂×C₈	C₄×Dic₆₁	1	2	yes	—
170	C₄²	—	61	4	no	—
171	C₄²	—	61	4	no	—
172	C₄²	—	61	2	no	—
173	C₄²	—	61	2	no	—
174	C₄²	—	122	4	no	—
175	C₄²	—	122	1	no	—
176	C₄²	—	122	1	no	—
177	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
178	C₄²	#11758224	1	1	yes	characteristic
179	C₄²	#11758226	1	1	yes	characteristic
180	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
181	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	61	4	no	—
184	C₂×C₄	—	61	4	no	—
185	C₂×C₄	—	61	4	no	—
186	C₂×C₄	—	61	4	no	—
187	C₂×C₄	—	122	2	no	—
188	C₂×C₄	—	122	2	no	—
189	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
190	C₂×C₄	#14519735	1	1	yes	characteristic
191	C₂×C₄	#14519739	1	1	yes	characteristic
192	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
193	C₂×C₄	#14519742	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₆₁	1	1	yes	characteristic
195	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
196	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
197	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
198	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
199	C₂×C₄	#14519796	1	1	yes	characteristic
200	C₂×C₄	#14519800	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	4	4	no	—
203	C₈	—	122	4	no	—
204	C₈	—	244	4	no	—
205	C₈	C₈×Dic₆₁	1	4	yes	complemented
206	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
207	C₂²	C₄⋊C₈×D₆₁	1	2	yes	—
208	C₂²	#107×D₆₁	1	1	yes	characteristic
209	C₂²	#16839799	1	1	yes	characteristic
210	C₂²	#108×D₆₁	1	2	yes	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	61	8	no	—
214	C₄	—	122	4	no	—
215	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
216	C₄	M₁₆×Dic₆₁	1	1	yes	characteristic
217	C₄	#16839814	1	2	yes	—
218	C₄	#16839844	1	1	yes	characteristic
219	C₄	D₈×Dic₆₁	1	1	yes	characteristic
220	C₄	#16839856	1	1	yes	characteristic
221	C₄	Q₁₆×Dic₆₁	1	1	yes	characteristic
222	C₂	#335×D₆₁	1	2	yes	—
223	C₂	#21444566	1	2	yes	—
224	C₂	C₄⋊C₈×Dic₆₁	1	1	yes	characteristic
225	C₂	#107×Dic₆₁	1	1	yes	characteristic
226	C₂	#108×Dic₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	C₄×#335
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
13. #16839515	C₄	semidirect
16. C₆₁×#335	C₄	semidirect
29. C₈×Dic₆₁	C₈	semidirect
89. C₈×C₆₁	C₄×C₈	semidirect
108. Dic₆₁	#335	direct (N × H)
129. #335	Dic₆₁	direct (N × H)
130. C₆₁	C₄×#335	semidirect
205. C₈	C₈×Dic₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₆₁	2
C₄×C₈×Dic₆₁	2
C₂×#16839515	2
#21443653	2
C₂×C₆₁×#335	2
C₄×#335	61

Subgroups — #335×Dic61 #29571200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #335×Dic₆₁ #29571200