Subgroups — C₆₁×#34182 #29537615

Order 15616.

2858 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#519	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2351	C₂	1	8	yes	maximal, complemented
4	Q₈×C₄⋊C₄×C₆₁	C₂	1	8	yes	maximal, complemented
5	C₆₁×#2316	C₂	1	6	yes	maximal
6	C₆₁×#2285	C₂	1	6	yes	maximal
7	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×#519	—	2	16	no	—
9	C₂×C₄×Q₈×C₆₁	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₆₁	C₂²	1	16	yes	—
11	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
13	C₂×#117×C₆₁	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
15	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
16	C₆₁×#389	C₂²	1	48	yes	—
17	C₆₁×#384	C₂²	1	12	yes	—
18	C₆₁×#382	C₂²	1	12	yes	—
19	C₆₁×#380	C₂²	1	4	yes	—
20	C₆₁×#380	C₂²	1	12	yes	—
21	C₄×C₄⋊C₄×C₆₁	C₂²	1	12	yes	—
22	C₄×C₄⋊C₄×C₆₁	C₂²	1	12	yes	—
23	C₂²×Q₈×C₆₁	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₆₁	D₄	1	8	yes	—
25	C₂²×Q₈×C₆₁	Q₈	1	8	yes	—
26	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
30	C₄×Q₈×C₆₁	—	2	64	no	—
31	C₄×Q₈×C₆₁	C₂×C₄	1	32	yes	—
32	C₄×Q₈×C₆₁	C₂×C₄	1	32	yes	—
33	#117×C₆₁	—	2	48	no	—
34	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	24	yes	—
35	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
36	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
37	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
38	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
39	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	24	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	24	yes	—
42	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
44	#95×C₆₁	C₂³	1	8	yes	—
45	#95×C₆₁	C₂³	1	24	yes	—
46	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₆₁	C₄⋊C₄	1	64	yes	complemented
48	C₂×Q₈×C₆₁	C₂²×C₄	1	16	yes	—
49	C₂×Q₈×C₆₁	C₂×D₄	1	16	yes	—
50	C₂×Q₈×C₆₁	C₂×Q₈	1	16	yes	—
51	C₂²×C₄×C₆₁	—	2	4	no	—
52	C₂²×C₄×C₆₁	—	4	6	no	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₆₁	C₂×Q₈	1	12	yes	—
58	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
59	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
62	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₆₁	—	2	48	no	—
64	C₄⋊C₄×C₆₁	—	2	16	no	—
65	C₄⋊C₄×C₆₁	—	4	24	no	—
66	C₄⋊C₄×C₆₁	C₂²×C₄	1	48	yes	—
67	C₂²⋊C₄×C₆₁	—	2	16	no	—
68	C₄²×C₆₁	—	2	48	no	—
69	C₄²×C₆₁	C₂²×C₄	1	24	yes	—
70	C₄²×C₆₁	C₂²×C₄	1	24	yes	—
71	C₂³×C₆₁	—	2	2	no	—
72	C₂³×C₆₁	—	2	2	no	—
73	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
74	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
75	C₂³×C₆₁	C₂²×Q₈	1	2	yes	—
76	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
77	Q₈×C₆₁	C₂×C₄⋊C₄	1	64	yes	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	12	no	—
80	C₂×C₄×C₆₁	—	2	8	no	—
81	C₂×C₄×C₆₁	—	2	16	no	—
82	C₂×C₄×C₆₁	—	2	16	no	—
83	C₂×C₄×C₆₁	—	2	16	no	—
84	C₂×C₄×C₆₁	—	4	12	no	—
85	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	48	yes	—
86	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
87	C₂×C₄×C₆₁	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₆₁	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
90	C₂×C₄×C₆₁	C₂²×D₄	1	12	yes	—
91	C₂×C₄×C₆₁	C₂²×Q₈	1	12	yes	—
92	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
93	#34182	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₆₁	—	2	2	no	—
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	—	2	4	no	—
97	C₂²×C₆₁	—	2	4	no	—
98	C₂²×C₆₁	C₂²×C₄⋊C₄	1	4	yes	—
99	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
100	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₆₁	C₂³×Q₈	1	1	yes	characteristic
102	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
103	C₂²×C₆₁	#585	1	2	yes	—
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	—	2	12	no	—
106	C₄×C₆₁	—	4	16	no	—
107	C₄×C₆₁	C₂²×C₄⋊C₄	1	24	yes	—
108	C₄×C₆₁	#519	1	4	yes	complemented
109	C₂×#519	C₂×C₆₁	1	2	yes	—
110	C₂²×C₄×Q₈	C₂×C₆₁	1	1	yes	characteristic
111	#2351	C₂×C₆₁	1	8	yes	complemented
112	Q₈×C₄⋊C₄	C₂×C₆₁	1	8	yes	complemented
113	#2316	C₂×C₆₁	1	6	yes	—
114	#2285	C₂×C₆₁	1	6	yes	—
115	C₂×C₆₁	—	2	4	no	—
116	C₂×C₆₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₆₁	C₂×#519	1	1	yes	characteristic
118	C₂×C₆₁	#3428	1	1	yes	characteristic
119	C₂×C₆₁	#3497	1	2	yes	—
120	C₂×C₆₁	#3502	1	2	yes	—
121	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
122	#519	—	2	16	no	—
123	C₂×C₄×Q₈	C₄×C₆₁	1	16	yes	complemented
124	C₂×C₄×Q₈	C₂²×C₆₁	1	16	yes	—
125	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
126	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
127	C₂×#117	C₂²×C₆₁	1	12	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₆₁	1	3	yes	—
129	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
130	#389	C₂²×C₆₁	1	48	yes	—
131	#384	C₂²×C₆₁	1	12	yes	—
132	#382	C₂²×C₆₁	1	12	yes	—
133	#380	C₂²×C₆₁	1	12	yes	—
134	#380	C₂²×C₆₁	1	4	yes	—
135	C₄×C₄⋊C₄	C₂²×C₆₁	1	12	yes	—
136	C₄×C₄⋊C₄	C₂²×C₆₁	1	12	yes	—
137	C₆₁	#34182	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₈	C₂×C₄×C₆₁	1	8	yes	—
139	C₂²×Q₈	D₄×C₆₁	1	8	yes	—
140	C₂²×Q₈	Q₈×C₆₁	1	8	yes	—
141	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
142	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
143	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
144	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
145	C₄×Q₈	—	2	64	no	—
146	C₄×Q₈	C₂×C₄×C₆₁	1	32	yes	—
147	C₄×Q₈	C₂×C₄×C₆₁	1	32	yes	—
148	#117	—	2	48	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	24	yes	—
150	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
151	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
152	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
153	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
154	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
155	C₂×C₄²	C₂×C₄×C₆₁	1	24	yes	—
156	C₂×C₄²	C₂³×C₆₁	1	24	yes	—
157	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
158	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
159	#95	C₂³×C₆₁	1	8	yes	—
160	#95	C₂³×C₆₁	1	24	yes	—
161	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
162	C₂×Q₈	C₄⋊C₄×C₆₁	1	64	yes	complemented
163	C₂×Q₈	C₂²×C₄×C₆₁	1	16	yes	—
164	C₂×Q₈	C₂×D₄×C₆₁	1	16	yes	—
165	C₂×Q₈	C₂×Q₈×C₆₁	1	16	yes	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	6	no	—
168	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₆₁	1	12	yes	—
170	C₂²×C₄	C₂²×C₄×C₆₁	1	12	yes	—
171	C₂²×C₄	C₂×D₄×C₆₁	1	12	yes	—
172	C₂²×C₄	C₂×Q₈×C₆₁	1	12	yes	—
173	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
174	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
175	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
176	C₂²×C₄	C₂⁴×C₆₁	1	6	yes	—
177	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
178	C₄⋊C₄	—	2	48	no	—
179	C₄⋊C₄	—	2	16	no	—
180	C₄⋊C₄	—	4	24	no	—
181	C₄⋊C₄	C₂²×C₄×C₆₁	1	48	yes	—
182	C₂²⋊C₄	—	2	16	no	—
183	C₄²	—	2	48	no	—
184	C₄²	C₂²×C₄×C₆₁	1	24	yes	—
185	C₄²	C₂²×C₄×C₆₁	1	24	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
189	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
190	C₂³	C₂²×Q₈×C₆₁	1	2	yes	—
191	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
192	Q₈	C₂×C₄⋊C₄×C₆₁	1	64	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	4	12	no	—
200	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	48	yes	—
201	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
202	C₂×C₄	C₂³×C₄×C₆₁	1	12	yes	—
203	C₂×C₄	C₂³×C₄×C₆₁	1	12	yes	—
204	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
205	C₂×C₄	C₂²×D₄×C₆₁	1	12	yes	—
206	C₂×C₄	C₂²×Q₈×C₆₁	1	12	yes	—
207	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂²×C₄⋊C₄×C₆₁	1	4	yes	—
213	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
214	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
215	C₂²	C₂³×Q₈×C₆₁	1	1	yes	characteristic
216	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
217	C₂²	C₆₁×#585	1	2	yes	—
218	C₄	—	2	2	no	—
219	C₄	—	2	12	no	—
220	C₄	—	4	16	no	—
221	C₄	C₂²×C₄⋊C₄×C₆₁	1	24	yes	—
222	C₄	C₆₁×#519	1	4	yes	complemented
223	C₂	—	2	4	no	—
224	C₂	C₂³×C₄⋊C₄×C₆₁	1	1	yes	characteristic
225	C₂	C₂×C₆₁×#519	1	1	yes	characteristic
226	C₂	C₆₁×#3428	1	1	yes	characteristic
227	C₂	C₆₁×#3497	1	2	yes	—
228	C₂	C₆₁×#3502	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂⁴×C₆₁	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2351	C₂	semidirect
4. Q₈×C₄⋊C₄×C₆₁	C₂	semidirect
9. C₂×C₄×Q₈×C₆₁	C₄	semidirect
47. C₂×Q₈×C₆₁	C₄⋊C₄	semidirect
93. #34182	C₆₁	direct (N × H)
108. C₄×C₆₁	#519	semidirect
111. #2351	C₂×C₆₁	semidirect
112. Q₈×C₄⋊C₄	C₂×C₆₁	semidirect
123. C₂×C₄×Q₈	C₄×C₆₁	semidirect
137. C₆₁	#34182	direct (N × H)
162. C₂×Q₈	C₄⋊C₄×C₆₁	semidirect
222. C₄	C₆₁×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#519	2
C₂²×C₄×Q₈×C₆₁	2
C₆₁×#2351	2
Q₈×C₄⋊C₄×C₆₁	2
C₆₁×#2316	2
C₆₁×#2285	2
#34182	61

Subgroups — C61×#34182 #29537615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#34182 #29537615