Subgroups — C₂²×C₆₁×#364 #29536832

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#364	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	—	2	6	no	—
8	C₂×M₃₂×C₆₁	C₂²	1	12	yes	—
9	C₂²×C₁₆×C₆₁	—	2	2	no	—
10	C₂×C₄×C₈×C₆₁	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	6	yes	—
12	C₆₁×#364	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₆₁	—	2	1	no	—
14	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	C₈	1	6	yes	—
16	C₂²×C₈×C₆₁	C₈	1	2	yes	—
17	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₆₁	D₄	1	3	yes	—
20	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	Q₈	1	3	yes	—
23	C₂²×C₈×C₆₁	C₂³	1	3	yes	—
24	C₂×C₄²×C₆₁	—	2	3	no	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
26	M₃₂×C₆₁	—	2	12	no	—
27	M₃₂×C₆₁	C₂³	1	8	yes	—
28	C₂×C₁₆×C₆₁	—	2	12	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	12	yes	—
30	C₄×C₈×C₆₁	C₂³	1	4	yes	—
31	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	—	2	8	no	—
33	C₂²×C₄×C₆₁	—	2	6	no	—
34	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	M₁₆	1	3	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₆₁	C₄⋊C₄	1	3	yes	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	6	yes	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	3	yes	—
43	C₂×C₈×C₆₁	C₂×C₈	1	12	yes	—
44	C₂×C₈×C₆₁	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
48	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
49	C₂×C₈×C₆₁	C₂×Q₈	1	3	yes	—
50	C₂×C₈×C₆₁	C₂×Q₈	1	3	yes	—
51	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₄²×C₆₁	—	2	6	no	—
53	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
54	C₁₆×C₆₁	—	2	8	no	—
55	C₂³×C₆₁	—	2	4	no	—
56	C₂³×C₆₁	C₄⋊C₈	1	3	yes	—
57	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
58	C₂³×C₆₁	C₂×M₁₆	1	3	yes	—
59	C₂×C₄×C₆₁	—	2	24	no	—
60	C₂×C₄×C₆₁	—	2	4	no	—
61	C₂×C₄×C₆₁	C₄⋊C₈	1	6	yes	—
62	C₂×C₄×C₆₁	C₄⋊C₈	1	3	yes	—
63	C₂×C₄×C₆₁	C₄⋊C₈	1	3	yes	—
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
69	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
70	C₈×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
71	C₈×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
72	C₈×C₆₁	C₂²×C₈	1	8	yes	—
73	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
74	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×#364	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₆₁	—	2	6	no	—
77	C₂²×C₆₁	—	2	6	no	—
78	C₂²×C₆₁	#364	1	4	yes	direct product factor
79	C₂²×C₆₁	C₂×C₄⋊C₈	1	3	yes	—
80	C₂²×C₆₁	C₂×C₄⋊C₈	1	3	yes	—
81	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
82	C₄×C₆₁	—	2	8	no	—
83	C₄×C₆₁	C₂×C₄⋊C₈	1	3	yes	—
84	C₄×C₆₁	C₂×C₄⋊C₈	1	3	yes	—
85	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
87	C₂²×M₃₂	C₂×C₆₁	1	2	yes	—
88	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
89	C₂×#364	C₂×C₆₁	1	12	yes	direct product factor
90	C₂×C₆₁	—	2	1	no	—
91	C₂×C₆₁	—	2	3	no	—
92	C₂×C₆₁	C₂×#364	1	6	yes	direct product factor
93	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
94	C₂³×C₈	C₄×C₆₁	1	1	yes	characteristic
95	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
96	C₂²×C₄²	C₄×C₆₁	1	1	yes	characteristic
97	C₂×M₃₂	—	2	6	no	—
98	C₂×M₃₂	C₂²×C₆₁	1	12	yes	—
99	C₂²×C₁₆	—	2	2	no	—
100	C₂×C₄×C₈	C₄×C₆₁	1	6	yes	—
101	C₂×C₄×C₈	C₂²×C₆₁	1	6	yes	—
102	#364	C₂²×C₆₁	1	16	yes	direct product factor
103	C₆₁	C₂²×#364	1	1	yes	characteristic, direct product factor, Sylow
104	C₂³×C₄	—	2	1	no	—
105	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
106	C₂²×C₈	C₈×C₆₁	1	6	yes	—
107	C₂²×C₈	C₈×C₆₁	1	2	yes	—
108	C₂²×C₈	C₂×C₄×C₆₁	1	3	yes	—
109	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
110	C₂²×C₈	D₄×C₆₁	1	3	yes	—
111	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
112	C₂²×C₈	Q₈×C₆₁	1	1	yes	characteristic
113	C₂²×C₈	Q₈×C₆₁	1	3	yes	—
114	C₂²×C₈	C₂³×C₆₁	1	3	yes	—
115	C₂×C₄²	—	2	3	no	—
116	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
117	M₃₂	—	2	12	no	—
118	M₃₂	C₂³×C₆₁	1	8	yes	—
119	C₂×C₁₆	—	2	12	no	—
120	C₄×C₈	C₂×C₄×C₆₁	1	12	yes	—
121	C₄×C₈	C₂³×C₆₁	1	4	yes	—
122	C₂⁴	M₁₆×C₆₁	1	1	yes	characteristic
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	6	no	—
125	C₂²×C₄	C₄⋊C₄×C₆₁	1	3	yes	—
126	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₈×C₆₁	1	3	yes	—
128	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
129	C₂²×C₄	M₁₆×C₆₁	1	3	yes	—
130	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
131	C₂×C₈	C₄⋊C₄×C₆₁	1	6	yes	—
132	C₂×C₈	C₄⋊C₄×C₆₁	1	3	yes	—
133	C₂×C₈	C₄⋊C₄×C₆₁	1	3	yes	—
134	C₂×C₈	C₂×C₈×C₆₁	1	12	yes	—
135	C₂×C₈	C₂×C₈×C₆₁	1	12	yes	—
136	C₂×C₈	C₂²×C₄×C₆₁	1	3	yes	—
137	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
138	C₂×C₈	C₂×D₄×C₆₁	1	3	yes	—
139	C₂×C₈	C₂×D₄×C₆₁	1	3	yes	—
140	C₂×C₈	C₂×Q₈×C₆₁	1	3	yes	—
141	C₂×C₈	C₂×Q₈×C₆₁	1	3	yes	—
142	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
143	C₄²	—	2	6	no	—
144	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
145	C₁₆	—	2	8	no	—
146	C₂³	—	2	4	no	—
147	C₂³	C₄⋊C₈×C₆₁	1	3	yes	—
148	C₂³	C₄⋊C₈×C₆₁	1	1	yes	characteristic
149	C₂³	C₂×M₁₆×C₆₁	1	3	yes	—
150	C₂×C₄	—	2	24	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	C₄⋊C₈×C₆₁	1	6	yes	—
153	C₂×C₄	C₄⋊C₈×C₆₁	1	3	yes	—
154	C₂×C₄	C₄⋊C₈×C₆₁	1	3	yes	—
155	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	3	yes	—
156	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	3	yes	—
157	C₂×C₄	C₂²×C₈×C₆₁	1	3	yes	—
158	C₂×C₄	C₂²×C₈×C₆₁	1	3	yes	—
159	C₂×C₄	C₂×M₁₆×C₆₁	1	3	yes	—
160	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
161	C₈	C₂×C₄⋊C₄×C₆₁	1	3	yes	—
162	C₈	C₂×C₄⋊C₄×C₆₁	1	3	yes	—
163	C₈	C₂²×C₈×C₆₁	1	8	yes	—
164	C₈	C₂²×D₄×C₆₁	1	1	yes	characteristic
165	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
166	C₂²	—	2	6	no	—
167	C₂²	—	2	6	no	—
168	C₂²	C₆₁×#364	1	4	yes	direct product factor
169	C₂²	C₂×C₄⋊C₈×C₆₁	1	3	yes	—
170	C₂²	C₂×C₄⋊C₈×C₆₁	1	3	yes	—
171	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
172	C₄	—	2	8	no	—
173	C₄	C₂×C₄⋊C₈×C₆₁	1	3	yes	—
174	C₄	C₂×C₄⋊C₈×C₆₁	1	3	yes	—
175	C₄	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
176	C₄	C₂³×C₈×C₆₁	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×C₆₁×#364	1	6	yes	direct product factor
180	C₂	C₂²×C₄⋊C₈×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₆₁	D₄
commutator	C₄	C₂³×C₈×C₆₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₄⋊C₈
prime radical	C₂⁴×C₆₁	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₆₁×#364	C₂	direct (N × H)
12. C₆₁×#364	C₂²	direct (N × H)
75. C₂²×#364	C₆₁	direct (N × H)
78. C₂²×C₆₁	#364	direct (N × H)
89. C₂×#364	C₂×C₆₁	direct (N × H)
92. C₂×C₆₁	C₂×#364	direct (N × H)
102. #364	C₂²×C₆₁	direct (N × H)
103. C₆₁	C₂²×#364	direct (N × H)
168. C₂²	C₆₁×#364	direct (N × H)
179. C₂	C₂×C₆₁×#364	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₁	2
C₂²×C₄×C₈×C₆₁	2
C₂×C₆₁×#364	2
C₂²×#364	61

Subgroups — C22×C61×#364 #29536832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#364 #29536832