Subgroups — C₆₁×#7174 #29510607

Order 15616.

262 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₆₁	—	2	1	no	—
5	C₆₁×#431	—	2	1	no	—
6	C₂×#106×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
9	C₆₁×#346	—	2	1	no	—
10	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
11	C₆₁×#340	—	2	1	no	—
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#334	C₄	1	2	yes	complemented
14	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₂×C₄²×C₆₁	—	4	1	no	—
18	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
19	M₃₂×C₆₁	—	2	4	no	—
20	C₂×C₁₆×C₆₁	—	2	2	no	—
21	#106×C₆₁	—	2	2	no	—
22	#106×C₆₁	C₈	1	4	yes	complemented
23	C₄⋊C₈×C₆₁	—	2	2	no	—
24	C₂²⋊C₈×C₆₁	—	4	1	no	—
25	C₄×C₈×C₆₁	—	4	1	no	—
26	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	#95×C₆₁	—	4	1	no	—
31	C₂²×C₄×C₆₁	—	4	2	no	—
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
43	C₄⋊C₄×C₆₁	—	2	4	no	—
44	C₄⋊C₄×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	—	8	2	no	—
46	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
50	C₁₆×C₆₁	—	2	4	no	—
51	C₂³×C₆₁	#102	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	—	4	4	no	—
55	C₂×C₄×C₆₁	—	8	2	no	—
56	C₂×C₄×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#104	1	2	yes	—
63	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	—	2	1	no	—
67	C₈×C₆₁	—	8	2	no	—
68	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
69	C₈×C₆₁	#97	1	1	yes	characteristic
70	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
71	#7174	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₆₁	—	2	1	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	D₄⋊C₈	1	2	yes	—
75	C₂²×C₆₁	#340	1	1	yes	characteristic
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	—	2	1	no	—
79	C₄×C₆₁	—	8	4	no	—
80	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
81	C₄×C₆₁	#336	1	1	yes	characteristic
82	C₄×C₆₁	#339	1	1	yes	characteristic
83	C₄×C₆₁	#365	1	2	yes	—
84	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic
85	C₄×#106	C₂×C₆₁	1	1	yes	characteristic
86	#1564	C₂×C₆₁	1	1	yes	characteristic
87	C₂×C₆₁	—	2	2	no	—
88	C₂×C₆₁	#1279	1	1	yes	characteristic
89	C₂×C₆₁	#1383	1	1	yes	characteristic
90	C₂×C₆₁	#1384	1	1	yes	characteristic
91	C₂×M₃₂	—	2	1	no	—
92	#431	—	2	1	no	—
93	C₂×#106	C₄×C₆₁	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
95	C₄×C₄⋊C₄	—	2	1	no	—
96	#346	—	2	1	no	—
97	C₄×C₁₆	C₄×C₆₁	1	2	yes	complemented
98	#340	—	2	1	no	—
99	#335	—	2	1	no	—
100	#334	C₄×C₆₁	1	2	yes	complemented
101	C₆₁	#7174	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
103	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	—	4	1	no	—
106	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
107	M₃₂	—	2	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#106	—	2	2	no	—
110	#106	C₈×C₆₁	1	4	yes	complemented
111	C₄⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
115	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
116	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
118	#95	—	4	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
121	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
122	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
128	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
129	C₂×C₈	C₂×C₈×C₆₁	1	2	yes	—
130	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄²	—	8	2	no	—
134	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
135	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
136	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
137	C₄²	Q₁₆×C₆₁	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	#102×C₆₁	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
147	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
148	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
149	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
150	C₂×C₄	#104×C₆₁	1	2	yes	—
151	C₂×C₄	#106×C₆₁	1	1	yes	characteristic
152	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	8	2	no	—
156	C₈	C₄×C₈×C₆₁	1	1	yes	characteristic
157	C₈	#97×C₆₁	1	1	yes	characteristic
158	C₈	C₄⋊C₈×C₆₁	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₆₁×D₄⋊C₈	1	2	yes	—
162	C₂²	C₆₁×#340	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	8	4	no	—
167	C₄	C₆₁×Q₈⋊C₈	1	2	yes	—
168	C₄	C₆₁×#336	1	1	yes	characteristic
169	C₄	C₆₁×#339	1	1	yes	characteristic
170	C₄	C₆₁×#365	1	2	yes	—
171	C₂	—	2	2	no	—
172	C₂	C₆₁×#1279	1	1	yes	characteristic
173	C₂	C₆₁×#1383	1	1	yes	characteristic
174	C₂	C₆₁×#1384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#106
commutator	C₈	C₄×C₈×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₆₁	C₄	semidirect
13. C₆₁×#334	C₄	semidirect
22. #106×C₆₁	C₈	semidirect
71. #7174	C₆₁	direct (N × H)
97. C₄×C₁₆	C₄×C₆₁	semidirect
100. #334	C₄×C₆₁	semidirect
101. C₆₁	#7174	direct (N × H)
110. #106	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₆₁	2
C₄×#106×C₆₁	2
C₆₁×#1564	2
#7174	61

Subgroups — C61×#7174 #29510607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7174 #29510607