Subgroups — C₈×#95×C₆₁ #29514128

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₆₁	C₂	1	3	yes	maximal
2	C₂×C₆₁×#336	C₂	1	3	yes	maximal
3	C₄×#95×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₆₁	C₂²	1	3	yes	—
5	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
7	C₂×C₄×C₈×C₆₁	—	2	12	no	—
8	C₂×C₄×C₈×C₆₁	C₄	1	12	yes	complemented
9	C₂×#95×C₆₁	C₄	1	4	yes	—
10	C₆₁×#336	C₄	1	12	yes	complemented
11	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	—	2	6	no	—
14	C₂²×C₈×C₆₁	—	2	3	no	—
15	C₂²×C₈×C₆₁	—	2	3	no	—
16	C₂²×C₈×C₆₁	C₂×C₄	1	12	yes	—
17	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₆₁	D₄	1	6	yes	—
20	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
21	C₂×C₄²×C₆₁	—	2	6	no	—
22	C₂×C₄²×C₆₁	—	2	6	no	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
25	C₄×C₈×C₆₁	—	2	24	no	—
26	#95×C₆₁	C₈	1	16	yes	direct product factor
27	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	—	2	12	no	—
29	C₂²×C₄×C₆₁	—	2	12	no	—
30	C₂²×C₄×C₆₁	C₄²	1	12	yes	—
31	C₂²×C₄×C₆₁	C₂×C₈	1	24	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	#41	1	3	yes	—
38	C₂×C₈×C₆₁	—	2	6	no	—
39	C₂×C₈×C₆₁	—	2	6	no	—
40	C₂×C₈×C₆₁	—	2	12	no	—
41	C₂×C₈×C₆₁	—	2	12	no	—
42	C₂×C₈×C₆₁	C₄²	1	12	yes	—
43	C₂×C₈×C₆₁	C₄²	1	4	yes	—
44	C₂×C₈×C₆₁	C₂²⋊C₄	1	12	yes	—
45	C₂×C₈×C₆₁	C₄⋊C₄	1	12	yes	—
46	C₄²×C₆₁	—	2	6	no	—
47	C₄²×C₆₁	—	2	6	no	—
48	C₄²×C₆₁	—	2	12	no	—
49	C₂³×C₆₁	C₂×C₄²	1	3	yes	—
50	C₂³×C₆₁	C₄×D₄	1	3	yes	—
51	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
52	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
53	C₂³×C₆₁	#131	1	4	yes	—
54	C₂×C₄×C₆₁	—	2	12	no	—
55	C₂×C₄×C₆₁	—	2	12	no	—
56	C₂×C₄×C₆₁	—	2	48	no	—
57	C₂×C₄×C₆₁	C₄×C₈	1	24	yes	complemented
58	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₆₁	#117	1	3	yes	—
63	C₂×C₄×C₆₁	#117	1	3	yes	—
64	C₂×C₄×C₆₁	C₄×D₄	1	3	yes	—
65	C₂×C₄×C₆₁	C₄×D₄	1	6	yes	—
66	C₂×C₄×C₆₁	C₄×Q₈	1	3	yes	—
67	C₈×C₆₁	—	2	12	no	—
68	C₈×C₆₁	#95	1	8	yes	direct product factor
69	C₈×#95	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₆₁	C₄³	1	1	yes	characteristic
71	C₂²×C₆₁	C₄×C₂²⋊C₄	1	3	yes	—
72	C₂²×C₆₁	C₄×C₄⋊C₄	1	3	yes	—
73	C₂²×C₆₁	C₂×C₄×C₈	1	6	yes	—
74	C₂²×C₆₁	#405	1	6	yes	—
75	C₂²×C₆₁	C₈×D₄	1	12	yes	—
76	C₂²×C₆₁	C₈×Q₈	1	4	yes	—
77	C₄×C₆₁	—	2	24	no	—
78	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
79	C₄×C₆₁	#376	1	1	yes	characteristic
80	C₄×C₆₁	C₄×C₂²⋊C₄	1	3	yes	—
81	C₄×C₆₁	C₄×C₄⋊C₄	1	3	yes	—
82	C₂²×C₄×C₈	C₂×C₆₁	1	3	yes	—
83	C₂×#336	C₂×C₆₁	1	3	yes	—
84	C₄×#95	C₂×C₆₁	1	1	yes	characteristic
85	C₂×C₆₁	C₄×#95	1	1	yes	characteristic
86	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
87	C₂×C₆₁	#1726	1	1	yes	characteristic
88	C₂×C₆₁	C₈×C₂²⋊C₄	1	6	yes	—
89	C₂×C₆₁	C₈×C₄⋊C₄	1	6	yes	—
90	C₂³×C₈	C₂²×C₆₁	1	3	yes	—
91	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
92	C₂²×C₄²	C₂²×C₆₁	1	3	yes	—
93	C₂×C₄×C₈	—	2	12	no	—
94	C₂×C₄×C₈	C₄×C₆₁	1	12	yes	complemented
95	C₂×#95	C₄×C₆₁	1	4	yes	—
96	#336	C₄×C₆₁	1	12	yes	complemented
97	C₆₁	C₈×#95	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³×C₄	C₂×C₄×C₆₁	1	6	yes	—
99	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
100	C₂²×C₈	—	2	6	no	—
101	C₂²×C₈	—	2	3	no	—
102	C₂²×C₈	—	2	3	no	—
103	C₂²×C₈	C₂×C₄×C₆₁	1	12	yes	—
104	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
105	C₂²×C₈	C₂×C₄×C₆₁	1	6	yes	—
106	C₂²×C₈	D₄×C₆₁	1	6	yes	—
107	C₂²×C₈	Q₈×C₆₁	1	2	yes	—
108	C₂×C₄²	—	2	6	no	—
109	C₂×C₄²	—	2	6	no	—
110	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
111	C₂×C₄²	C₂×C₄×C₆₁	1	6	yes	—
112	C₄×C₈	—	2	24	no	—
113	#95	C₈×C₆₁	1	16	yes	direct product factor
114	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	C₄²×C₆₁	1	12	yes	—
118	C₂²×C₄	C₂×C₈×C₆₁	1	24	yes	—
119	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
120	C₂²×C₄	C₂²×C₄×C₆₁	1	3	yes	—
121	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
122	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
123	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
124	C₂²×C₄	#41×C₆₁	1	3	yes	—
125	C₂×C₈	—	2	6	no	—
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	—	2	12	no	—
128	C₂×C₈	—	2	12	no	—
129	C₂×C₈	C₄²×C₆₁	1	12	yes	—
130	C₂×C₈	C₄²×C₆₁	1	4	yes	—
131	C₂×C₈	C₂²⋊C₄×C₆₁	1	12	yes	—
132	C₂×C₈	C₄⋊C₄×C₆₁	1	12	yes	—
133	C₄²	—	2	6	no	—
134	C₄²	—	2	6	no	—
135	C₄²	—	2	12	no	—
136	C₂³	C₂×C₄²×C₆₁	1	3	yes	—
137	C₂³	C₄×D₄×C₆₁	1	3	yes	—
138	C₂³	C₄×Q₈×C₆₁	1	1	yes	characteristic
139	C₂³	C₂²×C₈×C₆₁	1	4	yes	—
140	C₂³	#131×C₆₁	1	4	yes	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	2	12	no	—
143	C₂×C₄	—	2	48	no	—
144	C₂×C₄	C₄×C₈×C₆₁	1	24	yes	complemented
145	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄²×C₆₁	1	3	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	3	yes	—
148	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	3	yes	—
149	C₂×C₄	#117×C₆₁	1	3	yes	—
150	C₂×C₄	#117×C₆₁	1	3	yes	—
151	C₂×C₄	C₄×D₄×C₆₁	1	3	yes	—
152	C₂×C₄	C₄×D₄×C₆₁	1	6	yes	—
153	C₂×C₄	C₄×Q₈×C₆₁	1	3	yes	—
154	C₈	—	2	12	no	—
155	C₈	#95×C₆₁	1	8	yes	direct product factor
156	C₂²	C₄³×C₆₁	1	1	yes	characteristic
157	C₂²	C₄×C₂²⋊C₄×C₆₁	1	3	yes	—
158	C₂²	C₄×C₄⋊C₄×C₆₁	1	3	yes	—
159	C₂²	C₂×C₄×C₈×C₆₁	1	6	yes	—
160	C₂²	C₆₁×#405	1	6	yes	—
161	C₂²	C₈×D₄×C₆₁	1	12	yes	—
162	C₂²	C₈×Q₈×C₆₁	1	4	yes	—
163	C₄	—	2	24	no	—
164	C₄	C₂×#95×C₆₁	1	1	yes	characteristic
165	C₄	C₆₁×#376	1	1	yes	characteristic
166	C₄	C₄×C₂²⋊C₄×C₆₁	1	3	yes	—
167	C₄	C₄×C₄⋊C₄×C₆₁	1	3	yes	—
168	C₂	C₄×#95×C₆₁	1	1	yes	characteristic
169	C₂	C₄²×C₈×C₆₁	1	1	yes	characteristic
170	C₂	C₆₁×#1726	1	1	yes	characteristic
171	C₂	C₈×C₂²⋊C₄×C₆₁	1	6	yes	—
172	C₂	C₈×C₄⋊C₄×C₆₁	1	6	yes	—

	subgroup	G/H
center	C₂³×C₈×C₆₁	C₂²
commutator	C₂	C₄²×C₈×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
socle, prime radical	C₂⁴×C₆₁	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂×C₄×C₈×C₆₁	C₄	semidirect
10. C₆₁×#336	C₄	semidirect
26. #95×C₆₁	C₈	direct (N × H)
57. C₂×C₄×C₆₁	C₄×C₈	semidirect
68. C₈×C₆₁	#95	direct (N × H)
69. C₈×#95	C₆₁	direct (N × H)
94. C₂×C₄×C₈	C₄×C₆₁	semidirect
96. #336	C₄×C₆₁	semidirect
97. C₆₁	C₈×#95	direct (N × H)
113. #95	C₈×C₆₁	direct (N × H)
144. C₂×C₄	C₄×C₈×C₆₁	semidirect
155. C₈	#95×C₆₁	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×C₄×C₈×C₆₁	2
C₂×C₆₁×#336	2
C₄×#95×C₆₁	2
C₈×#95	61

Subgroups — C8×#95×C61 #29514128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#95×C₆₁ #29514128