Subgroups — C₂×C₆₁×#3556 #29566193

Order 15616.

6338 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3556	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₆₁×#564	C₂	1	4	yes	maximal, complemented
3	C₂×C₆₁×#557	C₂	1	12	yes	maximal, complemented
4	C₂×C₆₁×#541	C₂	1	12	yes	maximal
5	C₂×C₆₁×#533	C₂	1	3	yes	maximal
6	C₆₁×#564	C₂²	1	64	yes	complemented
7	C₆₁×#557	C₂²	1	192	yes	complemented
8	C₆₁×#541	C₂²	1	192	yes	—
9	C₆₁×#533	C₂²	1	48	yes	—
10	C₂×C₄⋊Q₈×C₆₁	C₂²	1	6	yes	—
11	C₂×C₄⋊Q₈×C₆₁	C₂²	1	12	yes	—
12	C₂×#126×C₆₁	C₂²	1	16	yes	—
13	C₂×#125×C₆₁	C₂²	1	6	yes	—
14	C₂×#125×C₆₁	C₂²	1	12	yes	—
15	C₂×#125×C₆₁	C₂²	1	48	yes	—
16	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	12	yes	—
17	C₂×C₄×Q₈×C₆₁	C₂²	1	12	yes	—
18	C₂×C₄×Q₈×C₆₁	C₂²	1	24	yes	—
19	C₂×#117×C₆₁	C₂²	1	6	yes	—
20	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
22	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
23	C₄⋊Q₈×C₆₁	C₂³	1	48	yes	—
24	C₄⋊Q₈×C₆₁	C₂³	1	96	yes	—
25	#126×C₆₁	C₂³	1	128	yes	—
26	#125×C₆₁	C₂³	1	48	yes	—
27	#125×C₆₁	C₂³	1	96	yes	—
28	#125×C₆₁	C₂³	1	384	yes	—
29	C₂²⋊Q₈×C₆₁	C₂³	1	96	yes	—
30	C₄×Q₈×C₆₁	C₂³	1	96	yes	—
31	C₄×Q₈×C₆₁	C₂³	1	192	yes	—
32	#117×C₆₁	C₂³	1	48	yes	—
33	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄×C₆₁	C₂³	1	48	yes	—
36	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄×C₆₁	C₂³	1	12	yes	—
38	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	12	yes	—
40	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂³	1	16	yes	—
43	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	—	2	48	no	—
45	C₂×Q₈×C₆₁	C₂⁴	1	48	yes	—
46	C₂²×C₄×C₆₁	—	2	12	no	—
47	C₂²×C₄×C₆₁	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₆₁	C₂⁴	1	24	yes	—
49	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
51	C₄⋊C₄×C₆₁	C₂⁴	1	48	yes	—
52	C₄⋊C₄×C₆₁	C₂⁴	1	48	yes	—
53	C₄⋊C₄×C₆₁	C₂⁴	1	96	yes	—
54	C₄⋊C₄×C₆₁	C₂⁴	1	192	yes	—
55	C₂²⋊C₄×C₆₁	C₂⁴	1	48	yes	—
56	C₄²×C₆₁	C₂⁴	1	12	yes	—
57	C₄²×C₆₁	C₂⁴	1	4	yes	—
58	C₄²×C₆₁	C₂⁴	1	48	yes	—
59	C₄²×C₆₁	C₂⁴	1	64	yes	—
60	C₂³×C₆₁	—	2	6	no	—
61	C₂³×C₆₁	C₂⁵	1	2	yes	—
62	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
63	Q₈×C₆₁	—	2	96	no	—
64	C₂×C₄×C₆₁	—	2	6	no	—
65	C₂×C₄×C₆₁	—	2	48	no	—
66	C₂×C₄×C₆₁	—	2	24	no	—
67	C₂×C₄×C₆₁	2₋¹⁺⁴	1	12	yes	—
68	C₂×C₄×C₆₁	C₂⁵	1	48	yes	—
69	C₂×C₄×C₆₁	C₂⁵	1	6	yes	—
70	C₂×C₄×C₆₁	C₂⁵	1	6	yes	—
71	C₂×#3556	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₆₁	—	2	6	no	—
73	C₂²×C₆₁	—	4	4	no	—
74	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	6	yes	—
75	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
76	C₄×C₆₁	—	2	6	no	—
77	C₄×C₆₁	—	2	48	no	—
78	C₄×C₆₁	C₂×2₋¹⁺⁴	1	12	yes	—
79	#3556	C₂×C₆₁	1	32	yes	direct product factor
80	C₂×#564	C₂×C₆₁	1	4	yes	complemented
81	C₂×#557	C₂×C₆₁	1	12	yes	complemented
82	C₂×#541	C₂×C₆₁	1	12	yes	—
83	C₂×#533	C₂×C₆₁	1	3	yes	—
84	C₂×C₆₁	—	4	2	no	—
85	C₂×C₆₁	#3556	1	4	yes	direct product factor
86	C₂×C₆₁	C₂²×2₋¹⁺⁴	1	3	yes	—
87	#564	C₂²×C₆₁	1	64	yes	complemented
88	#557	C₂²×C₆₁	1	192	yes	complemented
89	#541	C₂²×C₆₁	1	192	yes	—
90	#533	C₂²×C₆₁	1	48	yes	—
91	C₂×C₄⋊Q₈	C₂²×C₆₁	1	6	yes	—
92	C₂×C₄⋊Q₈	C₂²×C₆₁	1	12	yes	—
93	C₂×#126	C₂²×C₆₁	1	16	yes	—
94	C₂×#125	C₂²×C₆₁	1	6	yes	—
95	C₂×#125	C₂²×C₆₁	1	12	yes	—
96	C₂×#125	C₂²×C₆₁	1	48	yes	—
97	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	12	yes	—
98	C₂×C₄×Q₈	C₂²×C₆₁	1	12	yes	—
99	C₂×C₄×Q₈	C₂²×C₆₁	1	24	yes	—
100	C₂×#117	C₂²×C₆₁	1	6	yes	—
101	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
102	C₆₁	C₂×#3556	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
104	C₂³×C₄	C₂³×C₆₁	1	3	yes	—
105	C₄⋊Q₈	C₂³×C₆₁	1	48	yes	—
106	C₄⋊Q₈	C₂³×C₆₁	1	96	yes	—
107	#126	C₂³×C₆₁	1	128	yes	—
108	#125	C₂³×C₆₁	1	48	yes	—
109	#125	C₂³×C₆₁	1	96	yes	—
110	#125	C₂³×C₆₁	1	384	yes	—
111	C₂²⋊Q₈	C₂³×C₆₁	1	96	yes	—
112	C₄×Q₈	C₂³×C₆₁	1	96	yes	—
113	C₄×Q₈	C₂³×C₆₁	1	192	yes	—
114	#117	C₂³×C₆₁	1	48	yes	—
115	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
116	C₂×C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
117	C₂×C₄⋊C₄	C₂³×C₆₁	1	24	yes	—
118	C₂×C₄⋊C₄	C₂³×C₆₁	1	48	yes	—
119	C₂×C₂²⋊C₄	C₂³×C₆₁	1	12	yes	—
120	C₂×C₄²	C₂³×C₆₁	1	12	yes	—
121	C₂×C₄²	C₂³×C₆₁	1	3	yes	—
122	C₂×C₄²	C₂³×C₆₁	1	16	yes	—
123	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
124	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
125	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
126	C₂×Q₈	—	2	48	no	—
127	C₂×Q₈	C₂⁴×C₆₁	1	48	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂⁴×C₆₁	1	24	yes	—
130	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
131	C₂²×C₄	C₂⁴×C₆₁	1	12	yes	—
132	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
133	C₄⋊C₄	C₂⁴×C₆₁	1	48	yes	—
134	C₄⋊C₄	C₂⁴×C₆₁	1	48	yes	—
135	C₄⋊C₄	C₂⁴×C₆₁	1	96	yes	—
136	C₄⋊C₄	C₂⁴×C₆₁	1	192	yes	—
137	C₂²⋊C₄	C₂⁴×C₆₁	1	48	yes	—
138	C₄²	C₂⁴×C₆₁	1	12	yes	—
139	C₄²	C₂⁴×C₆₁	1	4	yes	—
140	C₄²	C₂⁴×C₆₁	1	48	yes	—
141	C₄²	C₂⁴×C₆₁	1	64	yes	—
142	C₂³	—	2	6	no	—
143	C₂³	C₂⁵×C₆₁	1	2	yes	—
144	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
145	Q₈	—	2	96	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	48	no	—
148	C₂×C₄	—	2	24	no	—
149	C₂×C₄	2₋¹⁺⁴×C₆₁	1	12	yes	—
150	C₂×C₄	C₂⁵×C₆₁	1	48	yes	—
151	C₂×C₄	C₂⁵×C₆₁	1	6	yes	—
152	C₂×C₄	C₂⁵×C₆₁	1	6	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	6	yes	—
156	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
157	C₄	—	2	6	no	—
158	C₄	—	2	48	no	—
159	C₄	C₂×2₋¹⁺⁴×C₆₁	1	12	yes	—
160	C₂	—	4	2	no	—
161	C₂	C₆₁×#3556	1	4	yes	direct product factor
162	C₂	C₂²×2₋¹⁺⁴×C₆₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×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
1. C₆₁×#3556	C₂	direct (N × H)
2. C₂×C₆₁×#564	C₂	semidirect
3. C₂×C₆₁×#557	C₂	semidirect
6. C₆₁×#564	C₂²	semidirect
7. C₆₁×#557	C₂²	semidirect
71. C₂×#3556	C₆₁	direct (N × H)
79. #3556	C₂×C₆₁	direct (N × H)
80. C₂×#564	C₂×C₆₁	semidirect
81. C₂×#557	C₂×C₆₁	semidirect
85. C₂×C₆₁	#3556	direct (N × H)
87. #564	C₂²×C₆₁	semidirect
88. #557	C₂²×C₆₁	semidirect
102. C₆₁	C₂×#3556	direct (N × H)
161. C₂	C₆₁×#3556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3556	2
C₂×C₆₁×#564	2
C₂×C₆₁×#557	2
C₂×C₆₁×#541	2
C₂×C₆₁×#533	2
C₂×#3556	61

Subgroups — C2×C61×#3556 #29566193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3556 #29566193