Subgroups — C₆₁×#3487 #21443528

Order 7808.

1536 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	C₂×2₋¹⁺⁴×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#550	C₂	1	8	yes	maximal, complemented
4	C₆₁×#548	C₂	1	8	yes	maximal, complemented
5	C₆₁×#535	C₂	1	6	yes	maximal, complemented
6	C₆₁×#525	C₂	1	6	yes	maximal, complemented
7	C₆₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴×C₆₁	—	2	8	no	—
9	C₂×#41×C₆₁	—	2	8	no	—
10	C₂×#41×C₆₁	C₂²	1	6	yes	—
11	C₂×#41×C₆₁	C₂²	1	8	yes	—
12	C₂×#41×C₆₁	C₂²	1	4	yes	complemented
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₆₁	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₆₁	C₂²	1	3	yes	—
16	C₂³×C₄×C₆₁	C₂²	1	3	yes	—
17	#127×C₆₁	C₂²	1	12	yes	complemented
18	#124×C₆₁	C₂²	1	12	yes	complemented
19	#123×C₆₁	C₂²	1	4	yes	—
20	C₂²⋊Q₈×C₆₁	C₂²	1	12	yes	complemented
21	#121×C₆₁	C₂²	1	24	yes	complemented
22	#121×C₆₁	C₂²	1	12	yes	complemented
23	#120×C₆₁	C₂²	1	12	yes	—
24	C₄×Q₈×C₆₁	C₂²	1	8	yes	complemented
25	C₄×D₄×C₆₁	C₂²	1	24	yes	complemented
26	#117×C₆₁	C₂²	1	6	yes	—
27	C₂⁴×C₆₁	C₂³	1	3	yes	—
28	#41×C₆₁	—	2	32	no	—
29	#41×C₆₁	—	2	16	no	—
30	#41×C₆₁	—	2	24	no	—
31	C₂×Q₈×C₆₁	—	2	16	no	—
32	C₂×Q₈×C₆₁	D₄	1	8	yes	complemented
33	C₂×Q₈×C₆₁	C₂³	1	6	yes	—
34	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
35	C₂×D₄×C₆₁	—	2	12	no	—
36	C₂×D₄×C₆₁	C₂³	1	4	yes	—
37	C₂×D₄×C₆₁	C₂³	1	12	yes	—
38	C₂×D₄×C₆₁	C₂³	1	6	yes	—
39	C₂×D₄×C₆₁	C₂³	1	24	yes	—
40	C₂²×C₄×C₆₁	—	2	12	no	—
41	C₂²×C₄×C₆₁	C₂³	1	12	yes	—
42	C₂²×C₄×C₆₁	C₂³	1	12	yes	—
43	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
45	C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
46	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
47	C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
49	C₂²⋊C₄×C₆₁	C₂³	1	24	yes	—
50	C₄²×C₆₁	C₂³	1	12	yes	—
51	C₂³×C₆₁	—	2	6	no	—
52	C₂³×C₆₁	—	4	6	no	—
53	C₂³×C₆₁	C₂⁴	1	6	yes	—
54	C₂³×C₆₁	C₂⁴	1	4	yes	—
55	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
56	Q₈×C₆₁	—	2	12	no	—
57	Q₈×C₆₁	C₂×D₄	1	16	yes	—
58	D₄×C₆₁	—	2	8	no	—
59	D₄×C₆₁	—	2	24	no	—
60	D₄×C₆₁	—	2	12	no	—
61	D₄×C₆₁	—	4	24	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	24	no	—
64	C₂×C₄×C₆₁	—	2	24	no	—
65	C₂×C₄×C₆₁	C₂×D₄	1	12	yes	—
66	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
67	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
68	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
69	C₂×C₄×C₆₁	C₂⁴	1	6	yes	—
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	—	2	4	no	—
72	C₂²×C₆₁	—	2	6	no	—
73	C₂²×C₆₁	—	2	8	no	—
74	C₂²×C₆₁	—	2	12	no	—
75	C₂²×C₆₁	—	4	12	no	—
76	C₂²×C₆₁	C₂²×D₄	1	2	yes	—
77	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
78	C₄×C₆₁	—	2	2	no	—
79	C₄×C₆₁	—	2	4	no	—
80	C₄×C₆₁	—	2	8	no	—
81	C₄×C₆₁	C₂²×D₄	1	12	yes	—
82	#3487	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₆₁	—	2	2	no	—
84	C₂×C₆₁	—	4	4	no	—
85	C₂×C₆₁	—	4	6	no	—
86	C₂×C₆₁	C₂³×D₄	1	1	yes	characteristic
87	C₂×C₆₁	#585	1	2	yes	—
88	C₂×2₋¹⁺⁴	C₂×C₆₁	1	1	yes	characteristic, complemented
89	C₂²×#41	C₂×C₆₁	1	1	yes	characteristic, complemented
90	#550	C₂×C₆₁	1	8	yes	complemented
91	#548	C₂×C₆₁	1	8	yes	complemented
92	#535	C₂×C₆₁	1	6	yes	complemented
93	#525	C₂×C₆₁	1	6	yes	complemented
94	#519	C₂×C₆₁	1	1	yes	characteristic, complemented
95	C₆₁	#3487	1	1	yes	characteristic, direct product factor, Sylow
96	2₋¹⁺⁴	—	2	8	no	—
97	C₂×#41	—	2	8	no	—
98	C₂×#41	C₂²×C₆₁	1	6	yes	—
99	C₂×#41	C₂²×C₆₁	1	8	yes	—
100	C₂×#41	C₂²×C₆₁	1	4	yes	complemented
101	C₂²×Q₈	C₂²×C₆₁	1	1	yes	characteristic
102	C₂²×Q₈	C₂²×C₆₁	1	4	yes	complemented
103	C₂²×D₄	C₂²×C₆₁	1	3	yes	—
104	C₂³×C₄	C₂²×C₆₁	1	3	yes	—
105	#127	C₂²×C₆₁	1	12	yes	complemented
106	#124	C₂²×C₆₁	1	12	yes	complemented
107	#123	C₂²×C₆₁	1	4	yes	—
108	C₂²⋊Q₈	C₂²×C₆₁	1	12	yes	complemented
109	#121	C₂²×C₆₁	1	12	yes	complemented
110	#121	C₂²×C₆₁	1	24	yes	complemented
111	#120	C₂²×C₆₁	1	12	yes	—
112	C₄×Q₈	C₂²×C₆₁	1	8	yes	complemented
113	C₄×D₄	C₂²×C₆₁	1	24	yes	complemented
114	#117	C₂²×C₆₁	1	6	yes	—
115	C₂⁴	C₂³×C₆₁	1	3	yes	—
116	#41	—	2	32	no	—
117	#41	—	2	16	no	—
118	#41	—	2	24	no	—
119	C₂×Q₈	—	2	16	no	—
120	C₂×Q₈	D₄×C₆₁	1	8	yes	complemented
121	C₂×Q₈	C₂³×C₆₁	1	6	yes	—
122	C₂×Q₈	C₂³×C₆₁	1	4	yes	—
123	C₂×D₄	—	2	12	no	—
124	C₂×D₄	C₂³×C₆₁	1	4	yes	—
125	C₂×D₄	C₂³×C₆₁	1	12	yes	—
126	C₂×D₄	C₂³×C₆₁	1	6	yes	—
127	C₂×D₄	C₂³×C₆₁	1	24	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂³×C₆₁	1	12	yes	—
130	C₂²×C₄	C₂³×C₆₁	1	12	yes	—
131	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₆₁	1	3	yes	—
133	C₄⋊C₄	C₂³×C₆₁	1	12	yes	—
134	C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
135	C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
136	C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
137	C₂²⋊C₄	C₂³×C₆₁	1	24	yes	—
138	C₄²	C₂³×C₆₁	1	12	yes	—
139	C₂³	—	2	6	no	—
140	C₂³	—	4	6	no	—
141	C₂³	C₂⁴×C₆₁	1	6	yes	—
142	C₂³	C₂⁴×C₆₁	1	4	yes	—
143	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
144	Q₈	—	2	12	no	—
145	Q₈	C₂×D₄×C₆₁	1	16	yes	—
146	D₄	—	2	8	no	—
147	D₄	—	2	24	no	—
148	D₄	—	2	12	no	—
149	D₄	—	4	24	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	24	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	C₂×D₄×C₆₁	1	12	yes	—
154	C₂×C₄	C₂⁴×C₆₁	1	8	yes	—
155	C₂×C₄	C₂⁴×C₆₁	1	4	yes	—
156	C₂×C₄	C₂⁴×C₆₁	1	2	yes	—
157	C₂×C₄	C₂⁴×C₆₁	1	6	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	6	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	12	no	—
163	C₂²	—	4	12	no	—
164	C₂²	C₂²×D₄×C₆₁	1	2	yes	—
165	C₂²	C₂⁵×C₆₁	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	8	no	—
169	C₄	C₂²×D₄×C₆₁	1	12	yes	—
170	C₂	—	2	2	no	—
171	C₂	—	4	6	no	—
172	C₂	—	4	4	no	—
173	C₂	C₂³×D₄×C₆₁	1	1	yes	characteristic
174	C₂	C₆₁×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₆₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₆₁	C₂	semidirect
2. C₂²×#41×C₆₁	C₂	semidirect
3. C₆₁×#550	C₂	semidirect
4. C₆₁×#548	C₂	semidirect
5. C₆₁×#535	C₂	semidirect
6. C₆₁×#525	C₂	semidirect
7. C₆₁×#519	C₂	semidirect
12. C₂×#41×C₆₁	C₂²	semidirect
14. C₂²×Q₈×C₆₁	C₂²	semidirect
17. #127×C₆₁	C₂²	semidirect
18. #124×C₆₁	C₂²	semidirect
20. C₂²⋊Q₈×C₆₁	C₂²	semidirect
21. #121×C₆₁	C₂²	semidirect
22. #121×C₆₁	C₂²	semidirect
24. C₄×Q₈×C₆₁	C₂²	semidirect
25. C₄×D₄×C₆₁	C₂²	semidirect
32. C₂×Q₈×C₆₁	D₄	semidirect
82. #3487	C₆₁	direct (N × H)
88. C₂×2₋¹⁺⁴	C₂×C₆₁	semidirect
89. C₂²×#41	C₂×C₆₁	semidirect
90. #550	C₂×C₆₁	semidirect
91. #548	C₂×C₆₁	semidirect
92. #535	C₂×C₆₁	semidirect
93. #525	C₂×C₆₁	semidirect
94. #519	C₂×C₆₁	semidirect
95. C₆₁	#3487	direct (N × H)
100. C₂×#41	C₂²×C₆₁	semidirect
102. C₂²×Q₈	C₂²×C₆₁	semidirect
105. #127	C₂²×C₆₁	semidirect
106. #124	C₂²×C₆₁	semidirect
108. C₂²⋊Q₈	C₂²×C₆₁	semidirect
109. #121	C₂²×C₆₁	semidirect
110. #121	C₂²×C₆₁	semidirect
112. C₄×Q₈	C₂²×C₆₁	semidirect
113. C₄×D₄	C₂²×C₆₁	semidirect
120. C₂×Q₈	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₆₁	2
C₂²×#41×C₆₁	2
C₆₁×#550	2
C₆₁×#548	2
C₆₁×#535	2
C₆₁×#525	2
C₆₁×#519	2
#3487	61

Subgroups — C61×#3487 #21443528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3487 #21443528