Subgroups — C₂²×C₄₁×#499 #23836854

Order 10496.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄₁×#499	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#107×C₄₁	C₂	1	2	yes	maximal
4	C₂²×#106×C₄₁	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#125×C₄₁	—	2	12	no	—
9	C₂×#125×C₄₁	C₂²	1	24	yes	—
10	C₂²×C₄⋊C₄×C₄₁	—	2	4	no	—
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#499	C₂²	1	64	yes	direct product factor
14	C₂×#107×C₄₁	C₂²	1	24	yes	—
15	C₂×#106×C₄₁	C₂²	1	24	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	12	yes	—
17	C₂³×C₄×C₄₁	—	2	4	no	—
18	C₂³×C₄×C₄₁	D₄	1	2	yes	—
19	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₄₁	Q₈	1	8	yes	—
21	C₂²×C₈×C₄₁	Q₈	1	8	yes	—
22	C₂²×C₈×C₄₁	C₂³	1	6	yes	—
23	C₂²×C₈×C₄₁	C₂³	1	6	yes	—
24	#125×C₄₁	—	2	48	no	—
25	#125×C₄₁	C₂³	1	32	yes	—
26	C₂×C₄⋊C₄×C₄₁	—	2	48	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
28	C₂×C₄⋊C₄×C₄₁	C₂³	1	24	yes	—
29	C₂×C₄²×C₄₁	D₄	1	6	yes	—
30	C₂×C₄²×C₄₁	C₂³	1	6	yes	—
31	#107×C₄₁	C₂³	1	32	yes	—
32	#106×C₄₁	C₂³	1	32	yes	—
33	C₄×C₈×C₄₁	C₂³	1	16	yes	—
34	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₁	—	2	8	no	—
36	C₂²×C₄×C₄₁	—	2	24	no	—
37	C₂²×C₄×C₄₁	—	4	16	no	—
38	C₂²×C₄×C₄₁	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₄₁	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₄₁	C₂⁴	1	3	yes	—
43	C₂×C₈×C₄₁	C₂×Q₈	1	24	yes	—
44	C₂×C₈×C₄₁	C₂×Q₈	1	24	yes	—
45	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
46	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
47	C₄⋊C₄×C₄₁	—	2	64	no	—
48	C₄⋊C₄×C₄₁	—	2	24	no	—
49	C₄⋊C₄×C₄₁	C₂⁴	1	16	yes	—
50	C₄²×C₄₁	C₂×D₄	1	12	yes	—
51	C₄²×C₄₁	C₂⁴	1	4	yes	—
52	C₂³×C₄₁	C₄⋊Q₈	1	4	yes	—
53	C₂³×C₄₁	#135	1	8	yes	—
54	C₂³×C₄₁	C₂²×D₄	1	3	yes	—
55	C₂×C₄×C₄₁	—	2	24	no	—
56	C₂×C₄×C₄₁	—	2	16	no	—
57	C₂×C₄×C₄₁	—	4	48	no	—
58	C₂×C₄×C₄₁	C₄⋊Q₈	1	12	yes	—
59	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
60	C₂×C₄×C₄₁	C₂²×D₄	1	3	yes	—
61	C₂×C₄×C₄₁	C₂²×Q₈	1	6	yes	—
62	C₂×C₄×C₄₁	C₂²×Q₈	1	6	yes	—
63	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
64	C₈×C₄₁	C₂²×Q₈	1	8	yes	—
65	C₈×C₄₁	C₂²×Q₈	1	8	yes	—
66	C₂²×#499	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₄₁	#499	1	16	yes	direct product factor
68	C₂²×C₄₁	C₂×C₄⋊Q₈	1	6	yes	—
69	C₂²×C₄₁	C₂×#135	1	12	yes	—
70	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₄₁	—	2	8	no	—
72	C₄×C₄₁	—	4	16	no	—
73	C₄×C₄₁	C₂×C₄⋊Q₈	1	6	yes	—
74	C₄×C₄₁	C₂³×Q₈	1	1	yes	characteristic
75	C₄×C₄₁	C₂³×Q₈	1	1	yes	characteristic
76	C₂²×#125	C₂×C₄₁	1	2	yes	—
77	C₂×#499	C₂×C₄₁	1	24	yes	direct product factor
78	C₂²×#107	C₂×C₄₁	1	2	yes	—
79	C₂²×#106	C₂×C₄₁	1	2	yes	—
80	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
81	C₂×C₄₁	C₂×#499	1	12	yes	direct product factor
82	C₂×C₄₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄₁	C₂²×#135	1	2	yes	—
84	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
86	C₂×#125	—	2	12	no	—
87	C₂×#125	C₂²×C₄₁	1	24	yes	—
88	C₂²×C₄⋊C₄	—	2	4	no	—
89	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
90	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
91	#499	C₂²×C₄₁	1	64	yes	direct product factor
92	C₂×#107	C₂²×C₄₁	1	24	yes	—
93	C₂×#106	C₂²×C₄₁	1	24	yes	—
94	C₂×C₄×C₈	C₂²×C₄₁	1	12	yes	—
95	C₄₁	C₂²×#499	1	1	yes	characteristic, direct product factor, Sylow
96	C₂³×C₄	—	2	4	no	—
97	C₂³×C₄	D₄×C₄₁	1	2	yes	—
98	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₄₁	1	8	yes	—
100	C₂²×C₈	Q₈×C₄₁	1	8	yes	—
101	C₂²×C₈	C₂³×C₄₁	1	6	yes	—
102	C₂²×C₈	C₂³×C₄₁	1	6	yes	—
103	#125	—	2	48	no	—
104	#125	C₂³×C₄₁	1	32	yes	—
105	C₂×C₄⋊C₄	—	2	48	no	—
106	C₂×C₄⋊C₄	—	2	12	no	—
107	C₂×C₄⋊C₄	C₂³×C₄₁	1	24	yes	—
108	C₂×C₄²	D₄×C₄₁	1	6	yes	—
109	C₂×C₄²	C₂³×C₄₁	1	6	yes	—
110	#107	C₂³×C₄₁	1	32	yes	—
111	#106	C₂³×C₄₁	1	32	yes	—
112	C₄×C₈	C₂³×C₄₁	1	16	yes	—
113	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	24	no	—
116	C₂²×C₄	—	4	16	no	—
117	C₂²×C₄	C₂×D₄×C₄₁	1	12	yes	—
118	C₂²×C₄	C₂×D₄×C₄₁	1	3	yes	—
119	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
120	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
121	C₂²×C₄	C₂⁴×C₄₁	1	3	yes	—
122	C₂×C₈	C₂×Q₈×C₄₁	1	24	yes	—
123	C₂×C₈	C₂×Q₈×C₄₁	1	24	yes	—
124	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
125	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
126	C₄⋊C₄	—	2	64	no	—
127	C₄⋊C₄	—	2	24	no	—
128	C₄⋊C₄	C₂⁴×C₄₁	1	16	yes	—
129	C₄²	C₂×D₄×C₄₁	1	12	yes	—
130	C₄²	C₂⁴×C₄₁	1	4	yes	—
131	C₂³	C₄⋊Q₈×C₄₁	1	4	yes	—
132	C₂³	#135×C₄₁	1	8	yes	—
133	C₂³	C₂²×D₄×C₄₁	1	3	yes	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	48	no	—
137	C₂×C₄	C₄⋊Q₈×C₄₁	1	12	yes	—
138	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
139	C₂×C₄	C₂²×D₄×C₄₁	1	3	yes	—
140	C₂×C₄	C₂²×Q₈×C₄₁	1	6	yes	—
141	C₂×C₄	C₂²×Q₈×C₄₁	1	6	yes	—
142	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
143	C₈	C₂²×Q₈×C₄₁	1	8	yes	—
144	C₈	C₂²×Q₈×C₄₁	1	8	yes	—
145	C₂²	C₄₁×#499	1	16	yes	direct product factor
146	C₂²	C₂×C₄⋊Q₈×C₄₁	1	6	yes	—
147	C₂²	C₂×#135×C₄₁	1	12	yes	—
148	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	4	16	no	—
151	C₄	C₂×C₄⋊Q₈×C₄₁	1	6	yes	—
152	C₄	C₂³×Q₈×C₄₁	1	1	yes	characteristic
153	C₄	C₂³×Q₈×C₄₁	1	1	yes	characteristic
154	C₂	C₂×C₄₁×#499	1	12	yes	direct product factor
155	C₂	C₂²×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
156	C₂	C₂²×#135×C₄₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#499	C₂	direct (N × H)
13. C₄₁×#499	C₂²	direct (N × H)
66. C₂²×#499	C₄₁	direct (N × H)
67. C₂²×C₄₁	#499	direct (N × H)
77. C₂×#499	C₂×C₄₁	direct (N × H)
81. C₂×C₄₁	C₂×#499	direct (N × H)
91. #499	C₂²×C₄₁	direct (N × H)
95. C₄₁	C₂²×#499	direct (N × H)
145. C₂²	C₄₁×#499	direct (N × H)
154. C₂	C₂×C₄₁×#499	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×#125×C₄₁	2
C₂×C₄₁×#499	2
C₂²×#107×C₄₁	2
C₂²×#106×C₄₁	2
C₂²×C₄×C₈×C₄₁	2
C₂²×#499	41

Subgroups — C22×C41×#499 #23836854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#499 #23836854