Subgroups — C₂×C₄₁×#431 #19106697

Order 5248.

528 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#431	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₈×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₄₁	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₄₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₄₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₄₁	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₄₁	C₂²	1	4	yes	—
10	C₂×C₄²×C₄₁	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₄₁	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₄₁	C₄	1	4	yes	—
13	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₄₁	C₂²	1	4	yes	—
15	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₁	C₂²	1	8	yes	complemented
17	C₂²⋊C₈×C₄₁	C₂²	1	8	yes	—
18	C₄×C₈×C₄₁	C₂²	1	8	yes	complemented
19	C₂⁴×C₄₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₁	—	2	4	no	—
21	C₂²×C₄×C₄₁	—	2	4	no	—
22	C₂²×C₄×C₄₁	C₈	1	8	yes	complemented
23	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₁	—	2	8	no	—
34	C₂×C₈×C₄₁	C₂³	1	8	yes	—
35	C₄²×C₄₁	C₂×C₄	1	4	yes	—
36	C₄²×C₄₁	C₂×C₄	1	4	yes	—
37	C₄²×C₄₁	C₂×C₄	1	2	yes	—
38	C₄²×C₄₁	C₂×C₄	1	2	yes	—
39	C₄²×C₄₁	C₂³	1	2	yes	—
40	C₄²×C₄₁	C₂³	1	2	yes	—
41	C₂³×C₄₁	—	2	2	no	—
42	C₂³×C₄₁	—	2	2	no	—
43	C₂³×C₄₁	C₂×C₈	1	4	yes	—
44	C₂³×C₄₁	C₂²×C₄	1	2	yes	—
45	C₂³×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₁	—	2	2	no	—
47	C₂×C₄×C₄₁	—	2	2	no	—
48	C₂×C₄×C₄₁	—	2	4	no	—
49	C₂×C₄×C₄₁	—	2	8	no	—
50	C₂×C₄×C₄₁	—	2	4	no	—
51	C₂×C₄×C₄₁	C₂×C₈	1	4	yes	—
52	C₂×C₄×C₄₁	C₂×C₈	1	4	yes	—
53	C₂×C₄×C₄₁	C₂×C₈	1	16	yes	complemented
54	C₂×C₄×C₄₁	M₁₆	1	4	yes	—
55	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₄₁	#41	1	4	yes	—
62	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₄₁	—	2	8	no	—
64	C₂²×C₄₁	—	2	2	no	—
65	C₂²×C₄₁	—	2	2	no	—
66	C₂²×C₄₁	—	2	8	no	—
67	C₂²×C₄₁	#117	1	2	yes	—
68	C₂²×C₄₁	C₂²×C₈	1	4	yes	—
69	C₂²×C₄₁	C₂²×C₈	1	2	yes	—
70	C₂²×C₄₁	C₂×M₁₆	1	2	yes	—
71	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
72	C₄×C₄₁	—	2	2	no	—
73	C₄×C₄₁	—	2	2	no	—
74	C₄×C₄₁	#117	1	2	yes	—
75	C₄×C₄₁	C₂²×C₈	1	4	yes	—
76	C₄×C₄₁	C₂²×C₈	1	4	yes	—
77	C₄×C₄₁	C₂×M₁₆	1	4	yes	—
78	C₄×C₄₁	C₂×#41	1	2	yes	—
79	C₂×#431	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₄₁	—	2	4	no	—
81	C₂×C₄₁	#431	1	4	yes	direct product factor
82	C₂×C₄₁	C₂×#117	1	1	yes	characteristic
83	C₂×C₄₁	C₂³×C₈	1	1	yes	characteristic
84	C₂×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
85	C₂²×C₄²	C₂×C₄₁	1	1	yes	characteristic
86	#431	C₂×C₄₁	1	8	yes	direct product factor
87	C₂×C₄⋊C₈	C₂×C₄₁	1	2	yes	complemented
88	C₂×C₂²⋊C₈	C₂×C₄₁	1	2	yes	—
89	C₂×C₄×C₈	C₂×C₄₁	1	2	yes	complemented
90	C₄₁	C₂×#431	1	1	yes	characteristic, direct product factor, Sylow
91	C₂³×C₄	C₄×C₄₁	1	1	yes	characteristic
92	C₂³×C₄	C₄×C₄₁	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₄₁	1	4	yes	—
95	C₂×C₄²	C₄×C₄₁	1	1	yes	characteristic
96	C₂×C₄²	C₄×C₄₁	1	4	yes	—
97	C₂×C₄²	C₄×C₄₁	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₄₁	1	4	yes	—
101	C₄⋊C₈	C₂²×C₄₁	1	8	yes	complemented
102	C₂²⋊C₈	C₂²×C₄₁	1	8	yes	—
103	C₄×C₈	C₂²×C₄₁	1	8	yes	complemented
104	C₂⁴	C₂×C₄×C₄₁	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	C₈×C₄₁	1	8	yes	complemented
108	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
110	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄×C₄₁	1	4	yes	—
112	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×C₄₁	1	4	yes	—
115	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
117	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
118	C₂×C₈	—	2	8	no	—
119	C₂×C₈	C₂³×C₄₁	1	8	yes	—
120	C₄²	C₂×C₄×C₄₁	1	4	yes	—
121	C₄²	C₂×C₄×C₄₁	1	4	yes	—
122	C₄²	C₂×C₄×C₄₁	1	2	yes	—
123	C₄²	C₂×C₄×C₄₁	1	2	yes	—
124	C₄²	C₂³×C₄₁	1	2	yes	—
125	C₄²	C₂³×C₄₁	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	C₂×C₈×C₄₁	1	4	yes	—
129	C₂³	C₂²×C₄×C₄₁	1	2	yes	—
130	C₂³	C₂²×C₄×C₄₁	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×C₈×C₄₁	1	4	yes	—
137	C₂×C₄	C₂×C₈×C₄₁	1	4	yes	—
138	C₂×C₄	C₂×C₈×C₄₁	1	16	yes	complemented
139	C₂×C₄	M₁₆×C₄₁	1	4	yes	—
140	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
141	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
142	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
144	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
145	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
146	C₂×C₄	#41×C₄₁	1	4	yes	—
147	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
148	C₈	—	2	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	8	no	—
152	C₂²	#117×C₄₁	1	2	yes	—
153	C₂²	C₂²×C₈×C₄₁	1	4	yes	—
154	C₂²	C₂²×C₈×C₄₁	1	2	yes	—
155	C₂²	C₂×M₁₆×C₄₁	1	2	yes	—
156	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	#117×C₄₁	1	2	yes	—
160	C₄	C₂²×C₈×C₄₁	1	4	yes	—
161	C₄	C₂²×C₈×C₄₁	1	4	yes	—
162	C₄	C₂×M₁₆×C₄₁	1	4	yes	—
163	C₄	C₂×#41×C₄₁	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	C₄₁×#431	1	4	yes	direct product factor
166	C₂	C₂×#117×C₄₁	1	1	yes	characteristic
167	C₂	C₂³×C₈×C₄₁	1	1	yes	characteristic
168	C₂	C₂²×M₁₆×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₄₁	C₂²
commutator	C₂	C₂³×C₈×C₄₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂²×C₄
prime radical	C₂⁴×C₄₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#431	C₂	direct (N × H)
3. C₂×C₄⋊C₈×C₄₁	C₂	semidirect
5. C₂×C₄×C₈×C₄₁	C₂	semidirect
16. C₄⋊C₈×C₄₁	C₂²	semidirect
18. C₄×C₈×C₄₁	C₂²	semidirect
22. C₂²×C₄×C₄₁	C₈	semidirect
53. C₂×C₄×C₄₁	C₂×C₈	semidirect
79. C₂×#431	C₄₁	direct (N × H)
81. C₂×C₄₁	#431	direct (N × H)
86. #431	C₂×C₄₁	direct (N × H)
87. C₂×C₄⋊C₈	C₂×C₄₁	semidirect
89. C₂×C₄×C₈	C₂×C₄₁	semidirect
90. C₄₁	C₂×#431	direct (N × H)
101. C₄⋊C₈	C₂²×C₄₁	semidirect
103. C₄×C₈	C₂²×C₄₁	semidirect
107. C₂²×C₄	C₈×C₄₁	semidirect
138. C₂×C₄	C₂×C₈×C₄₁	semidirect
165. C₂	C₄₁×#431	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×C₄²×C₄₁	2
C₄₁×#431	2
C₂×C₄⋊C₈×C₄₁	2
C₂×C₂²⋊C₈×C₄₁	2
C₂×C₄×C₈×C₄₁	2
C₂×#431	41

Subgroups — C2×C41×#431 #19106697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#431 #19106697