Subgroups — C₂×C₄×#97×C₄₁ #23796527

Order 10496.

1790 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₄₁	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈×C₄₁	C₂	1	2	yes	maximal
3	C₂×C₄³×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄×#97×C₄₁	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₈×C₄₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₄₁	C₄	1	4	yes	—
7	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#97×C₄₁	C₄	1	32	yes	direct product factor
10	C₂×#97×C₄₁	C₂²	1	16	yes	—
11	C₂×C₄×C₈×C₄₁	C₄	1	16	yes	complemented
12	C₂×C₄×C₈×C₄₁	C₂²	1	8	yes	—
13	C₄³×C₄₁	C₄	1	4	yes	—
14	C₄³×C₄₁	C₂²	1	4	yes	—
15	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₄₁	C₂×C₄	1	16	yes	—
19	C₂²×C₈×C₄₁	C₂×C₄	1	32	yes	—
20	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
21	C₂×C₄²×C₄₁	—	2	16	no	—
22	C₂×C₄²×C₄₁	C₂×C₄	1	16	yes	—
23	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₄₁	C₂×C₄	1	16	yes	—
26	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
28	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
29	#97×C₄₁	C₂×C₄	1	64	yes	direct product factor
30	C₄×C₈×C₄₁	C₂×C₄	1	32	yes	complemented
31	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₁	—	2	16	no	—
33	C₂²×C₄×C₄₁	C₄²	1	16	yes	—
34	C₂²×C₄×C₄₁	C₄²	1	8	yes	—
35	C₂²×C₄×C₄₁	M₁₆	1	16	yes	—
36	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₄₁	C₄²	1	64	yes	complemented
43	C₂×C₈×C₄₁	C₂²×C₄	1	16	yes	—
44	C₂×C₈×C₄₁	C₂²×C₄	1	32	yes	—
45	C₄²×C₄₁	—	2	32	no	—
46	C₄²×C₄₁	C₄²	1	16	yes	—
47	C₄²×C₄₁	C₄²	1	8	yes	—
48	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
49	C₄²×C₄₁	C₂²×C₄	1	16	yes	—
50	C₂³×C₄₁	C₂×C₄²	1	2	yes	—
51	C₂³×C₄₁	C₂×C₄²	1	4	yes	—
52	C₂³×C₄₁	C₂×M₁₆	1	8	yes	—
53	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₁	—	2	16	no	—
55	C₂×C₄×C₄₁	—	2	32	no	—
56	C₂×C₄×C₄₁	#97	1	32	yes	direct product factor
57	C₂×C₄×C₄₁	C₂×C₄²	1	16	yes	—
58	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
59	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
60	C₂×C₄×C₄₁	C₂×C₄²	1	16	yes	—
61	C₂×C₄×C₄₁	C₂×C₄²	1	8	yes	—
62	C₂×C₄×C₄₁	C₂×M₁₆	1	16	yes	—
63	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
64	C₂×C₄×C₄₁	C₂³×C₄	1	4	yes	—
65	C₈×C₄₁	C₂×C₄²	1	32	yes	complemented
66	C₂×C₄×#97	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₄₁	C₄³	1	4	yes	—
68	C₂²×C₄₁	C₂×#97	1	8	yes	—
69	C₂²×C₄₁	C₄×M₁₆	1	16	yes	—
70	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
71	C₂²×C₄₁	C₂²×C₄²	1	2	yes	—
72	C₂²×C₄₁	C₂²×M₁₆	1	4	yes	—
73	C₄×C₄₁	—	2	16	no	—
74	C₄×C₄₁	C₄³	1	4	yes	—
75	C₄×C₄₁	C₂×#97	1	16	yes	direct product factor
76	C₄×C₄₁	C₂²×C₄²	1	4	yes	—
77	C₂²×#97	C₂×C₄₁	1	4	yes	—
78	C₂²×C₄×C₈	C₂×C₄₁	1	2	yes	—
79	C₂×C₄³	C₂×C₄₁	1	1	yes	characteristic
80	C₄×#97	C₂×C₄₁	1	8	yes	direct product factor
81	C₂×C₄₁	C₄×#97	1	8	yes	direct product factor
82	C₂×C₄₁	C₂×C₄³	1	1	yes	characteristic
83	C₂×C₄₁	C₂²×#97	1	2	yes	—
84	C₂×C₄₁	C₂×C₄×M₁₆	1	4	yes	—
85	C₂³×C₈	C₂²×C₄₁	1	4	yes	—
86	C₂²×C₄²	C₄×C₄₁	1	4	yes	—
87	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
88	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
89	C₂×#97	C₄×C₄₁	1	32	yes	direct product factor
90	C₂×#97	C₂²×C₄₁	1	16	yes	—
91	C₂×C₄×C₈	C₄×C₄₁	1	16	yes	complemented
92	C₂×C₄×C₈	C₂²×C₄₁	1	8	yes	—
93	C₄³	C₄×C₄₁	1	4	yes	—
94	C₄³	C₂²×C₄₁	1	4	yes	—
95	C₄₁	C₂×C₄×#97	1	1	yes	characteristic, direct product factor, Sylow
96	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
97	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
98	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₄₁	1	16	yes	—
100	C₂²×C₈	C₂×C₄×C₄₁	1	32	yes	—
101	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	C₂×C₄×C₄₁	1	16	yes	—
104	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
105	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
107	C₂×C₄²	C₂×C₄×C₄₁	1	16	yes	—
108	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
109	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
110	#97	C₂×C₄×C₄₁	1	64	yes	direct product factor
111	C₄×C₈	C₂×C₄×C₄₁	1	32	yes	complemented
112	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
113	C₂²×C₄	—	2	16	no	—
114	C₂²×C₄	C₄²×C₄₁	1	16	yes	—
115	C₂²×C₄	C₄²×C₄₁	1	8	yes	—
116	C₂²×C₄	M₁₆×C₄₁	1	16	yes	—
117	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
118	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
119	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
120	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
121	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
122	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
123	C₂×C₈	C₄²×C₄₁	1	64	yes	complemented
124	C₂×C₈	C₂²×C₄×C₄₁	1	16	yes	—
125	C₂×C₈	C₂²×C₄×C₄₁	1	32	yes	—
126	C₄²	—	2	32	no	—
127	C₄²	C₄²×C₄₁	1	16	yes	—
128	C₄²	C₄²×C₄₁	1	8	yes	—
129	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
130	C₄²	C₂²×C₄×C₄₁	1	16	yes	—
131	C₂³	C₂×C₄²×C₄₁	1	2	yes	—
132	C₂³	C₂×C₄²×C₄₁	1	4	yes	—
133	C₂³	C₂×M₁₆×C₄₁	1	8	yes	—
134	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	2	32	no	—
137	C₂×C₄	#97×C₄₁	1	32	yes	direct product factor
138	C₂×C₄	C₂×C₄²×C₄₁	1	16	yes	—
139	C₂×C₄	C₂×C₄²×C₄₁	1	16	yes	—
140	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
141	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
142	C₂×C₄	C₂×C₄²×C₄₁	1	8	yes	—
143	C₂×C₄	C₂×M₁₆×C₄₁	1	16	yes	—
144	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
145	C₂×C₄	C₂³×C₄×C₄₁	1	4	yes	—
146	C₈	C₂×C₄²×C₄₁	1	32	yes	complemented
147	C₂²	C₄³×C₄₁	1	4	yes	—
148	C₂²	C₂×#97×C₄₁	1	8	yes	—
149	C₂²	C₄×M₁₆×C₄₁	1	16	yes	—
150	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
151	C₂²	C₂²×C₄²×C₄₁	1	2	yes	—
152	C₂²	C₂²×M₁₆×C₄₁	1	4	yes	—
153	C₄	—	2	16	no	—
154	C₄	C₄³×C₄₁	1	4	yes	—
155	C₄	C₂×#97×C₄₁	1	16	yes	direct product factor
156	C₄	C₂²×C₄²×C₄₁	1	4	yes	—
157	C₂	C₄×#97×C₄₁	1	8	yes	direct product factor
158	C₂	C₂×C₄³×C₄₁	1	1	yes	characteristic
159	C₂	C₂²×#97×C₄₁	1	2	yes	—
160	C₂	C₂×C₄×M₁₆×C₄₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₄₁	C₂²
commutator	C₂	C₂×C₄³×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle, prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×#97×C₄₁	C₂	direct (N × H)
9. C₂×#97×C₄₁	C₄	direct (N × H)
11. C₂×C₄×C₈×C₄₁	C₄	semidirect
29. #97×C₄₁	C₂×C₄	direct (N × H)
30. C₄×C₈×C₄₁	C₂×C₄	semidirect
42. C₂×C₈×C₄₁	C₄²	semidirect
56. C₂×C₄×C₄₁	#97	direct (N × H)
65. C₈×C₄₁	C₂×C₄²	semidirect
66. C₂×C₄×#97	C₄₁	direct (N × H)
75. C₄×C₄₁	C₂×#97	direct (N × H)
80. C₄×#97	C₂×C₄₁	direct (N × H)
81. C₂×C₄₁	C₄×#97	direct (N × H)
89. C₂×#97	C₄×C₄₁	direct (N × H)
91. C₂×C₄×C₈	C₄×C₄₁	semidirect
95. C₄₁	C₂×C₄×#97	direct (N × H)
110. #97	C₂×C₄×C₄₁	direct (N × H)
111. C₄×C₈	C₂×C₄×C₄₁	semidirect
123. C₂×C₈	C₄²×C₄₁	semidirect
137. C₂×C₄	#97×C₄₁	direct (N × H)
146. C₈	C₂×C₄²×C₄₁	semidirect
155. C₄	C₂×#97×C₄₁	direct (N × H)
157. C₂	C₄×#97×C₄₁	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×#97×C₄₁	2
C₂²×C₄×C₈×C₄₁	2
C₂×C₄³×C₄₁	2
C₄×#97×C₄₁	2
C₂×C₄×#97	41

Subgroups — C2×C4×#97×C41 #23796527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#97×C₄₁ #23796527