Subgroups — C₄₁×#23200 #23799401

Order 10496.

878 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2950	C₂	1	4	yes	maximal
3	C₄₁×#1979	C₂	1	4	yes	maximal
4	C₄₁×#1886	C₂	1	4	yes	maximal
5	C₄₁×#1772	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
8	C₄₁×#519	C₂²	1	4	yes	—
9	C₄₁×#494	C₄	1	8	yes	complemented
10	C₄₁×#441	—	2	8	no	—
11	C₄×Q₁₆×C₄₁	—	2	8	no	—
12	C₂×#108×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#427	C₂²	1	2	yes	—
14	C₄₁×#419	C₂²	1	8	yes	—
15	C₂×#101×C₄₁	C₂²	1	4	yes	—
16	C₄₁×#405	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#339	C₂²	1	4	yes	—
19	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
20	C₂×Q₁₆×C₄₁	—	2	4	no	—
21	C₂×Q₁₆×C₄₁	—	2	4	no	—
22	C₂×Q₁₆×C₄₁	—	2	8	no	—
23	C₂×Q₁₆×C₄₁	C₂×C₄	1	16	yes	—
24	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
25	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
26	C₄⋊Q₈×C₄₁	—	4	2	no	—
27	C₄⋊Q₈×C₄₁	C₂×C₄	1	8	yes	—
28	C₄×Q₈×C₄₁	—	2	16	no	—
29	#117×C₄₁	—	4	4	no	—
30	#117×C₄₁	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
32	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
33	#108×C₄₁	—	2	4	no	—
34	#107×C₄₁	D₄	1	4	yes	—
35	#106×C₄₁	D₄	1	4	yes	—
36	#103×C₄₁	—	2	16	no	—
37	#101×C₄₁	—	4	4	no	—
38	#97×C₄₁	—	2	4	no	—
39	C₄×C₈×C₄₁	—	2	4	no	—
40	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×Q₈×C₄₁	—	2	4	no	—
42	C₂×Q₈×C₄₁	—	2	8	no	—
43	C₂×Q₈×C₄₁	—	2	8	no	—
44	C₂×Q₈×C₄₁	C₂²×C₄	1	8	yes	—
45	C₂²×C₄×C₄₁	—	2	1	no	—
46	C₂²×C₄×C₄₁	—	2	4	no	—
47	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₄₁	—	2	16	no	—
49	Q₁₆×C₄₁	—	2	16	no	—
50	M₁₆×C₄₁	—	2	4	no	—
51	M₁₆×C₄₁	—	2	4	no	—
52	C₂×C₈×C₄₁	—	2	4	no	—
53	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₁	#41	1	4	yes	—
57	C₄⋊C₄×C₄₁	—	4	8	no	—
58	C₄⋊C₄×C₄₁	—	4	8	no	—
59	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
60	C₂²⋊C₄×C₄₁	—	2	4	no	—
61	C₄²×C₄₁	—	2	4	no	—
62	C₄²×C₄₁	—	2	1	no	—
63	C₄²×C₄₁	—	4	8	no	—
64	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
66	Q₈×C₄₁	—	2	16	no	—
67	Q₈×C₄₁	—	4	8	no	—
68	C₂×C₄×C₄₁	—	2	4	no	—
69	C₂×C₄×C₄₁	—	2	2	no	—
70	C₂×C₄×C₄₁	—	2	8	no	—
71	C₂×C₄×C₄₁	—	2	8	no	—
72	C₂×C₄×C₄₁	—	4	4	no	—
73	C₂×C₄×C₄₁	—	4	4	no	—
74	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
77	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
78	C₈×C₄₁	—	4	4	no	—
79	C₈×C₄₁	C₄×D₄	1	4	yes	—
80	C₈×C₄₁	C₄×D₄	1	4	yes	—
81	#23200	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₄₁	—	2	2	no	—
83	C₂²×C₄₁	#518	1	1	yes	characteristic
84	C₂²×C₄₁	D₄²	1	1	yes	characteristic
85	C₂²×C₄₁	#559	1	1	yes	characteristic
86	C₄×C₄₁	—	2	4	no	—
87	C₄×C₄₁	—	4	8	no	—
88	C₄×C₄₁	—	4	8	no	—
89	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
90	C₄×C₄₁	#550	1	2	yes	—
91	C₂×#494	C₂×C₄₁	1	1	yes	characteristic
92	#2950	C₂×C₄₁	1	4	yes	—
93	#1979	C₂×C₄₁	1	4	yes	—
94	#1886	C₂×C₄₁	1	4	yes	—
95	#1772	C₂×C₄₁	1	2	yes	—
96	C₂×C₄₁	—	2	1	no	—
97	C₂×C₄₁	—	2	1	no	—
98	C₂×C₄₁	#2366	1	1	yes	characteristic
99	C₂×C₄₁	#3293	1	2	yes	—
100	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
102	#519	C₂²×C₄₁	1	4	yes	—
103	#494	C₄×C₄₁	1	8	yes	complemented
104	#441	—	2	8	no	—
105	C₄×Q₁₆	—	2	8	no	—
106	C₂×#108	C₂²×C₄₁	1	2	yes	—
107	#427	C₂²×C₄₁	1	2	yes	—
108	#419	C₂²×C₄₁	1	8	yes	—
109	C₂×#101	C₂²×C₄₁	1	4	yes	—
110	#405	C₂²×C₄₁	1	4	yes	—
111	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
112	#339	C₂²×C₄₁	1	4	yes	—
113	C₄₁	#23200	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	8	no	—
118	C₂×Q₁₆	C₂×C₄×C₄₁	1	16	yes	—
119	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
120	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
121	C₄⋊Q₈	—	4	2	no	—
122	C₄⋊Q₈	C₂×C₄×C₄₁	1	8	yes	—
123	C₄×Q₈	—	2	16	no	—
124	#117	—	4	4	no	—
125	#117	C₂³×C₄₁	1	4	yes	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
128	#108	—	2	4	no	—
129	#107	D₄×C₄₁	1	4	yes	—
130	#106	D₄×C₄₁	1	4	yes	—
131	#103	—	2	16	no	—
132	#101	—	4	4	no	—
133	#97	—	2	4	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	2	8	no	—
139	C₂×Q₈	C₂²×C₄×C₄₁	1	8	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
143	Q₁₆	—	2	16	no	—
144	Q₁₆	—	2	16	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
149	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
150	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
151	C₂×C₈	#41×C₄₁	1	4	yes	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	4	8	no	—
159	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
160	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
161	Q₈	—	2	16	no	—
162	Q₈	—	4	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
171	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
172	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
173	C₈	—	4	4	no	—
174	C₈	C₄×D₄×C₄₁	1	4	yes	—
175	C₈	C₄×D₄×C₄₁	1	4	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	C₄₁×#518	1	1	yes	characteristic
178	C₂²	D₄²×C₄₁	1	1	yes	characteristic
179	C₂²	C₄₁×#559	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	4	8	no	—
182	C₄	—	4	8	no	—
183	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
184	C₄	C₄₁×#550	1	2	yes	—
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	C₄₁×#2366	1	1	yes	characteristic
188	C₂	C₄₁×#3293	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂³×C₄₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄₁×#494	C₄	semidirect
81. #23200	C₄₁	direct (N × H)
103. #494	C₄×C₄₁	semidirect
113. C₄₁	#23200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#494	2
C₄₁×#2950	2
C₄₁×#1979	2
C₄₁×#1886	2
C₄₁×#1772	2
#23200	41

Subgroups — C41×#23200 #23799401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23200 #23799401