Subgroups — C₄₁×#20811 #23797012

Order 10496.

834 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#432	C₂	1	2	yes	maximal
2	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1843	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1832	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1798	C₂	1	4	yes	maximal
6	C₄₁×#1767	C₂	1	4	yes	maximal, complemented
7	C₂×#117×C₄₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#432	—	2	8	no	—
10	C₂×C₄⋊C₈×C₄₁	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₂×#97×C₄₁	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₄₁	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₄₁	C₄	1	4	yes	—
16	C₄₁×#376	C₂²	1	1	yes	characteristic
17	C₄₁×#376	C₂²	1	1	yes	characteristic
18	C₄³×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#336	C₂²	1	8	yes	—
20	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
23	#117×C₄₁	—	2	8	no	—
24	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	D₄	1	4	yes	—
29	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
30	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
31	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
34	C₄⋊C₈×C₄₁	—	2	8	no	—
35	C₄⋊C₈×C₄₁	—	2	16	no	—
36	C₂²⋊C₈×C₄₁	—	2	8	no	—
37	#97×C₄₁	—	2	8	no	—
38	#95×C₄₁	C₂×C₄	1	4	yes	—
39	#95×C₄₁	C₂×C₄	1	4	yes	—
40	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₁	—	2	4	no	—
42	C₂²×C₄×C₄₁	—	4	4	no	—
43	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₄₁	—	2	8	no	—
53	C₂×C₈×C₄₁	—	2	16	no	—
54	C₄⋊C₄×C₄₁	—	2	8	no	—
55	C₂²⋊C₄×C₄₁	—	2	8	no	—
56	C₄²×C₄₁	—	2	4	no	—
57	C₄²×C₄₁	—	4	8	no	—
58	C₄²×C₄₁	C₄⋊C₄	1	8	yes	—
59	C₄²×C₄₁	C₂×D₄	1	2	yes	—
60	C₄²×C₄₁	C₂×D₄	1	2	yes	—
61	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
62	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
63	C₄²×C₄₁	#41	1	8	yes	—
64	C₂³×C₄₁	—	2	2	no	—
65	C₂³×C₄₁	—	2	4	no	—
66	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₁	—	2	2	no	—
69	C₂×C₄×C₄₁	—	2	4	no	—
70	C₂×C₄×C₄₁	—	2	8	no	—
71	C₂×C₄×C₄₁	—	2	4	no	—
72	C₂×C₄×C₄₁	—	2	4	no	—
73	C₂×C₄×C₄₁	—	2	4	no	—
74	C₂×C₄×C₄₁	—	2	4	no	—
75	C₂×C₄×C₄₁	—	2	8	no	—
76	C₂×C₄×C₄₁	—	4	8	no	—
77	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₄₁	#117	1	8	yes	—
81	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
84	C₈×C₄₁	—	4	8	no	—
85	#20811	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₄₁	—	2	4	no	—
87	C₂²×C₄₁	—	2	2	no	—
88	C₂²×C₄₁	—	4	4	no	—
89	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₁	C₂×#117	1	2	yes	—
91	C₂²×C₄₁	#568	1	4	yes	—
92	C₄×C₄₁	—	2	4	no	—
93	C₄×C₄₁	—	2	4	no	—
94	C₄×C₄₁	—	2	8	no	—
95	C₄×C₄₁	—	2	2	no	—
96	C₄×C₄₁	—	4	4	no	—
97	C₄×C₄₁	#532	1	2	yes	—
98	C₄×C₄₁	#533	1	2	yes	—
99	C₂×#432	C₂×C₄₁	1	2	yes	—
100	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
101	#1843	C₂×C₄₁	1	2	yes	complemented
102	#1832	C₂×C₄₁	1	2	yes	complemented
103	#1798	C₂×C₄₁	1	4	yes	—
104	#1767	C₂×C₄₁	1	4	yes	complemented
105	C₂×C₄₁	—	4	2	no	—
106	C₂×C₄₁	#2285	1	1	yes	characteristic
107	C₂×C₄₁	#2905	1	2	yes	—
108	C₂×C₄₁	#2923	1	4	yes	—
109	C₂×#117	C₂²×C₄₁	1	2	yes	—
110	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
111	#432	—	2	8	no	—
112	C₂×C₄⋊C₈	C₂²×C₄₁	1	8	yes	—
113	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
114	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
115	C₂×#97	C₂²×C₄₁	1	4	yes	—
116	C₄×C₄⋊C₄	C₄×C₄₁	1	4	yes	—
117	C₄×C₂²⋊C₄	C₄×C₄₁	1	4	yes	—
118	#376	C₂²×C₄₁	1	1	yes	characteristic
119	#376	C₂²×C₄₁	1	1	yes	characteristic
120	C₄³	C₂²×C₄₁	1	2	yes	—
121	#336	C₂²×C₄₁	1	8	yes	—
122	C₄₁	#20811	1	1	yes	characteristic, direct product factor, Sylow
123	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
124	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
126	#117	—	2	8	no	—
127	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
129	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
130	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
131	C₂×C₄²	D₄×C₄₁	1	4	yes	—
132	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
133	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
134	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
137	C₄⋊C₈	—	2	8	no	—
138	C₄⋊C₈	—	2	16	no	—
139	C₂²⋊C₈	—	2	8	no	—
140	#97	—	2	8	no	—
141	#95	C₂×C₄×C₄₁	1	4	yes	—
142	#95	C₂×C₄×C₄₁	1	4	yes	—
143	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
147	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
153	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
154	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	16	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	4	8	no	—
161	C₄²	C₄⋊C₄×C₄₁	1	8	yes	—
162	C₄²	C₂×D₄×C₄₁	1	2	yes	—
163	C₄²	C₂×D₄×C₄₁	1	2	yes	—
164	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
165	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
166	C₄²	#41×C₄₁	1	8	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
170	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
181	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
183	C₂×C₄	#117×C₄₁	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
186	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
187	C₈	—	4	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
192	C₂²	C₂×#117×C₄₁	1	2	yes	—
193	C₂²	C₄₁×#568	1	4	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	8	no	—
198	C₄	—	4	4	no	—
199	C₄	C₄₁×#532	1	2	yes	—
200	C₄	C₄₁×#533	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	C₄₁×#2285	1	1	yes	characteristic
203	C₂	C₄₁×#2905	1	2	yes	—
204	C₂	C₄₁×#2923	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1843	C₂	semidirect
4. C₄₁×#1832	C₂	semidirect
6. C₄₁×#1767	C₂	semidirect
85. #20811	C₄₁	direct (N × H)
101. #1843	C₂×C₄₁	semidirect
102. #1832	C₂×C₄₁	semidirect
104. #1767	C₂×C₄₁	semidirect
122. C₄₁	#20811	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×#432	2
C₄×#117×C₄₁	2
C₄₁×#1843	2
C₄₁×#1832	2
C₄₁×#1798	2
C₄₁×#1767	2
#20811	41

Subgroups — C41×#20811 #23797012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20811 #23797012