Subgroups — C₂×C₄₁×#2656 #23810772

Order 10496.

4886 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₄₁	C₂	1	6	yes	maximal, complemented
2	C₄₁×#2656	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#399	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₁×#392	C₂	1	4	yes	maximal, complemented
5	C₂²×#95×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₄₁	C₂²	1	4	yes	—
7	C₂³×D₄×C₄₁	C₂²	1	6	yes	—
8	C₂⁴×C₄×C₄₁	C₂²	1	3	yes	—
9	C₂×#121×C₄₁	—	2	48	no	—
10	C₂×#121×C₄₁	C₂²	1	48	yes	complemented
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	6	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	12	yes	—
13	C₄₁×#399	C₂²	1	32	yes	complemented
14	C₄₁×#392	C₂²	1	32	yes	complemented
15	C₂×#95×C₄₁	C₂²	1	4	yes	—
16	C₂×#95×C₄₁	C₂²	1	8	yes	complemented
17	C₂⁵×C₄₁	C₂³	1	4	yes	—
18	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
19	C₂²×D₄×C₄₁	—	2	48	no	—
20	C₂²×D₄×C₄₁	—	2	24	no	—
21	C₂²×D₄×C₄₁	—	4	24	no	—
22	C₂²×D₄×C₄₁	C₂³	1	16	yes	—
23	C₂²×D₄×C₄₁	C₂³	1	24	yes	—
24	C₂³×C₄×C₄₁	—	4	12	no	—
25	C₂³×C₄×C₄₁	D₄	1	24	yes	complemented
26	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	6	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	12	yes	—
29	#121×C₄₁	—	2	192	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	24	no	—
31	C₂×C₄⋊C₄×C₄₁	C₂³	1	24	yes	—
32	C₂×C₂²⋊C₄×C₄₁	—	2	48	no	—
33	C₂×C₂²⋊C₄×C₄₁	C₂³	1	48	yes	—
34	#95×C₄₁	C₂³	1	16	yes	—
35	C₂⁴×C₄₁	—	2	6	no	—
36	C₂⁴×C₄₁	—	2	2	no	—
37	C₂⁴×C₄₁	—	2	24	no	—
38	C₂⁴×C₄₁	—	2	24	no	—
39	C₂⁴×C₄₁	—	2	8	no	—
40	C₂⁴×C₄₁	C₂×D₄	1	12	yes	—
41	C₂⁴×C₄₁	C₂⁴	1	8	yes	—
42	C₂⁴×C₄₁	C₂⁴	1	2	yes	—
43	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
44	C₂×D₄×C₄₁	—	2	96	no	—
45	C₂×D₄×C₄₁	—	2	48	no	—
46	C₂×D₄×C₄₁	—	4	48	no	—
47	C₂×D₄×C₄₁	—	4	96	no	—
48	C₂×D₄×C₄₁	—	4	64	no	—
49	C₂²×C₄×C₄₁	—	2	24	no	—
50	C₂²×C₄×C₄₁	—	2	24	no	—
51	C₂²×C₄×C₄₁	—	4	48	no	—
52	C₂²×C₄×C₄₁	—	4	24	no	—
53	C₂²×C₄×C₄₁	C₂×D₄	1	24	yes	—
54	C₂²×C₄×C₄₁	C₂×D₄	1	48	yes	complemented
55	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₄₁	C₂⁴	1	12	yes	—
57	C₄⋊C₄×C₄₁	—	2	48	no	—
58	C₂²⋊C₄×C₄₁	—	2	96	no	—
59	C₂³×C₄₁	—	2	24	no	—
60	C₂³×C₄₁	—	2	24	no	—
61	C₂³×C₄₁	—	2	6	no	—
62	C₂³×C₄₁	—	2	2	no	—
63	C₂³×C₄₁	—	2	24	no	—
64	C₂³×C₄₁	—	2	24	no	—
65	C₂³×C₄₁	—	2	8	no	—
66	C₂³×C₄₁	—	4	16	no	—
67	C₂³×C₄₁	—	4	48	no	—
68	C₂³×C₄₁	—	4	48	no	—
69	C₂³×C₄₁	#127	1	16	yes	—
70	C₂³×C₄₁	C₂²×D₄	1	12	yes	—
71	C₂³×C₄₁	C₂²×D₄	1	6	yes	—
72	C₂³×C₄₁	H(5,𝔽₂)	1	6	yes	—
73	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
74	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
75	D₄×C₄₁	—	4	96	no	—
76	D₄×C₄₁	—	4	64	no	—
77	C₂×C₄×C₄₁	—	2	48	no	—
78	C₂×C₄×C₄₁	—	2	24	no	—
79	C₂×C₄×C₄₁	—	2	24	no	—
80	C₂×C₄×C₄₁	—	4	48	no	—
81	C₂×C₄×C₄₁	—	4	16	no	—
82	C₂×C₄×C₄₁	C₂²×D₄	1	24	yes	—
83	C₂×#2656	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₄₁	—	2	32	no	—
85	C₂²×C₄₁	—	2	12	no	—
86	C₂²×C₄₁	—	2	12	no	—
87	C₂²×C₄₁	—	4	8	no	—
88	C₂²×C₄₁	—	4	24	no	—
89	C₂²×C₄₁	—	4	24	no	—
90	C₂²×C₄₁	—	8	32	no	—
91	C₂²×C₄₁	C₂×#127	1	8	yes	—
92	C₂²×C₄₁	C₂×#127	1	4	yes	—
93	C₂²×C₄₁	#534	1	12	yes	—
94	C₂²×C₄₁	#537	1	12	yes	—
95	C₂²×C₄₁	C₂³×D₄	1	3	yes	—
96	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	3	yes	—
97	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
98	C₄×C₄₁	—	2	24	no	—
99	C₄×C₄₁	—	4	8	no	—
100	C₂²×#121	C₂×C₄₁	1	6	yes	complemented
101	#2656	C₂×C₄₁	1	16	yes	direct product factor
102	C₂×#399	C₂×C₄₁	1	4	yes	complemented
103	C₂×#392	C₂×C₄₁	1	4	yes	complemented
104	C₂²×#95	C₂×C₄₁	1	1	yes	characteristic, complemented
105	C₂×C₄₁	—	2	8	no	—
106	C₂×C₄₁	—	8	8	no	—
107	C₂×C₄₁	#2656	1	8	yes	direct product factor
108	C₂×C₄₁	C₂²×#127	1	1	yes	characteristic
109	C₂×C₄₁	C₂×#534	1	3	yes	—
110	C₂×C₄₁	C₂×#537	1	3	yes	—
111	C₂³×D₄	C₂²×C₄₁	1	4	yes	—
112	C₂³×D₄	C₂²×C₄₁	1	6	yes	—
113	C₂⁴×C₄	C₂²×C₄₁	1	3	yes	—
114	C₂×#121	—	2	48	no	—
115	C₂×#121	C₂²×C₄₁	1	48	yes	complemented
116	C₂²×C₄⋊C₄	C₂²×C₄₁	1	6	yes	—
117	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	12	yes	—
118	#399	C₂²×C₄₁	1	32	yes	complemented
119	#392	C₂²×C₄₁	1	32	yes	complemented
120	C₂×#95	C₂²×C₄₁	1	4	yes	—
121	C₂×#95	C₂²×C₄₁	1	8	yes	complemented
122	C₄₁	C₂×#2656	1	1	yes	characteristic, direct product factor, Sylow
123	C₂⁵	C₂³×C₄₁	1	4	yes	—
124	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
125	C₂²×D₄	—	2	48	no	—
126	C₂²×D₄	—	2	24	no	—
127	C₂²×D₄	—	4	24	no	—
128	C₂²×D₄	C₂³×C₄₁	1	16	yes	—
129	C₂²×D₄	C₂³×C₄₁	1	24	yes	—
130	C₂³×C₄	—	4	12	no	—
131	C₂³×C₄	D₄×C₄₁	1	24	yes	complemented
132	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
133	C₂³×C₄	C₂³×C₄₁	1	6	yes	—
134	C₂³×C₄	C₂³×C₄₁	1	12	yes	—
135	#121	—	2	192	no	—
136	C₂×C₄⋊C₄	—	2	24	no	—
137	C₂×C₄⋊C₄	C₂³×C₄₁	1	24	yes	—
138	C₂×C₂²⋊C₄	—	2	48	no	—
139	C₂×C₂²⋊C₄	C₂³×C₄₁	1	48	yes	—
140	#95	C₂³×C₄₁	1	16	yes	—
141	C₂⁴	—	2	6	no	—
142	C₂⁴	—	2	2	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	—	2	24	no	—
145	C₂⁴	—	2	8	no	—
146	C₂⁴	C₂×D₄×C₄₁	1	12	yes	—
147	C₂⁴	C₂⁴×C₄₁	1	8	yes	—
148	C₂⁴	C₂⁴×C₄₁	1	2	yes	—
149	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
150	C₂×D₄	—	2	96	no	—
151	C₂×D₄	—	2	48	no	—
152	C₂×D₄	—	4	48	no	—
153	C₂×D₄	—	4	96	no	—
154	C₂×D₄	—	4	64	no	—
155	C₂²×C₄	—	2	24	no	—
156	C₂²×C₄	—	2	24	no	—
157	C₂²×C₄	—	4	48	no	—
158	C₂²×C₄	—	4	24	no	—
159	C₂²×C₄	C₂×D₄×C₄₁	1	24	yes	—
160	C₂²×C₄	C₂×D₄×C₄₁	1	48	yes	complemented
161	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₄₁	1	12	yes	—
163	C₄⋊C₄	—	2	48	no	—
164	C₂²⋊C₄	—	2	96	no	—
165	C₂³	—	2	24	no	—
166	C₂³	—	2	24	no	—
167	C₂³	—	2	6	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	2	24	no	—
171	C₂³	—	2	8	no	—
172	C₂³	—	4	16	no	—
173	C₂³	—	4	48	no	—
174	C₂³	—	4	48	no	—
175	C₂³	#127×C₄₁	1	16	yes	—
176	C₂³	C₂²×D₄×C₄₁	1	12	yes	—
177	C₂³	C₂²×D₄×C₄₁	1	6	yes	—
178	C₂³	H(5,𝔽₂)×C₄₁	1	6	yes	—
179	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
180	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
181	D₄	—	4	96	no	—
182	D₄	—	4	64	no	—
183	C₂×C₄	—	2	48	no	—
184	C₂×C₄	—	2	24	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	4	48	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	C₂²×D₄×C₄₁	1	24	yes	—
189	C₂²	—	2	32	no	—
190	C₂²	—	2	12	no	—
191	C₂²	—	2	12	no	—
192	C₂²	—	4	8	no	—
193	C₂²	—	4	24	no	—
194	C₂²	—	4	24	no	—
195	C₂²	—	8	32	no	—
196	C₂²	C₂×#127×C₄₁	1	8	yes	—
197	C₂²	C₂×#127×C₄₁	1	4	yes	—
198	C₂²	C₄₁×#534	1	12	yes	—
199	C₂²	C₄₁×#537	1	12	yes	—
200	C₂²	C₂³×D₄×C₄₁	1	3	yes	—
201	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	3	yes	—
202	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
203	C₄	—	2	24	no	—
204	C₄	—	4	8	no	—
205	C₂	—	2	8	no	—
206	C₂	—	8	8	no	—
207	C₂	C₄₁×#2656	1	8	yes	direct product factor
208	C₂	C₂²×#127×C₄₁	1	1	yes	characteristic
209	C₂	C₂×C₄₁×#534	1	3	yes	—
210	C₂	C₂×C₄₁×#537	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#121×C₄₁	C₂	semidirect
2. C₄₁×#2656	C₂	direct (N × H)
3. C₂×C₄₁×#399	C₂	semidirect
4. C₂×C₄₁×#392	C₂	semidirect
5. C₂²×#95×C₄₁	C₂	semidirect
10. C₂×#121×C₄₁	C₂²	semidirect
13. C₄₁×#399	C₂²	semidirect
14. C₄₁×#392	C₂²	semidirect
16. C₂×#95×C₄₁	C₂²	semidirect
25. C₂³×C₄×C₄₁	D₄	semidirect
54. C₂²×C₄×C₄₁	C₂×D₄	semidirect
83. C₂×#2656	C₄₁	direct (N × H)
100. C₂²×#121	C₂×C₄₁	semidirect
101. #2656	C₂×C₄₁	direct (N × H)
102. C₂×#399	C₂×C₄₁	semidirect
103. C₂×#392	C₂×C₄₁	semidirect
104. C₂²×#95	C₂×C₄₁	semidirect
107. C₂×C₄₁	#2656	direct (N × H)
115. C₂×#121	C₂²×C₄₁	semidirect
118. #399	C₂²×C₄₁	semidirect
119. #392	C₂²×C₄₁	semidirect
121. C₂×#95	C₂²×C₄₁	semidirect
122. C₄₁	C₂×#2656	direct (N × H)
131. C₂³×C₄	D₄×C₄₁	semidirect
160. C₂²×C₄	C₂×D₄×C₄₁	semidirect
207. C₂	C₄₁×#2656	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×#121×C₄₁	2
C₄₁×#2656	2
C₂×C₄₁×#399	2
C₂×C₄₁×#392	2
C₂²×#95×C₄₁	2
C₂×#2656	41

Subgroups — C2×C41×#2656 #23810772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2656 #23810772