Subgroups — C₄₁×#14168 #23790369

Order 10496.

1342 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#376	C₂	1	2	yes	maximal
3	C₄₁×#1441	C₂	1	4	yes	maximal
4	C₄₁×#1436	C₂	1	8	yes	maximal
5	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
6	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#382	C₄	1	8	yes	complemented
10	C₄₁×#376	—	2	8	no	—
11	C₂×#95×C₄₁	C₂²	1	8	yes	—
12	C₂×#95×C₄₁	C₂²	1	4	yes	—
13	C₂×#95×C₄₁	C₂²	1	4	yes	—
14	C₂×#95×C₄₁	C₂²	1	4	yes	—
15	C₂×#95×C₄₁	C₂²	1	4	yes	—
16	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
17	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
18	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
19	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
21	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
22	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
23	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₄₁	—	2	8	no	—
26	C₂×C₄²×C₄₁	—	2	16	no	—
27	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₁	D₄	1	4	yes	—
29	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
30	#95×C₄₁	—	2	8	no	—
31	#95×C₄₁	—	2	8	no	—
32	#95×C₄₁	—	2	16	no	—
33	#95×C₄₁	—	4	4	no	—
34	#95×C₄₁	C₂×C₄	1	16	yes	—
35	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₄₁	—	2	8	no	—
37	C₂²×C₄×C₄₁	—	2	2	no	—
38	C₂²×C₄×C₄₁	—	2	2	no	—
39	C₂²×C₄×C₄₁	—	2	8	no	—
40	C₂²×C₄×C₄₁	—	2	8	no	—
41	C₂²×C₄×C₄₁	—	2	8	no	—
42	C₂²×C₄×C₄₁	—	2	8	no	—
43	C₂²×C₄×C₄₁	—	2	8	no	—
44	C₂²×C₄×C₄₁	—	2	8	no	—
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	—	2	8	no	—
47	C₂²×C₄×C₄₁	—	2	4	no	—
48	C₂²×C₄×C₄₁	—	2	8	no	—
49	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
50	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₁	#41	1	16	yes	—
57	C₄⋊C₄×C₄₁	—	4	8	no	—
58	C₄⋊C₄×C₄₁	—	4	16	no	—
59	C₄²×C₄₁	—	2	32	no	—
60	C₄²×C₄₁	C₄⋊C₄	1	16	yes	complemented
61	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
62	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
63	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
64	C₂³×C₄₁	C₂×#41	1	8	yes	—
65	C₂³×C₄₁	H(5,𝔽₂)	1	2	yes	—
66	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
67	C₂×C₄×C₄₁	—	2	8	no	—
68	C₂×C₄×C₄₁	—	2	2	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	—	2	4	no	—
73	C₂×C₄×C₄₁	—	2	4	no	—
74	C₂×C₄×C₄₁	—	2	16	no	—
75	C₂×C₄×C₄₁	—	2	8	no	—
76	C₂×C₄×C₄₁	—	2	16	no	—
77	C₂×C₄×C₄₁	—	2	16	no	—
78	C₂×C₄×C₄₁	—	2	16	no	—
79	C₂×C₄×C₄₁	—	4	8	no	—
80	C₂×C₄×C₄₁	—	4	8	no	—
81	C₂×C₄×C₄₁	—	4	4	no	—
82	C₂×C₄×C₄₁	—	4	4	no	—
83	C₂×C₄×C₄₁	—	4	8	no	—
84	C₂×C₄×C₄₁	—	4	8	no	—
85	C₂×C₄×C₄₁	—	4	8	no	—
86	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
88	#14168	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₁	C₄×#41	1	4	yes	—
91	C₂²×C₄₁	#518	1	1	yes	characteristic
92	C₂²×C₄₁	#519	1	1	yes	characteristic
93	C₂²×C₄₁	#525	1	4	yes	—
94	C₂²×C₄₁	#533	1	4	yes	—
95	C₂²×C₄₁	#537	1	2	yes	—
96	C₂²×C₄₁	#544	1	2	yes	—
97	C₂²×C₄₁	#551	1	4	yes	—
98	C₂²×C₄₁	#552	1	8	yes	—
99	C₂²×C₄₁	#555	1	4	yes	—
100	C₄×C₄₁	—	2	4	no	—
101	C₄×C₄₁	—	2	8	no	—
102	C₄×C₄₁	—	4	16	no	—
103	C₄×C₄₁	—	4	8	no	—
104	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
105	C₂×#376	C₂×C₄₁	1	2	yes	—
106	#1441	C₂×C₄₁	1	4	yes	—
107	#1436	C₂×C₄₁	1	8	yes	—
108	C₂×C₄₁	#2296	1	2	yes	—
109	C₂×C₄₁	#2316	1	1	yes	characteristic
110	C₂×C₄₁	#2371	1	2	yes	—
111	C₂×C₄₁	#2372	1	2	yes	—
112	C₂×C₄₁	#2609	1	4	yes	—
113	C₂×C₄₁	#2612	1	4	yes	—
114	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
116	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
117	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
118	#382	C₄×C₄₁	1	8	yes	complemented
119	#376	—	2	8	no	—
120	C₂×#95	C₂²×C₄₁	1	8	yes	—
121	C₂×#95	C₂²×C₄₁	1	4	yes	—
122	C₂×#95	C₂²×C₄₁	1	4	yes	—
123	C₂×#95	C₂²×C₄₁	1	4	yes	—
124	C₂×#95	C₂²×C₄₁	1	4	yes	—
125	C₄₁	#14168	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
127	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
128	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
129	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
130	C₂×C₄⋊C₄	—	2	16	no	—
131	C₂×C₄⋊C₄	—	4	2	no	—
132	C₂×C₄⋊C₄	—	4	2	no	—
133	C₂×C₄⋊C₄	—	4	4	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
135	C₂×C₄²	—	2	8	no	—
136	C₂×C₄²	—	2	16	no	—
137	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
138	C₂×C₄²	D₄×C₄₁	1	4	yes	—
139	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
140	#95	—	2	8	no	—
141	#95	—	2	8	no	—
142	#95	—	2	16	no	—
143	#95	—	4	4	no	—
144	#95	C₂×C₄×C₄₁	1	16	yes	—
145	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
162	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
164	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
165	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
166	C₂²×C₄	#41×C₄₁	1	16	yes	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	16	no	—
169	C₄²	—	2	32	no	—
170	C₄²	C₄⋊C₄×C₄₁	1	16	yes	complemented
171	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
172	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
173	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
174	C₂³	C₂×#41×C₄₁	1	8	yes	—
175	C₂³	H(5,𝔽₂)×C₄₁	1	2	yes	—
176	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	—	4	8	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
198	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
199	C₂²	C₄×#41×C₄₁	1	4	yes	—
200	C₂²	C₄₁×#518	1	1	yes	characteristic
201	C₂²	C₄₁×#519	1	1	yes	characteristic
202	C₂²	C₄₁×#525	1	4	yes	—
203	C₂²	C₄₁×#533	1	4	yes	—
204	C₂²	C₄₁×#537	1	2	yes	—
205	C₂²	C₄₁×#544	1	2	yes	—
206	C₂²	C₄₁×#551	1	4	yes	—
207	C₂²	C₄₁×#552	1	8	yes	—
208	C₂²	C₄₁×#555	1	4	yes	—
209	C₄	—	2	8	no	—
210	C₄	—	2	4	no	—
211	C₄	—	4	16	no	—
212	C₄	—	4	8	no	—
213	C₂	C₄₁×#2296	1	2	yes	—
214	C₂	C₄₁×#2316	1	1	yes	characteristic
215	C₂	C₄₁×#2371	1	2	yes	—
216	C₂	C₄₁×#2372	1	2	yes	—
217	C₂	C₄₁×#2609	1	4	yes	—
218	C₂	C₄₁×#2612	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×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
9. C₄₁×#382	C₄	semidirect
60. C₄²×C₄₁	C₄⋊C₄	semidirect
88. #14168	C₄₁	direct (N × H)
118. #382	C₄×C₄₁	semidirect
125. C₄₁	#14168	direct (N × H)
170. C₄²	C₄⋊C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#382	2
C₂×C₄₁×#376	2
C₄₁×#1441	2
C₄₁×#1436	2
#14168	41

Subgroups — C41×#14168 #23790369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14168 #23790369