Subgroups — C₄₁×#2349 #19106129

Order 5248.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#389	C₂	1	2	yes	maximal, complemented
4	C₄₁×#385	C₂	1	4	yes	maximal, complemented
5	C₄₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#380	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#95×C₄₁	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₄₁	C₂²	1	4	yes	—
11	C₂³×C₄×C₄₁	C₂²	1	2	yes	—
12	C₄×D₄×C₄₁	—	2	8	no	—
13	C₄×D₄×C₄₁	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
22	#95×C₄₁	C₂²	1	1	yes	characteristic
23	#95×C₄₁	C₂²	1	2	yes	—
24	#95×C₄₁	C₂²	1	4	yes	—
25	#95×C₄₁	C₂²	1	1	yes	characteristic
26	#95×C₄₁	C₂²	1	2	yes	—
27	C₂⁴×C₄₁	C₂³	1	2	yes	—
28	C₂×D₄×C₄₁	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₄₁	D₄	1	4	yes	—
30	C₂×D₄×C₄₁	Q₈	1	4	yes	—
31	C₂²×C₄×C₄₁	—	2	8	no	—
32	C₂²×C₄×C₄₁	—	2	8	no	—
33	C₂²×C₄×C₄₁	—	4	4	no	—
34	C₂²×C₄×C₄₁	—	4	4	no	—
35	C₂²×C₄×C₄₁	C₂×C₄	1	8	yes	complemented
36	C₂²×C₄×C₄₁	C₂³	1	4	yes	—
37	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₄⋊C₄×C₄₁	—	2	4	no	—
45	C₄⋊C₄×C₄₁	—	4	4	no	—
46	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	complemented
47	C₂²⋊C₄×C₄₁	—	2	8	no	—
48	C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
49	C₄²×C₄₁	—	2	4	no	—
50	C₄²×C₄₁	C₂×C₄	1	4	yes	complemented
51	C₂³×C₄₁	—	2	4	no	—
52	C₂³×C₄₁	—	2	4	no	—
53	C₂³×C₄₁	C₂²×C₄	1	4	yes	—
54	C₂³×C₄₁	C₂×D₄	1	4	yes	—
55	C₂³×C₄₁	C₂×Q₈	1	4	yes	—
56	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
57	D₄×C₄₁	C₄⋊C₄	1	16	yes	complemented
58	C₂×C₄×C₄₁	—	2	2	no	—
59	C₂×C₄×C₄₁	—	2	2	no	—
60	C₂×C₄×C₄₁	—	2	4	no	—
61	C₂×C₄×C₄₁	—	2	2	no	—
62	C₂×C₄×C₄₁	—	2	2	no	—
63	C₂×C₄×C₄₁	—	2	2	no	—
64	C₂×C₄×C₄₁	—	2	4	no	—
65	C₂×C₄×C₄₁	—	2	2	no	—
66	C₂×C₄×C₄₁	—	2	4	no	—
67	C₂×C₄×C₄₁	—	2	2	no	—
68	C₂×C₄×C₄₁	—	2	4	no	—
69	C₂×C₄×C₄₁	—	4	8	no	—
70	C₂×C₄×C₄₁	—	4	16	no	—
71	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₄₁	#41	1	4	yes	—
78	C₂²×C₄₁	—	2	4	no	—
79	C₂²×C₄₁	—	2	4	no	—
80	C₂²×C₄₁	—	2	8	no	—
81	C₂²×C₄₁	—	2	8	no	—
82	C₂²×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
83	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₄₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₄₁	C₂×#41	1	2	yes	—
87	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
88	C₄×C₄₁	—	2	2	no	—
89	C₄×C₄₁	—	2	2	no	—
90	C₄×C₄₁	—	2	4	no	—
91	C₄×C₄₁	—	2	4	no	—
92	C₄×C₄₁	—	4	2	no	—
93	C₄×C₄₁	—	4	4	no	—
94	C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
95	#2349	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₄₁	—	2	8	no	—
97	C₂×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄₁	C₄×#41	1	1	yes	characteristic
99	C₂×C₄₁	#518	1	1	yes	characteristic
100	C₂×C₄₁	#546	1	2	yes	—
101	C₂×C₄₁	#554	1	2	yes	—
102	C₂×C₄×D₄	C₂×C₄₁	1	2	yes	—
103	C₂×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic
104	#389	C₂×C₄₁	1	2	yes	complemented
105	#385	C₂×C₄₁	1	4	yes	complemented
106	#382	C₂×C₄₁	1	1	yes	characteristic, complemented
107	#380	C₂×C₄₁	1	2	yes	complemented
108	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
109	C₂×#95	C₂×C₄₁	1	2	yes	complemented
110	C₄₁	#2349	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₄₁	1	4	yes	—
113	C₂³×C₄	C₂²×C₄₁	1	2	yes	—
114	C₄×D₄	—	2	8	no	—
115	C₄×D₄	C₄×C₄₁	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
119	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
122	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
123	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
124	#95	C₂²×C₄₁	1	1	yes	characteristic
125	#95	C₂²×C₄₁	1	2	yes	—
126	#95	C₂²×C₄₁	1	4	yes	—
127	#95	C₂²×C₄₁	1	1	yes	characteristic
128	#95	C₂²×C₄₁	1	2	yes	—
129	C₂⁴	C₂³×C₄₁	1	2	yes	—
130	C₂×D₄	C₂×C₄×C₄₁	1	4	yes	—
131	C₂×D₄	D₄×C₄₁	1	4	yes	—
132	C₂×D₄	Q₈×C₄₁	1	4	yes	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	C₂×C₄×C₄₁	1	8	yes	complemented
138	C₂²×C₄	C₂³×C₄₁	1	4	yes	—
139	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
140	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
143	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	complemented
149	C₂²⋊C₄	—	2	8	no	—
150	C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
151	C₄²	—	2	4	no	—
152	C₄²	C₂×C₄×C₄₁	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×C₄×C₄₁	1	4	yes	—
156	C₂³	C₂×D₄×C₄₁	1	4	yes	—
157	C₂³	C₂×Q₈×C₄₁	1	4	yes	—
158	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
159	D₄	C₄⋊C₄×C₄₁	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	16	no	—
173	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
176	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
178	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
179	C₂×C₄	#41×C₄₁	1	4	yes	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	8	no	—
183	C₂²	—	2	8	no	—
184	C₂²	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
185	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
186	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
187	C₂²	C₂²×Q₈×C₄₁	1	1	yes	characteristic
188	C₂²	C₂×#41×C₄₁	1	2	yes	—
189	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	2	no	—
196	C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
199	C₂	C₄×#41×C₄₁	1	1	yes	characteristic
200	C₂	C₄₁×#518	1	1	yes	characteristic
201	C₂	C₄₁×#546	1	2	yes	—
202	C₂	C₄₁×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁴×C₄₁
prime radical	C₂²×D₄×C₄₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#389	C₂	semidirect
4. C₄₁×#385	C₂	semidirect
5. C₄₁×#382	C₂	semidirect
6. C₄₁×#380	C₂	semidirect
7. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
8. C₂×#95×C₄₁	C₂	semidirect
13. C₄×D₄×C₄₁	C₄	semidirect
35. C₂²×C₄×C₄₁	C₂×C₄	semidirect
46. C₄⋊C₄×C₄₁	C₂×C₄	semidirect
48. C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
50. C₄²×C₄₁	C₂×C₄	semidirect
57. D₄×C₄₁	C₄⋊C₄	semidirect
95. #2349	C₄₁	direct (N × H)
104. #389	C₂×C₄₁	semidirect
105. #385	C₂×C₄₁	semidirect
106. #382	C₂×C₄₁	semidirect
107. #380	C₂×C₄₁	semidirect
108. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
109. C₂×#95	C₂×C₄₁	semidirect
110. C₄₁	#2349	direct (N × H)
115. C₄×D₄	C₄×C₄₁	semidirect
137. C₂²×C₄	C₂×C₄×C₄₁	semidirect
148. C₄⋊C₄	C₂×C₄×C₄₁	semidirect
150. C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
152. C₄²	C₂×C₄×C₄₁	semidirect
159. D₄	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×C₄×D₄×C₄₁	2
C₄₁×#389	2
C₄₁×#385	2
C₄₁×#382	2
C₄₁×#380	2
C₄×C₄⋊C₄×C₄₁	2
C₂×#95×C₄₁	2
#2349	41

Subgroups — C41×#2349 #19106129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2349 #19106129