Subgroups — C₄₁×#1736 #19105516

Order 5248.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#340	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₄₁	—	2	1	no	—
6	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₄₁	—	2	2	no	—
8	C₂×M₁₆×C₄₁	—	2	2	no	—
9	C₂×M₁₆×C₄₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
11	#117×C₄₁	—	2	2	no	—
12	#117×C₄₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
15	C₂×C₄⋊C₄×C₄₁	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
19	C₂×C₄²×C₄₁	—	2	1	no	—
20	C₂⁴×C₄₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₁	—	2	4	no	—
22	C₂²×C₄×C₄₁	—	2	2	no	—
23	C₂²×C₄×C₄₁	—	2	2	no	—
24	C₂²×C₄×C₄₁	—	4	4	no	—
25	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₁	D₄	1	2	yes	—
30	C₂²×C₄×C₄₁	D₄	1	2	yes	—
31	C₂²×C₄×C₄₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₄₁	—	2	8	no	—
34	C₂×C₈×C₄₁	—	2	2	no	—
35	C₂×C₈×C₄₁	—	2	2	no	—
36	C₂×C₈×C₄₁	C₂×C₄	1	4	yes	complemented
37	C₄⋊C₄×C₄₁	—	2	4	no	—
38	C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
41	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	complemented
42	C₂²⋊C₄×C₄₁	—	2	4	no	—
43	C₄²×C₄₁	—	2	4	no	—
44	C₂³×C₄₁	—	2	2	no	—
45	C₂³×C₄₁	—	2	2	no	—
46	C₂³×C₄₁	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₄₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₄₁	C₄⋊C₄	1	2	yes	—
49	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₁	—	2	4	no	—
51	C₂×C₄×C₄₁	—	2	4	no	—
52	C₂×C₄×C₄₁	—	4	8	no	—
53	C₂×C₄×C₄₁	—	4	2	no	—
54	C₂×C₄×C₄₁	—	4	4	no	—
55	C₂×C₄×C₄₁	—	4	2	no	—
56	C₂×C₄×C₄₁	C₄²	1	4	yes	—
57	C₂×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
60	C₂×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
69	C₈×C₄₁	—	2	4	no	—
70	C₂²×C₄₁	—	2	4	no	—
71	C₂²×C₄₁	—	2	2	no	—
72	C₂²×C₄₁	—	2	2	no	—
73	C₂²×C₄₁	—	2	4	no	—
74	C₂²×C₄₁	#95	1	4	yes	—
75	C₂²×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₄₁	#137	1	2	yes	—
80	C₄×C₄₁	—	4	4	no	—
81	C₄×C₄₁	—	4	4	no	—
82	C₄×C₄₁	#95	1	4	yes	—
83	C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
84	C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	#1736	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₄₁	—	2	4	no	—
89	C₂×C₄₁	C₂×#95	1	1	yes	characteristic
90	C₂×C₄₁	#417	1	2	yes	—
91	C₂×C₄₁	#418	1	1	yes	characteristic
92	C₂×C₄₁	#419	1	1	yes	characteristic
93	C₂×C₄₁	#428	1	2	yes	—
94	C₂²×M₁₆	C₂×C₄₁	1	1	yes	characteristic
95	C₂×#117	C₂×C₄₁	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
97	#340	C₂×C₄₁	1	4	yes	complemented
98	C₄₁	#1736	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₄	—	2	1	no	—
100	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	C₄×C₄₁	1	4	yes	complemented
104	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
105	#117	—	2	2	no	—
106	#117	C₄×C₄₁	1	4	yes	complemented
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	C₄×C₄₁	1	4	yes	complemented
110	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₄²	—	2	1	no	—
114	C₂⁴	D₄×C₄₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
120	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
121	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
122	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
123	C₂²×C₄	D₄×C₄₁	1	2	yes	—
124	C₂²×C₄	D₄×C₄₁	1	2	yes	—
125	C₂²×C₄	Q₈×C₄₁	1	2	yes	—
126	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂×C₄×C₄₁	1	4	yes	complemented
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
135	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂²⋊C₄×C₄₁	1	2	yes	—
141	C₂³	C₂²⋊C₄×C₄₁	1	2	yes	—
142	C₂³	C₄⋊C₄×C₄₁	1	2	yes	—
143	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₄²×C₄₁	1	4	yes	—
151	C₂×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
152	C₂×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
153	C₂×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
155	C₂×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
156	C₂×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
157	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
160	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
162	C₂×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	#95×C₄₁	1	4	yes	—
169	C₂²	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
170	C₂²	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
171	C₂²	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
172	C₂²	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
173	C₂²	#137×C₄₁	1	2	yes	—
174	C₄	—	4	4	no	—
175	C₄	—	4	4	no	—
176	C₄	#95×C₄₁	1	4	yes	—
177	C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
178	C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
179	C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
180	C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×#95×C₄₁	1	1	yes	characteristic
183	C₂	C₄₁×#417	1	2	yes	—
184	C₂	C₄₁×#418	1	1	yes	characteristic
185	C₂	C₄₁×#419	1	1	yes	characteristic
186	C₂	C₄₁×#428	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#340	C₂	semidirect
9. C₂×M₁₆×C₄₁	C₄	semidirect
12. #117×C₄₁	C₄	semidirect
15. C₂×C₄⋊C₄×C₄₁	C₄	semidirect
36. C₂×C₈×C₄₁	C₂×C₄	semidirect
41. C₄⋊C₄×C₄₁	C₂×C₄	semidirect
87. #1736	C₄₁	direct (N × H)
97. #340	C₂×C₄₁	semidirect
98. C₄₁	#1736	direct (N × H)
103. C₂×M₁₆	C₄×C₄₁	semidirect
106. #117	C₄×C₄₁	semidirect
109. C₂×C₄⋊C₄	C₄×C₄₁	semidirect
130. C₂×C₈	C₂×C₄×C₄₁	semidirect
135. C₄⋊C₄	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₄₁	2
C₂×#117×C₄₁	2
C₂²×C₄⋊C₄×C₄₁	2
C₄₁×#340	2
#1736	41

Subgroups — C41×#1736 #19105516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1736 #19105516