Subgroups — C₄×C₄₁×#367 #23788874

Order 10496.

366 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#367	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₄₁	C₂	1	2	yes	maximal
4	C₄₁×#1380	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×#107×C₄₁	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
10	C₄₁×#367	C₄	1	8	yes	direct product factor
11	C₄×C₁₆×C₄₁	C₄	1	4	yes	complemented
12	C₄₁×#340	—	2	2	no	—
13	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
16	C₂×C₄²×C₄₁	—	4	2	no	—
17	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₄₁	—	2	2	no	—
19	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₄₁	C₂×C₄	1	4	yes	—
22	#107×C₄₁	—	2	4	no	—
23	#107×C₄₁	C₂×C₄	1	8	yes	—
24	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₁	Q₈	1	2	yes	—
26	#95×C₄₁	—	4	2	no	—
27	C₂²×C₄×C₄₁	—	4	4	no	—
28	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₄₁	—	2	2	no	—
32	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₄₁	#41	1	2	yes	—
37	C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₄⋊C₄×C₄₁	—	4	4	no	—
39	C₄²×C₄₁	—	8	4	no	—
40	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
41	C₄²×C₄₁	Q₁₆	1	2	yes	—
42	C₁₆×C₄₁	—	2	2	no	—
43	C₁₆×C₄₁	C₄²	1	4	yes	complemented
44	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₄₁	—	2	2	no	—
46	C₂×C₄×C₄₁	—	4	8	no	—
47	C₂×C₄×C₄₁	—	8	4	no	—
48	C₂×C₄×C₄₁	—	8	4	no	—
49	C₂×C₄×C₄₁	#107	1	2	yes	—
50	C₂×C₄×C₄₁	#107	1	2	yes	—
51	C₂×C₄×C₄₁	SD₃₂	1	4	yes	—
52	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₁	#135	1	2	yes	—
56	C₈×C₄₁	—	2	2	no	—
57	C₈×C₄₁	C₂×C₄²	1	1	yes	characteristic
58	C₈×C₄₁	#117	1	1	yes	characteristic
59	C₈×C₄₁	C₄×Q₈	1	2	yes	—
60	C₄×#367	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₄₁	C₂×#107	1	1	yes	characteristic
62	C₂²×C₄₁	C₄×D₈	1	2	yes	—
63	C₂²×C₄₁	C₂×SD₃₂	1	2	yes	—
64	C₂²×C₄₁	#508	1	2	yes	—
65	C₄×C₄₁	—	2	2	no	—
66	C₄×C₄₁	—	8	8	no	—
67	C₄×C₄₁	#367	1	4	yes	direct product factor
68	C₄×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₄₁	#427	1	1	yes	characteristic
70	C₄×C₄₁	C₄×Q₁₆	1	2	yes	—
71	C₂×#367	C₂×C₄₁	1	2	yes	—
72	C₂×C₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic
73	C₄×#107	C₂×C₄₁	1	2	yes	—
74	#1380	C₂×C₄₁	1	2	yes	—
75	C₂×C₄₁	C₄×#107	1	1	yes	characteristic
76	C₂×C₄₁	C₂×#367	1	1	yes	characteristic
77	C₂×C₄₁	#2158	1	1	yes	characteristic
78	C₂×C₄₁	C₄×SD₃₂	1	4	yes	—
79	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
81	C₂×#107	C₂²×C₄₁	1	4	yes	—
82	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
83	C₄×C₄⋊C₄	—	2	2	no	—
84	#367	C₄×C₄₁	1	8	yes	direct product factor
85	C₄×C₁₆	C₄×C₄₁	1	4	yes	complemented
86	#340	—	2	2	no	—
87	C₄₁	C₄×#367	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
89	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄²	—	4	2	no	—
92	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
95	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
96	C₂×C₁₆	C₂×C₄×C₄₁	1	4	yes	—
97	#107	—	2	4	no	—
98	#107	C₂×C₄×C₄₁	1	8	yes	—
99	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
100	C₄×C₈	Q₈×C₄₁	1	2	yes	—
101	#95	—	4	2	no	—
102	C₂²×C₄	—	4	4	no	—
103	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
104	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
108	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
109	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
110	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
111	C₂×C₈	#41×C₄₁	1	2	yes	—
112	C₄⋊C₄	—	2	8	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄²	—	8	4	no	—
115	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
116	C₄²	Q₁₆×C₄₁	1	2	yes	—
117	C₁₆	—	2	2	no	—
118	C₁₆	C₄²×C₄₁	1	4	yes	complemented
119	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	8	4	no	—
123	C₂×C₄	—	8	4	no	—
124	C₂×C₄	#107×C₄₁	1	2	yes	—
125	C₂×C₄	#107×C₄₁	1	2	yes	—
126	C₂×C₄	SD₃₂×C₄₁	1	4	yes	—
127	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
129	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
130	C₂×C₄	#135×C₄₁	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	C₂×C₄²×C₄₁	1	1	yes	characteristic
133	C₈	#117×C₄₁	1	1	yes	characteristic
134	C₈	C₄×Q₈×C₄₁	1	2	yes	—
135	C₂²	C₂×#107×C₄₁	1	1	yes	characteristic
136	C₂²	C₄×D₈×C₄₁	1	2	yes	—
137	C₂²	C₂×SD₃₂×C₄₁	1	2	yes	—
138	C₂²	C₄₁×#508	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	8	no	—
141	C₄	C₄₁×#367	1	4	yes	direct product factor
142	C₄	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
143	C₄	C₄₁×#427	1	1	yes	characteristic
144	C₄	C₄×Q₁₆×C₄₁	1	2	yes	—
145	C₂	C₄×#107×C₄₁	1	1	yes	characteristic
146	C₂	C₂×C₄₁×#367	1	1	yes	characteristic
147	C₂	C₄₁×#2158	1	1	yes	characteristic
148	C₂	C₄×SD₃₂×C₄₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	D₈
commutator	C₈	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄₁×#367	C₄	direct (N × H)
11. C₄×C₁₆×C₄₁	C₄	semidirect
43. C₁₆×C₄₁	C₄²	semidirect
60. C₄×#367	C₄₁	direct (N × H)
67. C₄×C₄₁	#367	direct (N × H)
84. #367	C₄×C₄₁	direct (N × H)
85. C₄×C₁₆	C₄×C₄₁	semidirect
87. C₄₁	C₄×#367	direct (N × H)
118. C₁₆	C₄²×C₄₁	semidirect
141. C₄	C₄₁×#367	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×#367	2
C₂×C₄×C₁₆×C₄₁	2
C₄×#107×C₄₁	2
C₄₁×#1380	2
C₄×#367	41

Subgroups — C4×C41×#367 #23788874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×#367 #23788874