Subgroups — C₂²×#135×C₂₄₁ #44505244

Order 30848.

1504 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₄₁	C₂	1	2	yes	maximal, complemented
2	C₂×#135×C₂₄₁	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Q₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₁₆×C₂₄₁	C₂	1	2	yes	maximal, complemented
5	C₂²×D₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₄₁	—	2	12	no	—
8	C₂×#41×C₂₄₁	C₂²	1	24	yes	complemented
9	C₂²×Q₈×C₂₄₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₄₁	—	2	2	no	—
11	C₂²×D₄×C₂₄₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₄₁	—	2	2	no	—
13	C₂³×C₄×C₂₄₁	C₂²	1	1	yes	characteristic
14	#135×C₂₄₁	C₂²	1	64	yes	direct product factor
15	C₂×Q₁₆×C₂₄₁	C₂²	1	12	yes	complemented
16	C₂×SD₁₆×C₂₄₁	C₂²	1	24	yes	complemented
17	C₂×D₈×C₂₄₁	C₂²	1	12	yes	complemented
18	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₄₁	C₂²	1	12	yes	complemented
21	C₂⁴×C₂₄₁	—	2	2	no	—
22	C₂⁴×C₂₄₁	D₄	1	1	yes	characteristic
23	#41×C₂₄₁	—	2	48	no	—
24	#41×C₂₄₁	C₂³	1	32	yes	complemented
25	C₂×Q₈×C₂₄₁	—	2	6	no	—
26	C₂×Q₈×C₂₄₁	C₂³	1	12	yes	—
27	C₂×D₄×C₂₄₁	—	2	24	no	—
28	C₂×D₄×C₂₄₁	—	2	6	no	—
29	C₂×D₄×C₂₄₁	C₂³	1	12	yes	—
30	C₂²×C₄×C₂₄₁	—	2	2	no	—
31	C₂²×C₄×C₂₄₁	—	2	24	no	—
32	C₂²×C₄×C₂₄₁	D₄	1	6	yes	—
33	C₂²×C₄×C₂₄₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₄₁	C₂³	1	6	yes	—
36	Q₁₆×C₂₄₁	C₂³	1	16	yes	complemented
37	SD₁₆×C₂₄₁	C₂³	1	32	yes	complemented
38	D₈×C₂₄₁	C₂³	1	16	yes	complemented
39	C₂×C₈×C₂₄₁	C₂³	1	6	yes	—
40	C₂×C₈×C₂₄₁	C₂³	1	6	yes	—
41	C₂×C₈×C₂₄₁	C₂³	1	16	yes	complemented
42	C₂³×C₂₄₁	—	2	4	no	—
43	C₂³×C₂₄₁	—	2	12	no	—
44	C₂³×C₂₄₁	—	4	8	no	—
45	C₂³×C₂₄₁	C₂×D₄	1	6	yes	—
46	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₂₄₁	—	2	12	no	—
48	Q₈×C₂₄₁	C₂⁴	1	8	yes	—
49	D₄×C₂₄₁	—	2	32	no	—
50	D₄×C₂₄₁	—	2	12	no	—
51	D₄×C₂₄₁	C₂⁴	1	8	yes	—
52	C₂×C₄×C₂₄₁	—	2	12	no	—
53	C₂×C₄×C₂₄₁	—	2	32	no	—
54	C₂×C₄×C₂₄₁	C₂×D₄	1	6	yes	—
55	C₂×C₄×C₂₄₁	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₂₄₁	C₂×D₄	1	12	yes	—
57	C₂×C₄×C₂₄₁	C₂⁴	1	3	yes	—
58	C₂×C₄×C₂₄₁	C₂⁴	1	4	yes	—
59	C₈×C₂₄₁	C₂⁴	1	4	yes	—
60	C₈×C₂₄₁	C₂⁴	1	4	yes	—
61	C₂²×C₂₄₁	—	2	12	no	—
62	C₂²×C₂₄₁	—	2	8	no	—
63	C₂²×C₂₄₁	—	4	24	no	—
64	C₂²×C₂₄₁	#135	1	4	yes	direct product factor
65	C₂²×C₂₄₁	C₂²×D₄	1	4	yes	—
66	C₂²×C₂₄₁	C₂²×D₄	1	3	yes	—
67	C₄×C₂₄₁	—	2	8	no	—
68	C₄×C₂₄₁	C₂²×D₄	1	4	yes	—
69	C₄×C₂₄₁	C₂²×D₄	1	3	yes	—
70	C₄×C₂₄₁	C₂⁵	1	1	yes	characteristic
71	C₂×C₂₄₁	—	2	4	no	—
72	C₂×C₂₄₁	—	4	8	no	—
73	C₂×C₂₄₁	C₂×#135	1	6	yes	direct product factor
74	C₂×C₂₄₁	C₂³×D₄	1	1	yes	characteristic
75	C₂₄₁	C₂²×#135	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×#135	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×#41	C₂×C₂₄₁	1	2	yes	complemented
78	C₂×#135	C₂×C₂₄₁	1	24	yes	direct product factor
79	C₂²×Q₁₆	C₂×C₂₄₁	1	1	yes	characteristic, complemented
80	C₂²×SD₁₆	C₂×C₂₄₁	1	2	yes	complemented
81	C₂²×D₈	C₂×C₂₄₁	1	1	yes	characteristic, complemented
82	C₂³×C₈	C₂×C₂₄₁	1	1	yes	characteristic, complemented
83	C₂×#41	—	2	12	no	—
84	C₂×#41	C₂²×C₂₄₁	1	24	yes	complemented
85	C₂²×Q₈	C₂²×C₂₄₁	1	2	yes	—
86	C₂²×D₄	—	2	2	no	—
87	C₂²×D₄	C₂²×C₂₄₁	1	2	yes	—
88	C₂³×C₄	—	2	2	no	—
89	C₂³×C₄	C₂²×C₂₄₁	1	1	yes	characteristic
90	#135	C₂²×C₂₄₁	1	64	yes	direct product factor
91	C₂×Q₁₆	C₂²×C₂₄₁	1	12	yes	complemented
92	C₂×SD₁₆	C₂²×C₂₄₁	1	24	yes	complemented
93	C₂×D₈	C₂²×C₂₄₁	1	12	yes	complemented
94	C₂²×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₂₄₁	1	12	yes	complemented
97	C₂⁴	—	2	2	no	—
98	C₂⁴	D₄×C₂₄₁	1	1	yes	characteristic
99	#41	—	2	48	no	—
100	#41	C₂³×C₂₄₁	1	32	yes	complemented
101	C₂×Q₈	—	2	6	no	—
102	C₂×Q₈	C₂³×C₂₄₁	1	12	yes	—
103	C₂×D₄	—	2	24	no	—
104	C₂×D₄	—	2	6	no	—
105	C₂×D₄	C₂³×C₂₄₁	1	12	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	24	no	—
108	C₂²×C₄	D₄×C₂₄₁	1	6	yes	—
109	C₂²×C₄	D₄×C₂₄₁	1	1	yes	characteristic
110	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
111	C₂²×C₄	C₂³×C₂₄₁	1	6	yes	—
112	Q₁₆	C₂³×C₂₄₁	1	16	yes	complemented
113	SD₁₆	C₂³×C₂₄₁	1	32	yes	complemented
114	D₈	C₂³×C₂₄₁	1	16	yes	complemented
115	C₂×C₈	C₂³×C₂₄₁	1	6	yes	—
116	C₂×C₈	C₂³×C₂₄₁	1	6	yes	—
117	C₂×C₈	C₂³×C₂₄₁	1	16	yes	complemented
118	C₂³	—	2	4	no	—
119	C₂³	—	2	12	no	—
120	C₂³	—	4	8	no	—
121	C₂³	C₂×D₄×C₂₄₁	1	6	yes	—
122	C₂³	C₂×D₄×C₂₄₁	1	1	yes	characteristic
123	Q₈	—	2	12	no	—
124	Q₈	C₂⁴×C₂₄₁	1	8	yes	—
125	D₄	—	2	32	no	—
126	D₄	—	2	12	no	—
127	D₄	C₂⁴×C₂₄₁	1	8	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	—	2	32	no	—
130	C₂×C₄	C₂×D₄×C₂₄₁	1	6	yes	—
131	C₂×C₄	C₂×D₄×C₂₄₁	1	3	yes	—
132	C₂×C₄	C₂×D₄×C₂₄₁	1	12	yes	—
133	C₂×C₄	C₂⁴×C₂₄₁	1	3	yes	—
134	C₂×C₄	C₂⁴×C₂₄₁	1	4	yes	—
135	C₈	C₂⁴×C₂₄₁	1	4	yes	—
136	C₈	C₂⁴×C₂₄₁	1	4	yes	—
137	C₂²	—	2	12	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	4	24	no	—
140	C₂²	#135×C₂₄₁	1	4	yes	direct product factor
141	C₂²	C₂²×D₄×C₂₄₁	1	4	yes	—
142	C₂²	C₂²×D₄×C₂₄₁	1	3	yes	—
143	C₄	—	2	8	no	—
144	C₄	C₂²×D₄×C₂₄₁	1	4	yes	—
145	C₄	C₂²×D₄×C₂₄₁	1	3	yes	—
146	C₄	C₂⁵×C₂₄₁	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	—	4	8	no	—
149	C₂	C₂×#135×C₂₄₁	1	6	yes	direct product factor
150	C₂	C₂³×D₄×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₄₁	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₄₁
socle	C₂³×C₂₄₁	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₄₁ ◁ C₂³×C₄×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₄₁	C₂	semidirect
2. C₂×#135×C₂₄₁	C₂	direct (N × H)
3. C₂²×Q₁₆×C₂₄₁	C₂	semidirect
4. C₂²×SD₁₆×C₂₄₁	C₂	semidirect
5. C₂²×D₈×C₂₄₁	C₂	semidirect
6. C₂³×C₈×C₂₄₁	C₂	semidirect
8. C₂×#41×C₂₄₁	C₂²	semidirect
14. #135×C₂₄₁	C₂²	direct (N × H)
15. C₂×Q₁₆×C₂₄₁	C₂²	semidirect
16. C₂×SD₁₆×C₂₄₁	C₂²	semidirect
17. C₂×D₈×C₂₄₁	C₂²	semidirect
20. C₂²×C₈×C₂₄₁	C₂²	semidirect
24. #41×C₂₄₁	C₂³	semidirect
36. Q₁₆×C₂₄₁	C₂³	semidirect
37. SD₁₆×C₂₄₁	C₂³	semidirect
38. D₈×C₂₄₁	C₂³	semidirect
41. C₂×C₈×C₂₄₁	C₂³	semidirect
64. C₂²×C₂₄₁	#135	direct (N × H)
73. C₂×C₂₄₁	C₂×#135	direct (N × H)
75. C₂₄₁	C₂²×#135	direct (N × H)
76. C₂²×#135	C₂₄₁	direct (N × H)
77. C₂²×#41	C₂×C₂₄₁	semidirect
78. C₂×#135	C₂×C₂₄₁	direct (N × H)
79. C₂²×Q₁₆	C₂×C₂₄₁	semidirect
80. C₂²×SD₁₆	C₂×C₂₄₁	semidirect
81. C₂²×D₈	C₂×C₂₄₁	semidirect
82. C₂³×C₈	C₂×C₂₄₁	semidirect
84. C₂×#41	C₂²×C₂₄₁	semidirect
90. #135	C₂²×C₂₄₁	direct (N × H)
91. C₂×Q₁₆	C₂²×C₂₄₁	semidirect
92. C₂×SD₁₆	C₂²×C₂₄₁	semidirect
93. C₂×D₈	C₂²×C₂₄₁	semidirect
96. C₂²×C₈	C₂²×C₂₄₁	semidirect
100. #41	C₂³×C₂₄₁	semidirect
112. Q₁₆	C₂³×C₂₄₁	semidirect
113. SD₁₆	C₂³×C₂₄₁	semidirect
114. D₈	C₂³×C₂₄₁	semidirect
117. C₂×C₈	C₂³×C₂₄₁	semidirect
140. C₂²	#135×C₂₄₁	direct (N × H)
149. C₂	C₂×#135×C₂₄₁	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×#41×C₂₄₁	2
C₂×#135×C₂₄₁	2
C₂²×Q₁₆×C₂₄₁	2
C₂²×SD₁₆×C₂₄₁	2
C₂²×D₈×C₂₄₁	2
C₂³×C₈×C₂₄₁	2
C₂²×#135	241

Subgroups — C22×#135×C241 #44505244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#135×C₂₄₁ #44505244