Subgroups — C₂³×C₁₁×#124 #14639853

Order 2816.

14334 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₁×#124	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₁₁×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶×C₁₁	C₂²	1	2	yes	—
7	C₂³×Q₈×C₁₁	—	2	16	no	—
8	C₂³×Q₈×C₁₁	C₂²	1	28	yes	complemented
9	C₂³×D₄×C₁₁	—	2	16	no	—
10	C₂³×D₄×C₁₁	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×#124	C₂²	1	448	yes	direct product factor
15	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	112	yes	complemented
16	C₂²×C₄²×C₁₁	C₂²	1	28	yes	complemented
17	C₂⁵×C₁₁	—	2	16	no	—
18	C₂⁵×C₁₁	—	2	32	no	—
19	C₂⁵×C₁₁	C₂³	1	28	yes	—
20	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₁₁	—	2	224	no	—
22	C₂²×Q₈×C₁₁	C₂³	1	112	yes	complemented
23	C₂²×D₄×C₁₁	—	2	224	no	—
24	C₂²×D₄×C₁₁	C₂³	1	112	yes	—
25	C₂³×C₄×C₁₁	—	2	16	no	—
26	C₂³×C₄×C₁₁	—	2	16	no	—
27	C₂³×C₄×C₁₁	D₄	1	16	yes	—
28	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
29	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
30	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
31	C₁₁×#124	C₂³	1	512	yes	direct product factor
32	C₂×C₁₁×C₂²⋊C₄	C₂³	1	448	yes	complemented
33	C₂×C₄²×C₁₁	C₂³	1	112	yes	complemented
34	C₂³×#124	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂⁴×C₁₁	—	2	112	no	—
36	C₂⁴×C₁₁	—	2	224	no	—
37	C₂⁴×C₁₁	—	4	128	no	—
38	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
39	C₂⁴×C₁₁	#41	1	16	yes	—
40	C₂⁴×C₁₁	C₂⁴	1	56	yes	—
41	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
42	C₂×Q₈×C₁₁	—	2	448	no	—
43	C₂×Q₈×C₁₁	C₂⁴	1	64	yes	complemented
44	C₂×D₄×C₁₁	—	2	448	no	—
45	C₂×D₄×C₁₁	C₂⁴	1	64	yes	—
46	C₂²×C₄×C₁₁	—	2	112	no	—
47	C₂²×C₄×C₁₁	—	2	112	no	—
48	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	—
49	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
50	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
51	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
52	C₁₁×C₂²⋊C₄	C₂⁴	1	256	yes	complemented
53	C₄²×C₁₁	C₂⁴	1	64	yes	complemented
54	C₂⁴×Q₈	C₂×C₁₁	1	1	yes	characteristic, complemented
55	C₂⁴×D₄	C₂×C₁₁	1	1	yes	characteristic
56	C₂²×#124	C₂×C₁₁	1	56	yes	direct product factor
57	C₂³×C₂²⋊C₄	C₂×C₁₁	1	4	yes	complemented
58	C₂³×C₄²	C₂×C₁₁	1	1	yes	characteristic, complemented
59	C₂³×C₁₁	—	2	112	no	—
60	C₂³×C₁₁	—	2	224	no	—
61	C₂³×C₁₁	—	4	448	no	—
62	C₂³×C₁₁	#124	1	64	yes	direct product factor
63	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
64	C₂³×C₁₁	C₂×#41	1	56	yes	—
65	C₂³×C₁₁	C₂⁵	1	16	yes	—
66	C₂³×C₁₁	C₂⁵	1	7	yes	—
67	Q₈×C₁₁	—	2	128	no	—
68	D₄×C₁₁	—	2	128	no	—
69	C₂×C₄×C₁₁	—	2	112	no	—
70	C₂×C₄×C₁₁	—	2	112	no	—
71	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	—
72	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
73	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
74	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
75	C₂⁶	C₂²×C₁₁	1	2	yes	—
76	C₂³×Q₈	—	2	16	no	—
77	C₂³×Q₈	C₂²×C₁₁	1	28	yes	complemented
78	C₂³×D₄	—	2	16	no	—
79	C₂³×D₄	C₂²×C₁₁	1	28	yes	—
80	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
81	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
82	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
83	C₂×#124	C₂²×C₁₁	1	448	yes	direct product factor
84	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	112	yes	complemented
85	C₂²×C₄²	C₂²×C₁₁	1	28	yes	complemented
86	C₂²×C₁₁	—	2	16	no	—
87	C₂²×C₁₁	—	2	32	no	—
88	C₂²×C₁₁	—	4	224	no	—
89	C₂²×C₁₁	C₂×#124	1	112	yes	direct product factor
90	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
91	C₂²×C₁₁	C₂²×#41	1	28	yes	—
92	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
93	C₄×C₁₁	—	2	16	no	—
94	C₄×C₁₁	—	2	16	no	—
95	C₄×C₁₁	C₂³×D₄	1	16	yes	—
96	C₂⁵	—	2	16	no	—
97	C₂⁵	—	2	32	no	—
98	C₂⁵	C₂³×C₁₁	1	28	yes	—
99	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
100	C₂²×Q₈	—	2	224	no	—
101	C₂²×Q₈	C₂³×C₁₁	1	112	yes	complemented
102	C₂²×D₄	—	2	224	no	—
103	C₂²×D₄	C₂³×C₁₁	1	112	yes	—
104	C₂³×C₄	—	2	16	no	—
105	C₂³×C₄	—	2	16	no	—
106	C₂³×C₄	D₄×C₁₁	1	16	yes	—
107	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
108	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
109	C₂³×C₄	C₂³×C₁₁	1	14	yes	—
110	#124	C₂³×C₁₁	1	512	yes	direct product factor
111	C₂×C₂²⋊C₄	C₂³×C₁₁	1	448	yes	complemented
112	C₂×C₄²	C₂³×C₁₁	1	112	yes	complemented
113	C₂×C₁₁	—	4	16	no	—
114	C₂×C₁₁	C₂²×#124	1	28	yes	direct product factor
115	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
116	C₂×C₁₁	C₂³×#41	1	2	yes	—
117	C₂⁴	—	2	112	no	—
118	C₂⁴	—	2	224	no	—
119	C₂⁴	—	4	128	no	—
120	C₂⁴	C₂×D₄×C₁₁	1	8	yes	—
121	C₂⁴	C₁₁×#41	1	16	yes	—
122	C₂⁴	C₂⁴×C₁₁	1	56	yes	—
123	C₂⁴	C₂⁴×C₁₁	1	7	yes	—
124	C₂×Q₈	—	2	448	no	—
125	C₂×Q₈	C₂⁴×C₁₁	1	64	yes	complemented
126	C₂×D₄	—	2	448	no	—
127	C₂×D₄	C₂⁴×C₁₁	1	64	yes	—
128	C₂²×C₄	—	2	112	no	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	C₂×D₄×C₁₁	1	112	yes	—
131	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
132	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
133	C₂²×C₄	C₂⁴×C₁₁	1	28	yes	—
134	C₂²⋊C₄	C₂⁴×C₁₁	1	256	yes	complemented
135	C₄²	C₂⁴×C₁₁	1	64	yes	complemented
136	C₁₁	C₂³×#124	1	1	yes	characteristic, direct product factor, Sylow
137	C₂³	—	2	112	no	—
138	C₂³	—	2	224	no	—
139	C₂³	—	4	448	no	—
140	C₂³	C₁₁×#124	1	64	yes	direct product factor
141	C₂³	C₂²×D₄×C₁₁	1	28	yes	—
142	C₂³	C₂×C₁₁×#41	1	56	yes	—
143	C₂³	C₂⁵×C₁₁	1	16	yes	—
144	C₂³	C₂⁵×C₁₁	1	7	yes	—
145	Q₈	—	2	128	no	—
146	D₄	—	2	128	no	—
147	C₂×C₄	—	2	112	no	—
148	C₂×C₄	—	2	112	no	—
149	C₂×C₄	C₂²×D₄×C₁₁	1	112	yes	—
150	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
151	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
152	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	32	no	—
155	C₂²	—	4	224	no	—
156	C₂²	C₂×C₁₁×#124	1	112	yes	direct product factor
157	C₂²	C₂³×D₄×C₁₁	1	14	yes	—
158	C₂²	C₂²×C₁₁×#41	1	28	yes	—
159	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
160	C₄	—	2	16	no	—
161	C₄	—	2	16	no	—
162	C₄	C₂³×D₄×C₁₁	1	16	yes	—
163	C₂	—	4	16	no	—
164	C₂	C₂²×C₁₁×#124	1	28	yes	direct product factor
165	C₂	C₂⁴×D₄×C₁₁	1	1	yes	characteristic
166	C₂	C₂³×C₁₁×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
prime radical	C₂⁴×D₄×C₁₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₁₁	C₂	semidirect
3. C₂²×C₁₁×#124	C₂	direct (N × H)
4. C₂³×C₁₁×C₂²⋊C₄	C₂	semidirect
5. C₂³×C₄²×C₁₁	C₂	semidirect
8. C₂³×Q₈×C₁₁	C₂²	semidirect
14. C₂×C₁₁×#124	C₂²	direct (N × H)
15. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
16. C₂²×C₄²×C₁₁	C₂²	semidirect
22. C₂²×Q₈×C₁₁	C₂³	semidirect
31. C₁₁×#124	C₂³	direct (N × H)
32. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
33. C₂×C₄²×C₁₁	C₂³	semidirect
34. C₂³×#124	C₁₁	direct (N × H)
43. C₂×Q₈×C₁₁	C₂⁴	semidirect
52. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
53. C₄²×C₁₁	C₂⁴	semidirect
54. C₂⁴×Q₈	C₂×C₁₁	semidirect
56. C₂²×#124	C₂×C₁₁	direct (N × H)
57. C₂³×C₂²⋊C₄	C₂×C₁₁	semidirect
58. C₂³×C₄²	C₂×C₁₁	semidirect
62. C₂³×C₁₁	#124	direct (N × H)
77. C₂³×Q₈	C₂²×C₁₁	semidirect
83. C₂×#124	C₂²×C₁₁	direct (N × H)
84. C₂²×C₂²⋊C₄	C₂²×C₁₁	semidirect
85. C₂²×C₄²	C₂²×C₁₁	semidirect
89. C₂²×C₁₁	C₂×#124	direct (N × H)
101. C₂²×Q₈	C₂³×C₁₁	semidirect
110. #124	C₂³×C₁₁	direct (N × H)
111. C₂×C₂²⋊C₄	C₂³×C₁₁	semidirect
112. C₂×C₄²	C₂³×C₁₁	semidirect
114. C₂×C₁₁	C₂²×#124	direct (N × H)
125. C₂×Q₈	C₂⁴×C₁₁	semidirect
134. C₂²⋊C₄	C₂⁴×C₁₁	semidirect
135. C₄²	C₂⁴×C₁₁	semidirect
136. C₁₁	C₂³×#124	direct (N × H)
140. C₂³	C₁₁×#124	direct (N × H)
156. C₂²	C₂×C₁₁×#124	direct (N × H)
164. C₂	C₂²×C₁₁×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₁	2
C₂⁴×D₄×C₁₁	2
C₂²×C₁₁×#124	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂³×C₄²×C₁₁	2
C₂³×#124	11

Subgroups — C23×C11×#124 #14639853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×#124 #14639853