Subgroups — C₂³×#66879 #15617155

Order 2816.

4126 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#66879	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₁₁	C₄	1	2	yes	—
7	C₂³×C₈×C₁₁	C₄	1	16	yes	complemented
8	C₂³×C₈×C₁₁	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₁₁	C₄	1	14	yes	—
10	C₂²×C₄×Dic₁₁	C₂²	1	14	yes	—
11	C₂³×C₁₁⋊C₈	C₄	1	16	yes	complemented
12	C₂³×C₁₁⋊C₈	C₂²	1	14	yes	—
13	C₂×#66879	C₂²	1	112	yes	direct product factor
14	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₁	C₂×C₄	1	7	yes	—
16	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
17	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
19	C₂³×Dic₁₁	—	2	16	no	—
20	C₂³×Dic₁₁	C₂×C₄	1	28	yes	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	112	yes	complemented
22	C₂²×C₈×C₁₁	C₂³	1	28	yes	—
23	C₂×C₄×Dic₁₁	C₂×C₄	1	84	yes	—
24	C₂×C₄×Dic₁₁	C₂³	1	28	yes	—
25	C₂²×C₁₁⋊C₈	C₂×C₄	1	112	yes	complemented
26	C₂²×C₁₁⋊C₈	C₂³	1	28	yes	—
27	#66879	C₂³	1	64	yes	direct product factor
28	C₂³×#97	—	11	1	no	maximal, Sylow
29	C₂⁴×C₁₁	C₄²	1	8	yes	—
30	C₂⁴×C₁₁	M₁₆	1	16	yes	—
31	C₂⁴×C₁₁	C₂²×C₄	1	7	yes	—
32	C₂²×C₄×C₁₁	C₄²	1	56	yes	—
33	C₂²×C₄×C₁₁	C₂²×C₄	1	21	yes	—
34	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
35	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
36	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
37	C₂²×Dic₁₁	—	2	112	no	—
38	C₂²×Dic₁₁	C₂²×C₄	1	56	yes	—
39	C₂×C₈×C₁₁	C₂²×C₄	1	112	yes	complemented
40	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
41	C₄×Dic₁₁	C₂²×C₄	1	56	yes	—
42	C₄×Dic₁₁	C₂⁴	1	8	yes	—
43	C₂×C₁₁⋊C₈	C₂²×C₄	1	112	yes	complemented
44	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
45	C₂⁴×C₈	—	11	1	no	—
46	C₂⁴×C₈	D₁₁	1	1	yes	characteristic
47	C₂³×C₄²	—	11	1	no	—
48	C₂²×#97	—	11	28	no	—
49	C₂³×C₁₁	#97	1	64	yes	complemented
50	C₂³×C₁₁	C₂×C₄²	1	28	yes	—
51	C₂³×C₁₁	C₂×M₁₆	1	56	yes	—
52	C₂³×C₁₁	C₂³×C₄	1	7	yes	—
53	C₂×C₄×C₁₁	C₂×C₄²	1	84	yes	—
54	C₂×C₄×C₁₁	C₂³×C₄	1	14	yes	—
55	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₁₁	C₂³×C₄	1	14	yes	—
57	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
58	C₂×Dic₁₁	—	2	112	no	—
59	C₂×Dic₁₁	C₂³×C₄	1	16	yes	—
60	C₈×C₁₁	C₂³×C₄	1	16	yes	complemented
61	C₁₁⋊C₈	C₂³×C₄	1	16	yes	complemented
62	C₂⁴×C₄	—	11	2	no	—
63	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
64	C₂³×C₈	—	11	14	no	—
65	C₂³×C₈	—	11	16	no	—
66	C₂³×C₈	Dic₁₁	1	16	yes	complemented
67	C₂³×C₈	C₂×D₁₁	1	14	yes	—
68	C₂²×C₄²	—	11	14	no	—
69	C₂²×C₄²	—	11	14	no	—
70	C₂×#97	—	11	112	no	—
71	C₂²×C₁₁	C₂×#97	1	112	yes	complemented
72	C₂²×C₁₁	C₂²×C₄²	1	14	yes	—
73	C₂²×C₁₁	C₂²×M₁₆	1	28	yes	—
74	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
75	C₄×C₁₁	C₂²×C₄²	1	14	yes	—
76	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
77	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
78	Dic₁₁	—	2	16	no	—
79	C₂⁵	C₄×D₁₁	1	1	yes	characteristic
80	C₂³×C₄	—	11	28	no	—
81	C₂³×C₄	—	22	16	no	—
82	C₂³×C₄	C₄×D₁₁	1	7	yes	—
83	C₂³×C₄	C₄×D₁₁	1	8	yes	—
84	C₂³×C₄	C₂×Dic₁₁	1	8	yes	—
85	C₂³×C₄	C₂²×D₁₁	1	7	yes	—
86	C₂²×C₈	—	11	112	no	—
87	C₂²×C₈	—	11	28	no	—
88	C₂²×C₈	C₂×Dic₁₁	1	112	yes	complemented
89	C₂²×C₈	C₂²×D₁₁	1	28	yes	—
90	C₂×C₄²	—	11	84	no	—
91	C₂×C₄²	—	11	28	no	—
92	#97	—	11	64	no	—
93	C₂×C₁₁	C₂²×#97	1	28	yes	complemented
94	C₂×C₁₁	C₂³×C₄²	1	1	yes	characteristic
95	C₂×C₁₁	C₂³×M₁₆	1	2	yes	—
96	C₂⁴	#4313	1	16	yes	—
97	C₂⁴	C₄×Dic₁₁	1	8	yes	—
98	C₂⁴	C₂×C₄×D₁₁	1	7	yes	—
99	C₂²×C₄	—	11	56	no	—
100	C₂²×C₄	—	22	112	no	—
101	C₂²×C₄	C₄×Dic₁₁	1	56	yes	—
102	C₂²×C₄	C₂×C₄×D₁₁	1	21	yes	—
103	C₂²×C₄	C₂×C₄×D₁₁	1	28	yes	—
104	C₂²×C₄	C₂²×Dic₁₁	1	28	yes	—
105	C₂²×C₄	C₂³×D₁₁	1	7	yes	—
106	C₂×C₈	—	11	112	no	—
107	C₂×C₈	—	11	8	no	—
108	C₂×C₈	C₂²×Dic₁₁	1	112	yes	complemented
109	C₂×C₈	C₂³×D₁₁	1	8	yes	—
110	C₄²	—	11	56	no	—
111	C₄²	—	11	8	no	—
112	C₁₁	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
113	C₂³	#66879	1	64	yes	direct product factor
114	C₂³	C₂×#4313	1	56	yes	—
115	C₂³	C₂×C₄×Dic₁₁	1	28	yes	—
116	C₂³	C₂²×C₄×D₁₁	1	7	yes	—
117	C₂×C₄	—	11	16	no	—
118	C₂×C₄	—	22	112	no	—
119	C₂×C₄	C₂×C₄×Dic₁₁	1	84	yes	—
120	C₂×C₄	C₂²×C₄×D₁₁	1	7	yes	—
121	C₂×C₄	C₂²×C₄×D₁₁	1	14	yes	—
122	C₂×C₄	C₂³×Dic₁₁	1	14	yes	—
123	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
124	C₈	—	11	16	no	—
125	C₈	C₂³×Dic₁₁	1	16	yes	complemented
126	C₂²	C₂×#66879	1	112	yes	direct product factor
127	C₂²	C₂²×#4313	1	28	yes	—
128	C₂²	C₂²×C₄×Dic₁₁	1	14	yes	—
129	C₂²	C₂³×C₄×D₁₁	1	1	yes	characteristic
130	C₄	—	22	16	no	—
131	C₄	C₂²×C₄×Dic₁₁	1	14	yes	—
132	C₄	C₂³×C₄×D₁₁	1	1	yes	characteristic
133	C₄	C₂⁴×Dic₁₁	1	1	yes	characteristic
134	C₂	C₂²×#66879	1	28	yes	direct product factor
135	C₂	C₂³×C₄×Dic₁₁	1	1	yes	characteristic
136	C₂	C₂³×#4313	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₁
commutator	C₂×C₁₁	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂⁴×C₈×C₁₁	C₂
socle, prime radical	C₂⁵×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂⁴×C₈	D₁₁
hypocenter	C₁₁	C₂³×#97
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#66879	C₂	direct (N × H)
7. C₂³×C₈×C₁₁	C₄	semidirect
11. C₂³×C₁₁⋊C₈	C₄	semidirect
13. C₂×#66879	C₂²	direct (N × H)
21. C₂²×C₈×C₁₁	C₂×C₄	semidirect
25. C₂²×C₁₁⋊C₈	C₂×C₄	semidirect
27. #66879	C₂³	direct (N × H)
39. C₂×C₈×C₁₁	C₂²×C₄	semidirect
43. C₂×C₁₁⋊C₈	C₂²×C₄	semidirect
49. C₂³×C₁₁	#97	semidirect
60. C₈×C₁₁	C₂³×C₄	semidirect
61. C₁₁⋊C₈	C₂³×C₄	semidirect
66. C₂³×C₈	Dic₁₁	semidirect
71. C₂²×C₁₁	C₂×#97	semidirect
88. C₂²×C₈	C₂×Dic₁₁	semidirect
93. C₂×C₁₁	C₂²×#97	semidirect
108. C₂×C₈	C₂²×Dic₁₁	semidirect
112. C₁₁	C₂³×#97	semidirect
113. C₂³	#66879	direct (N × H)
125. C₈	C₂³×Dic₁₁	semidirect
126. C₂²	C₂×#66879	direct (N × H)
134. C₂	C₂²×#66879	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×C₁₁⋊C₈	2
C₂³×C₄×Dic₁₁	2
C₂²×#66879	2
C₂⁴×C₈×C₁₁	2
C₂³×#97	11

Subgroups — C23×#66879 #15617155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66879 #15617155