Subgroups — C₂³×C₄×Dic₂₆ #16769488

Order 3328.

12446 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₆	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#6101	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₃	C₂	1	2	yes	maximal
6	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
12	C₂³×Dic₂₆	C₄	1	32	yes	direct product factor
13	C₂³×Dic₂₆	C₂²	1	28	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	28	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	—
16	C₂²×#6101	C₂²	1	56	yes	—
17	C₂²×C₄×Dic₁₃	C₂²	1	56	yes	—
18	C₂×C₄×Dic₂₆	C₂²	1	448	yes	direct product factor
19	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	—	2	8	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
22	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
23	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
26	C₂³×Dic₁₃	—	2	16	no	—
27	C₂³×Dic₁₃	C₂×C₄	1	32	yes	—
28	C₂³×Dic₁₃	C₂³	1	28	yes	—
29	C₂³×Dic₁₃	C₂³	1	28	yes	—
30	C₂²×Dic₂₆	C₂×C₄	1	448	yes	direct product factor
31	C₂²×Dic₂₆	C₂³	1	112	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	112	yes	—
33	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	—
34	C₂×#6101	C₂³	1	224	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	224	yes	—
36	C₄×Dic₂₆	C₂³	1	512	yes	direct product factor
37	C₂³×C₄×Q₈	—	13	1	no	maximal, Sylow
38	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
39	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
40	C₂⁴×C₁₃	#41	1	8	yes	—
41	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
42	C₂²×C₄×C₁₃	—	2	56	no	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	112	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	112	yes	complemented
45	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
46	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
48	C₂²×Dic₁₃	—	2	112	no	—
49	C₂²×Dic₁₃	C₂²×C₄	1	224	yes	—
50	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
51	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
52	C₂×Dic₂₆	C₂²×C₄	1	896	yes	direct product factor
53	C₂×Dic₂₆	C₂⁴	1	64	yes	—
54	C₄²×C₁₃	C₂⁴	1	64	yes	—
55	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
56	#6101	C₂⁴	1	128	yes	—
57	C₄×Dic₁₃	C₂⁴	1	128	yes	—
58	C₂⁴×Q₈	—	13	1	no	—
59	C₂²×C₄×Q₈	—	13	56	no	—
60	C₂³×C₄⋊C₄	—	13	2	no	—
61	C₂³×C₄⋊C₄	—	13	1	no	—
62	C₂³×C₄²	—	13	2	no	—
63	C₂³×C₄²	D₁₃	1	1	yes	characteristic
64	C₂³×C₁₃	C₄×Q₈	1	64	yes	complemented
65	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
66	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
67	C₂³×C₁₃	C₂×#41	1	28	yes	—
68	C₂³×C₁₃	C₂⁵	1	7	yes	—
69	C₂×C₄×C₁₃	—	2	56	no	—
70	C₂×C₄×C₁₃	C₂³×C₄	1	112	yes	—
71	C₂×C₄×C₁₃	C₂²×Q₈	1	112	yes	complemented
72	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
73	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
74	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
75	C₂×Dic₁₃	—	2	112	no	—
76	C₂×Dic₁₃	C₂³×C₄	1	224	yes	—
77	C₂×Dic₁₃	C₂⁵	1	16	yes	—
78	C₂×Dic₁₃	C₂⁵	1	16	yes	—
79	Dic₂₆	C₂³×C₄	1	256	yes	direct product factor
80	C₂³×Q₈	—	13	32	no	—
81	C₂³×Q₈	—	13	28	no	—
82	C₂⁴×C₄	—	13	2	no	—
83	C₂⁴×C₄	—	13	2	no	—
84	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
85	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
86	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
87	C₂×C₄×Q₈	—	13	448	no	—
88	C₂²×C₄⋊C₄	—	13	56	no	—
89	C₂²×C₄⋊C₄	—	13	28	no	—
90	C₂²×C₄²	—	13	56	no	—
91	C₂²×C₄²	C₂×D₁₃	1	28	yes	—
92	C₂²×C₁₃	C₂×C₄×Q₈	1	112	yes	complemented
93	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
94	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
95	C₂²×C₁₃	C₂²×#41	1	14	yes	—
96	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
97	C₄×C₁₃	—	2	8	no	—
98	C₄×C₁₃	C₂⁴×C₄	1	16	yes	—
99	C₄×C₁₃	C₂³×Q₈	1	16	yes	complemented
100	Dic₁₃	—	2	16	no	—
101	Dic₁₃	C₂⁴×C₄	1	32	yes	—
102	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
103	C₂²×Q₈	—	13	448	no	—
104	C₂²×Q₈	—	13	112	no	—
105	C₂³×C₄	—	2	8	no	—
106	C₂³×C₄	—	13	32	no	—
107	C₂³×C₄	—	13	28	no	—
108	C₂³×C₄	—	13	28	no	—
109	C₂³×C₄	—	26	16	no	—
110	C₂³×C₄	Dic₂₆	1	16	yes	direct product factor
111	C₂³×C₄	C₄×D₁₃	1	16	yes	—
112	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
113	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
114	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
115	C₄×Q₈	—	13	512	no	—
116	C₂×C₄⋊C₄	—	13	224	no	—
117	C₂×C₄⋊C₄	—	13	112	no	—
118	C₂×C₄²	—	13	224	no	—
119	C₂×C₄²	C₂²×D₁₃	1	112	yes	—
120	C₂×C₁₃	C₂²×C₄×Q₈	1	28	yes	complemented
121	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
122	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
123	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₂₆	1	8	yes	—
125	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
126	C₂⁴	#6127	1	8	yes	—
127	C₂⁴	C₂³×D₁₃	1	7	yes	—
128	C₂×Q₈	—	13	896	no	—
129	C₂×Q₈	—	13	64	no	—
130	C₂²×C₄	—	2	56	no	—
131	C₂²×C₄	—	13	224	no	—
132	C₂²×C₄	—	13	56	no	—
133	C₂²×C₄	—	13	56	no	—
134	C₂²×C₄	—	26	112	no	—
135	C₂²×C₄	C₂×Dic₂₆	1	112	yes	direct product factor
136	C₂²×C₄	C₂×C₄×D₁₃	1	112	yes	—
137	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
138	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
139	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
140	C₄⋊C₄	—	13	128	no	—
141	C₄⋊C₄	—	13	64	no	—
142	C₄²	—	13	128	no	—
143	C₄²	C₂³×D₁₃	1	64	yes	—
144	C₁₃	C₂³×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₄×Dic₂₆	1	64	yes	direct product factor
146	C₂³	C₂²×Dic₂₆	1	28	yes	—
147	C₂³	C₂²×C₄×D₁₃	1	28	yes	—
148	C₂³	C₂×#6127	1	28	yes	—
149	C₂³	C₂⁴×D₁₃	1	7	yes	—
150	Q₈	—	13	256	no	—
151	C₂×C₄	—	2	56	no	—
152	C₂×C₄	—	13	224	no	—
153	C₂×C₄	—	13	16	no	—
154	C₂×C₄	—	13	16	no	—
155	C₂×C₄	—	26	112	no	—
156	C₂×C₄	C₂²×Dic₂₆	1	112	yes	direct product factor
157	C₂×C₄	C₂²×C₄×D₁₃	1	112	yes	—
158	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
159	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
160	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
161	C₂²	C₂×C₄×Dic₂₆	1	112	yes	direct product factor
162	C₂²	C₂³×Dic₂₆	1	14	yes	—
163	C₂²	C₂³×C₄×D₁₃	1	14	yes	—
164	C₂²	C₂²×#6127	1	14	yes	—
165	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
166	C₄	—	2	8	no	—
167	C₄	—	13	32	no	—
168	C₄	—	26	16	no	—
169	C₄	C₂³×Dic₂₆	1	16	yes	direct product factor
170	C₄	C₂³×C₄×D₁₃	1	16	yes	—
171	C₂	C₂²×C₄×Dic₂₆	1	28	yes	direct product factor
172	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
173	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
174	C₂	C₂³×#6127	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle, prime radical	C₂⁵×C₁₃	C₂³
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂³×C₄×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₂₆	C₂	direct (N × H)
12. C₂³×Dic₂₆	C₄	direct (N × H)
18. C₂×C₄×Dic₂₆	C₂²	direct (N × H)
22. C₂³×C₄×C₁₃	Q₈	semidirect
30. C₂²×Dic₂₆	C₂×C₄	direct (N × H)
36. C₄×Dic₂₆	C₂³	direct (N × H)
44. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
52. C₂×Dic₂₆	C₂²×C₄	direct (N × H)
64. C₂³×C₁₃	C₄×Q₈	semidirect
71. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
79. Dic₂₆	C₂³×C₄	direct (N × H)
92. C₂²×C₁₃	C₂×C₄×Q₈	semidirect
99. C₄×C₁₃	C₂³×Q₈	semidirect
110. C₂³×C₄	Dic₂₆	direct (N × H)
120. C₂×C₁₃	C₂²×C₄×Q₈	semidirect
135. C₂²×C₄	C₂×Dic₂₆	direct (N × H)
144. C₁₃	C₂³×C₄×Q₈	semidirect
145. C₂³	C₄×Dic₂₆	direct (N × H)
156. C₂×C₄	C₂²×Dic₂₆	direct (N × H)
161. C₂²	C₂×C₄×Dic₂₆	direct (N × H)
169. C₄	C₂³×Dic₂₆	direct (N × H)
171. C₂	C₂²×C₄×Dic₂₆	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×Dic₂₆	2
C₂²×C₄×Dic₂₆	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×C₄×Dic₁₃	2
C₂³×C₄²×C₁₃	2
C₂³×C₄×Q₈	13

Subgroups — C23×C4×Dic26 #16769488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Dic₂₆ #16769488