Subgroups — C₂³×#88054 #16726526

Order 3328.

3870 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88054	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₁₃	C₄	1	1	yes	characteristic
7	C₂⁴×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	—	2	8	no	—
9	C₂³×C₈×C₁₃	C₂²	1	14	yes	—
10	C₂²×C₄×Dic₁₃	C₄	1	14	yes	—
11	C₂²×C₄×Dic₁₃	C₂²	1	14	yes	—
12	C₂³×#1055	—	2	8	no	—
13	C₂³×#1055	C₂²	1	14	yes	—
14	C₂×#88054	C₂²	1	112	yes	direct product factor
15	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
17	C₂³×C₄×C₁₃	D₄	1	8	yes	—
18	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
20	C₂³×Dic₁₃	—	2	8	no	—
21	C₂³×Dic₁₃	C₈	1	16	yes	complemented
22	C₂³×Dic₁₃	C₂×C₄	1	14	yes	—
23	C₂³×Dic₁₃	C₂×C₄	1	14	yes	—
24	C₂²×C₈×C₁₃	—	2	56	no	—
25	C₂²×C₈×C₁₃	C₂³	1	28	yes	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	84	yes	—
27	C₂×C₄×Dic₁₃	C₂³	1	28	yes	—
28	C₂²×#1055	—	2	56	no	—
29	C₂²×#1055	C₂³	1	28	yes	—
30	#88054	C₂³	1	64	yes	direct product factor
31	C₂³×C₄⋊C₈	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
33	C₂⁴×C₁₃	C₂×C₈	1	8	yes	—
34	C₂⁴×C₁₃	M₁₆	1	8	yes	—
35	C₂⁴×C₁₃	C₂²×C₄	1	7	yes	—
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	56	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	28	yes	—
39	C₂²×C₄×C₁₃	C₂×Q₈	1	28	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
41	C₂²×Dic₁₃	—	2	56	no	—
42	C₂²×Dic₁₃	C₂×C₈	1	112	yes	complemented
43	C₂²×Dic₁₃	C₂²×C₄	1	28	yes	—
44	C₂²×Dic₁₃	C₂²×C₄	1	28	yes	—
45	C₂×C₈×C₁₃	—	2	56	no	—
46	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
47	C₄×Dic₁₃	C₂²×C₄	1	56	yes	—
48	C₄×Dic₁₃	C₂⁴	1	8	yes	—
49	C₂×#1055	—	2	56	no	—
50	C₂×#1055	C₂⁴	1	8	yes	—
51	C₂⁴×C₈	—	13	1	no	—
52	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
53	C₂³×C₄²	—	13	1	no	—
54	C₂²×C₄⋊C₈	—	13	28	no	—
55	C₂³×C₁₃	C₄⋊C₈	1	64	yes	complemented
56	C₂³×C₁₃	C₂×C₄⋊C₄	1	28	yes	—
57	C₂³×C₁₃	C₂²×C₈	1	28	yes	—
58	C₂³×C₁₃	C₂×M₁₆	1	28	yes	—
59	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	84	yes	—
61	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
62	C₂×C₄×C₁₃	C₂²×D₄	1	14	yes	—
63	C₂×C₄×C₁₃	C₂²×Q₈	1	14	yes	—
64	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
65	C₂×Dic₁₃	—	2	56	no	—
66	C₂×Dic₁₃	C₂²×C₈	1	112	yes	complemented
67	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
68	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
69	C₈×C₁₃	—	2	8	no	—
70	#1055	—	2	8	no	—
71	C₂⁴×C₄	—	13	1	no	—
72	C₂⁴×C₄	—	13	1	no	—
73	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
74	C₂³×C₈	—	2	8	no	—
75	C₂³×C₈	—	13	14	no	—
76	C₂³×C₈	—	26	8	no	—
77	C₂³×C₈	C₂×D₁₃	1	14	yes	—
78	C₂²×C₄²	—	13	14	no	—
79	C₂²×C₄²	—	13	14	no	—
80	C₂×C₄⋊C₈	—	13	112	no	—
81	C₂²×C₁₃	C₂×C₄⋊C₈	1	112	yes	complemented
82	C₂²×C₁₃	C₂²×C₄⋊C₄	1	14	yes	—
83	C₂²×C₁₃	C₂³×C₈	1	14	yes	—
84	C₂²×C₁₃	C₂²×M₁₆	1	14	yes	—
85	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
86	C₄×C₁₃	C₂²×C₄⋊C₄	1	14	yes	—
87	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
88	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
89	Dic₁₃	—	2	8	no	—
90	Dic₁₃	C₂³×C₈	1	16	yes	complemented
91	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
92	C₂³×C₄	—	13	16	no	—
93	C₂³×C₄	—	13	14	no	—
94	C₂³×C₄	—	13	14	no	—
95	C₂³×C₄	—	26	8	no	—
96	C₂³×C₄	Dic₂₆	1	8	yes	—
97	C₂³×C₄	C₄×D₁₃	1	7	yes	—
98	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
99	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
100	C₂²×C₈	—	2	56	no	—
101	C₂²×C₈	—	13	28	no	—
102	C₂²×C₈	—	26	56	no	—
103	C₂²×C₈	C₂²×D₁₃	1	28	yes	—
104	C₂×C₄²	—	13	84	no	—
105	C₂×C₄²	—	13	28	no	—
106	C₄⋊C₈	—	13	64	no	—
107	C₂×C₁₃	C₂²×C₄⋊C₈	1	28	yes	complemented
108	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
110	C₂×C₁₃	C₂³×M₁₆	1	1	yes	characteristic
111	C₂⁴	C₈×D₁₃	1	8	yes	—
112	C₂⁴	#6094	1	8	yes	—
113	C₂⁴	#6101	1	8	yes	—
114	C₂⁴	C₂×C₄×D₁₃	1	7	yes	—
115	C₂²×C₄	—	13	112	no	—
116	C₂²×C₄	—	13	28	no	—
117	C₂²×C₄	—	13	28	no	—
118	C₂²×C₄	—	26	56	no	—
119	C₂²×C₄	#6101	1	56	yes	—
120	C₂²×C₄	C₂×Dic₂₆	1	28	yes	—
121	C₂²×C₄	C₂×C₄×D₁₃	1	21	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₃	1	28	yes	—
123	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
124	C₂×C₈	—	2	56	no	—
125	C₂×C₈	—	13	8	no	—
126	C₂×C₈	—	26	56	no	—
127	C₂×C₈	C₂³×D₁₃	1	8	yes	—
128	C₄²	—	13	56	no	—
129	C₄²	—	13	8	no	—
130	C₁₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
131	C₂³	#88054	1	64	yes	direct product factor
132	C₂³	C₂×C₈×D₁₃	1	28	yes	—
133	C₂³	C₂×#6094	1	28	yes	—
134	C₂³	C₂×#6101	1	28	yes	—
135	C₂³	C₂²×C₄×D₁₃	1	7	yes	—
136	C₂×C₄	—	13	112	no	—
137	C₂×C₄	—	13	8	no	—
138	C₂×C₄	—	13	8	no	—
139	C₂×C₄	—	26	56	no	—
140	C₂×C₄	C₂×#6101	1	84	yes	—
141	C₂×C₄	C₂²×Dic₂₆	1	14	yes	—
142	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
143	C₂×C₄	C₂²×C₂²⋊D₁₃	1	14	yes	—
144	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
145	C₈	—	2	8	no	—
146	C₈	—	26	8	no	—
147	C₂²	C₂×#88054	1	112	yes	direct product factor
148	C₂²	C₂²×C₈×D₁₃	1	14	yes	—
149	C₂²	C₂²×#6094	1	14	yes	—
150	C₂²	C₂²×#6101	1	14	yes	—
151	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
152	C₄	—	13	16	no	—
153	C₄	—	26	8	no	—
154	C₄	C₂²×#6101	1	14	yes	—
155	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
156	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂	C₂²×#88054	1	28	yes	direct product factor
158	C₂	C₂³×#6101	1	1	yes	characteristic
159	C₂	C₂³×C₈×D₁₃	1	1	yes	characteristic
160	C₂	C₂³×#6094	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle, prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#88054	C₂	direct (N × H)
14. C₂×#88054	C₂²	direct (N × H)
21. C₂³×Dic₁₃	C₈	semidirect
30. #88054	C₂³	direct (N × H)
42. C₂²×Dic₁₃	C₂×C₈	semidirect
55. C₂³×C₁₃	C₄⋊C₈	semidirect
66. C₂×Dic₁₃	C₂²×C₈	semidirect
81. C₂²×C₁₃	C₂×C₄⋊C₈	semidirect
90. Dic₁₃	C₂³×C₈	semidirect
107. C₂×C₁₃	C₂²×C₄⋊C₈	semidirect
130. C₁₃	C₂³×C₄⋊C₈	semidirect
131. C₂³	#88054	direct (N × H)
147. C₂²	C₂×#88054	direct (N × H)
157. C₂	C₂²×#88054	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×#1055	2
C₂³×C₄×Dic₁₃	2
C₂²×#88054	2
C₂⁴×C₈×C₁₃	2
C₂³×C₄⋊C₈	13

Subgroups — C23×#88054 #16726526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88054 #16726526