Subgroups — C₂³×#88047 #16726563

Order 3328.

3614 conjugacy classes of subgroups.

The subgroup lattice has 160 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×#88047	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	—	2	1	no	—
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂⁴×Dic₁₃	—	2	1	no	—
8	C₂²×C₁₃×C₄⋊C₄	—	2	7	no	—
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	14	yes	—
10	C₂²×C₄⋊Dic₁₃	—	2	7	no	—
11	C₂²×C₄⋊Dic₁₃	C₂²	1	14	yes	—
12	C₂³×#1055	C₄	1	16	yes	complemented
13	C₂³×#1055	C₂²	1	14	yes	—
14	C₂×#88047	C₂²	1	112	yes	direct product factor
15	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	—	2	14	no	—
17	C₂³×C₄×C₁₃	—	4	8	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₁₃	D₄	1	7	yes	—
20	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
22	C₂³×Dic₁₃	—	2	14	no	—
23	C₂³×Dic₁₃	—	4	8	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	42	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂³	1	28	yes	—
26	C₂×C₄⋊Dic₁₃	—	2	42	no	—
27	C₂×C₄⋊Dic₁₃	C₂³	1	28	yes	—
28	C₂²×#1055	C₂×C₄	1	112	yes	complemented
29	C₂²×#1055	C₂³	1	28	yes	—
30	#88047	C₂³	1	64	yes	direct product factor
31	C₂³×#107	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
33	C₂⁴×C₁₃	D₈	1	8	yes	—
34	C₂⁴×C₁₃	Q₁₆	1	8	yes	—
35	C₂⁴×C₁₃	C₂×D₄	1	7	yes	—
36	C₂²×C₄×C₁₃	—	2	28	no	—
37	C₂²×C₄×C₁₃	—	4	56	no	—
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	56	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	28	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
41	C₂²×C₄×C₁₃	C₂×Q₈	1	28	yes	—
42	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
43	C₂²×Dic₁₃	—	2	28	no	—
44	C₂²×Dic₁₃	—	4	56	no	—
45	C₁₃×C₄⋊C₄	—	2	28	no	—
46	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
47	C₄⋊Dic₁₃	—	2	28	no	—
48	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
49	C₂×#1055	C₂²×C₄	1	112	yes	complemented
50	C₂×#1055	C₂⁴	1	8	yes	—
51	C₂⁴×C₈	—	13	1	no	—
52	C₂³×C₄⋊C₄	—	13	1	no	—
53	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
54	C₂²×#107	—	13	28	no	—
55	C₂³×C₁₃	#107	1	64	yes	complemented
56	C₂³×C₁₃	C₂×C₄⋊C₄	1	28	yes	—
57	C₂³×C₁₃	C₂×D₈	1	28	yes	—
58	C₂³×C₁₃	C₂×Q₁₆	1	28	yes	—
59	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
60	C₂×C₄×C₁₃	—	2	8	no	—
61	C₂×C₄×C₁₃	—	4	56	no	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	84	yes	—
63	C₂×C₄×C₁₃	C₂³×C₄	1	14	yes	—
64	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
65	C₂×C₄×C₁₃	C₂²×Q₈	1	14	yes	—
66	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
67	C₂×Dic₁₃	—	2	8	no	—
68	C₂×Dic₁₃	—	4	56	no	—
69	#1055	C₂³×C₄	1	16	yes	complemented
70	C₂⁴×C₄	—	2	1	no	—
71	C₂⁴×C₄	—	26	1	no	—
72	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
73	C₂³×C₈	—	13	16	no	—
74	C₂³×C₈	—	13	14	no	—
75	C₂²×C₄⋊C₄	—	2	7	no	—
76	C₂²×C₄⋊C₄	—	13	14	no	—
77	C₂²×C₄⋊C₄	—	26	7	no	—
78	C₂²×C₄⋊C₄	C₂×D₁₃	1	14	yes	—
79	C₂×#107	—	13	112	no	—
80	C₂²×C₁₃	C₂×#107	1	112	yes	complemented
81	C₂²×C₁₃	C₂²×C₄⋊C₄	1	14	yes	—
82	C₂²×C₁₃	C₂²×D₈	1	14	yes	—
83	C₂²×C₁₃	C₂²×Q₁₆	1	14	yes	—
84	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
85	C₄×C₁₃	—	4	8	no	—
86	C₄×C₁₃	C₂²×C₄⋊C₄	1	14	yes	—
87	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
88	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
89	Dic₁₃	—	4	8	no	—
90	C₂⁵	C₂²⋊D₁₃	1	1	yes	characteristic
91	C₂³×C₄	—	2	14	no	—
92	C₂³×C₄	—	4	8	no	—
93	C₂³×C₄	—	26	14	no	—
94	C₂³×C₄	—	52	8	no	—
95	C₂³×C₄	Dic₂₆	1	8	yes	—
96	C₂³×C₄	C₄×D₁₃	1	8	yes	—
97	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
98	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
99	C₂²×C₈	—	13	112	no	—
100	C₂²×C₈	—	13	28	no	—
101	C₂×C₄⋊C₄	—	2	42	no	—
102	C₂×C₄⋊C₄	—	13	28	no	—
103	C₂×C₄⋊C₄	—	26	42	no	—
104	C₂×C₄⋊C₄	C₂²×D₁₃	1	28	yes	—
105	#107	—	13	64	no	—
106	C₂×C₁₃	C₂²×#107	1	28	yes	complemented
107	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
109	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
110	C₂⁴	#6101	1	8	yes	—
111	C₂⁴	#6104	1	8	yes	—
112	C₂⁴	#6107	1	8	yes	—
113	C₂⁴	C₂×C₂²⋊D₁₃	1	7	yes	—
114	C₂²×C₄	—	2	28	no	—
115	C₂²×C₄	—	4	56	no	—
116	C₂²×C₄	—	26	28	no	—
117	C₂²×C₄	—	52	56	no	—
118	C₂²×C₄	#6101	1	56	yes	—
119	C₂²×C₄	C₂×Dic₂₆	1	28	yes	—
120	C₂²×C₄	C₂×C₄×D₁₃	1	28	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
122	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
123	C₂×C₈	—	13	112	no	—
124	C₂×C₈	—	13	8	no	—
125	C₄⋊C₄	—	2	28	no	—
126	C₄⋊C₄	—	13	8	no	—
127	C₄⋊C₄	—	26	28	no	—
128	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
129	C₁₃	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
130	C₂³	#88047	1	64	yes	direct product factor
131	C₂³	C₂×#6101	1	28	yes	—
132	C₂³	C₂×#6104	1	28	yes	—
133	C₂³	C₂×#6107	1	28	yes	—
134	C₂³	C₂²×C₂²⋊D₁₃	1	7	yes	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	4	56	no	—
137	C₂×C₄	—	26	8	no	—
138	C₂×C₄	—	52	56	no	—
139	C₂×C₄	C₂×#6101	1	84	yes	—
140	C₂×C₄	C₂²×Dic₂₆	1	14	yes	—
141	C₂×C₄	C₂²×C₄×D₁₃	1	14	yes	—
142	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
143	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
144	C₈	—	13	16	no	—
145	C₂²	C₂×#88047	1	112	yes	direct product factor
146	C₂²	C₂²×#6101	1	14	yes	—
147	C₂²	C₂²×#6104	1	14	yes	—
148	C₂²	C₂²×#6107	1	14	yes	—
149	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
150	C₄	—	4	8	no	—
151	C₄	—	52	8	no	—
152	C₄	C₂²×#6101	1	14	yes	—
153	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
154	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
155	C₂	C₂²×#88047	1	28	yes	direct product factor
156	C₂	C₂³×#6101	1	1	yes	characteristic
157	C₂	C₂³×#6104	1	1	yes	characteristic
158	C₂	C₂³×#6107	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₁₃	D₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#107
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#88047	C₂	direct (N × H)
12. C₂³×#1055	C₄	semidirect
14. C₂×#88047	C₂²	direct (N × H)
28. C₂²×#1055	C₂×C₄	semidirect
30. #88047	C₂³	direct (N × H)
49. C₂×#1055	C₂²×C₄	semidirect
55. C₂³×C₁₃	#107	semidirect
69. #1055	C₂³×C₄	semidirect
80. C₂²×C₁₃	C₂×#107	semidirect
106. C₂×C₁₃	C₂²×#107	semidirect
129. C₁₃	C₂³×#107	semidirect
130. C₂³	#88047	direct (N × H)
145. C₂²	C₂×#88047	direct (N × H)
155. C₂	C₂²×#88047	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×#88047	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×#107	13

Subgroups — C23×#88047 #16726563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88047 #16726563