Subgroups — C₈×#88043 #15752272

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114582	C₂	1	1	yes	maximal, characteristic
2	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718046	C₂	1	2	yes	maximal
4	C₂×C₈×#1055	C₂	1	2	yes	maximal
5	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88043	C₄	1	4	yes	—
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	2	yes	—
13	#11718116	C₄	1	4	yes	—
14	#11718046	C₄	1	8	yes	complemented
15	C₈×#1055	C₄	1	8	yes	complemented
16	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	4	yes	—
23	C₂²×#1055	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₁₃	—	2	8	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	#88043	C₈	1	16	yes	direct product factor
28	C₄×#1055	C₂×C₄	1	4	yes	—
29	C₄×#1055	C₂×C₄	1	4	yes	—
30	C₄×#1055	C₂×C₄	1	4	yes	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	C₈×#97	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	4	no	—
38	C₂×C₈×C₁₃	C₄²	1	4	yes	—
39	C₂×C₈×C₁₃	C₄²	1	4	yes	—
40	C₂×C₈×C₁₃	M₁₆	1	8	yes	—
41	C₄²×C₁₃	—	2	4	no	—
42	C₄²×C₁₃	—	2	4	no	—
43	C₄²×C₁₃	C₂×C₈	1	8	yes	—
44	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
45	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×#1055	—	2	8	no	—
47	C₂×#1055	C₄²	1	8	yes	—
48	C₂×#1055	C₄²	1	8	yes	—
49	C₂×#1055	C₂×C₈	1	16	yes	—
50	#1744	—	13	1	no	—
51	C₄×#97	—	13	1	no	—
52	C₄²×C₈	D₁₃	1	1	yes	characteristic
53	C₂×#322	—	13	2	no	—
54	C₂×C₈²	—	13	2	no	—
55	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
59	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
63	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
64	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
65	C₂×C₄×C₁₃	#131	1	4	yes	—
66	C₈×C₁₃	—	2	4	no	—
67	C₈×C₁₃	#97	1	8	yes	complemented
68	#1055	—	2	8	no	—
69	#1055	C₄×C₈	1	16	yes	complemented
70	C₂×#97	—	13	4	no	—
71	C₂×C₄×C₈	—	13	2	no	—
72	C₂×C₄×C₈	—	13	2	no	—
73	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
76	C₄³	C₂×D₁₃	1	1	yes	characteristic
77	#336	—	13	4	no	—
78	#322	—	13	8	no	—
79	C₈²	—	13	8	no	—
80	C₂²×C₁₃	C₄³	1	1	yes	characteristic
81	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
82	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
83	C₂²×C₁₃	#405	1	2	yes	—
84	C₄×C₁₃	—	2	8	no	—
85	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
86	C₄×C₁₃	C₂×#97	1	2	yes	—
87	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
88	C₄×C₁₃	#405	1	4	yes	—
89	C₂²×C₈	—	13	4	no	—
90	C₂²×C₈	—	13	4	no	—
91	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
92	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
93	C₂×C₄²	C₄×D₁₃	1	2	yes	—
94	C₂×C₄²	C₄×D₁₃	1	2	yes	—
95	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
96	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
97	#97	—	13	16	no	—
98	C₄×C₈	—	2	8	no	—
99	C₄×C₈	—	13	4	no	—
100	C₄×C₈	—	13	4	no	—
101	C₄×C₈	—	13	4	no	—
102	C₄×C₈	—	13	4	no	—
103	C₄×C₈	C₄×D₁₃	1	4	yes	—
104	C₄×C₈	C₄×D₁₃	1	4	yes	—
105	C₂×C₁₃	C₈×M₁₆	1	4	yes	—
106	C₂×C₁₃	C₄²×C₈	1	1	yes	characteristic
107	C₂×C₁₃	C₄×#97	1	1	yes	characteristic
108	C₂×C₁₃	#1726	1	1	yes	characteristic
109	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
110	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	13	8	no	—
115	C₂×C₈	—	13	16	no	—
116	C₂×C₈	—	13	8	no	—
117	C₂×C₈	—	26	8	no	—
118	C₂×C₈	#6099	1	8	yes	—
119	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
120	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
121	C₄²	—	2	4	no	—
122	C₄²	—	2	4	no	—
123	C₄²	C₈×D₁₃	1	8	yes	—
124	C₄²	C₂×C₄×D₁₃	1	2	yes	—
125	C₄²	C₂×C₄×D₁₃	1	2	yes	—
126	C₁₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
127	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₈×Dic₁₃	1	8	yes	—
131	C₂×C₄	C₄²×D₁₃	1	2	yes	—
132	C₂×C₄	C₄²×D₁₃	1	2	yes	—
133	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
134	C₂×C₄	#88155	1	4	yes	—
135	C₂×C₄	C₂×#6099	1	4	yes	—
136	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	13	16	no	—
140	C₈	—	26	8	no	—
141	C₈	#88043	1	8	yes	direct product factor
142	C₂²	C₄×#6099	1	2	yes	—
143	C₂²	C₄²×Dic₁₃	1	1	yes	characteristic
144	C₂²	C₂×C₈×Dic₁₃	1	2	yes	—
145	C₂²	#11718762	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₄×C₈×D₁₃	1	4	yes	—
148	C₄	#11718350	1	4	yes	—
149	C₄	C₂×#88043	1	2	yes	—
150	C₄	C₄×#6099	1	2	yes	—
151	C₂	C₈×#6099	1	4	yes	—
152	C₂	C₄×C₈×Dic₁₃	1	1	yes	characteristic
153	C₂	C₄×#88043	1	1	yes	characteristic
154	C₂	#13114522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	C₈×#97
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
14. #11718046	C₄	semidirect
15. C₈×#1055	C₄	semidirect
27. #88043	C₈	direct (N × H)
67. C₈×C₁₃	#97	semidirect
69. #1055	C₄×C₈	semidirect
126. C₁₃	C₈×#97	semidirect
141. C₈	#88043	direct (N × H)

Maximal subgroups

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

subgroup	index
#13114582	2
C₄×#88043	2
C₂×#11718046	2
C₂×C₈×#1055	2
C₄²×C₈×C₁₃	2
C₈×#97	13

Subgroups — C8×#88043 #15752272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#88043 #15752272