Subgroups — C₈×#88106 #16786580

Order 3328.

354 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	C₄×#88106	C₂	1	2	yes	maximal
2	#13129748	C₂	1	2	yes	maximal, complemented
3	C₈×#6092	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×#88106	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×#1055	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
9	#11718231	C₄	1	4	yes	—
10	C₄×#6092	C₂²	1	4	yes	—
11	C₈×#1055	C₄	1	1	yes	characteristic
12	C₈×#1055	C₄	1	1	yes	characteristic
13	C₈×#1055	C₂²	1	1	yes	characteristic
14	C₈×#1055	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₈	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	C₈	1	2	yes	—
18	C₂²×#1055	C₂×C₄	1	2	yes	—
19	#88106	C₈	1	16	yes	direct product factor
20	C₂×#6092	C₂×C₄	1	8	yes	—
21	C₄×C₈×C₁₃	C₈	1	2	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#1055	C₈	1	2	yes	—
25	C₄×#1055	C₂×C₄	1	1	yes	characteristic
26	C₄×#1055	C₂×C₄	1	2	yes	—
27	C₄×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	C₂×C₄	1	1	yes	characteristic
29	C₄×#1055	C₂³	1	1	yes	characteristic
30	C₈×M₃₂	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	—	2	4	no	—
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
37	C₄²×C₁₃	C₄²	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×#1055	—	2	4	no	—
42	C₂×#1055	C₄²	1	2	yes	—
43	C₂×#1055	C₂×C₈	1	2	yes	—
44	C₂×#1055	C₂×C₈	1	8	yes	—
45	C₂×#1055	C₂×C₈	1	2	yes	—
46	C₂×#1055	C₂²×C₄	1	2	yes	—
47	#6092	C₂×C₈	1	16	yes	complemented
48	C₄×M₃₂	—	13	2	no	—
49	C₂×C₈²	—	13	1	no	—
50	#1311	—	13	2	no	—
51	C₈×C₁₆	—	13	2	no	—
52	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
59	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	M₃₂	1	4	yes	complemented
64	#1055	—	2	2	no	—
65	#1055	C₄×C₈	1	4	yes	—
66	#1055	M₃₂	1	4	yes	complemented
67	#1055	C₂²×C₈	1	4	yes	—
68	C₂×M₃₂	—	13	4	no	—
69	C₂×C₄×C₈	—	13	1	no	—
70	C₂×C₄×C₈	—	13	1	no	—
71	C₂×C₄×C₈	#261	1	1	yes	characteristic
72	#346	—	13	4	no	—
73	C₄×C₁₆	—	13	4	no	—
74	C₈²	—	13	1	no	—
75	C₈²	—	13	1	no	—
76	C₈²	—	13	1	no	—
77	C₈²	—	13	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	C₈²	1	2	yes	—
81	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	C₈²	1	2	yes	—
85	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
86	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
87	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
88	C₂²×C₈	—	13	2	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	#1057	1	2	yes	—
91	C₂×C₄²	C₂×#261	1	1	yes	characteristic
92	M₃₂	—	13	16	no	—
93	C₂×C₁₆	—	13	8	no	—
94	C₄×C₈	—	13	2	no	—
95	C₄×C₈	—	13	2	no	—
96	C₄×C₈	—	13	1	no	—
97	C₄×C₈	—	13	1	no	—
98	C₄×C₈	—	13	1	no	—
99	C₄×C₈	—	13	1	no	—
100	C₄×C₈	#1057	1	2	yes	—
101	C₄×C₈	C₂×#261	1	1	yes	characteristic
102	C₄×C₈	C₂×#261	1	1	yes	characteristic
103	C₂×C₁₃	—	2	2	no	—
104	C₂×C₁₃	C₂×C₈²	1	1	yes	characteristic
105	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
106	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
107	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₄×#261	1	1	yes	characteristic
109	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	13	8	no	—
112	C₂×C₈	—	13	2	no	—
113	C₂×C₈	—	13	2	no	—
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	26	4	no	—
117	C₂×C₈	C₂×#1057	1	2	yes	—
118	C₂×C₈	C₂×#1057	1	2	yes	—
119	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
120	C₄²	C₄×#261	1	1	yes	characteristic
121	C₄²	C₂×#1057	1	1	yes	characteristic
122	C₄²	C₂²×#261	1	1	yes	characteristic
123	C₁₆	—	13	16	no	—
124	C₁₃	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₄×#1057	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×#261	1	4	yes	—
129	C₂×C₄	C₄×#1057	1	1	yes	characteristic
130	C₂×C₄	C₄×#1057	1	1	yes	characteristic
131	C₂×C₄	C₄×#1057	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
134	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	13	4	no	—
137	C₈	—	13	4	no	—
138	C₈	—	13	4	no	—
139	C₈	—	26	2	no	—
140	C₈	#88106	1	4	yes	direct product factor
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₈×#1057	1	2	yes	—
144	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	C₈×#1057	1	2	yes	—
148	C₄	D₁₃⋊M₃₂	1	2	yes	—
149	C₄	C₂×C₈×#261	1	2	yes	—
150	C₄	C₂×#88106	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	C₂×C₈×#1057	1	1	yes	characteristic
153	C₂	C₄×#88106	1	1	yes	characteristic
154	C₂	#13131042	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₈²
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₈×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129748	C₂	semidirect
3. C₈×#6092	C₂	semidirect
19. #88106	C₈	direct (N × H)
47. #6092	C₂×C₈	semidirect
63. C₈×C₁₃	M₃₂	semidirect
66. #1055	M₃₂	semidirect
124. C₁₃	C₈×M₃₂	semidirect
140. C₈	#88106	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#88106	2
#13129748	2
C₈×#6092	2
C₂×C₈×#1055	2
C₈×M₃₂	13

Subgroups — C8×#88106 #16786580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#88106 #16786580