Subgroups — C₄²×C₁₃×D₈ #15706836

Order 3328.

1350 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	3	yes	maximal
2	C₄²×D₄×C₁₃	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×#107	C₂	1	3	yes	maximal, complemented
4	C₄×C₁₃×#102	C₂	1	6	yes	maximal, complemented
5	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	6	yes	—
8	C₂²×C₄²×C₁₃	—	2	2	no	—
9	C₄×C₁₃×D₈	C₄	1	24	yes	direct product factor
10	C₂×C₁₃×#107	C₂²	1	3	yes	—
11	C₂×C₁₃×#102	C₂²	1	6	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
15	C₄×C₁₃×C₂²⋊C₄	—	2	6	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#340	C₂²	1	6	yes	—
18	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₁₃	—	2	6	no	—
20	C₂×C₁₃×D₈	C₂×C₄	1	12	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₄×D₄×C₁₃	—	2	12	no	—
24	C₄×D₄×C₁₃	C₂×C₄	1	24	yes	complemented
25	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
26	C₂×C₁₃×C₂²⋊C₄	—	2	6	no	—
27	C₂×C₄²×C₁₃	—	2	6	no	—
28	C₂×C₄²×C₁₃	—	4	8	no	—
29	C₂×C₄²×C₁₃	D₄	1	3	yes	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
32	C₁₃×#107	C₂×C₄	1	12	yes	complemented
33	C₁₃×#102	C₂×C₄	1	24	yes	complemented
34	C₄×C₈×C₁₃	—	2	6	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	complemented
36	C₁₃×#95	—	2	6	no	—
37	C₄²×D₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	—	2	6	no	—
40	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
41	C₂²×C₄×C₁₃	—	2	6	no	—
42	C₂²×C₄×C₁₃	—	2	24	no	—
43	C₂²×C₄×C₁₃	—	4	12	no	—
44	C₂²×C₄×C₁₃	—	4	12	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₁₃×D₈	C₄²	1	16	yes	direct product factor
49	C₂×C₈×C₁₃	—	2	6	no	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
52	C₁₃×C₄⋊C₄	—	2	6	no	—
53	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
54	C₁₃×C₂²⋊C₄	—	2	24	no	—
55	C₄²×C₁₃	—	2	6	no	—
56	C₄²×C₁₃	—	4	12	no	—
57	C₄²×C₁₃	D₈	1	4	yes	direct product factor
58	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
59	C₄²×C₁₃	#41	1	6	yes	—
60	C₂×C₄×D₈	C₂×C₁₃	1	3	yes	—
61	C₄²×D₄	C₂×C₁₃	1	2	yes	complemented
62	C₄×#107	C₂×C₁₃	1	3	yes	complemented
63	C₄×#102	C₂×C₁₃	1	6	yes	complemented
64	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
65	C₂³×C₁₃	—	2	12	no	—
66	C₂³×C₁₃	—	4	2	no	—
67	C₂³×C₁₃	—	4	6	no	—
68	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
69	D₄×C₁₃	—	2	12	no	—
70	D₄×C₁₃	C₂×C₄²	1	8	yes	complemented
71	C₂×C₄×C₁₃	—	2	6	no	—
72	C₂×C₄×C₁₃	—	2	12	no	—
73	C₂×C₄×C₁₃	—	4	6	no	—
74	C₂×C₄×C₁₃	—	4	6	no	—
75	C₂×C₄×C₁₃	—	4	48	no	—
76	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
77	C₂×C₄×C₁₃	C₄×D₄	1	6	yes	—
78	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
79	C₂×C₄×C₁₃	#135	1	6	yes	—
80	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
81	C₂×C₄×C₁₃	C₂×#41	1	3	yes	—
82	C₈×C₁₃	—	2	6	no	—
83	C₈×C₁₃	C₂×C₄²	1	4	yes	complemented
84	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
85	C₂×C₄×D₄	C₂²×C₁₃	1	6	yes	—
86	C₂²×C₄²	—	2	2	no	—
87	C₄×D₈	C₄×C₁₃	1	24	yes	direct product factor
88	C₂×#107	C₂²×C₁₃	1	3	yes	—
89	C₂×#102	C₂²×C₁₃	1	6	yes	—
90	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
91	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
92	C₄×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
93	C₄×C₂²⋊C₄	—	2	6	no	—
94	C₄³	C₂²×C₁₃	1	1	yes	characteristic
95	#340	C₂²×C₁₃	1	6	yes	—
96	C₂²×C₁₃	—	2	8	no	—
97	C₂²×C₁₃	—	4	12	no	—
98	C₂²×C₁₃	—	4	12	no	—
99	C₂²×C₁₃	C₂×C₄×D₄	1	3	yes	—
100	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×#135	1	3	yes	—
102	C₄×C₁₃	—	2	6	no	—
103	C₄×C₁₃	—	4	12	no	—
104	C₄×C₁₃	C₄×D₈	1	12	yes	direct product factor
105	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₁₃	C₄×#41	1	3	yes	—
107	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
108	C₂³×C₄	—	2	6	no	—
109	C₂×D₈	C₂×C₄×C₁₃	1	12	yes	—
110	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
111	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
112	C₄×D₄	—	2	12	no	—
113	C₄×D₄	C₂×C₄×C₁₃	1	24	yes	complemented
114	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
115	C₂×C₂²⋊C₄	—	2	6	no	—
116	C₂×C₄²	—	2	6	no	—
117	C₂×C₄²	—	4	8	no	—
118	C₂×C₄²	D₄×C₁₃	1	3	yes	—
119	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
121	#107	C₂×C₄×C₁₃	1	12	yes	complemented
122	#102	C₂×C₄×C₁₃	1	24	yes	complemented
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	C₂×C₄×C₁₃	1	12	yes	complemented
125	#95	—	2	6	no	—
126	C₂×C₁₃	—	4	8	no	—
127	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
128	C₂×C₁₃	C₂×C₄×D₈	1	3	yes	—
129	C₂×C₁₃	C₄×#135	1	3	yes	—
130	C₂⁴	—	2	2	no	—
131	C₂×D₄	—	2	6	no	—
132	C₂×D₄	C₂²×C₄×C₁₃	1	12	yes	—
133	C₂²×C₄	—	2	6	no	—
134	C₂²×C₄	—	2	24	no	—
135	C₂²×C₄	—	4	12	no	—
136	C₂²×C₄	—	4	12	no	—
137	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
138	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
139	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
140	D₈	C₄²×C₁₃	1	16	yes	direct product factor
141	C₂×C₈	—	2	6	no	—
142	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
143	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
144	C₄⋊C₄	—	2	6	no	—
145	C₄⋊C₄	C₂²×C₄×C₁₃	1	12	yes	—
146	C₂²⋊C₄	—	2	24	no	—
147	C₄²	—	2	6	no	—
148	C₄²	—	4	12	no	—
149	C₄²	C₁₃×D₈	1	4	yes	direct product factor
150	C₄²	C₂²×C₄×C₁₃	1	6	yes	—
151	C₄²	C₁₃×#41	1	6	yes	—
152	C₁₃	C₄²×D₈	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³	—	2	12	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	4	6	no	—
156	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
157	D₄	—	2	12	no	—
158	D₄	C₂×C₄²×C₁₃	1	8	yes	complemented
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	4	6	no	—
162	C₂×C₄	—	4	6	no	—
163	C₂×C₄	—	4	48	no	—
164	C₂×C₄	C₄×D₄×C₁₃	1	6	yes	—
165	C₂×C₄	C₄×D₄×C₁₃	1	6	yes	—
166	C₂×C₄	C₂×C₁₃×D₈	1	6	yes	—
167	C₂×C₄	C₁₃×#135	1	6	yes	—
168	C₂×C₄	C₂³×C₄×C₁₃	1	3	yes	—
169	C₂×C₄	C₂×C₁₃×#41	1	3	yes	—
170	C₈	—	2	6	no	—
171	C₈	C₂×C₄²×C₁₃	1	4	yes	complemented
172	C₂²	—	2	8	no	—
173	C₂²	—	4	12	no	—
174	C₂²	—	4	12	no	—
175	C₂²	C₂×C₄×D₄×C₁₃	1	3	yes	—
176	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
177	C₂²	C₂×C₁₃×#135	1	3	yes	—
178	C₄	—	2	6	no	—
179	C₄	—	4	12	no	—
180	C₄	C₄×C₁₃×D₈	1	12	yes	direct product factor
181	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
182	C₄	C₄×C₁₃×#41	1	3	yes	—
183	C₂	—	4	8	no	—
184	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
185	C₂	C₂×C₄×C₁₃×D₈	1	3	yes	—
186	C₂	C₄×C₁₃×#135	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃	D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×D₄×C₁₃	C₂	semidirect
3. C₄×C₁₃×#107	C₂	semidirect
4. C₄×C₁₃×#102	C₂	semidirect
5. C₄²×C₈×C₁₃	C₂	semidirect
9. C₄×C₁₃×D₈	C₄	direct (N × H)
24. C₄×D₄×C₁₃	C₂×C₄	semidirect
32. C₁₃×#107	C₂×C₄	semidirect
33. C₁₃×#102	C₂×C₄	semidirect
35. C₄×C₈×C₁₃	C₂×C₄	semidirect
37. C₄²×D₈	C₁₃	direct (N × H)
48. C₁₃×D₈	C₄²	direct (N × H)
57. C₄²×C₁₃	D₈	direct (N × H)
61. C₄²×D₄	C₂×C₁₃	semidirect
62. C₄×#107	C₂×C₁₃	semidirect
63. C₄×#102	C₂×C₁₃	semidirect
64. C₄²×C₈	C₂×C₁₃	semidirect
70. D₄×C₁₃	C₂×C₄²	semidirect
83. C₈×C₁₃	C₂×C₄²	semidirect
87. C₄×D₈	C₄×C₁₃	direct (N × H)
104. C₄×C₁₃	C₄×D₈	direct (N × H)
113. C₄×D₄	C₂×C₄×C₁₃	semidirect
121. #107	C₂×C₄×C₁₃	semidirect
122. #102	C₂×C₄×C₁₃	semidirect
124. C₄×C₈	C₂×C₄×C₁₃	semidirect
140. D₈	C₄²×C₁₃	direct (N × H)
149. C₄²	C₁₃×D₈	direct (N × H)
152. C₁₃	C₄²×D₈	direct (N × H)
158. D₄	C₂×C₄²×C₁₃	semidirect
171. C₈	C₂×C₄²×C₁₃	semidirect
180. C₄	C₄×C₁₃×D₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₃×D₈	2
C₄²×D₄×C₁₃	2
C₄×C₁₃×#107	2
C₄×C₁₃×#102	2
C₄²×C₈×C₁₃	2
C₄²×D₈	13

Subgroups — C42×C13×D8 #15706836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₁₃×D₈ #15706836