Subgroups — C₂×#578×C₁₅₁ #34066915

Order 19328.

1408 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₁₅₁	C₂	1	2	yes	maximal, complemented
2	#578×C₁₅₁	C₂	1	16	yes	maximal, direct product factor
3	C₂×#137×C₁₅₁	C₂	1	6	yes	maximal, complemented
4	C₂×#135×C₁₅₁	C₂	1	3	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₅₁	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	2₋¹⁺⁴×C₁₅₁	—	2	8	no	—
8	2₋¹⁺⁴×C₁₅₁	C₂²	1	16	yes	complemented
9	C₂×#41×C₁₅₁	—	2	6	no	—
10	C₂×#41×C₁₅₁	C₂²	1	6	yes	—
11	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₅₁	—	2	2	no	—
13	C₂²×Q₈×C₁₅₁	C₂²	1	6	yes	—
14	#137×C₁₅₁	C₂²	1	48	yes	complemented
15	#135×C₁₅₁	C₂²	1	24	yes	complemented
16	C₂×Q₁₆×C₁₅₁	C₂²	1	3	yes	—
17	C₂×Q₁₆×C₁₅₁	C₂²	1	6	yes	—
18	C₂×Q₁₆×C₁₅₁	C₂²	1	24	yes	complemented
19	C₂×SD₁₆×C₁₅₁	C₂²	1	6	yes	—
20	C₂×D₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	#131×C₁₅₁	C₂²	1	8	yes	complemented
22	C₂×M₁₆×C₁₅₁	C₂²	1	3	yes	—
23	C₂²×C₈×C₁₅₁	C₂²	1	3	yes	—
24	#41×C₁₅₁	—	2	48	no	—
25	#41×C₁₅₁	—	2	12	no	—
26	#41×C₁₅₁	D₄	1	4	yes	—
27	#41×C₁₅₁	C₂³	1	24	yes	—
28	#41×C₁₅₁	C₂³	1	4	yes	—
29	C₂×Q₈×C₁₅₁	—	2	16	no	—
30	C₂×Q₈×C₁₅₁	—	2	6	no	—
31	C₂×Q₈×C₁₅₁	—	2	12	no	—
32	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₅₁	C₂³	1	6	yes	—
34	C₂×Q₈×C₁₅₁	C₂³	1	24	yes	—
35	C₂×D₄×C₁₅₁	—	2	6	no	—
36	C₂×D₄×C₁₅₁	D₄	1	3	yes	—
37	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₅₁	—	2	6	no	—
39	C₂²×C₄×C₁₅₁	—	2	6	no	—
40	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
41	Q₁₆×C₁₅₁	C₂³	1	12	yes	—
42	Q₁₆×C₁₅₁	C₂³	1	24	yes	—
43	SD₁₆×C₁₅₁	C₂³	1	24	yes	—
44	D₈×C₁₅₁	C₂³	1	4	yes	—
45	M₁₆×C₁₅₁	C₂³	1	12	yes	—
46	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₅₁	C₂³	1	3	yes	—
48	C₂×C₈×C₁₅₁	C₂³	1	12	yes	—
49	C₂³×C₁₅₁	—	2	2	no	—
50	C₂³×C₁₅₁	C₂×D₄	1	3	yes	—
51	Q₈×C₁₅₁	—	2	24	no	—
52	Q₈×C₁₅₁	—	2	6	no	—
53	Q₈×C₁₅₁	C₂×D₄	1	4	yes	—
54	Q₈×C₁₅₁	C₂⁴	1	12	yes	—
55	D₄×C₁₅₁	—	2	24	no	—
56	D₄×C₁₅₁	—	2	2	no	—
57	D₄×C₁₅₁	C₂×D₄	1	12	yes	—
58	D₄×C₁₅₁	C₂⁴	1	4	yes	—
59	C₂×C₄×C₁₅₁	—	2	24	no	—
60	C₂×C₄×C₁₅₁	—	2	6	no	—
61	C₂×C₄×C₁₅₁	—	2	24	no	—
62	C₂×C₄×C₁₅₁	C₂×D₄	1	6	yes	—
63	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
64	C₂×C₄×C₁₅₁	C₂⁴	1	6	yes	—
65	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
66	C₈×C₁₅₁	C₂⁴	1	2	yes	—
67	C₈×C₁₅₁	C₂⁴	1	6	yes	—
68	C₂²×C₁₅₁	—	2	6	no	—
69	C₂²×C₁₅₁	—	2	4	no	—
70	C₂²×C₁₅₁	—	4	4	no	—
71	C₂²×C₁₅₁	C₂²×D₄	1	6	yes	—
72	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₅₁	—	2	12	no	—
74	C₄×C₁₅₁	C₂²×D₄	1	6	yes	—
75	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₁₅₁	C₂⁵	1	1	yes	characteristic
77	C₂×C₁₅₁	—	2	6	no	—
78	C₂×C₁₅₁	—	4	4	no	—
79	C₂×C₁₅₁	#578	1	2	yes	direct product factor
80	C₂×C₁₅₁	C₂³×D₄	1	1	yes	characteristic
81	C₁₅₁	C₂×#578	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×#578	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×2₋¹⁺⁴	C₂×C₁₅₁	1	2	yes	complemented
84	#578	C₂×C₁₅₁	1	16	yes	direct product factor
85	C₂×#137	C₂×C₁₅₁	1	6	yes	complemented
86	C₂×#135	C₂×C₁₅₁	1	3	yes	complemented
87	C₂²×Q₁₆	C₂×C₁₅₁	1	3	yes	complemented
88	C₂×#131	C₂×C₁₅₁	1	1	yes	characteristic, complemented
89	2₋¹⁺⁴	—	2	8	no	—
90	2₋¹⁺⁴	C₂²×C₁₅₁	1	16	yes	complemented
91	C₂×#41	—	2	6	no	—
92	C₂×#41	C₂²×C₁₅₁	1	6	yes	—
93	C₂×#41	C₂²×C₁₅₁	1	1	yes	characteristic
94	C₂²×Q₈	—	2	2	no	—
95	C₂²×Q₈	C₂²×C₁₅₁	1	6	yes	—
96	#137	C₂²×C₁₅₁	1	48	yes	complemented
97	#135	C₂²×C₁₅₁	1	24	yes	complemented
98	C₂×Q₁₆	C₂²×C₁₅₁	1	3	yes	—
99	C₂×Q₁₆	C₂²×C₁₅₁	1	6	yes	—
100	C₂×Q₁₆	C₂²×C₁₅₁	1	24	yes	complemented
101	C₂×SD₁₆	C₂²×C₁₅₁	1	6	yes	—
102	C₂×D₈	C₂²×C₁₅₁	1	1	yes	characteristic
103	#131	C₂²×C₁₅₁	1	8	yes	complemented
104	C₂×M₁₆	C₂²×C₁₅₁	1	3	yes	—
105	C₂²×C₈	C₂²×C₁₅₁	1	3	yes	—
106	#41	—	2	48	no	—
107	#41	—	2	12	no	—
108	#41	D₄×C₁₅₁	1	4	yes	—
109	#41	C₂³×C₁₅₁	1	24	yes	—
110	#41	C₂³×C₁₅₁	1	4	yes	—
111	C₂×Q₈	—	2	16	no	—
112	C₂×Q₈	—	2	6	no	—
113	C₂×Q₈	—	2	12	no	—
114	C₂×Q₈	D₄×C₁₅₁	1	1	yes	characteristic
115	C₂×Q₈	C₂³×C₁₅₁	1	6	yes	—
116	C₂×Q₈	C₂³×C₁₅₁	1	24	yes	—
117	C₂×D₄	—	2	6	no	—
118	C₂×D₄	D₄×C₁₅₁	1	3	yes	—
119	C₂×D₄	C₂³×C₁₅₁	1	2	yes	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	—	2	6	no	—
122	C₂²×C₄	C₂³×C₁₅₁	1	3	yes	—
123	Q₁₆	C₂³×C₁₅₁	1	12	yes	—
124	Q₁₆	C₂³×C₁₅₁	1	24	yes	—
125	SD₁₆	C₂³×C₁₅₁	1	24	yes	—
126	D₈	C₂³×C₁₅₁	1	4	yes	—
127	M₁₆	C₂³×C₁₅₁	1	12	yes	—
128	C₂×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
129	C₂×C₈	C₂³×C₁₅₁	1	3	yes	—
130	C₂×C₈	C₂³×C₁₅₁	1	12	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	C₂×D₄×C₁₅₁	1	3	yes	—
133	Q₈	—	2	24	no	—
134	Q₈	—	2	6	no	—
135	Q₈	C₂×D₄×C₁₅₁	1	4	yes	—
136	Q₈	C₂⁴×C₁₅₁	1	12	yes	—
137	D₄	—	2	24	no	—
138	D₄	—	2	2	no	—
139	D₄	C₂×D₄×C₁₅₁	1	12	yes	—
140	D₄	C₂⁴×C₁₅₁	1	4	yes	—
141	C₂×C₄	—	2	6	no	—
142	C₂×C₄	—	2	24	no	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	C₂×D₄×C₁₅₁	1	3	yes	—
145	C₂×C₄	C₂×D₄×C₁₅₁	1	6	yes	—
146	C₂×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
147	C₂×C₄	C₂⁴×C₁₅₁	1	6	yes	—
148	C₈	C₂⁴×C₁₅₁	1	2	yes	—
149	C₈	C₂⁴×C₁₅₁	1	6	yes	—
150	C₂²	—	2	6	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	4	4	no	—
153	C₂²	C₂²×D₄×C₁₅₁	1	6	yes	—
154	C₂²	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
155	C₄	—	2	12	no	—
156	C₄	C₂²×D₄×C₁₅₁	1	6	yes	—
157	C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
158	C₄	C₂⁵×C₁₅₁	1	1	yes	characteristic
159	C₂	—	2	6	no	—
160	C₂	—	4	4	no	—
161	C₂	#578×C₁₅₁	1	2	yes	direct product factor
162	C₂	C₂³×D₄×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₁₅₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×#41×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₁₅₁	C₂	semidirect
2. #578×C₁₅₁	C₂	direct (N × H)
3. C₂×#137×C₁₅₁	C₂	semidirect
4. C₂×#135×C₁₅₁	C₂	semidirect
5. C₂²×Q₁₆×C₁₅₁	C₂	semidirect
6. C₂×#131×C₁₅₁	C₂	semidirect
8. 2₋¹⁺⁴×C₁₅₁	C₂²	semidirect
14. #137×C₁₅₁	C₂²	semidirect
15. #135×C₁₅₁	C₂²	semidirect
18. C₂×Q₁₆×C₁₅₁	C₂²	semidirect
21. #131×C₁₅₁	C₂²	semidirect
79. C₂×C₁₅₁	#578	direct (N × H)
81. C₁₅₁	C₂×#578	direct (N × H)
82. C₂×#578	C₁₅₁	direct (N × H)
83. C₂×2₋¹⁺⁴	C₂×C₁₅₁	semidirect
84. #578	C₂×C₁₅₁	direct (N × H)
85. C₂×#137	C₂×C₁₅₁	semidirect
86. C₂×#135	C₂×C₁₅₁	semidirect
87. C₂²×Q₁₆	C₂×C₁₅₁	semidirect
88. C₂×#131	C₂×C₁₅₁	semidirect
90. 2₋¹⁺⁴	C₂²×C₁₅₁	semidirect
96. #137	C₂²×C₁₅₁	semidirect
97. #135	C₂²×C₁₅₁	semidirect
100. C₂×Q₁₆	C₂²×C₁₅₁	semidirect
103. #131	C₂²×C₁₅₁	semidirect
161. C₂	#578×C₁₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₁₅₁	2
#578×C₁₅₁	2
C₂×#137×C₁₅₁	2
C₂×#135×C₁₅₁	2
C₂²×Q₁₆×C₁₅₁	2
C₂×#131×C₁₅₁	2
C₂×#578	151

Subgroups — C2×#578×C151 #34066915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#578×C₁₅₁ #34066915