Subgroups — C₂²×#334×C₁₅₁ #50322708

Order 38656.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₁₅₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₅₁	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₁₅₁	—	2	2	no	—
5	C₂³×C₈×C₁₅₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₅₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₅₁	—	2	6	no	—
9	C₂×C₄⋊C₈×C₁₅₁	C₂²	1	12	yes	—
10	C₂×C₄×C₈×C₁₅₁	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₁₅₁	C₂²	1	6	yes	—
12	#334×C₁₅₁	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₅₁	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₅₁	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₅₁	—	2	4	no	—
17	C₂²×C₈×C₁₅₁	—	2	12	no	—
18	C₂²×C₈×C₁₅₁	—	4	8	no	—
19	C₂²×C₈×C₁₅₁	C₈	1	8	yes	complemented
20	C₂²×C₈×C₁₅₁	C₂×C₄	1	6	yes	—
21	C₂²×C₈×C₁₅₁	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₁₅₁	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₁₅₁	D₄	1	3	yes	—
24	C₂×C₄²×C₁₅₁	Q₈	1	3	yes	—
25	C₂×C₄²×C₁₅₁	C₂³	1	3	yes	—
26	C₄⋊C₈×C₁₅₁	—	2	12	no	—
27	C₄⋊C₈×C₁₅₁	C₂³	1	8	yes	—
28	C₄×C₈×C₁₅₁	C₂×C₄	1	12	yes	—
29	C₄×C₈×C₁₅₁	C₂³	1	4	yes	—
30	C₂⁴×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₁	—	2	4	no	—
32	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	3	yes	—
33	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₁₅₁	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₁₅₁	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₁₅₁	M₁₆	1	4	yes	—
37	C₂²×C₄×C₁₅₁	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₁₅₁	C₂²×C₄	1	3	yes	—
39	C₂²×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₅₁	C₂×Q₈	1	3	yes	—
41	C₂×C₈×C₁₅₁	—	2	12	no	—
42	C₂×C₈×C₁₅₁	—	2	8	no	—
43	C₂×C₈×C₁₅₁	—	4	24	no	—
44	C₂×C₈×C₁₅₁	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₁₅₁	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₁₅₁	C₂²×C₄	1	4	yes	—
47	C₄²×C₁₅₁	C₄⋊C₄	1	6	yes	—
48	C₄²×C₁₅₁	C₂²×C₄	1	3	yes	—
49	C₄²×C₁₅₁	C₂×D₄	1	3	yes	—
50	C₄²×C₁₅₁	C₂×Q₈	1	3	yes	—
51	C₄²×C₁₅₁	C₂⁴	1	1	yes	characteristic
52	C₂³×C₁₅₁	C₄⋊C₈	1	4	yes	—
53	C₂³×C₁₅₁	#106	1	4	yes	—
54	C₂³×C₁₅₁	#108	1	4	yes	—
55	C₂³×C₁₅₁	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₁₅₁	—	2	6	no	—
57	C₂×C₄×C₁₅₁	—	2	6	no	—
58	C₂×C₄×C₁₅₁	C₄⋊C₈	1	12	yes	—
59	C₂×C₄×C₁₅₁	#106	1	12	yes	—
60	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	3	yes	—
61	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₁₅₁	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₁₅₁	C₂²×C₈	1	6	yes	—
64	C₂×C₄×C₁₅₁	C₂×M₁₆	1	6	yes	—
65	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	12	yes	—
66	C₂×C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
69	C₈×C₁₅₁	—	2	4	no	—
70	C₈×C₁₅₁	—	4	8	no	—
71	C₈×C₁₅₁	C₂²×C₈	1	8	yes	complemented
72	C₂²×C₁₅₁	#334	1	16	yes	direct product factor
73	C₂²×C₁₅₁	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²×C₁₅₁	C₂×#106	1	6	yes	—
75	C₂²×C₁₅₁	C₂×#108	1	6	yes	—
76	C₂²×C₁₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₅₁	—	2	3	no	—
78	C₄×C₁₅₁	—	2	1	no	—
79	C₄×C₁₅₁	C₂×C₄⋊C₈	1	6	yes	—
80	C₄×C₁₅₁	C₂×#106	1	6	yes	—
81	C₄×C₁₅₁	C₂³×C₈	1	1	yes	characteristic
82	C₄×C₁₅₁	C₂²×M₁₆	1	1	yes	characteristic
83	C₄×C₁₅₁	C₂²×SD₁₆	1	2	yes	—
84	C₂×C₁₅₁	C₂×#334	1	12	yes	direct product factor
85	C₂×C₁₅₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₁₅₁	C₂²×#106	1	1	yes	characteristic
87	C₂×C₁₅₁	C₂²×#108	1	1	yes	characteristic
88	C₂²×#334	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₅₁	C₂²×#334	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₄⋊C₈	C₂×C₁₅₁	1	2	yes	—
91	C₂²×C₄×C₈	C₂×C₁₅₁	1	1	yes	characteristic
92	C₂×#334	C₂×C₁₅₁	1	12	yes	direct product factor
93	C₂³×C₈	—	2	2	no	—
94	C₂³×C₈	C₄×C₁₅₁	1	1	yes	characteristic
95	C₂³×C₈	C₄×C₁₅₁	1	1	yes	characteristic
96	C₂²×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	2	6	no	—
98	C₂×C₄⋊C₈	C₂²×C₁₅₁	1	12	yes	—
99	C₂×C₄×C₈	C₄×C₁₅₁	1	6	yes	—
100	C₂×C₄×C₈	C₂²×C₁₅₁	1	6	yes	—
101	#334	C₂²×C₁₅₁	1	16	yes	direct product factor
102	C₂³×C₄	C₂×C₄×C₁₅₁	1	1	yes	characteristic
103	C₂³×C₄	D₄×C₁₅₁	1	1	yes	characteristic
104	C₂³×C₄	Q₈×C₁₅₁	1	1	yes	characteristic
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	12	no	—
107	C₂²×C₈	—	4	8	no	—
108	C₂²×C₈	C₈×C₁₅₁	1	8	yes	complemented
109	C₂²×C₈	C₂×C₄×C₁₅₁	1	6	yes	—
110	C₂²×C₈	C₂×C₄×C₁₅₁	1	6	yes	—
111	C₂×C₄²	C₂×C₄×C₁₅₁	1	3	yes	—
112	C₂×C₄²	D₄×C₁₅₁	1	3	yes	—
113	C₂×C₄²	Q₈×C₁₅₁	1	3	yes	—
114	C₂×C₄²	C₂³×C₁₅₁	1	3	yes	—
115	C₄⋊C₈	—	2	12	no	—
116	C₄⋊C₈	C₂³×C₁₅₁	1	8	yes	—
117	C₄×C₈	C₂×C₄×C₁₅₁	1	12	yes	—
118	C₄×C₈	C₂³×C₁₅₁	1	4	yes	—
119	C₂⁴	C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	3	yes	—
122	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	3	yes	—
123	C₂²×C₄	C₄⋊C₄×C₁₅₁	1	3	yes	—
124	C₂²×C₄	C₂×C₈×C₁₅₁	1	4	yes	—
125	C₂²×C₄	M₁₆×C₁₅₁	1	4	yes	—
126	C₂²×C₄	SD₁₆×C₁₅₁	1	8	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₅₁	1	3	yes	—
128	C₂²×C₄	C₂×D₄×C₁₅₁	1	3	yes	—
129	C₂²×C₄	C₂×Q₈×C₁₅₁	1	3	yes	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	24	no	—
133	C₂×C₈	C₂×C₈×C₁₅₁	1	24	yes	complemented
134	C₂×C₈	C₂²×C₄×C₁₅₁	1	4	yes	—
135	C₂×C₈	C₂²×C₄×C₁₅₁	1	4	yes	—
136	C₄²	C₄⋊C₄×C₁₅₁	1	6	yes	—
137	C₄²	C₂²×C₄×C₁₅₁	1	3	yes	—
138	C₄²	C₂×D₄×C₁₅₁	1	3	yes	—
139	C₄²	C₂×Q₈×C₁₅₁	1	3	yes	—
140	C₄²	C₂⁴×C₁₅₁	1	1	yes	characteristic
141	C₂³	C₄⋊C₈×C₁₅₁	1	4	yes	—
142	C₂³	#106×C₁₅₁	1	4	yes	—
143	C₂³	#108×C₁₅₁	1	4	yes	—
144	C₂³	C₂×C₄⋊C₄×C₁₅₁	1	3	yes	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	C₄⋊C₈×C₁₅₁	1	12	yes	—
148	C₂×C₄	#106×C₁₅₁	1	12	yes	—
149	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	3	yes	—
150	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	3	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₁₅₁	1	3	yes	—
152	C₂×C₄	C₂²×C₈×C₁₅₁	1	6	yes	—
153	C₂×C₄	C₂×M₁₆×C₁₅₁	1	6	yes	—
154	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	12	yes	—
155	C₂×C₄	C₂³×C₄×C₁₅₁	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
157	C₂×C₄	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	4	8	no	—
160	C₈	C₂²×C₈×C₁₅₁	1	8	yes	complemented
161	C₂²	#334×C₁₅₁	1	16	yes	direct product factor
162	C₂²	C₂×C₄⋊C₈×C₁₅₁	1	6	yes	—
163	C₂²	C₂×#106×C₁₅₁	1	6	yes	—
164	C₂²	C₂×#108×C₁₅₁	1	6	yes	—
165	C₂²	C₂²×C₄⋊C₄×C₁₅₁	1	1	yes	characteristic
166	C₄	—	2	3	no	—
167	C₄	—	2	1	no	—
168	C₄	C₂×C₄⋊C₈×C₁₅₁	1	6	yes	—
169	C₄	C₂×#106×C₁₅₁	1	6	yes	—
170	C₄	C₂³×C₈×C₁₅₁	1	1	yes	characteristic
171	C₄	C₂²×M₁₆×C₁₅₁	1	1	yes	characteristic
172	C₄	C₂²×SD₁₆×C₁₅₁	1	2	yes	—
173	C₂	C₂×#334×C₁₅₁	1	12	yes	direct product factor
174	C₂	C₂²×C₄⋊C₈×C₁₅₁	1	1	yes	characteristic
175	C₂	C₂²×#106×C₁₅₁	1	1	yes	characteristic
176	C₂	C₂²×#108×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₅₁	D₄
commutator	C₄	C₂³×C₈×C₁₅₁
Frattini, cosocle	C₄²	C₂⁴×C₁₅₁
socle, prime radical	C₂⁴×C₁₅₁	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₅₁ ◁ C₂²×C₄²×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#334×C₁₅₁	C₂	direct (N × H)
12. #334×C₁₅₁	C₂²	direct (N × H)
19. C₂²×C₈×C₁₅₁	C₈	semidirect
44. C₂×C₈×C₁₅₁	C₂×C₈	semidirect
71. C₈×C₁₅₁	C₂²×C₈	semidirect
72. C₂²×C₁₅₁	#334	direct (N × H)
84. C₂×C₁₅₁	C₂×#334	direct (N × H)
88. C₂²×#334	C₁₅₁	direct (N × H)
89. C₁₅₁	C₂²×#334	direct (N × H)
92. C₂×#334	C₂×C₁₅₁	direct (N × H)
101. #334	C₂²×C₁₅₁	direct (N × H)
108. C₂²×C₈	C₈×C₁₅₁	semidirect
133. C₂×C₈	C₂×C₈×C₁₅₁	semidirect
160. C₈	C₂²×C₈×C₁₅₁	semidirect
161. C₂²	#334×C₁₅₁	direct (N × H)
173. C₂	C₂×#334×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
C₂²×C₄⋊C₈×C₁₅₁	2
C₂²×C₄×C₈×C₁₅₁	2
C₂×#334×C₁₅₁	2
C₂²×#334	151

Subgroups — C22×#334×C151 #50322708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#334×C₁₅₁ #50322708