Subgroups — #72234335

Order 2048.

896 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	#72234311	C₂	1	4	yes	maximal
2	C₄×#7052	C₂	1	1	yes	maximal, characteristic
3	C₄×#7329	C₂	1	2	yes	maximal
4	C₄²×C₃₂	C₄	1	4	yes	—
5	C₂×#7329	C₄	1	4	yes	—
6	C₂×#7329	C₂²	1	4	yes	—
7	C₄×C₈×C₁₆	C₄	1	2	yes	—
8	C₄×C₈×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#7052	C₄	1	2	yes	—
10	C₂×#7052	C₂²	1	2	yes	—
11	C₂×C₄×C₃₂	C₈	1	8	yes	—
12	C₂×C₄×C₃₂	C₈	1	8	yes	—
13	C₂×C₄×C₃₂	C₂×C₄	1	8	yes	—
14	C₄²×C₁₆	C₈	1	4	yes	—
15	C₄²×C₁₆	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₁₆	C₈	1	4	yes	—
17	C₂×C₈×C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₆	C₂³	1	1	yes	characteristic
21	C₄×C₈²	C₂×C₄	1	1	yes	characteristic
22	#7329	—	2	8	no	—
23	#7329	—	2	8	no	—
24	#7052	—	2	4	no	—
25	#7052	—	2	4	no	—
26	C₂²×C₃₂	C₂×C₈	1	16	yes	—
27	C₂×C₄×C₁₆	—	2	8	no	—
28	C₂×C₄×C₁₆	C₄²	1	2	yes	—
29	C₂×C₄×C₁₆	C₂×C₈	1	8	yes	—
30	C₂×C₄×C₁₆	C₂×C₈	1	4	yes	—
31	C₂×C₄×C₁₆	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₆	C₂²×C₄	1	2	yes	—
33	C₄²×C₈	C₄²	1	1	yes	characteristic
34	C₄²×C₈	C₂×C₈	1	2	yes	—
35	C₂×C₈²	C₄²	1	1	yes	characteristic
36	C₂×C₈²	C₂×C₈	1	2	yes	—
37	C₂×C₈²	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₂	—	2	16	no	—
39	C₄×C₃₂	—	2	8	no	—
40	C₄×C₃₂	—	2	8	no	—
41	C₄×C₃₂	C₁₆	1	32	yes	complemented
42	C₈×C₁₆	—	2	4	no	—
43	C₈×C₁₆	—	2	4	no	—
44	C₈×C₁₆	C₁₆	1	16	yes	—
45	C₈×C₁₆	M₁₆	1	8	yes	—
46	C₈×C₁₆	M₁₆	1	4	yes	—
47	C₈×C₁₆	#41	1	4	yes	—
48	C₂²×C₁₆	—	2	8	no	—
49	C₂²×C₁₆	C₄×C₈	1	4	yes	—
50	C₂²×C₁₆	C₂²×C₈	1	4	yes	—
51	C₂×C₄×C₈	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₈	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₈	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₈	C₄×C₈	1	2	yes	—
55	C₂×C₄×C₈	M₃₂	1	8	yes	—
56	C₂×C₄×C₈	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂²×C₈	1	2	yes	—
58	C₄³	C₄×C₈	1	1	yes	characteristic
59	C₂×C₃₂	—	2	16	no	—
60	C₂×C₃₂	—	2	16	no	—
61	C₂×C₃₂	C₂×C₁₆	1	32	yes	—
62	C₄×C₁₆	—	2	8	no	—
63	C₄×C₁₆	—	4	16	no	—
64	C₄×C₁₆	#97	1	4	yes	—
65	C₄×C₁₆	#97	1	4	yes	—
66	C₄×C₁₆	C₂×C₁₆	1	16	yes	—
67	C₄×C₁₆	C₂×C₁₆	1	16	yes	—
68	C₄×C₁₆	M₃₂	1	16	yes	—
69	C₄×C₁₆	#117	1	4	yes	—
70	C₄×C₁₆	C₂×M₁₆	1	4	yes	—
71	C₈²	#97	1	4	yes	—
72	C₈²	C₂×C₁₆	1	8	yes	—
73	C₈²	#117	1	2	yes	—
74	C₈²	C₂×M₁₆	1	2	yes	—
75	C₂²×C₈	C₈²	1	1	yes	characteristic
76	C₂²×C₈	C₈²	1	1	yes	characteristic
77	C₂²×C₈	#346	1	4	yes	—
78	C₂²×C₈	C₂×C₄×C₈	1	2	yes	—
79	C₂²×C₈	C₂×M₃₂	1	4	yes	—
80	C₂×C₄²	C₈²	1	2	yes	—
81	C₂×C₄²	#346	1	4	yes	—
82	C₂×C₄²	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂×C₁₆	—	2	16	no	—
84	C₂×C₁₆	—	4	16	no	—
85	C₂×C₁₆	#322	1	8	yes	—
86	C₂×C₁₆	C₄×C₁₆	1	8	yes	—
87	C₂×C₁₆	#346	1	8	yes	—
88	C₂×C₁₆	#431	1	8	yes	—
89	C₂×C₁₆	C₂²×C₁₆	1	8	yes	—
90	C₂×C₁₆	C₂×M₃₂	1	8	yes	—
91	C₄×C₈	—	2	16	no	—
92	C₄×C₈	#322	1	4	yes	—
93	C₄×C₈	C₄×C₁₆	1	8	yes	—
94	C₄×C₈	C₄×C₁₆	1	4	yes	—
95	C₄×C₈	C₄×C₁₆	1	4	yes	—
96	C₄×C₈	#346	1	8	yes	—
97	C₄×C₈	#376	1	1	yes	characteristic
98	C₄×C₈	C₂×#97	1	2	yes	—
99	C₄×C₈	C₂×#97	1	1	yes	characteristic
100	C₄×C₈	#431	1	4	yes	—
101	C₄×C₈	C₂²×C₁₆	1	4	yes	—
102	C₃₂	—	2	32	no	—
103	C₂²×C₄	#1311	1	4	yes	—
104	C₂²×C₄	C₂×C₈²	1	1	yes	characteristic
105	C₂²×C₄	C₂×#346	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	2	8	no	—
108	C₂×C₈	C₈×C₁₆	1	2	yes	—
109	C₂×C₈	C₈×C₁₆	1	4	yes	—
110	C₂×C₈	C₈×C₁₆	1	2	yes	—
111	C₂×C₈	#1311	1	4	yes	—
112	C₂×C₈	#1312	1	8	yes	—
113	C₂×C₈	C₂×#322	1	2	yes	—
114	C₂×C₈	#1744	1	2	yes	—
115	C₂×C₈	C₂×C₄×C₁₆	1	4	yes	—
116	C₂×C₈	C₂×#346	1	4	yes	—
117	C₂×C₈	#2162	1	8	yes	—
118	C₄²	C₈×C₁₆	1	4	yes	—
119	C₄²	#1311	1	4	yes	—
120	C₄²	#1398	1	16	yes	—
121	C₄²	C₂×#322	1	1	yes	characteristic
122	C₄²	#1744	1	1	yes	characteristic
123	C₄²	C₂×C₄×C₁₆	1	2	yes	—
124	C₁₆	—	4	16	no	—
125	C₁₆	#1312	1	16	yes	—
126	C₁₆	#2162	1	16	yes	—
127	C₂³	C₂×#1311	1	1	yes	characteristic
128	C₂×C₄	#7052	1	8	yes	—
129	C₂×C₄	#7330	1	16	yes	—
130	C₂×C₄	#8131	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×C₁₆	1	1	yes	characteristic
132	C₂×C₄	C₂×C₈×C₁₆	1	2	yes	—
133	C₂×C₄	C₂×#1311	1	2	yes	—
134	C₂×C₄	C₂×#1312	1	2	yes	—
135	C₂×C₄	#12590	1	2	yes	—
136	C₂×C₄	C₂×#1398	1	8	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	2	8	no	—
139	C₈	C₁₆²	1	4	yes	—
140	C₈	#7052	1	4	yes	—
141	C₈	C₂×#1312	1	4	yes	—
142	C₈	#12590	1	4	yes	—
143	C₂²	C₂×#7052	1	2	yes	—
144	C₂²	#128203	1	1	yes	characteristic
145	C₂²	C₂×#7330	1	4	yes	—
146	C₄	#93666	1	16	yes	—
147	C₄	C₂×C₁₆²	1	1	yes	characteristic
148	C₄	C₂×#7052	1	1	yes	characteristic
149	C₄	#128203	1	2	yes	—
150	C₄	#149502	1	4	yes	—
151	C₄	#151879	1	4	yes	—
152	C₂	#72234237	1	2	yes	—
153	C₂	C₂×#93666	1	4	yes	—
154	C₂	#72234260	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₈	C₄²
commutator	C₄	C₂×C₁₆²
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle, prime radical	C₂³	C₂×#1311
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
41. C₄×C₃₂	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72234311	2
C₄×#7052	2
C₄×#7329	2