Subgroups — C₂²×#7332 #72231868

Order 1024.

973 conjugacy classes of subgroups.

The subgroup lattice has 137 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1397	C₂	1	2	yes	maximal
2	C₂²×#1311	C₂	1	1	yes	maximal, characteristic
3	C₂×#7332	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₃₂	C₄	1	4	yes	—
5	C₂²×C₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈²	C₄	1	1	yes	characteristic
8	C₂×#1397	C₄	1	12	yes	—
9	C₂×#1397	C₂²	1	12	yes	—
10	C₂×#1311	C₄	1	6	yes	—
11	C₂×#1311	C₂²	1	6	yes	—
12	#7332	C₂²	1	16	yes	direct product factor
13	C₂³×C₁₆	C₈	1	2	yes	—
14	C₂³×C₁₆	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₈	C₈	1	2	yes	—
16	C₂²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃₂	—	2	16	no	—
18	C₂²×C₃₂	C₂×C₄	1	24	yes	—
19	C₂×C₄×C₁₆	C₈	1	6	yes	—
20	C₂×C₄×C₁₆	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₁₆	C₂×C₄	1	3	yes	—
22	C₂×C₄×C₁₆	C₂×C₄	1	6	yes	—
23	C₂×C₄×C₁₆	C₂³	1	3	yes	—
24	C₂×C₈²	C₈	1	6	yes	—
25	C₂×C₈²	C₂×C₄	1	6	yes	—
26	#1397	C₂×C₄	1	24	yes	—
27	#1397	C₂³	1	8	yes	—
28	#1311	C₂×C₄	1	12	yes	—
29	#1311	C₂³	1	4	yes	—
30	C₂³×C₈	—	2	2	no	—
31	C₂³×C₈	C₄²	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄²	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₁₆	C₁₆	1	16	yes	—
35	C₂²×C₁₆	C₄²	1	6	yes	—
36	C₂²×C₁₆	C₂×C₈	1	12	yes	—
37	C₂²×C₁₆	M₁₆	1	16	yes	—
38	C₂²×C₁₆	C₂²×C₄	1	6	yes	—
39	C₂×C₄×C₈	—	2	6	no	—
40	C₂×C₄×C₈	C₄²	1	3	yes	—
41	C₂×C₄×C₈	C₂×C₈	1	12	yes	—
42	C₂×C₄×C₈	C₂×C₈	1	3	yes	—
43	C₂×C₄×C₈	C₂×C₈	1	3	yes	—
44	C₂×C₄×C₈	C₂²×C₄	1	3	yes	—
45	C₂×C₃₂	—	2	48	no	—
46	C₂×C₃₂	C₂²×C₄	1	16	yes	—
47	C₄×C₁₆	C₄²	1	6	yes	—
48	C₄×C₁₆	C₂×C₈	1	12	yes	—
49	C₄×C₁₆	C₂²×C₄	1	6	yes	—
50	C₄×C₁₆	C₂²×C₄	1	3	yes	—
51	C₄×C₁₆	C₂²×C₄	1	4	yes	—
52	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
53	C₈²	C₂×C₈	1	12	yes	—
54	C₈²	C₂²×C₄	1	4	yes	—
55	C₂³×C₄	C₄×C₈	1	1	yes	characteristic
56	C₂³×C₄	M₃₂	1	2	yes	—
57	C₂²×C₈	—	2	12	no	—
58	C₂²×C₈	—	4	8	no	—
59	C₂²×C₈	C₄×C₈	1	3	yes	—
60	C₂²×C₈	C₄×C₈	1	3	yes	—
61	C₂²×C₈	#97	1	8	yes	—
62	C₂²×C₈	C₂×C₁₆	1	8	yes	—
63	C₂²×C₈	C₂×C₄²	1	3	yes	—
64	C₂²×C₈	C₂²×C₈	1	3	yes	—
65	C₂×C₄²	C₄×C₈	1	3	yes	—
66	C₂×C₄²	M₃₂	1	6	yes	—
67	C₂×C₄²	C₂²×C₈	1	3	yes	—
68	C₂×C₁₆	#97	1	24	yes	—
69	C₂×C₁₆	C₂×C₁₆	1	48	yes	—
70	C₂×C₁₆	C₂×C₄²	1	6	yes	—
71	C₂×C₁₆	C₂²×C₈	1	8	yes	—
72	C₂×C₁₆	C₂×M₁₆	1	24	yes	—
73	C₂×C₁₆	C₂³×C₄	1	2	yes	—
74	C₄×C₈	—	2	12	no	—
75	C₄×C₈	C₄×C₈	1	6	yes	—
76	C₄×C₈	C₂×C₄²	1	3	yes	—
77	C₄×C₈	C₂²×C₈	1	8	yes	—
78	C₄×C₈	C₂²×C₈	1	3	yes	—
79	C₄×C₈	C₂²×C₈	1	3	yes	—
80	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
81	C₃₂	—	2	16	no	—
82	C₂⁴	#346	1	1	yes	characteristic
83	C₂²×C₄	—	2	8	no	—
84	C₂²×C₄	#322	1	4	yes	—
85	C₂²×C₄	C₄×C₁₆	1	4	yes	—
86	C₂²×C₄	#346	1	6	yes	—
87	C₂²×C₄	#346	1	3	yes	—
88	C₂²×C₄	C₂×C₄×C₈	1	3	yes	—
89	C₂²×C₄	C₂×M₃₂	1	6	yes	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	4	24	no	—
92	C₂×C₈	#322	1	12	yes	—
93	C₂×C₈	C₄×C₁₆	1	12	yes	—
94	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
95	C₂×C₈	C₂×C₄×C₈	1	3	yes	—
96	C₂×C₈	C₂×#97	1	12	yes	—
97	C₂×C₈	C₂²×C₁₆	1	12	yes	—
98	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
99	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
100	C₄²	#346	1	6	yes	—
101	C₄²	C₂×C₄×C₈	1	3	yes	—
102	C₄²	C₂×M₃₂	1	6	yes	—
103	C₄²	C₂³×C₈	1	1	yes	characteristic
104	C₁₆	C₂×#97	1	12	yes	—
105	C₁₆	C₂²×C₁₆	1	16	yes	—
106	C₁₆	C₂²×M₁₆	1	4	yes	—
107	C₂³	#1312	1	4	yes	—
108	C₂³	#1398	1	8	yes	—
109	C₂³	C₂×#346	1	3	yes	—
110	C₂×C₄	—	2	12	no	—
111	C₂×C₄	—	2	12	no	—
112	C₂×C₄	#1312	1	12	yes	—
113	C₂×C₄	C₂×#322	1	6	yes	—
114	C₂×C₄	C₂×C₄×C₁₆	1	6	yes	—
115	C₂×C₄	C₂×#346	1	6	yes	—
116	C₂×C₄	C₂×#346	1	3	yes	—
117	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×M₃₂	1	2	yes	—
119	C₈	—	4	8	no	—
120	C₈	C₂×#322	1	6	yes	—
121	C₈	C₂×C₄×C₁₆	1	6	yes	—
122	C₈	C₂²×#97	1	2	yes	—
123	C₈	C₂³×C₁₆	1	2	yes	—
124	C₂²	#7332	1	16	yes	direct product factor
125	C₂²	C₂×#1312	1	6	yes	—
126	C₂²	C₂×#1398	1	12	yes	—
127	C₂²	C₂²×#346	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	6	no	—
130	C₄	C₂×#1312	1	6	yes	—
131	C₄	C₂²×#322	1	1	yes	characteristic
132	C₄	C₂²×C₄×C₁₆	1	1	yes	characteristic
133	C₂	C₂×#7332	1	12	yes	direct product factor
134	C₂	C₂²×#1312	1	1	yes	characteristic
135	C₂	C₂²×#1398	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₄²
commutator	C₄	C₂²×C₄×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle, prime radical	C₂⁴	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#7332	C₂	direct (N × H)
12. #7332	C₂²	direct (N × H)
124. C₂²	#7332	direct (N × H)
133. C₂	C₂×#7332	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×#1397	2
C₂²×#1311	2
C₂×#7332	2

Subgroups — C22×#7332 #72231868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#7332 #72231868