Subgroups — C₂²×C₄×#11924031 #34029774

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924031	C₂	1	2	yes	maximal
2	C₂×C₄×#11924031	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₄	1	16	yes	direct product factor
5	C₂²×#11924031	C₂²	1	12	yes	—
6	C₄×#11924031	C₂²	1	16	yes	direct product factor
7	C₂³×#10596831	C₄	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₄	1	6	yes	—
10	C₂×C₄×#10596831	C₂²	1	6	yes	—
11	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	C₂×C₄	1	48	yes	direct product factor
13	C₂×#11924031	C₂³	1	8	yes	—
14	C₂²×#10596831	C₈	1	8	yes	—
15	C₂²×#10596831	C₂×C₄	1	8	yes	—
16	C₂²×#10596831	C₂×C₄	1	3	yes	—
17	C₂²×#10596831	C₂×C₄	1	6	yes	—
18	C₂²×#10596831	C₂³	1	3	yes	—
19	C₄×#10596831	C₂×C₄	1	12	yes	—
20	C₄×#10596831	C₂³	1	4	yes	—
21	C₂³×C₄×C₇₃	C₈	1	2	yes	—
22	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	C₈	1	6	yes	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
25	#11924031	C₂²×C₄	1	16	yes	direct product factor
26	C₂×#10596831	C₄²	1	12	yes	—
27	C₂×#10596831	C₂×C₈	1	24	yes	—
28	C₂×#10596831	C₂²×C₄	1	3	yes	—
29	C₂×#10596831	C₂²×C₄	1	12	yes	—
30	C₂×#10596831	C₂²×C₄	1	4	yes	—
31	C₂×#10596831	C₂⁴	1	1	yes	characteristic
32	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₁₆	1	16	yes	complemented
34	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
35	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
36	C₂²×C₄×C₇₃	C₂×C₈	1	12	yes	—
37	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
39	C₄²×C₇₃	C₂×C₈	1	12	yes	—
40	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
41	C₂³×C₇₃	C₄×C₈	1	4	yes	—
42	C₂³×C₇₃	C₂×C₁₆	1	8	yes	—
43	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
44	C₂×C₄×C₇₃	C₄×C₈	1	12	yes	—
45	C₂×C₄×C₇₃	C₂×C₁₆	1	48	yes	complemented
46	C₂×C₄×C₇₃	C₂×C₄²	1	6	yes	—
47	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	8	yes	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
50	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
51	#10596831	C₂×C₄²	1	6	yes	—
52	#10596831	C₂²×C₈	1	8	yes	—
53	#10596831	C₂³×C₄	1	2	yes	—
54	C₂²×C₇₃	C₄×C₁₆	1	16	yes	complemented
55	C₂²×C₇₃	C₂×C₄×C₈	1	6	yes	—
56	C₂²×C₇₃	C₂²×C₁₆	1	12	yes	—
57	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
58	C₄×C₇₃	C₂×C₄×C₈	1	6	yes	—
59	C₄×C₇₃	C₂²×C₁₆	1	16	yes	complemented
60	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
61	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₆	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	C₂×C₄×C₁₆	1	12	yes	complemented
64	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
65	C₂×C₇₃	C₂³×C₁₆	1	2	yes	—
66	C₂³×C₁₆	—	73	2	no	—
67	C₂²×C₄×C₈	—	73	1	no	—
68	C₂×C₄×C₁₆	—	73	12	no	—
69	C₇₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₈	—	73	1	no	—
71	C₂³×C₈	—	73	1	no	—
72	C₂²×C₄²	#64408	1	1	yes	characteristic
73	C₂²×C₁₆	—	73	16	no	—
74	C₂²×C₁₆	—	73	12	no	—
75	C₂×C₄×C₈	—	73	6	no	—
76	C₂×C₄×C₈	—	73	6	no	—
77	C₄×C₁₆	—	73	16	no	—
78	C₂³×C₄	#10596833	1	2	yes	—
79	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
80	C₂²×C₈	—	73	8	no	—
81	C₂²×C₈	—	73	8	no	—
82	C₂²×C₈	—	73	6	no	—
83	C₂²×C₈	—	73	3	no	—
84	C₂²×C₈	—	73	3	no	—
85	C₂×C₄²	#10596833	1	6	yes	—
86	C₂×C₄²	C₂×#64408	1	6	yes	—
87	C₂×C₁₆	—	73	8	no	—
88	C₂×C₁₆	—	73	48	no	—
89	C₄×C₈	—	73	12	no	—
90	C₄×C₈	—	73	4	no	—
91	C₂⁴	C₂×#10596833	1	1	yes	characteristic
92	C₂²×C₄	#11924031	1	16	yes	direct product factor
93	C₂²×C₄	C₄×#64408	1	4	yes	—
94	C₂²×C₄	C₂×#10596833	1	3	yes	—
95	C₂²×C₄	C₂×#10596833	1	12	yes	—
96	C₂²×C₄	#11924036	1	4	yes	—
97	C₂²×C₄	C₂²×#64408	1	3	yes	—
98	C₂×C₈	—	73	24	no	—
99	C₂×C₈	—	73	3	no	—
100	C₂×C₈	—	73	4	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	12	no	—
103	C₂×C₈	—	73	12	no	—
104	C₄²	C₂×#10596833	1	12	yes	—
105	C₄²	C₂²×#64408	1	4	yes	—
106	C₁₆	—	73	16	no	—
107	C₂³	C₄×#10596833	1	4	yes	—
108	C₂³	C₂×#11924031	1	8	yes	—
109	C₂³	C₂²×#10596833	1	3	yes	—
110	C₂×C₄	C₄×#10596833	1	12	yes	—
111	C₂×C₄	C₂×#11924031	1	48	yes	direct product factor
112	C₂×C₄	C₂×C₄×#64408	1	6	yes	—
113	C₂×C₄	C₂²×#10596833	1	8	yes	—
114	C₂×C₄	C₂²×#10596833	1	3	yes	—
115	C₂×C₄	C₂×#11924036	1	6	yes	—
116	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
117	C₈	—	73	2	no	—
118	C₈	—	73	6	no	—
119	C₈	—	73	8	no	—
120	C₂²	C₄×#11924031	1	16	yes	direct product factor
121	C₂²	C₂×C₄×#10596833	1	6	yes	—
122	C₂²	C₂²×#11924031	1	12	yes	—
123	C₂²	C₂³×#10596833	1	1	yes	characteristic
124	C₄	C₂×C₄×#10596833	1	6	yes	—
125	C₄	C₂²×#11924031	1	16	yes	direct product factor
126	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
127	C₄	C₂²×#11924036	1	1	yes	characteristic
128	C₂	C₂×C₄×#11924031	1	12	yes	direct product factor
129	C₂	C₂²×C₄×#10596833	1	1	yes	characteristic
130	C₂	C₂³×#11924031	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#64408
commutator, hypocenter	C₇₃	C₂²×C₄×C₁₆
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle, prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11924031	C₂	direct (N × H)
4. C₂²×#11924031	C₄	direct (N × H)
6. C₄×#11924031	C₂²	direct (N × H)
12. C₂×#11924031	C₂×C₄	direct (N × H)
25. #11924031	C₂²×C₄	direct (N × H)
33. C₂²×C₄×C₇₃	C₁₆	semidirect
45. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
54. C₂²×C₇₃	C₄×C₁₆	semidirect
59. C₄×C₇₃	C₂²×C₁₆	semidirect
63. C₂×C₇₃	C₂×C₄×C₁₆	semidirect
69. C₇₃	C₂²×C₄×C₁₆	semidirect
92. C₂²×C₄	#11924031	direct (N × H)
111. C₂×C₄	C₂×#11924031	direct (N × H)
120. C₂²	C₄×#11924031	direct (N × H)
125. C₄	C₂²×#11924031	direct (N × H)
128. C₂	C₂×C₄×#11924031	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924031	2
C₂×C₄×#11924031	2
C₂²×C₄×#10596831	2
C₂²×C₄×C₁₆	73

Subgroups — C22×C4×#11924031 #34029774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11924031 #34029774