Subgroups — C₂²×#17989645 #34032680

Order 18688.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989645	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#14556665	C₂	1	3	yes	maximal, complemented
3	C₂³×#11924031	C₂	1	3	yes	maximal, complemented
4	C₂²×#14556598	C₂	1	1	yes	maximal, characteristic
5	#17989645	C₂²	1	64	yes	direct product factor
6	C₂×#14556665	C₂²	1	36	yes	complemented
7	C₂²×#11924031	C₂²	1	1	yes	characteristic
8	C₂²×#11924031	C₂²	1	3	yes	—
9	C₂²×#11924031	C₂²	1	36	yes	complemented
10	C₂×#14556598	C₄	1	12	yes	—
11	C₂×#14556598	C₂²	1	12	yes	—
12	C₂²×#11924012	C₄	1	3	yes	—
13	C₂³×#10596831	C₂²	1	3	yes	—
14	C₂²×#41×C₇₃	C₄	1	1	yes	characteristic
15	#14556665	C₂³	1	48	yes	complemented
16	C₂×#11924031	C₂³	1	6	yes	—
17	C₂×#11924031	C₂³	1	18	yes	—
18	C₂×#11924031	C₂³	1	48	yes	complemented
19	#14556598	C₂×C₄	1	48	yes	—
20	#14556598	C₂³	1	16	yes	—
21	C₂×#11924012	C₂×C₄	1	36	yes	—
22	C₂²×#10596831	C₂×C₄	1	3	yes	—
23	C₂²×#10596831	C₂×C₄	1	18	yes	—
24	C₂²×#10596831	C₂³	1	1	yes	characteristic
25	C₂²×#10596831	C₂³	1	18	yes	—
26	C₂×#41×C₇₃	C₈	1	12	yes	—
27	C₂×#41×C₇₃	C₂×C₄	1	12	yes	—
28	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
29	C₂²×D₄×C₇₃	C₈	1	3	yes	—
30	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
31	#11924031	C₂⁴	1	4	yes	—
32	#11924031	C₂⁴	1	12	yes	—
33	#11924012	C₂²×C₄	1	48	yes	—
34	C₂×#10596831	C₂²×C₄	1	3	yes	—
35	C₂×#10596831	C₂²×C₄	1	18	yes	—
36	C₂×#10596831	C₂²×C₄	1	36	yes	—
37	C₂×#10596831	C₂⁴	1	3	yes	—
38	C₂×#10596831	C₂⁴	1	12	yes	—
39	C₂⁴×C₇₃	C₂×C₈	1	3	yes	—
40	#41×C₇₃	C₂×C₈	1	48	yes	—
41	#41×C₇₃	C₂²×C₄	1	16	yes	—
42	C₂×Q₈×C₇₃	C₂×C₈	1	12	yes	—
43	C₂×D₄×C₇₃	C₂×C₈	1	36	yes	—
44	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₇₃	C₂×C₈	1	18	yes	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	18	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₇₃	—	2	12	no	—
49	C₂³×C₇₃	C₂²×C₈	1	18	yes	—
50	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	Q₈×C₇₃	C₂²×C₈	1	16	yes	—
52	D₄×C₇₃	C₂²×C₈	1	48	yes	—
53	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₇₃	C₂²×C₈	1	18	yes	—
55	C₂×C₄×C₇₃	C₂²×C₈	1	36	yes	—
56	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
57	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
58	#10596831	C₂³×C₄	1	3	yes	—
59	#10596831	C₂³×C₄	1	12	yes	—
60	#10596831	C₂⁵	1	1	yes	characteristic
61	C₂²×C₇₃	—	2	36	no	—
62	C₂²×C₇₃	#504	1	4	yes	complemented
63	C₂²×C₇₃	C₂³×C₈	1	12	yes	—
64	C₂²×C₇₃	C₂³×C₈	1	3	yes	—
65	C₄×C₇₃	C₂³×C₈	1	3	yes	—
66	C₄×C₇₃	C₂³×C₈	1	12	yes	—
67	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
68	C₂²×#504	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	—	2	12	no	—
70	C₂×C₇₃	C₂×#504	1	6	yes	complemented
71	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
72	C₂²×#131	—	73	1	no	—
73	C₂×#504	—	73	24	no	—
74	C₂²×M₃₂	—	73	3	no	—
75	C₂³×C₁₆	—	73	3	no	—
76	C₇₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
77	C₂²×#41	#64408	1	1	yes	characteristic
78	C₂×#131	—	73	12	no	—
79	C₂×#131	—	73	12	no	—
80	C₂²×M₁₆	—	73	3	no	—
81	C₂³×C₈	—	73	3	no	—
82	#504	—	73	64	no	—
83	C₂×M₃₂	—	73	36	no	—
84	C₂²×C₁₆	—	73	1	no	—
85	C₂²×C₁₆	—	73	3	no	—
86	C₂²×C₁₆	—	73	36	no	—
87	C₂×#41	#10596833	1	12	yes	—
88	C₂×#41	C₂×#64408	1	12	yes	—
89	C₂²×Q₈	#10596833	1	1	yes	characteristic
90	C₂²×D₄	#10596833	1	3	yes	—
91	C₂³×C₄	C₂×#64408	1	3	yes	—
92	#131	—	73	48	no	—
93	#131	—	73	16	no	—
94	C₂×M₁₆	—	73	36	no	—
95	C₂²×C₈	—	73	18	no	—
96	C₂²×C₈	—	73	18	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	73	3	no	—
99	M₃₂	—	73	48	no	—
100	C₂×C₁₆	—	73	6	no	—
101	C₂×C₁₆	—	73	18	no	—
102	C₂×C₁₆	—	73	48	no	—
103	C₂⁴	C₂×#10596833	1	3	yes	—
104	#41	C₂×#10596833	1	48	yes	—
105	#41	C₂²×#64408	1	16	yes	—
106	C₂×Q₈	C₂×#10596833	1	12	yes	—
107	C₂×D₄	C₂×#10596833	1	36	yes	—
108	C₂²×C₄	C₂×#10596833	1	3	yes	—
109	C₂²×C₄	C₂×#10596833	1	18	yes	—
110	C₂²×C₄	C₂²×#64408	1	18	yes	—
111	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
112	M₁₆	—	73	48	no	—
113	C₂×C₈	—	73	3	no	—
114	C₂×C₈	—	73	18	no	—
115	C₂×C₈	—	73	36	no	—
116	C₂×C₈	—	73	3	no	—
117	C₂×C₈	—	73	12	no	—
118	C₁₆	—	73	4	no	—
119	C₁₆	—	73	12	no	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×#10596833	1	18	yes	—
122	C₂³	C₂²×#10596833	1	1	yes	characteristic
123	Q₈	C₂²×#10596833	1	16	yes	—
124	D₄	C₂²×#10596833	1	48	yes	—
125	C₂×C₄	C₂²×#10596833	1	3	yes	—
126	C₂×C₄	C₂²×#10596833	1	18	yes	—
127	C₂×C₄	C₂²×#10596833	1	36	yes	—
128	C₂×C₄	C₂³×#64408	1	3	yes	—
129	C₂×C₄	C₂³×#64408	1	12	yes	—
130	C₈	—	73	1	no	—
131	C₈	—	73	3	no	—
132	C₈	—	73	12	no	—
133	C₂²	—	2	36	no	—
134	C₂²	#17989645	1	4	yes	direct product factor
135	C₂²	C₂³×#10596833	1	12	yes	—
136	C₂²	C₂³×#10596833	1	3	yes	—
137	C₄	C₂³×#10596833	1	3	yes	—
138	C₄	C₂³×#10596833	1	12	yes	—
139	C₄	C₂⁴×#64408	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×#17989645	1	6	yes	direct product factor
142	C₂	C₂⁴×#10596833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#64408
Fitting, prime radical	C₂²×#41×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
cosocle	#10596831	C₂⁵
hypercenter	C₂²×#41	#64408
hypocenter	C₇₃	C₂²×#504
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#17989645	C₂	direct (N × H)
2. C₂²×#14556665	C₂	semidirect
3. C₂³×#11924031	C₂	semidirect
5. #17989645	C₂²	direct (N × H)
6. C₂×#14556665	C₂²	semidirect
9. C₂²×#11924031	C₂²	semidirect
15. #14556665	C₂³	semidirect
18. C₂×#11924031	C₂³	semidirect
62. C₂²×C₇₃	#504	semidirect
70. C₂×C₇₃	C₂×#504	semidirect
76. C₇₃	C₂²×#504	semidirect
134. C₂²	#17989645	direct (N × H)
141. C₂	C₂×#17989645	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×#17989645	2
C₂²×#14556665	2
C₂³×#11924031	2
C₂²×#14556598	2
C₂²×#504	73

Subgroups — C22×#17989645 #34032680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989645 #34032680