Subgroups — C₂×#22624842 #34009639

Order 18688.

366 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989473	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
3	#22624842	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	C₄	1	2	yes	—
6	C₂×C₄×#10596833	C₄	1	2	yes	—
7	#17989473	C₄	1	2	yes	—
8	#17989473	C₂²	1	2	yes	—
9	C₈×#10596833	C₄	1	2	yes	—
10	C₈×#10596833	C₂²	1	2	yes	—
11	C₂²×#11924003	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₇₃	C₄	1	2	yes	—
14	C₁₆×Dic₇₃	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596833	C₈	1	4	yes	—
17	C₂×#11924031	C₈	1	8	yes	complemented
18	C₂×#11924031	C₂×C₄	1	4	yes	—
19	C₄×#10596833	C₈	1	4	yes	—
20	C₄×#10596833	C₂×C₄	1	4	yes	—
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924003	C₈	1	4	yes	complemented
24	C₂×#11924003	C₂×C₄	1	2	yes	—
25	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₈	1	4	yes	complemented
31	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×#10596833	—	2	8	no	—
33	C₂×#10596833	C₂×C₈	1	8	yes	—
34	#11924031	C₂×C₈	1	8	yes	complemented
35	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
36	C₂×#10596831	C₄²	1	1	yes	characteristic
37	C₂×#10596831	C₂×C₈	1	4	yes	—
38	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
41	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
42	#11924003	C₂×C₈	1	4	yes	complemented
43	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
49	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
55	C₂×Dic₇₃	M₃₂	1	8	yes	—
56	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
57	#10596833	—	2	8	no	—
58	C₈×C₇₃	C₄×C₈	1	2	yes	—
59	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	#10596831	C₄×C₈	1	2	yes	—
62	#10596831	C₂²×C₈	1	2	yes	—
63	C₂²×C₇₃	C₈²	1	2	yes	—
64	C₂²×C₇₃	#346	1	4	yes	—
65	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₇₃	C₈²	1	2	yes	—
67	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
69	Dic₇₃	#346	1	4	yes	—
70	Dic₇₃	C₂×M₃₂	1	4	yes	—
71	C₂×#1311	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	#1311	1	4	yes	complemented
73	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
74	C₂×C₇₃	C₂×#346	1	2	yes	—
75	C₂×C₄×C₁₆	—	73	1	no	—
76	C₂×C₄×C₁₆	—	73	1	no	—
77	C₂×C₈²	—	73	1	no	—
78	#1311	—	73	4	no	—
79	C₇₃	C₂×#1311	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₁₆	—	73	2	no	—
81	C₂²×C₁₆	—	73	1	no	—
82	C₂²×C₁₆	#64408	1	1	yes	characteristic
83	C₂×C₄×C₈	—	73	2	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	C₄×C₁₆	—	73	2	no	—
86	C₄×C₁₆	—	73	2	no	—
87	C₄×C₁₆	—	73	2	no	—
88	C₄×C₁₆	—	73	2	no	—
89	C₈²	—	73	2	no	—
90	C₈²	—	73	2	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂²×C₈	—	73	4	no	—
93	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
94	C₂×C₄²	—	73	1	no	—
95	C₂×C₁₆	—	73	4	no	—
96	C₂×C₁₆	—	73	8	no	—
97	C₂×C₁₆	—	73	4	no	—
98	C₂×C₁₆	—	73	2	no	—
99	C₂×C₁₆	#10596833	1	4	yes	complemented
100	C₂×C₁₆	C₂×#64408	1	2	yes	—
101	C₄×C₈	—	73	4	no	—
102	C₄×C₈	—	73	4	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₂²×C₄	—	73	2	no	—
108	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
109	C₂×C₈	—	73	4	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	8	no	—
113	C₂×C₈	—	146	8	no	—
114	C₂×C₈	C₄×#64408	1	1	yes	characteristic
115	C₂×C₈	C₂×#10596833	1	2	yes	—
116	C₂×C₈	#11924036	1	2	yes	—
117	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
118	C₄²	—	73	2	no	—
119	C₄²	—	73	1	no	—
120	C₄²	—	73	1	no	—
121	C₁₆	—	73	8	no	—
122	C₁₆	—	73	4	no	—
123	C₁₆	C₂×#10596833	1	4	yes	complemented
124	C₂³	C₈×#64408	1	1	yes	characteristic
125	C₂×C₄	—	73	8	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	C₈×#64408	1	1	yes	characteristic
129	C₂×C₄	C₈×#64408	1	2	yes	—
130	C₂×C₄	C₄×#10596833	1	2	yes	—
131	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
132	C₈	—	73	2	no	—
133	C₈	—	73	2	no	—
134	C₈	—	146	8	no	—
135	C₈	C₄×#10596833	1	2	yes	—
136	C₈	C₂²×#10596833	1	1	yes	characteristic
137	C₈	C₂×#11924036	1	1	yes	characteristic
138	C₂²	C₈×#10596833	1	2	yes	—
139	C₂²	#17989474	1	4	yes	—
140	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
141	C₄	—	73	4	no	—
142	C₄	—	73	4	no	—
143	C₄	C₈×#10596833	1	2	yes	—
144	C₄	C₂×C₈×#64408	1	1	yes	characteristic
145	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
146	C₂	#22624842	1	4	yes	direct product factor
147	C₂	C₂×C₈×#10596833	1	1	yes	characteristic
148	C₂	C₂×#17989474	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₈²
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	C₂×#1311
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22624842	C₂	direct (N × H)
17. C₂×#11924031	C₈	semidirect
23. C₂×#11924003	C₈	semidirect
30. C₂×C₁₆×C₇₃	C₈	semidirect
34. #11924031	C₂×C₈	semidirect
42. #11924003	C₂×C₈	semidirect
48. C₁₆×C₇₃	C₂×C₈	semidirect
72. C₂×C₇₃	#1311	semidirect
79. C₇₃	C₂×#1311	semidirect
99. C₂×C₁₆	#10596833	semidirect
123. C₁₆	C₂×#10596833	semidirect
146. C₂	#22624842	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×#17989473	2
C₂×C₈×#10596833	2
#22624842	2
C₂×C₁₆×Dic₇₃	2
C₂×#1311	73

Subgroups — C2×#22624842 #34009639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624842 #34009639