Subgroups — C₂²×#17989508 #34030055

Order 18688.

862 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	C₂³×#11924031	C₂	1	2	yes	maximal
2	C₂×#17989508	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	—	2	8	no	—
5	C₂²×#11924031	C₂²	1	12	yes	—
6	#17989508	C₂²	1	16	yes	direct product factor
7	C₂³×#10596831	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	6	yes	—
10	C₂×C₈×Dic₇₃	C₂²	1	6	yes	—
11	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	—	2	24	no	—
13	C₂×#11924031	C₂³	1	8	yes	—
14	C₂³×Dic₇₃	C₈	1	1	yes	characteristic
15	C₂³×Dic₇₃	C₈	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	3	yes	—
17	C₂²×#10596831	D₄	1	4	yes	—
18	C₂²×#10596831	Q₈	1	4	yes	—
19	C₂²×#10596831	C₂³	1	3	yes	—
20	C₂×C₄×Dic₇₃	C₈	1	6	yes	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
22	C₈×Dic₇₃	C₂×C₄	1	12	yes	—
23	C₈×Dic₇₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	—	2	4	no	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	6	yes	—
27	#11924031	—	2	8	no	—
28	C₂²×Dic₇₃	—	2	4	no	—
29	C₂²×Dic₇₃	C₁₆	1	8	yes	complemented
30	C₂²×Dic₇₃	C₂×C₈	1	6	yes	—
31	C₂²×Dic₇₃	C₂×C₈	1	6	yes	—
32	C₂×#10596831	C₄⋊C₄	1	12	yes	—
33	C₂×#10596831	C₂²×C₄	1	3	yes	—
34	C₂×#10596831	C₂×D₄	1	6	yes	—
35	C₂×#10596831	C₂×Q₈	1	6	yes	—
36	C₂×#10596831	C₂⁴	1	1	yes	characteristic
37	C₄×Dic₇₃	C₂×C₈	1	12	yes	—
38	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
39	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₇₃	M₁₆	1	4	yes	—
43	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₇₃	—	2	12	no	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
46	C₂³×C₇₃	C₄⋊C₈	1	4	yes	—
47	C₂³×C₇₃	C₂×C₁₆	1	4	yes	—
48	C₂³×C₇₃	M₃₂	1	4	yes	—
49	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₇₃	C₄⋊C₈	1	12	yes	—
51	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
52	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
53	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
54	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	12	no	—
56	C₂×Dic₇₃	C₂×C₁₆	1	24	yes	complemented
57	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
58	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
59	C₈×C₇₃	—	2	4	no	—
60	#10596831	C₂×C₄⋊C₄	1	6	yes	—
61	#10596831	C₂²×D₄	1	1	yes	characteristic
62	#10596831	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₇₃	C₄⋊C₁₆	1	16	yes	complemented
64	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
65	C₂²×C₇₃	C₂²×C₁₆	1	6	yes	—
66	C₂²×C₇₃	C₂×M₃₂	1	6	yes	—
67	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
69	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
71	Dic₇₃	—	2	4	no	—
72	Dic₇₃	C₂²×C₁₆	1	8	yes	complemented
73	C₂²×C₄⋊C₁₆	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	C₂×C₄⋊C₁₆	1	12	yes	complemented
75	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
77	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
78	C₂³×C₁₆	—	73	2	no	—
79	C₂²×C₄×C₈	—	73	1	no	—
80	C₂×C₄⋊C₁₆	—	73	12	no	—
81	C₇₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	73	1	no	—
83	C₂³×C₈	#64408	1	1	yes	characteristic
84	C₂²×C₄²	—	73	1	no	—
85	C₂²×C₁₆	—	73	12	no	—
86	C₂²×C₁₆	—	146	8	no	—
87	C₂×C₄×C₈	—	73	6	no	—
88	C₂×C₄×C₈	—	73	6	no	—
89	C₄⋊C₁₆	—	73	16	no	—
90	C₂³×C₄	—	73	1	no	—
91	C₂³×C₄	—	73	1	no	—
92	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	—	73	4	no	—
95	C₂²×C₈	—	73	4	no	—
96	C₂²×C₈	—	73	3	no	—
97	C₂²×C₈	—	73	3	no	—
98	C₂²×C₈	C₂×#64408	1	6	yes	—
99	C₂×C₄²	—	73	6	no	—
100	C₂×C₄²	—	73	6	no	—
101	C₂×C₁₆	—	73	8	no	—
102	C₂×C₁₆	—	146	24	no	—
103	C₄×C₈	—	73	12	no	—
104	C₄×C₈	—	73	4	no	—
105	C₂⁴	#11924036	1	1	yes	characteristic
106	C₂²×C₄	—	73	6	no	—
107	C₂²×C₄	—	73	8	no	—
108	C₂²×C₄	—	73	6	no	—
109	C₂²×C₄	—	146	4	no	—
110	C₂²×C₄	C₄⋊#64408	1	4	yes	—
111	C₂²×C₄	#11924036	1	3	yes	—
112	C₂²×C₄	#11924037	1	4	yes	—
113	C₂²×C₄	C₂²×#64408	1	3	yes	—
114	C₂×C₈	—	2	12	no	—
115	C₂×C₈	—	73	3	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	12	no	—
118	C₂×C₈	—	73	6	no	—
119	C₂×C₈	—	73	6	no	—
120	C₂×C₈	C₂²×#64408	1	4	yes	—
121	C₄²	—	73	12	no	—
122	C₄²	—	73	4	no	—
123	C₁₆	—	146	8	no	—
124	C₂³	#14556658	1	4	yes	—
125	C₂³	#14556664	1	4	yes	—
126	C₂³	#14556666	1	4	yes	—
127	C₂³	C₂×#11924036	1	3	yes	—
128	C₂×C₄	—	73	24	no	—
129	C₂×C₄	—	73	4	no	—
130	C₂×C₄	—	73	4	no	—
131	C₂×C₄	—	146	12	no	—
132	C₂×C₄	#14556658	1	12	yes	—
133	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
134	C₂×C₄	C₂×#11924036	1	3	yes	—
135	C₂×C₄	C₂×#11924037	1	6	yes	—
136	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	73	1	no	—
139	C₈	—	73	6	no	—
140	C₈	—	73	1	no	—
141	C₂²	#17989508	1	16	yes	direct product factor
142	C₂²	C₂×#14556658	1	6	yes	—
143	C₂²	C₂×#14556664	1	6	yes	—
144	C₂²	C₂×#14556666	1	6	yes	—
145	C₂²	C₂²×#11924036	1	1	yes	characteristic
146	C₄	—	73	8	no	—
147	C₄	—	146	4	no	—
148	C₄	C₂×#14556658	1	6	yes	—
149	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
150	C₄	C₂²×#11924037	1	1	yes	characteristic
151	C₂	C₂×#17989508	1	12	yes	direct product factor
152	C₂	C₂²×#14556658	1	1	yes	characteristic
153	C₂	C₂²×#14556664	1	1	yes	characteristic
154	C₂	C₂²×#14556666	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#17989508	C₂	direct (N × H)
6. #17989508	C₂²	direct (N × H)
29. C₂²×Dic₇₃	C₁₆	semidirect
56. C₂×Dic₇₃	C₂×C₁₆	semidirect
63. C₂²×C₇₃	C₄⋊C₁₆	semidirect
72. Dic₇₃	C₂²×C₁₆	semidirect
74. C₂×C₇₃	C₂×C₄⋊C₁₆	semidirect
81. C₇₃	C₂²×C₄⋊C₁₆	semidirect
141. C₂²	#17989508	direct (N × H)
151. C₂	C₂×#17989508	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×#17989508	2
C₂²×C₈×Dic₇₃	2
C₂²×C₄⋊C₁₆	73

Subgroups — C22×#17989508 #34030055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989508 #34030055