Subgroups — C₂²×#64408⋊C₁₆ #34034661

Order 18688.

1022 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556686	C₂	1	2	yes	maximal
2	C₂×#64408⋊C₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₄	1	12	yes	complemented
5	C₂×#14556686	C₂²	1	12	yes	—
6	C₂²×#11924040	C₄	1	4	yes	complemented
7	#64408⋊C₁₆	C₂²	1	16	yes	direct product factor
8	C₂²×#11924036	C₂²	1	1	yes	characteristic
9	C₂²×C₄×#64408	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	6	yes	—
11	C₂×C₈×#64408	C₂²	1	6	yes	—
12	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
13	#14556686	C₂×C₄	1	24	yes	complemented
14	#14556686	C₂³	1	8	yes	—
15	C₂×#11924040	C₂×C₄	1	24	yes	complemented
16	C₂³×#64408	C₈	1	2	yes	—
17	C₂×#11924036	C₂×C₄	1	6	yes	—
18	C₂×#11924036	C₂×C₄	1	3	yes	—
19	C₂×#11924036	C₂³	1	3	yes	—
20	C₂²×#10596833	C₂×C₄	1	2	yes	—
21	C₂×C₄×#64408	C₈	1	6	yes	—
22	C₂×C₄×#64408	C₂×C₄	1	6	yes	—
23	C₈×#64408	C₂×C₄	1	12	yes	—
24	C₈×#64408	C₂³	1	4	yes	—
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₈	1	1	yes	characteristic
27	C₂×C₈×D₇₃	C₈	1	6	yes	—
28	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
29	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
30	#11924040	C₂²×C₄	1	16	yes	complemented
31	C₂²×#64408	C₁₆	1	16	yes	complemented
32	C₂²×#64408	C₂×C₈	1	12	yes	—
33	#11924036	C₄²	1	6	yes	—
34	#11924036	C₂²×C₄	1	3	yes	—
35	#11924036	C₂²×C₄	1	6	yes	—
36	#11924036	C₂⁴	1	1	yes	characteristic
37	C₂×#10596833	C₄²	1	6	yes	—
38	C₂×#10596833	C₂²×C₄	1	6	yes	—
39	C₄×#64408	C₂×C₈	1	12	yes	—
40	C₄×#64408	C₂²×C₄	1	4	yes	—
41	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₄×D₇₃	C₄²	1	3	yes	—
44	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
45	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
46	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
47	C₂×#10596831	C₂×C₈	1	6	yes	—
48	C₈×D₇₃	C₂×C₈	1	12	yes	—
49	C₈×D₇₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×C₈	1	6	yes	—
52	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
53	C₂²×D₇₃	C₄×C₈	1	3	yes	—
54	C₂²×D₇₃	C₂×C₁₆	1	8	yes	—
55	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
56	C₂×#64408	C₂×C₁₆	1	48	yes	complemented
57	C₂×#64408	C₂²×C₈	1	8	yes	—
58	C₂×C₄×C₇₃	C₄×C₈	1	3	yes	—
59	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
60	C₂×Dic₇₃	C₄×C₈	1	3	yes	—
61	C₂×Dic₇₃	C₂×C₄²	1	3	yes	—
62	C₄×D₇₃	C₄×C₈	1	6	yes	—
63	C₄×D₇₃	C₂×C₄²	1	3	yes	—
64	C₄×D₇₃	C₂²×C₈	1	3	yes	—
65	C₄×D₇₃	C₂²×C₈	1	3	yes	—
66	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
67	#10596833	C₂×C₄²	1	6	yes	—
68	#10596833	C₂³×C₄	1	2	yes	—
69	C₈×C₇₃	C₂²×C₈	1	4	yes	—
70	#10596831	C₂²×C₈	1	4	yes	—
71	C₂²×C₇₃	C₄×C₁₆	1	4	yes	complemented
72	C₂²×C₇₃	C₂×C₄×C₈	1	3	yes	—
73	C₂×D₇₃	C₄×C₁₆	1	12	yes	complemented
74	C₂×D₇₃	C₂×C₄×C₈	1	3	yes	—
75	C₂×D₇₃	C₂²×C₁₆	1	12	yes	—
76	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
77	#64408	C₂²×C₁₆	1	16	yes	complemented
78	C₄×C₇₃	C₂×C₄×C₈	1	3	yes	—
79	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
80	Dic₇₃	C₂×C₄×C₈	1	3	yes	—
81	Dic₇₃	C₂²×C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₁₆	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	C₂×C₄×C₁₆	1	6	yes	complemented
84	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
85	D₇₃	C₂×C₄×C₁₆	1	6	yes	complemented
86	D₇₃	C₂³×C₁₆	1	2	yes	—
87	C₂³×C₁₆	—	73	2	no	—
88	C₂²×C₄×C₈	—	73	1	no	—
89	C₂×C₄×C₁₆	—	73	12	no	—
90	C₇₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₈	—	73	1	no	—
92	C₂³×C₈	—	73	1	no	—
93	C₂²×C₄²	—	73	1	no	—
94	C₂²×C₁₆	—	73	12	no	—
95	C₂²×C₁₆	—	73	4	no	—
96	C₂²×C₁₆	—	73	12	no	—
97	C₂×C₄×C₈	—	73	6	no	—
98	C₂×C₄×C₈	—	73	6	no	—
99	C₄×C₁₆	—	73	16	no	—
100	C₂³×C₄	—	73	2	no	—
101	C₂³×C₄	—	73	1	no	—
102	C₂²×C₈	—	73	2	no	—
103	C₂²×C₈	—	73	3	no	—
104	C₂²×C₈	—	73	6	no	—
105	C₂²×C₈	—	73	3	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	6	no	—
108	C₂²×C₈	—	73	6	no	—
109	C₂²×C₈	#10596842	1	1	yes	characteristic
110	C₂×C₄²	—	73	6	no	—
111	C₂×C₄²	—	73	6	no	—
112	C₂×C₁₆	—	73	24	no	—
113	C₂×C₁₆	—	73	24	no	—
114	C₂×C₁₆	—	73	8	no	—
115	C₄×C₈	—	73	4	no	—
116	C₄×C₈	—	73	12	no	—
117	C₂⁴	—	73	1	no	—
118	C₂²×C₄	—	73	16	no	—
119	C₂²×C₄	—	73	12	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	3	no	—
122	C₂²×C₄	—	73	3	no	—
123	C₂²×C₄	—	73	3	no	—
124	C₂²×C₄	—	73	3	no	—
125	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
126	C₂×C₈	—	73	6	no	—
127	C₂×C₈	—	73	6	no	—
128	C₂×C₈	—	73	6	no	—
129	C₂×C₈	—	73	3	no	—
130	C₂×C₈	—	73	6	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	6	no	—
133	C₂×C₈	—	73	12	no	—
134	C₂×C₈	—	73	4	no	—
135	C₂×C₈	C₂×#10596842	1	6	yes	—
136	C₄²	—	73	12	no	—
137	C₄²	—	73	4	no	—
138	C₁₆	—	73	16	no	—
139	C₂³	—	73	8	no	—
140	C₂³	—	73	3	no	—
141	C₂³	—	73	3	no	—
142	C₂³	C₄×#10596842	1	1	yes	characteristic
143	C₂×C₄	—	73	8	no	—
144	C₂×C₄	—	73	48	no	—
145	C₂×C₄	—	73	3	no	—
146	C₂×C₄	—	73	3	no	—
147	C₂×C₄	—	73	3	no	—
148	C₂×C₄	—	73	3	no	—
149	C₂×C₄	—	73	6	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	73	3	no	—
152	C₂×C₄	C₄×#10596842	1	3	yes	—
153	C₂×C₄	C₂²×#10596842	1	3	yes	—
154	C₈	—	73	2	no	—
155	C₈	—	73	6	no	—
156	C₈	—	73	4	no	—
157	C₈	C₂²×#10596842	1	4	yes	—
158	C₂²	—	73	12	no	—
159	C₂²	—	73	12	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	73	3	no	—
162	C₂²	#64408⋊C₁₆	1	4	yes	direct product factor
163	C₂²	C₂×C₄×#10596842	1	3	yes	—
164	C₄	—	73	16	no	—
165	C₄	—	73	1	no	—
166	C₄	—	73	3	no	—
167	C₄	C₂×C₄×#10596842	1	3	yes	—
168	C₄	C₂³×#10596842	1	1	yes	characteristic
169	C₂	—	73	2	no	—
170	C₂	—	73	6	no	—
171	C₂	C₂×#64408⋊C₁₆	1	6	yes	direct product factor
172	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	#10596842
commutator, hypocenter	C₇₃	C₂²×C₄×C₁₆
Frattini	C₄	C₂³×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂³×C₇₃	C₄×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	#11924036	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#64408⋊C₁₆	C₂	direct (N × H)
4. C₂×#14556686	C₄	semidirect
6. C₂²×#11924040	C₄	semidirect
7. #64408⋊C₁₆	C₂²	direct (N × H)
13. #14556686	C₂×C₄	semidirect
15. C₂×#11924040	C₂×C₄	semidirect
30. #11924040	C₂²×C₄	semidirect
31. C₂²×#64408	C₁₆	semidirect
56. C₂×#64408	C₂×C₁₆	semidirect
71. C₂²×C₇₃	C₄×C₁₆	semidirect
73. C₂×D₇₃	C₄×C₁₆	semidirect
77. #64408	C₂²×C₁₆	semidirect
83. C₂×C₇₃	C₂×C₄×C₁₆	semidirect
85. D₇₃	C₂×C₄×C₁₆	semidirect
90. C₇₃	C₂²×C₄×C₁₆	semidirect
162. C₂²	#64408⋊C₁₆	direct (N × H)
171. C₂	C₂×#64408⋊C₁₆	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×#14556686	2
C₂×#64408⋊C₁₆	2
C₂²×C₈×#64408	2
C₂²×C₄×C₁₆	73

Subgroups — C22×#64408⋊C16 #34034661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64408⋊C₁₆ #34034661