Subgroups — C₂×C₁₆×#10596842 #34033472

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64408⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#10596842	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#10596842	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556686	C₄	1	2	yes	—
6	C₂×C₄×#10596842	C₄	1	2	yes	—
7	#64408⋊C₁₆	C₄	1	2	yes	—
8	#64408⋊C₁₆	C₂²	1	2	yes	—
9	C₈×#10596842	C₄	1	2	yes	—
10	C₈×#10596842	C₂²	1	2	yes	—
11	C₂×#14556664	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₁₆×#64408	C₄	1	2	yes	—
14	C₁₆×#64408	C₂²	1	2	yes	—
15	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596842	C₈	1	4	yes	—
17	#14556686	C₈	1	4	yes	complemented
18	#14556686	C₂×C₄	1	4	yes	—
19	C₂×#11924040	C₈	1	4	yes	complemented
20	C₄×#10596842	C₈	1	4	yes	—
21	C₄×#10596842	C₂×C₄	1	4	yes	—
22	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
24	#14556664	C₈	1	2	yes	complemented
25	#14556664	C₂×C₄	1	2	yes	—
26	C₂×#11924031	C₈	1	2	yes	complemented
27	C₈×#64408	C₂×C₄	1	1	yes	characteristic
28	C₈×#64408	C₂×C₄	1	1	yes	characteristic
29	C₈×#64408	C₂×C₄	1	1	yes	characteristic
30	C₈×#64408	C₂³	1	1	yes	characteristic
31	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#11924003	C₈	1	1	yes	characteristic, complemented
33	C₁₆×D₇₃	C₈	1	2	yes	complemented
34	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic, complemented
36	C₂×#10596842	C₁₆	1	16	yes	direct product factor
37	C₂×#10596842	C₂×C₈	1	8	yes	—
38	#11924040	C₂×C₈	1	8	yes	complemented
39	C₂²×#64408	C₂×C₈	1	2	yes	—
40	#11924036	C₄²	1	1	yes	characteristic
41	#11924036	C₂×C₈	1	2	yes	—
42	#11924036	C₂²×C₄	1	1	yes	characteristic
43	C₂×#10596833	C₂×C₈	1	2	yes	—
44	C₄×#64408	C₄²	1	1	yes	characteristic
45	C₄×#64408	C₂×C₈	1	2	yes	—
46	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
47	#11924031	C₂×C₈	1	4	yes	complemented
48	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
49	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
50	C₈×D₇₃	C₄²	1	1	yes	characteristic
51	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
52	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
53	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
54	#11924003	C₂×C₈	1	2	yes	complemented
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₃	C₂×C₈	1	2	yes	complemented
57	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
58	C₂×#64408	C₄×C₈	1	2	yes	—
59	C₂×#64408	C₂×C₁₆	1	8	yes	—
60	C₂×#64408	C₂²×C₈	1	2	yes	—
61	#10596842	C₂×C₁₆	1	16	yes	direct product factor
62	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
63	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
64	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
65	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
66	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
67	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
68	#10596833	C₄×C₈	1	2	yes	—
69	#10596833	C₂²×C₈	1	2	yes	—
70	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
71	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	#10596831	C₄×C₈	1	1	yes	characteristic
73	#10596831	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₇₃	C₈²	1	1	yes	characteristic
75	C₂×D₇₃	C₈²	1	1	yes	characteristic
76	C₂×D₇₃	C₄×C₁₆	1	4	yes	—
77	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
78	#64408	C₄×C₁₆	1	4	yes	—
79	#64408	C₂²×C₁₆	1	4	yes	—
80	C₄×C₇₃	C₈²	1	1	yes	characteristic
81	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
82	Dic₇₃	C₈²	1	1	yes	characteristic
83	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂×C₈×C₁₆	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	C₈×C₁₆	1	2	yes	complemented
86	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
87	D₇₃	C₈×C₁₆	1	2	yes	complemented
88	D₇₃	C₂×C₄×C₁₆	1	2	yes	—
89	C₂×C₄×C₁₆	—	73	1	no	—
90	C₂×C₄×C₁₆	—	73	1	no	—
91	C₂×C₈²	—	73	1	no	—
92	C₈×C₁₆	—	73	4	no	—
93	C₇₃	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₁₆	—	73	1	no	—
95	C₂²×C₁₆	—	73	2	no	—
96	C₂²×C₁₆	—	73	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₂×C₄×C₈	—	73	2	no	—
99	C₄×C₁₆	—	73	2	no	—
100	C₄×C₁₆	—	73	2	no	—
101	C₄×C₁₆	—	73	2	no	—
102	C₄×C₁₆	—	73	2	no	—
103	C₈²	—	73	2	no	—
104	C₈²	—	73	2	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	4	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₂×C₁₆	—	73	4	no	—
110	C₂×C₁₆	—	73	4	no	—
111	C₂×C₁₆	—	73	4	no	—
112	C₂×C₁₆	—	73	2	no	—
113	C₂×C₁₆	—	73	2	no	—
114	C₂×C₁₆	—	73	2	no	—
115	C₂×C₁₆	—	73	1	no	—
116	C₂×C₁₆	—	73	2	no	—
117	C₂×C₁₆	—	73	2	no	—
118	C₂×C₁₆	#10596842	1	1	yes	characteristic, direct product factor
119	C₄×C₈	—	73	4	no	—
120	C₄×C₈	—	73	4	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₂²×C₄	—	73	2	no	—
126	C₂²×C₄	—	73	1	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	2	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	16	no	—
132	C₂×C₈	—	73	8	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
139	C₄²	—	73	1	no	—
140	C₄²	—	73	2	no	—
141	C₄²	—	73	1	no	—
142	C₁₆	—	73	4	no	—
143	C₁₆	—	73	8	no	—
144	C₁₆	—	73	2	no	—
145	C₁₆	C₂×#10596842	1	2	yes	direct product factor
146	C₂³	—	73	1	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	8	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
156	C₈	—	73	2	no	—
157	C₈	—	73	2	no	—
158	C₈	—	73	16	no	—
159	C₈	—	73	1	no	—
160	C₈	—	73	1	no	—
161	C₈	C₄×#10596842	1	1	yes	characteristic
162	C₈	C₂²×#10596842	1	1	yes	characteristic
163	C₂²	—	73	4	no	—
164	C₂²	—	73	1	no	—
165	C₂²	—	73	1	no	—
166	C₂²	C₈×#10596842	1	1	yes	characteristic
167	C₄	—	73	4	no	—
168	C₄	—	73	4	no	—
169	C₄	—	73	1	no	—
170	C₄	—	73	1	no	—
171	C₄	C₈×#10596842	1	1	yes	characteristic
172	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
173	C₂	—	73	2	no	—
174	C₂	—	73	2	no	—
175	C₂	C₁₆×#10596842	1	2	yes	direct product factor
176	C₂	C₂×C₈×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₆	#10596842
commutator, hypocenter	C₇₃	C₂×C₈×C₁₆
Frattini	C₈	C₂²×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂²×C₇₃	C₈²
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₈×#64408	C₂³
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₆×#10596842	C₂	direct (N × H)
17. #14556686	C₈	semidirect
19. C₂×#11924040	C₈	semidirect
24. #14556664	C₈	semidirect
26. C₂×#11924031	C₈	semidirect
32. C₂×#11924003	C₈	semidirect
33. C₁₆×D₇₃	C₈	semidirect
35. C₂×C₁₆×C₇₃	C₈	semidirect
36. C₂×#10596842	C₁₆	direct (N × H)
38. #11924040	C₂×C₈	semidirect
47. #11924031	C₂×C₈	semidirect
54. #11924003	C₂×C₈	semidirect
56. C₁₆×C₇₃	C₂×C₈	semidirect
61. #10596842	C₂×C₁₆	direct (N × H)
85. C₂×C₇₃	C₈×C₁₆	semidirect
87. D₇₃	C₈×C₁₆	semidirect
93. C₇₃	C₂×C₈×C₁₆	semidirect
118. C₂×C₁₆	#10596842	direct (N × H)
145. C₁₆	C₂×#10596842	direct (N × H)
175. C₂	C₁₆×#10596842	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×#64408⋊C₁₆	2
C₂×C₈×#10596842	2
C₁₆×#10596842	2
C₂×C₁₆×#64408	2
C₂×C₈×C₁₆	73

Subgroups — C2×C16×#10596842 #34033472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×#10596842 #34033472