Subgroups — C₂×C₈×D₁₇₂ #19143269

Order 5504.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₄₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	C₈×D₁₇₂	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₄₃)⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
9	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₈×D₄₃	—	2	8	no	—
11	C₂×C₈×D₄₃	C₂²	1	8	yes	complemented
12	C₄×D₁₇₂	C₄	1	4	yes	—
13	C₄×D₁₇₂	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₄₃)⋊C₄	C₄	1	2	yes	—
16	C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	complemented
17	(C₂×D₄₃)⋊C₈	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₃	C₂²	1	4	yes	complemented
22	C₂³×D₄₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	C₂×C₄	1	2	yes	—
27	C₂×D₁₇₂	C₈	1	8	yes	direct product factor
28	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₄₃	—	2	4	no	—
30	C₂×C₄×D₄₃	—	2	4	no	—
31	C₂×C₄×D₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₄₃	C₂³	1	4	yes	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	D₄	1	4	yes	complemented
35	C₂×C₈×C₄₃	C₂³	1	2	yes	—
36	C₂×C₈×C₄₃	C₂³	1	2	yes	—
37	C₄²×C₄₃	C₂×C₄	1	2	yes	—
38	C₄²×C₄₃	C₂³	1	2	yes	—
39	(C₂×D₄₃)⋊C₄	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
41	C₂×C₄₃⋊C₈	—	2	4	no	—
42	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
43	C₈×D₄₃	—	2	16	no	—
44	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×D₄₃	—	2	4	no	—
46	C₂²×D₄₃	—	2	4	no	—
47	C₂²×D₄₃	C₂×C₈	1	8	yes	complemented
48	C₂²×D₄₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	C₂×C₈	1	4	yes	complemented
51	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₃	#41	1	2	yes	—
56	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₄₃	—	2	4	no	—
58	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
59	D₁₇₂	C₂×C₈	1	16	yes	direct product factor
60	C₄×D₄₃	—	2	8	no	—
61	C₄×D₄₃	—	2	4	no	—
62	C₄×D₄₃	—	2	4	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	C₂×D₄	1	4	yes	complemented
65	C₄₃⋊C₈	—	2	4	no	—
66	C₂²×C₄₃	C₄×D₄	1	2	yes	—
67	C₂²×C₄₃	C₂²×C₈	1	2	yes	—
68	C₂²×C₄₃	#131	1	2	yes	—
69	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×D₄₃	—	2	4	no	—
71	C₂×D₄₃	—	2	4	no	—
72	C₂×D₄₃	—	2	16	no	—
73	C₂×D₄₃	C₂²×C₈	1	8	yes	complemented
74	C₄×C₄₃	—	2	2	no	—
75	C₄×C₄₃	C₄×D₄	1	2	yes	—
76	C₄×C₄₃	C₂²×C₈	1	4	yes	complemented
77	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
79	Dic₄₃	—	2	4	no	—
80	C₂×C₈×D₄	—	43	1	no	maximal, Sylow
81	C₂×C₄₃	C₈×D₄	1	4	yes	complemented
82	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₄₃	C₂³×C₈	1	1	yes	characteristic
84	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
85	D₄₃	—	2	8	no	—
86	C₂³×C₈	—	43	2	no	—
87	C₂×C₄×D₄	—	43	1	no	—
88	C₈×D₄	—	43	8	no	—
89	C₂×C₄⋊C₈	—	43	1	no	—
90	C₂×C₂²⋊C₈	—	43	2	no	—
91	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
92	C₄₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	43	1	no	—
94	C₂³×C₄	—	43	2	no	—
95	C₂²×C₈	—	43	2	no	—
96	C₂²×C₈	—	43	8	no	—
97	C₂²×C₈	—	86	8	no	—
98	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
100	C₄×D₄	—	43	4	no	—
101	C₄×D₄	—	43	4	no	—
102	C₂×C₄⋊C₄	—	43	1	no	—
103	C₂×C₂²⋊C₄	—	43	2	no	—
104	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
105	C₄⋊C₈	—	43	4	no	—
106	C₂²⋊C₈	—	43	8	no	—
107	C₄×C₈	C₂×D₄₃	1	4	yes	complemented
108	C₂⁴	—	43	2	no	—
109	C₂×D₄	—	43	8	no	—
110	C₂×D₄	—	43	4	no	—
111	C₂²×C₄	—	43	2	no	—
112	C₂²×C₄	—	43	4	no	—
113	C₂²×C₄	—	43	4	no	—
114	C₂²×C₄	—	86	4	no	—
115	C₂²×C₄	—	86	4	no	—
116	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	43	4	no	—
121	C₂×C₈	—	86	4	no	—
122	C₂×C₈	—	86	16	no	—
123	C₂×C₈	D₁₇₂	1	4	yes	direct product factor
124	C₂×C₈	C₂²×D₄₃	1	2	yes	—
125	C₂×C₈	C₂²×D₄₃	1	2	yes	—
126	C₄⋊C₄	—	43	4	no	—
127	C₂²⋊C₄	—	43	8	no	—
128	C₄²	C₄×D₄₃	1	2	yes	—
129	C₄²	C₂²×D₄₃	1	2	yes	—
130	C₂³	—	43	8	no	—
131	C₂³	—	43	4	no	—
132	C₂³	—	86	4	no	—
133	C₂³	—	86	4	no	—
134	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
135	D₄	—	43	16	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	43	4	no	—
138	C₂×C₄	—	86	4	no	—
139	C₂×C₄	—	86	8	no	—
140	C₂×C₄	—	86	4	no	—
141	C₂×C₄	—	86	4	no	—
142	C₂×C₄	C₈×D₄₃	1	4	yes	complemented
143	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
146	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
147	C₂×C₄	#10621898	1	2	yes	—
148	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	86	4	no	—
151	C₈	C₂×D₁₇₂	1	4	yes	direct product factor
152	C₂²	—	43	8	no	—
153	C₂²	—	86	4	no	—
154	C₂²	—	86	4	no	—
155	C₂²	—	86	16	no	—
156	C₂²	C₄×D₁₇₂	1	2	yes	—
157	C₂²	C₂×C₈×D₄₃	1	2	yes	—
158	C₂²	#13063331	1	2	yes	—
159	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	86	4	no	—
162	C₄	C₄×D₁₇₂	1	2	yes	—
163	C₄	C₂×C₈×D₄₃	1	4	yes	complemented
164	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
165	C₄	C₂×#10621898	1	1	yes	characteristic
166	C₂	—	86	8	no	—
167	C₂	C₈×D₁₇₂	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₁₇₂	1	1	yes	characteristic
169	C₂	C₂²×C₈×D₄₃	1	1	yes	characteristic
170	C₂	C₂×#13063331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄₃
commutator	C₂×C₄₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂×C₈×D₄
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₄₃	C₂	semidirect
3. C₈×D₁₇₂	C₂	direct (N × H)
4. C₂×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
5. C₂×(C₂×D₄₃)⋊C₈	C₂	semidirect
6. C₂×C₄×C₈×C₄₃	C₂	semidirect
11. C₂×C₈×D₄₃	C₂²	semidirect
16. C₄⋊(C₄₃⋊C₈)	C₂²	semidirect
17. (C₂×D₄₃)⋊C₈	C₂²	semidirect
21. C₄×C₈×C₄₃	C₂²	semidirect
27. C₂×D₁₇₂	C₈	direct (N × H)
34. C₂×C₈×C₄₃	D₄	semidirect
47. C₂²×D₄₃	C₂×C₈	semidirect
50. C₂×C₄×C₄₃	C₂×C₈	semidirect
59. D₁₇₂	C₂×C₈	direct (N × H)
64. C₈×C₄₃	C₂×D₄	semidirect
73. C₂×D₄₃	C₂²×C₈	semidirect
76. C₄×C₄₃	C₂²×C₈	semidirect
81. C₂×C₄₃	C₈×D₄	semidirect
91. C₂×C₄×C₈	D₄₃	semidirect
92. C₄₃	C₂×C₈×D₄	semidirect
107. C₄×C₈	C₂×D₄₃	semidirect
123. C₂×C₈	D₁₇₂	direct (N × H)
142. C₂×C₄	C₈×D₄₃	semidirect
151. C₈	C₂×D₁₇₂	direct (N × H)
163. C₄	C₂×C₈×D₄₃	semidirect
167. C₂	C₈×D₁₇₂	direct (N × H)

Subgroups — C2×C8×D172 #19143269

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₈×D₁₇₂ #19143269