Subgroups — C₂×C₁₆×C₁₇×C₃₂ #72195261

Order 17408.

890 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	C₂×C₁₆²×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₁₇×C₃₂	C₂	1	2	yes	maximal
3	C₁₆×C₁₇×C₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×C₃₂	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₄×C₁₇×C₃₂	C₄	1	4	yes	—
6	C₂×C₈×C₁₆×C₁₇	C₄	1	2	yes	—
7	C₂×C₈×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₈×C₁₇×C₃₂	C₄	1	4	yes	—
9	C₈×C₁₇×C₃₂	C₂²	1	4	yes	—
10	C₁₆²×C₁₇	C₄	1	2	yes	—
11	C₁₆²×C₁₇	C₂²	1	2	yes	—
12	C₂²×C₁₇×C₃₂	C₈	1	8	yes	—
13	C₂×C₄×C₁₆×C₁₇	C₈	1	4	yes	—
14	C₂×C₄×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
15	C₂×C₈²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₄×C₁₇×C₃₂	C₈	1	8	yes	—
17	C₄×C₁₇×C₃₂	C₂×C₄	1	8	yes	—
18	C₈×C₁₆×C₁₇	C₈	1	4	yes	—
19	C₈×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₈×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
21	C₈×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
22	C₈×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×C₁₆×C₁₇	C₁₆	1	8	yes	—
24	C₂²×C₁₆×C₁₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₈×C₁₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₇	C₂×C₈	1	2	yes	—
27	C₂×C₁₇×C₃₂	C₁₆	1	32	yes	direct product factor
28	C₂×C₁₇×C₃₂	C₂×C₈	1	16	yes	—
29	C₄×C₁₆×C₁₇	C₁₆	1	8	yes	—
30	C₄×C₁₆×C₁₇	C₄²	1	2	yes	—
31	C₄×C₁₆×C₁₇	C₂×C₈	1	4	yes	—
32	C₄×C₁₆×C₁₇	C₂×C₈	1	8	yes	—
33	C₄×C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
34	C₈²×C₁₇	C₄²	1	1	yes	characteristic
35	C₈²×C₁₇	C₂×C₈	1	2	yes	—
36	C₈²×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	C₄×C₈	1	2	yes	—
38	C₂²×C₈×C₁₇	C₂×C₁₆	1	4	yes	—
39	C₂×C₄²×C₁₇	C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₆×C₁₇	C₃₂	1	32	yes	direct product factor
41	C₂×C₁₆×C₁₇	C₄×C₈	1	4	yes	—
42	C₂×C₁₆×C₁₇	C₂×C₁₆	1	16	yes	—
43	C₂×C₁₆×C₁₇	C₂×C₁₆	1	16	yes	—
44	C₂×C₁₆×C₁₇	C₂²×C₈	1	4	yes	—
45	C₄×C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
46	C₄×C₈×C₁₇	C₄×C₈	1	2	yes	—
47	C₄×C₈×C₁₇	C₄×C₈	1	2	yes	—
48	C₄×C₈×C₁₇	C₂×C₁₆	1	4	yes	—
49	C₄×C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂²×C₈	1	2	yes	—
51	C₁₇×C₃₂	C₂×C₁₆	1	32	yes	direct product factor
52	C₂×C₈×C₃₂	C₂×C₁₇	1	2	yes	—
53	C₂×C₁₆²	C₂×C₁₇	1	1	yes	characteristic
54	C₁₆×C₃₂	C₂×C₁₇	1	4	yes	direct product factor
55	C₂²×C₄×C₁₇	C₈²	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₄×C₁₆	1	2	yes	—
57	C₂×C₈×C₁₇	C₈²	1	2	yes	—
58	C₂×C₈×C₁₇	C₄×C₁₆	1	4	yes	—
59	C₂×C₈×C₁₇	C₄×C₁₆	1	8	yes	—
60	C₂×C₈×C₁₇	C₂×C₃₂	1	16	yes	—
61	C₂×C₈×C₁₇	C₂×C₄×C₈	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²×C₁₆	1	4	yes	—
63	C₄²×C₁₇	C₈²	1	1	yes	characteristic
64	C₄²×C₁₇	C₄×C₁₆	1	2	yes	—
65	C₄²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₁₆×C₁₇	C₄×C₁₆	1	8	yes	—
67	C₁₆×C₁₇	C₂×C₃₂	1	32	yes	direct product factor
68	C₁₆×C₁₇	C₂²×C₁₆	1	8	yes	—
69	C₂×C₄×C₃₂	C₄×C₁₇	1	4	yes	—
70	C₂×C₈×C₁₆	C₄×C₁₇	1	2	yes	—
71	C₂×C₈×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
72	C₈×C₃₂	C₄×C₁₇	1	4	yes	—
73	C₈×C₃₂	C₂²×C₁₇	1	4	yes	—
74	C₁₆²	C₄×C₁₇	1	2	yes	—
75	C₁₆²	C₂²×C₁₇	1	2	yes	—
76	C₂³×C₁₇	C₈×C₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₈×C₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₈×C₁₆	1	2	yes	—
79	C₂×C₄×C₁₇	C₈×C₁₆	1	4	yes	—
80	C₂×C₄×C₁₇	C₄×C₃₂	1	8	yes	—
81	C₂×C₄×C₁₇	C₂×C₈²	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×C₄×C₁₆	1	2	yes	—
83	C₈×C₁₇	C₈×C₁₆	1	4	yes	—
84	C₈×C₁₇	C₄×C₃₂	1	8	yes	—
85	C₈×C₁₇	C₂×C₄×C₁₆	1	4	yes	—
86	C₈×C₁₇	C₂²×C₃₂	1	8	yes	—
87	C₂²×C₃₂	C₈×C₁₇	1	8	yes	—
88	C₂×C₄×C₁₆	C₈×C₁₇	1	4	yes	—
89	C₂×C₄×C₁₆	C₂×C₄×C₁₇	1	2	yes	—
90	C₂×C₈²	C₂×C₄×C₁₇	1	1	yes	characteristic
91	C₄×C₃₂	C₈×C₁₇	1	8	yes	—
92	C₄×C₃₂	C₂×C₄×C₁₇	1	8	yes	—
93	C₈×C₁₆	C₈×C₁₇	1	4	yes	—
94	C₈×C₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
95	C₈×C₁₆	C₂×C₄×C₁₇	1	2	yes	—
96	C₈×C₁₆	C₂×C₄×C₁₇	1	4	yes	—
97	C₈×C₁₆	C₂³×C₁₇	1	1	yes	characteristic
98	C₂²×C₁₇	C₁₆²	1	2	yes	—
99	C₂²×C₁₇	C₈×C₃₂	1	4	yes	—
100	C₂²×C₁₇	C₂×C₈×C₁₆	1	1	yes	characteristic
101	C₄×C₁₇	C₁₆²	1	2	yes	—
102	C₄×C₁₇	C₈×C₃₂	1	4	yes	—
103	C₄×C₁₇	C₂×C₈×C₁₆	1	2	yes	—
104	C₄×C₁₇	C₂×C₄×C₃₂	1	4	yes	—
105	C₂²×C₁₆	C₁₆×C₁₇	1	8	yes	—
106	C₂²×C₁₆	C₂×C₈×C₁₇	1	4	yes	—
107	C₂×C₄×C₈	C₄²×C₁₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×C₈×C₁₇	1	2	yes	—
109	C₂×C₃₂	C₁₆×C₁₇	1	32	yes	direct product factor
110	C₂×C₃₂	C₂×C₈×C₁₇	1	16	yes	—
111	C₄×C₁₆	C₁₆×C₁₇	1	8	yes	—
112	C₄×C₁₆	C₄²×C₁₇	1	2	yes	—
113	C₄×C₁₆	C₂×C₈×C₁₇	1	4	yes	—
114	C₄×C₁₆	C₂×C₈×C₁₇	1	8	yes	—
115	C₄×C₁₆	C₂²×C₄×C₁₇	1	2	yes	—
116	C₈²	C₄²×C₁₇	1	1	yes	characteristic
117	C₈²	C₂×C₈×C₁₇	1	2	yes	—
118	C₈²	C₂²×C₄×C₁₇	1	1	yes	characteristic
119	C₂×C₁₇	C₁₆×C₃₂	1	4	yes	direct product factor
120	C₂×C₁₇	C₂×C₁₆²	1	1	yes	characteristic
121	C₂×C₁₇	C₂×C₈×C₃₂	1	2	yes	—
122	C₂²×C₈	C₄×C₈×C₁₇	1	2	yes	—
123	C₂²×C₈	C₂×C₁₆×C₁₇	1	4	yes	—
124	C₂×C₄²	C₄×C₈×C₁₇	1	1	yes	characteristic
125	C₂×C₁₆	C₁₇×C₃₂	1	32	yes	direct product factor
126	C₂×C₁₆	C₄×C₈×C₁₇	1	4	yes	—
127	C₂×C₁₆	C₂×C₁₆×C₁₇	1	16	yes	—
128	C₂×C₁₆	C₂×C₁₆×C₁₇	1	16	yes	—
129	C₂×C₁₆	C₂²×C₈×C₁₇	1	4	yes	—
130	C₄×C₈	C₄×C₈×C₁₇	1	1	yes	characteristic
131	C₄×C₈	C₄×C₈×C₁₇	1	2	yes	—
132	C₄×C₈	C₄×C₈×C₁₇	1	2	yes	—
133	C₄×C₈	C₂×C₁₆×C₁₇	1	4	yes	—
134	C₄×C₈	C₂×C₄²×C₁₇	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₈×C₁₇	1	2	yes	—
136	C₃₂	C₂×C₁₆×C₁₇	1	32	yes	direct product factor
137	C₁₇	C₂×C₁₆×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	C₈²×C₁₇	1	1	yes	characteristic
139	C₂²×C₄	C₄×C₁₆×C₁₇	1	2	yes	—
140	C₂×C₈	C₈²×C₁₇	1	2	yes	—
141	C₂×C₈	C₄×C₁₆×C₁₇	1	4	yes	—
142	C₂×C₈	C₄×C₁₆×C₁₇	1	8	yes	—
143	C₂×C₈	C₂×C₁₇×C₃₂	1	16	yes	—
144	C₂×C₈	C₂×C₄×C₈×C₁₇	1	2	yes	—
145	C₂×C₈	C₂²×C₁₆×C₁₇	1	4	yes	—
146	C₄²	C₈²×C₁₇	1	1	yes	characteristic
147	C₄²	C₄×C₁₆×C₁₇	1	2	yes	—
148	C₄²	C₂×C₄×C₈×C₁₇	1	1	yes	characteristic
149	C₁₆	C₄×C₁₆×C₁₇	1	8	yes	—
150	C₁₆	C₂×C₁₇×C₃₂	1	32	yes	direct product factor
151	C₁₆	C₂²×C₁₆×C₁₇	1	8	yes	—
152	C₂³	C₈×C₁₆×C₁₇	1	1	yes	characteristic
153	C₂×C₄	C₈×C₁₆×C₁₇	1	1	yes	characteristic
154	C₂×C₄	C₈×C₁₆×C₁₇	1	2	yes	—
155	C₂×C₄	C₈×C₁₆×C₁₇	1	4	yes	—
156	C₂×C₄	C₄×C₁₇×C₃₂	1	8	yes	—
157	C₂×C₄	C₂×C₈²×C₁₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×C₁₆×C₁₇	1	2	yes	—
159	C₈	C₈×C₁₆×C₁₇	1	4	yes	—
160	C₈	C₄×C₁₇×C₃₂	1	8	yes	—
161	C₈	C₂×C₄×C₁₆×C₁₇	1	4	yes	—
162	C₈	C₂²×C₁₇×C₃₂	1	8	yes	—
163	C₂²	C₁₆²×C₁₇	1	2	yes	—
164	C₂²	C₈×C₁₇×C₃₂	1	4	yes	—
165	C₂²	C₂×C₈×C₁₆×C₁₇	1	1	yes	characteristic
166	C₄	C₁₆²×C₁₇	1	2	yes	—
167	C₄	C₈×C₁₇×C₃₂	1	4	yes	—
168	C₄	C₂×C₈×C₁₆×C₁₇	1	2	yes	—
169	C₄	C₂×C₄×C₁₇×C₃₂	1	4	yes	—
170	C₂	C₁₆×C₁₇×C₃₂	1	4	yes	direct product factor
171	C₂	C₂×C₈×C₁₇×C₃₂	1	2	yes	—
172	C₂	C₂×C₁₆²×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₁₆	C₂³×C₁₇
socle, prime radical	C₂³×C₁₇	C₈×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₆×C₁₇×C₃₂	C₂	direct (N × H)
4. C₂×C₁₆×C₃₂	C₁₇	direct (N × H)
27. C₂×C₁₇×C₃₂	C₁₆	direct (N × H)
40. C₂×C₁₆×C₁₇	C₃₂	direct (N × H)
51. C₁₇×C₃₂	C₂×C₁₆	direct (N × H)
54. C₁₆×C₃₂	C₂×C₁₇	direct (N × H)
67. C₁₆×C₁₇	C₂×C₃₂	direct (N × H)
109. C₂×C₃₂	C₁₆×C₁₇	direct (N × H)
119. C₂×C₁₇	C₁₆×C₃₂	direct (N × H)
125. C₂×C₁₆	C₁₇×C₃₂	direct (N × H)
136. C₃₂	C₂×C₁₆×C₁₇	direct (N × H)
137. C₁₇	C₂×C₁₆×C₃₂	direct (N × H)
150. C₁₆	C₂×C₁₇×C₃₂	direct (N × H)
170. C₂	C₁₆×C₁₇×C₃₂	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×C₁₆²×C₁₇	2
C₂×C₈×C₁₇×C₃₂	2
C₁₆×C₁₇×C₃₂	2
C₂×C₁₆×C₃₂	17

Subgroups — C2×C16×C17×C32 #72195261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×C₁₇×C₃₂ #72195261