Subgroups — C₂²×C₁₆×#595 #17978113

Order 4352.

1022 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587529	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#595	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63242	C₄	1	1	yes	characteristic
6	C₂²×C₄×#595	C₄	1	1	yes	characteristic
7	C₂×#10587529	C₄	1	6	yes	complemented
8	C₂×#10587529	C₂²	1	6	yes	—
9	C₂²×#63216	C₄	1	2	yes	complemented
10	C₂×C₈×#595	C₄	1	6	yes	—
11	C₂×C₈×#595	C₂²	1	6	yes	—
12	C₁₆×#595	C₂²	1	16	yes	direct product factor
13	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₂²×#63214	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₁₇	C₄	1	6	yes	complemented
16	C₂×C₁₆×D₁₇	C₂²	1	6	yes	—
17	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂³×#595	C₈	1	2	yes	—
19	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×#3624	C₈	1	2	yes	—
21	C₂×C₄×#595	C₈	1	6	yes	—
22	C₂×C₄×#595	C₂×C₄	1	6	yes	—
23	C₂×#63242	C₈	1	6	yes	—
24	C₂×#63242	C₂×C₄	1	6	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
28	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₁₇	C₂³	1	3	yes	—
31	C₂×#63216	C₂×C₄	1	12	yes	complemented
32	C₈×#595	C₂×C₄	1	12	yes	—
33	C₈×#595	C₂³	1	4	yes	—
34	#10587529	C₂×C₄	1	12	yes	complemented
35	#10587529	C₂³	1	4	yes	—
36	C₂×C₁₆×C₁₇	C₂×C₄	1	6	yes	complemented
37	C₂×#63214	C₂×C₄	1	6	yes	complemented
38	C₁₆×D₁₇	C₂×C₄	1	12	yes	complemented
39	C₁₆×D₁₇	C₂³	1	4	yes	—
40	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×#595	C₁₆	1	16	yes	direct product factor
42	C₂²×#595	C₂×C₈	1	12	yes	—
43	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
44	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₄²	1	3	yes	—
46	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
47	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₁₇	C₂²×C₄	1	3	yes	—
49	C₂×#3624	C₂×C₈	1	12	yes	—
50	C₄×#595	C₂×C₈	1	12	yes	—
51	C₄×#595	C₂²×C₄	1	4	yes	—
52	#63242	C₂×C₈	1	12	yes	—
53	#63242	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₁₇	C₄²	1	3	yes	—
55	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
56	C₂×#3622	C₄²	1	3	yes	—
57	C₂×#3622	C₂²×C₄	1	3	yes	—
58	C₈×D₁₇	C₄²	1	6	yes	—
59	C₈×D₁₇	C₂²×C₄	1	3	yes	—
60	C₈×D₁₇	C₂²×C₄	1	3	yes	—
61	C₈×D₁₇	C₂²×C₄	1	3	yes	—
62	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
63	#63216	C₂²×C₄	1	8	yes	complemented
64	C₁₆×C₁₇	C₂²×C₄	1	4	yes	complemented
65	#63214	C₂²×C₄	1	4	yes	complemented
66	C₂²×C₄×C₁₆	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
68	C₂²×D₁₇	C₄×C₈	1	3	yes	—
69	C₂²×D₁₇	C₂×C₁₆	1	8	yes	—
70	C₂²×D₁₇	C₂²×C₈	1	3	yes	—
71	C₂×#595	C₂×C₁₆	1	48	yes	direct product factor
72	C₂×#595	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₁₇	C₄×C₈	1	3	yes	—
74	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
75	C₂×Dic₁₇	C₄×C₈	1	3	yes	—
76	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
77	C₄×D₁₇	C₄×C₈	1	6	yes	—
78	C₄×D₁₇	C₂×C₄²	1	3	yes	—
79	C₄×D₁₇	C₂²×C₈	1	3	yes	—
80	C₄×D₁₇	C₂²×C₈	1	3	yes	—
81	C₄×D₁₇	C₂³×C₄	1	1	yes	characteristic
82	#3624	C₂²×C₈	1	8	yes	—
83	C₈×C₁₇	C₂×C₄²	1	3	yes	—
84	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
85	#3622	C₂×C₄²	1	3	yes	—
86	#3622	C₂³×C₄	1	1	yes	characteristic
87	C₂³×C₁₆	—	17	1	no	—
88	C₂³×C₁₆	—	17	1	no	—
89	C₂²×C₄×C₈	—	17	1	no	—
90	C₂×C₄×C₁₆	—	17	12	no	—
91	C₂²×C₁₇	C₄×C₁₆	1	4	yes	complemented
92	C₂²×C₁₇	C₂×C₄×C₈	1	3	yes	—
93	C₂×D₁₇	C₄×C₁₆	1	12	yes	complemented
94	C₂×D₁₇	C₂×C₄×C₈	1	3	yes	—
95	C₂×D₁₇	C₂²×C₁₆	1	12	yes	—
96	C₂×D₁₇	C₂³×C₈	1	1	yes	characteristic
97	#595	C₂²×C₁₆	1	16	yes	direct product factor
98	C₄×C₁₇	C₂×C₄×C₈	1	3	yes	—
99	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
100	Dic₁₇	C₂×C₄×C₈	1	3	yes	—
101	Dic₁₇	C₂³×C₈	1	1	yes	characteristic
102	C₂³×C₈	—	17	1	no	—
103	C₂³×C₈	—	17	1	no	—
104	C₂²×C₄²	—	17	1	no	—
105	C₂²×C₁₆	—	17	6	no	—
106	C₂²×C₁₆	—	17	2	no	—
107	C₂²×C₁₆	—	17	6	no	—
108	C₂²×C₁₆	—	17	1	no	—
109	C₂²×C₁₆	—	17	6	no	—
110	C₂²×C₁₆	—	17	6	no	—
111	C₂²×C₁₆	#595	1	1	yes	characteristic, direct product factor
112	C₂×C₄×C₈	—	17	6	no	—
113	C₂×C₄×C₈	—	17	6	no	—
114	C₄×C₁₆	—	17	16	no	—
115	C₂×C₁₇	C₂×C₄×C₁₆	1	6	yes	complemented
116	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
117	D₁₇	C₂×C₄×C₁₆	1	6	yes	complemented
118	D₁₇	C₂³×C₁₆	1	2	yes	—
119	C₂³×C₄	—	17	2	no	—
120	C₂³×C₄	—	17	1	no	—
121	C₂²×C₈	—	17	6	no	—
122	C₂²×C₈	—	17	6	no	—
123	C₂²×C₈	—	17	2	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	—	17	3	no	—
126	C₂²×C₈	—	17	3	no	—
127	C₂²×C₈	—	17	3	no	—
128	C₂²×C₈	—	17	3	no	—
129	C₂²×C₈	C₂×#595	1	1	yes	characteristic
130	C₂×C₄²	—	17	6	no	—
131	C₂×C₄²	—	17	6	no	—
132	C₂×C₁₆	—	17	4	no	—
133	C₂×C₁₆	—	17	12	no	—
134	C₂×C₁₆	—	17	12	no	—
135	C₂×C₁₆	—	17	6	no	—
136	C₂×C₁₆	—	17	12	no	—
137	C₂×C₁₆	—	17	4	no	—
138	C₂×C₁₆	C₂×#595	1	6	yes	direct product factor
139	C₄×C₈	—	17	4	no	—
140	C₄×C₈	—	17	12	no	—
141	C₁₇	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	—	17	1	no	—
143	C₂²×C₄	—	17	16	no	—
144	C₂²×C₄	—	17	12	no	—
145	C₂²×C₄	—	17	1	no	—
146	C₂²×C₄	—	17	3	no	—
147	C₂²×C₄	—	17	3	no	—
148	C₂²×C₄	—	17	3	no	—
149	C₂²×C₄	—	17	3	no	—
150	C₂²×C₄	C₄×#595	1	1	yes	characteristic
151	C₂×C₈	—	17	12	no	—
152	C₂×C₈	—	17	4	no	—
153	C₂×C₈	—	17	12	no	—
154	C₂×C₈	—	17	3	no	—
155	C₂×C₈	—	17	3	no	—
156	C₂×C₈	—	17	3	no	—
157	C₂×C₈	—	17	3	no	—
158	C₂×C₈	—	17	6	no	—
159	C₂×C₈	—	17	1	no	—
160	C₂×C₈	—	17	3	no	—
161	C₂×C₈	C₄×#595	1	3	yes	—
162	C₂×C₈	C₂²×#595	1	3	yes	—
163	C₄²	—	17	12	no	—
164	C₄²	—	17	4	no	—
165	C₁₆	—	17	8	no	—
166	C₁₆	—	17	4	no	—
167	C₁₆	C₂²×#595	1	4	yes	direct product factor
168	C₂³	—	17	3	no	—
169	C₂³	—	17	3	no	—
170	C₂³	—	17	8	no	—
171	C₂³	C₈×#595	1	1	yes	characteristic
172	C₂×C₄	—	17	8	no	—
173	C₂×C₄	—	17	48	no	—
174	C₂×C₄	—	17	3	no	—
175	C₂×C₄	—	17	3	no	—
176	C₂×C₄	—	17	3	no	—
177	C₂×C₄	—	17	3	no	—
178	C₂×C₄	—	17	6	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	17	3	no	—
181	C₂×C₄	C₈×#595	1	3	yes	—
182	C₂×C₄	C₂×C₄×#595	1	3	yes	—
183	C₈	—	17	8	no	—
184	C₈	—	17	1	no	—
185	C₈	—	17	3	no	—
186	C₈	C₂×C₄×#595	1	3	yes	—
187	C₈	C₂³×#595	1	1	yes	characteristic
188	C₂²	—	17	12	no	—
189	C₂²	—	17	12	no	—
190	C₂²	—	17	1	no	—
191	C₂²	—	17	3	no	—
192	C₂²	C₁₆×#595	1	4	yes	direct product factor
193	C₂²	C₂×C₈×#595	1	3	yes	—
194	C₄	—	17	16	no	—
195	C₄	—	17	1	no	—
196	C₄	—	17	3	no	—
197	C₄	C₂×C₈×#595	1	3	yes	—
198	C₄	C₂²×C₄×#595	1	1	yes	characteristic
199	C₂	—	17	2	no	—
200	C₂	—	17	6	no	—
201	C₂	C₂×C₁₆×#595	1	6	yes	direct product factor
202	C₂	C₂²×C₈×#595	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	#595
commutator, hypocenter	C₁₇	C₂²×C₄×C₁₆
Frattini	C₈	C₂³×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂³×C₁₇	C₄×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₈×D₁₇	C₂⁴
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×#595	C₂	direct (N × H)
7. C₂×#10587529	C₄	semidirect
9. C₂²×#63216	C₄	semidirect
12. C₁₆×#595	C₂²	direct (N × H)
14. C₂²×#63214	C₄	semidirect
15. C₂×C₁₆×D₁₇	C₄	semidirect
17. C₂²×C₁₆×C₁₇	C₄	semidirect
31. C₂×#63216	C₂×C₄	semidirect
34. #10587529	C₂×C₄	semidirect
36. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
37. C₂×#63214	C₂×C₄	semidirect
38. C₁₆×D₁₇	C₂×C₄	semidirect
41. C₂²×#595	C₁₆	direct (N × H)
63. #63216	C₂²×C₄	semidirect
64. C₁₆×C₁₇	C₂²×C₄	semidirect
65. #63214	C₂²×C₄	semidirect
71. C₂×#595	C₂×C₁₆	direct (N × H)
91. C₂²×C₁₇	C₄×C₁₆	semidirect
93. C₂×D₁₇	C₄×C₁₆	semidirect
97. #595	C₂²×C₁₆	direct (N × H)
111. C₂²×C₁₆	#595	direct (N × H)
115. C₂×C₁₇	C₂×C₄×C₁₆	semidirect
117. D₁₇	C₂×C₄×C₁₆	semidirect
138. C₂×C₁₆	C₂×#595	direct (N × H)
141. C₁₇	C₂²×C₄×C₁₆	semidirect
167. C₁₆	C₂²×#595	direct (N × H)
192. C₂²	C₁₆×#595	direct (N × H)
201. C₂	C₂×C₁₆×#595	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×#10587529	2
C₂²×C₈×#595	2
C₂×C₁₆×#595	2
C₂²×C₁₆×D₁₇	2
C₂²×C₄×C₁₆	17

Subgroups — C22×C16×#595 #17978113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×#595 #17978113