Subgroups — C₂³×D₁₇⋊M₁₆ #17981049

Order 4352.

13310 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63243	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₇⋊M₁₆	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#63247	C₂	1	2	yes	maximal, complemented
4	C₂³×#63242	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63243	C₂²	1	56	yes	complemented
7	C₂×D₁₇⋊M₁₆	C₂²	1	448	yes	direct product factor
8	C₂²×#63247	C₂²	1	56	yes	complemented
9	C₂²×#63242	C₂²	1	56	yes	complemented
10	C₂³×#3624	C₂²	1	4	yes	complemented
11	C₂⁵×D₁₇	C₄	1	1	yes	characteristic
12	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₁₇	C₄	1	28	yes	—
14	C₂³×C₄×D₁₇	C₄	1	1	yes	characteristic
15	C₂³×C₄×D₁₇	C₄	1	1	yes	characteristic
16	C₂³×C₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×D₁₇	C₂²	1	28	yes	—
18	C₂³×C₄×D₁₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₁₇	C₄	1	1	yes	characteristic
20	C₂⁵×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×D₁₇	—	2	16	no	—
22	C₂⁴×D₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×D₁₇	C₂×C₄	1	28	yes	—
24	C₂⁴×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₇	C₂×C₄	1	28	yes	—
26	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₁₇	C₂×C₄	1	14	yes	—
29	C₂³×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₁₇	C₂³	1	1	yes	characteristic
31	C₂³×Dic₁₇	C₂³	1	14	yes	—
32	C₂²×C₄×D₁₇	C₂×C₄	1	28	yes	—
33	C₂²×C₄×D₁₇	C₂×C₄	1	336	yes	—
34	C₂²×C₄×D₁₇	C₂×C₄	1	14	yes	—
35	C₂²×C₄×D₁₇	C₂×C₄	1	14	yes	—
36	C₂²×C₄×D₁₇	C₂×C₄	1	28	yes	—
37	C₂²×C₄×D₁₇	C₂³	1	14	yes	—
38	C₂²×C₄×D₁₇	C₂³	1	112	yes	—
39	C₂²×C₄×D₁₇	C₂³	1	14	yes	—
40	C₂×#63247	C₂³	1	224	yes	complemented
41	C₂²×#3624	C₂³	1	56	yes	complemented
42	D₁₇⋊M₁₆	C₂³	1	512	yes	direct product factor
43	C₂×#63243	C₂³	1	224	yes	complemented
44	C₂×#63242	C₂³	1	224	yes	complemented
45	C₂⁴×C₁₇	—	2	8	no	—
46	C₂⁴×C₁₇	C₂²×C₄	1	14	yes	—
47	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂³×D₁₇	—	2	8	no	—
49	C₂³×D₁₇	—	2	224	no	—
50	C₂³×D₁₇	M₁₆	1	16	yes	complemented
51	C₂³×D₁₇	C₂²×C₄	1	14	yes	—
52	C₂³×D₁₇	C₂²×C₄	1	112	yes	—
53	C₂³×D₁₇	C₂²×C₄	1	14	yes	—
54	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	—
55	C₂²×C₄×C₁₇	C₂²×C₄	1	14	yes	—
56	C₂²×C₄×C₁₇	C₂²×C₄	1	14	yes	—
57	C₂²×Dic₁₇	C₂²×C₄	1	84	yes	—
58	C₂²×Dic₁₇	C₂²×C₄	1	7	yes	—
59	C₂²×Dic₁₇	C₂²×C₄	1	14	yes	—
60	C₂²×Dic₁₇	C₂⁴	1	7	yes	—
61	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
62	C₂×C₄×D₁₇	C₂²×C₄	1	448	yes	—
63	C₂×C₄×D₁₇	C₂²×C₄	1	112	yes	—
64	C₂×C₄×D₁₇	C₂²×C₄	1	84	yes	—
65	C₂×C₄×D₁₇	C₂²×C₄	1	84	yes	—
66	C₂×C₄×D₁₇	C₂²×C₄	1	112	yes	—
67	C₂×C₄×D₁₇	C₂⁴	1	28	yes	—
68	C₂×C₄×D₁₇	C₂⁴	1	28	yes	—
69	C₂×C₄×D₁₇	C₂⁴	1	64	yes	—
70	#63247	C₂⁴	1	128	yes	complemented
71	C₂×#3624	C₂⁴	1	112	yes	complemented
72	#63243	C₂⁴	1	128	yes	complemented
73	#63242	C₂⁴	1	128	yes	complemented
74	C₂⁴×M₁₆	—	17	1	no	maximal, Sylow
75	C₂³×C₁₇	—	2	56	no	—
76	C₂³×C₁₇	C₂×M₁₆	1	8	yes	complemented
77	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
78	C₂³×C₁₇	C₂³×C₄	1	7	yes	—
79	C₂²×D₁₇	—	2	56	no	—
80	C₂²×D₁₇	—	2	448	no	—
81	C₂²×D₁₇	C₂×M₁₆	1	112	yes	complemented
82	C₂²×D₁₇	C₂³×C₄	1	28	yes	—
83	C₂²×D₁₇	C₂³×C₄	1	64	yes	—
84	C₂²×D₁₇	C₂³×C₄	1	28	yes	—
85	C₂×C₄×C₁₇	C₂³×C₄	1	64	yes	—
86	C₂×C₄×C₁₇	C₂³×C₄	1	28	yes	—
87	C₂×C₄×C₁₇	C₂³×C₄	1	28	yes	—
88	C₂×Dic₁₇	C₂³×C₄	1	56	yes	—
89	C₂×Dic₁₇	C₂³×C₄	1	21	yes	—
90	C₂×Dic₁₇	C₂³×C₄	1	28	yes	—
91	C₂×Dic₁₇	C₂⁵	1	7	yes	—
92	C₂×Dic₁₇	C₂⁵	1	8	yes	—
93	C₄×D₁₇	C₂³×C₄	1	56	yes	—
94	C₄×D₁₇	C₂³×C₄	1	64	yes	—
95	C₄×D₁₇	C₂³×C₄	1	56	yes	—
96	C₄×D₁₇	C₂³×C₄	1	64	yes	—
97	C₄×D₁₇	C₂⁵	1	8	yes	—
98	C₄×D₁₇	C₂⁵	1	8	yes	—
99	#3624	C₂⁵	1	32	yes	complemented
100	C₂⁵×C₄	—	17	1	no	—
101	C₂³×M₁₆	—	17	2	no	—
102	C₂³×M₁₆	—	17	56	no	—
103	C₂³×M₁₆	—	17	2	no	—
104	C₂⁴×C₈	—	17	2	no	—
105	C₂²×C₁₇	—	2	56	no	—
106	C₂²×C₁₇	C₂²×M₁₆	1	28	yes	complemented
107	C₂²×C₁₇	C₂⁴×C₄	1	8	yes	—
108	C₂²×C₁₇	C₂⁴×C₄	1	7	yes	—
109	C₂×D₁₇	—	2	56	no	—
110	C₂×D₁₇	—	2	128	no	—
111	C₂×D₁₇	C₂²×M₁₆	1	112	yes	complemented
112	C₂×D₁₇	C₂⁴×C₄	1	8	yes	—
113	C₂×D₁₇	C₂⁴×C₄	1	8	yes	—
114	C₄×C₁₇	C₂⁴×C₄	1	8	yes	—
115	C₄×C₁₇	C₂⁴×C₄	1	8	yes	—
116	Dic₁₇	C₂⁴×C₄	1	7	yes	—
117	Dic₁₇	C₂⁴×C₄	1	8	yes	—
118	Dic₁₇	C₂⁶	1	1	yes	characteristic
119	C₂⁶	—	17	1	no	—
120	C₂⁴×C₄	—	17	1	no	—
121	C₂⁴×C₄	—	17	1	no	—
122	C₂⁴×C₄	—	17	28	no	—
123	C₂⁴×C₄	—	17	1	no	—
124	C₂⁴×C₄	—	17	28	no	—
125	C₂⁴×C₄	—	17	1	no	—
126	C₂⁴×C₄	—	17	1	no	—
127	C₂⁴×C₄	#595	1	1	yes	characteristic
128	C₂²×M₁₆	—	17	56	no	—
129	C₂²×M₁₆	—	17	448	no	—
130	C₂²×M₁₆	—	17	56	no	—
131	C₂³×C₈	—	17	4	no	—
132	C₂³×C₈	—	17	56	no	—
133	C₂×C₁₇	—	2	8	no	—
134	C₂×C₁₇	C₂³×M₁₆	1	14	yes	complemented
135	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
136	D₁₇	—	2	8	no	—
137	D₁₇	C₂³×M₁₆	1	16	yes	complemented
138	C₂⁵	—	17	1	no	—
139	C₂⁵	—	17	28	no	—
140	C₂⁵	—	17	1	no	—
141	C₂⁵	—	34	16	no	—
142	C₂⁵	C₂×#595	1	1	yes	characteristic
143	C₂³×C₄	—	17	14	no	—
144	C₂³×C₄	—	17	1	no	—
145	C₂³×C₄	—	17	1	no	—
146	C₂³×C₄	—	17	14	no	—
147	C₂³×C₄	—	17	28	no	—
148	C₂³×C₄	—	17	336	no	—
149	C₂³×C₄	—	17	14	no	—
150	C₂³×C₄	—	17	14	no	—
151	C₂³×C₄	—	17	28	no	—
152	C₂³×C₄	—	17	14	no	—
153	C₂³×C₄	—	17	14	no	—
154	C₂³×C₄	—	17	112	no	—
155	C₂³×C₄	C₂×#595	1	1	yes	characteristic
156	C₂³×C₄	C₂×#595	1	28	yes	—
157	C₂³×C₄	C₂×#595	1	1	yes	characteristic
158	C₂×M₁₆	—	17	224	no	—
159	C₂×M₁₆	—	17	512	no	—
160	C₂×M₁₆	—	17	224	no	—
161	C₂²×C₈	—	17	224	no	—
162	C₂²×C₈	—	17	56	no	—
163	C₁₇	C₂⁴×M₁₆	1	1	yes	characteristic, complemented, Sylow
164	C₂⁴	—	2	8	no	—
165	C₂⁴	—	17	16	no	—
166	C₂⁴	—	17	14	no	—
167	C₂⁴	—	17	112	no	—
168	C₂⁴	—	17	14	no	—
169	C₂⁴	—	34	224	no	—
170	C₂⁴	—	34	8	no	—
171	C₂⁴	C₂²×#595	1	14	yes	—
172	C₂⁴	C₂²×#595	1	1	yes	characteristic
173	C₂²×C₄	—	17	84	no	—
174	C₂²×C₄	—	17	7	no	—
175	C₂²×C₄	—	17	14	no	—
176	C₂²×C₄	—	17	7	no	—
177	C₂²×C₄	—	17	28	no	—
178	C₂²×C₄	—	17	84	no	—
179	C₂²×C₄	—	17	112	no	—
180	C₂²×C₄	—	17	448	no	—
181	C₂²×C₄	—	17	84	no	—
182	C₂²×C₄	—	17	112	no	—
183	C₂²×C₄	—	17	28	no	—
184	C₂²×C₄	—	17	28	no	—
185	C₂²×C₄	—	17	64	no	—
186	C₂²×C₄	C₂²×#595	1	112	yes	—
187	C₂²×C₄	C₂²×#595	1	14	yes	—
188	C₂²×C₄	C₂²×#595	1	14	yes	—
189	M₁₆	—	17	128	no	—
190	M₁₆	—	17	128	no	—
191	C₂×C₈	—	17	128	no	—
192	C₂×C₈	—	17	112	no	—
193	C₂³	—	2	56	no	—
194	C₂³	—	17	112	no	—
195	C₂³	—	17	28	no	—
196	C₂³	—	17	64	no	—
197	C₂³	—	17	28	no	—
198	C₂³	—	34	448	no	—
199	C₂³	—	34	56	no	—
200	C₂³	D₁₇⋊M₁₆	1	8	yes	direct product factor
201	C₂³	C₂³×#595	1	28	yes	—
202	C₂³	C₂³×#595	1	7	yes	—
203	C₂×C₄	—	17	7	no	—
204	C₂×C₄	—	17	8	no	—
205	C₂×C₄	—	17	56	no	—
206	C₂×C₄	—	17	21	no	—
207	C₂×C₄	—	17	28	no	—
208	C₂×C₄	—	17	8	no	—
209	C₂×C₄	—	17	8	no	—
210	C₂×C₄	—	17	56	no	—
211	C₂×C₄	—	17	64	no	—
212	C₂×C₄	—	17	56	no	—
213	C₂×C₄	—	17	64	no	—
214	C₂×C₄	C₂³×#595	1	64	yes	—
215	C₂×C₄	C₂³×#595	1	28	yes	—
216	C₂×C₄	C₂³×#595	1	28	yes	—
217	C₈	—	17	32	no	—
218	C₂²	—	2	56	no	—
219	C₂²	—	17	112	no	—
220	C₂²	—	17	8	no	—
221	C₂²	—	17	8	no	—
222	C₂²	—	34	128	no	—
223	C₂²	—	34	56	no	—
224	C₂²	C₂×D₁₇⋊M₁₆	1	28	yes	direct product factor
225	C₂²	C₂⁴×#595	1	8	yes	—
226	C₂²	C₂⁴×#595	1	7	yes	—
227	C₄	—	17	1	no	—
228	C₄	—	17	7	no	—
229	C₄	—	17	8	no	—
230	C₄	C₂⁴×#595	1	8	yes	—
231	C₄	C₂⁴×#595	1	8	yes	—
232	C₂	—	2	8	no	—
233	C₂	—	17	16	no	—
234	C₂	—	34	8	no	—
235	C₂	C₂²×D₁₇⋊M₁₆	1	14	yes	direct product factor
236	C₂	C₂⁵×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#595
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂	C₂⁵×#595
Fitting	C₂⁴×C₄×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂²×C₄
prime radical	C₂⁵×D₁₇	C₄
cosocle	Dic₁₇	C₂⁶
hypercenter	C₂⁴×C₄	#595
hypocenter	C₁₇	C₂⁴×M₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#63243	C₂	semidirect
2. C₂²×D₁₇⋊M₁₆	C₂	direct (N × H)
3. C₂³×#63247	C₂	semidirect
4. C₂³×#63242	C₂	semidirect
6. C₂²×#63243	C₂²	semidirect
7. C₂×D₁₇⋊M₁₆	C₂²	direct (N × H)
8. C₂²×#63247	C₂²	semidirect
9. C₂²×#63242	C₂²	semidirect
10. C₂³×#3624	C₂²	semidirect
40. C₂×#63247	C₂³	semidirect
41. C₂²×#3624	C₂³	semidirect
42. D₁₇⋊M₁₆	C₂³	direct (N × H)
43. C₂×#63243	C₂³	semidirect
44. C₂×#63242	C₂³	semidirect
50. C₂³×D₁₇	M₁₆	semidirect
70. #63247	C₂⁴	semidirect
71. C₂×#3624	C₂⁴	semidirect
72. #63243	C₂⁴	semidirect
73. #63242	C₂⁴	semidirect
76. C₂³×C₁₇	C₂×M₁₆	semidirect
81. C₂²×D₁₇	C₂×M₁₆	semidirect
99. #3624	C₂⁵	semidirect
106. C₂²×C₁₇	C₂²×M₁₆	semidirect
111. C₂×D₁₇	C₂²×M₁₆	semidirect
134. C₂×C₁₇	C₂³×M₁₆	semidirect
137. D₁₇	C₂³×M₁₆	semidirect
163. C₁₇	C₂⁴×M₁₆	semidirect
200. C₂³	D₁₇⋊M₁₆	direct (N × H)
224. C₂²	C₂×D₁₇⋊M₁₆	direct (N × H)
235. C₂	C₂²×D₁₇⋊M₁₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#63243	2
C₂²×D₁₇⋊M₁₆	2
C₂³×#63247	2
C₂³×#63242	2
C₂⁴×C₄×D₁₇	2
C₂⁴×M₁₆	17

Subgroups — C23×D17⋊M16 #17981049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₇⋊M₁₆ #17981049