Subgroups — C₂³×#10587623 #17940954

Order 4352.

42862 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₁₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#10587623	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#63227	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	—	2	16	no	—
8	C₂⁵×D₁₇	—	2	32	no	—
9	C₂⁵×D₁₇	C₂²	1	2	yes	—
10	C₂⁵×D₁₇	C₂²	1	28	yes	complemented
11	C₂³×C₂²⋊D₁₇	—	2	32	no	—
12	C₂³×C₂²⋊D₁₇	C₂²	1	56	yes	complemented
13	C₂⁴×Dic₁₇	C₂²	1	2	yes	complemented
14	C₂×#10587623	C₂²	1	448	yes	direct product factor
15	C₂²×C₂²⋊Dic₁₇	C₂²	1	28	yes	complemented
16	C₂²×#63227	C₂²	1	56	yes	complemented
17	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
18	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄×C₁₇	—	2	16	no	—
20	C₂³×D₄×C₁₇	C₂²	1	28	yes	complemented
21	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
22	C₂⁵×C₁₇	—	2	16	no	—
23	C₂⁵×C₁₇	—	2	8	no	—
24	C₂⁵×C₁₇	—	2	16	no	—
25	C₂⁵×C₁₇	D₄	1	16	yes	complemented
26	C₂⁵×C₁₇	C₂³	1	14	yes	—
27	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
28	C₂⁵×C₁₇	C₂³	1	14	yes	complemented
29	C₂⁴×D₁₇	—	2	512	no	—
30	C₂⁴×D₁₇	—	2	16	no	—
31	C₂⁴×D₁₇	—	2	16	no	—
32	C₂⁴×D₁₇	—	2	224	no	—
33	C₂⁴×D₁₇	—	2	448	no	—
34	C₂⁴×D₁₇	D₄	1	32	yes	complemented
35	C₂⁴×D₁₇	C₂³	1	28	yes	—
36	C₂⁴×D₁₇	C₂³	1	112	yes	complemented
37	C₂²×D₄×C₁₇	—	2	224	no	—
38	C₂²×D₄×C₁₇	C₂³	1	112	yes	complemented
39	C₂³×C₄×C₁₇	—	2	8	no	—
40	C₂³×C₄×C₁₇	C₂³	1	14	yes	complemented
41	C₂²×C₂²⋊D₁₇	—	2	448	no	—
42	C₂²×C₂²⋊D₁₇	C₂³	1	224	yes	complemented
43	C₂³×Dic₁₇	—	2	16	no	—
44	C₂³×Dic₁₇	C₂³	1	28	yes	complemented
45	C₂×C₂²⋊Dic₁₇	C₂³	1	112	yes	complemented
46	#10587623	C₂³	1	512	yes	direct product factor
47	C₂×#63227	C₂³	1	224	yes	complemented
48	C₂⁴×C₁₇	—	2	128	no	—
49	C₂⁴×C₁₇	—	2	112	no	—
50	C₂⁴×C₁₇	—	2	112	no	—
51	C₂⁴×C₁₇	—	2	56	no	—
52	C₂⁴×C₁₇	—	4	64	no	—
53	C₂⁴×C₁₇	C₂×D₄	1	16	yes	—
54	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
55	C₂⁴×C₁₇	C₂×D₄	1	112	yes	complemented
56	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
57	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
58	C₂⁴×C₁₇	C₂⁴	1	28	yes	complemented
59	C₂³×D₁₇	—	2	3584	no	—
60	C₂³×D₁₇	—	2	256	no	—
61	C₂³×D₁₇	—	2	112	no	—
62	C₂³×D₁₇	—	2	112	no	—
63	C₂³×D₁₇	—	2	448	no	—
64	C₂³×D₁₇	—	2	896	no	—
65	C₂³×D₁₇	C₂×D₄	1	224	yes	complemented
66	C₂³×D₁₇	C₂⁴	1	56	yes	—
67	C₂³×D₁₇	C₂⁴	1	64	yes	complemented
68	C₂×D₄×C₁₇	—	2	448	no	—
69	C₂×D₄×C₁₇	C₂⁴	1	64	yes	complemented
70	C₂²×C₄×C₁₇	—	2	56	no	—
71	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	complemented
72	C₂×C₂²⋊D₁₇	—	2	896	no	—
73	C₂×C₂²⋊D₁₇	C₂⁴	1	128	yes	complemented
74	C₂²×Dic₁₇	—	2	112	no	—
75	C₂²×Dic₁₇	C₂⁴	1	56	yes	complemented
76	C₂²⋊Dic₁₇	C₂⁴	1	64	yes	complemented
77	#63227	C₂⁴	1	128	yes	complemented
78	C₂³×#120	—	17	1	no	maximal, Sylow
79	C₂³×C₁₇	—	2	448	no	—
80	C₂³×C₁₇	—	2	112	no	—
81	C₂³×C₁₇	—	2	112	no	—
82	C₂³×C₁₇	—	2	56	no	—
83	C₂³×C₁₇	—	4	224	no	—
84	C₂³×C₁₇	#120	1	64	yes	complemented
85	C₂³×C₁₇	C₂²×D₄	1	56	yes	—
86	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
87	C₂³×C₁₇	C₂²×D₄	1	112	yes	complemented
88	C₂³×C₁₇	C₂⁵	1	8	yes	—
89	C₂³×C₁₇	C₂⁵	1	7	yes	—
90	C₂³×C₁₇	C₂⁵	1	8	yes	complemented
91	C₂²×D₁₇	—	2	3584	no	—
92	C₂²×D₁₇	—	2	896	no	—
93	C₂²×D₁₇	—	2	112	no	—
94	C₂²×D₁₇	—	2	112	no	—
95	C₂²×D₁₇	—	2	128	no	—
96	C₂²×D₁₇	—	2	256	no	—
97	C₂²×D₁₇	C₂²×D₄	1	224	yes	complemented
98	C₂²×D₁₇	C₂⁵	1	16	yes	—
99	D₄×C₁₇	—	2	128	no	—
100	C₂×C₄×C₁₇	—	2	56	no	—
101	C₂×C₄×C₁₇	C₂⁵	1	8	yes	complemented
102	C₂²⋊D₁₇	—	2	256	no	—
103	C₂×Dic₁₇	—	2	112	no	—
104	C₂×Dic₁₇	C₂⁵	1	16	yes	complemented
105	C₂⁷	—	17	1	no	—
106	C₂⁴×D₄	—	17	2	no	—
107	C₂⁴×D₄	D₁₇	1	1	yes	characteristic, complemented
108	C₂²×#120	—	17	56	no	—
109	C₂³×C₂²⋊C₄	—	17	1	no	—
110	C₂³×C₂²⋊C₄	—	17	2	no	—
111	C₂²×C₁₇	—	2	16	no	—
112	C₂²×C₁₇	—	2	224	no	—
113	C₂²×C₁₇	—	2	16	no	—
114	C₂²×C₁₇	—	2	8	no	—
115	C₂²×C₁₇	—	4	112	no	—
116	C₂²×C₁₇	C₂×#120	1	112	yes	complemented
117	C₂²×C₁₇	C₂³×D₄	1	28	yes	—
118	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
119	C₂²×C₁₇	C₂³×D₄	1	16	yes	complemented
120	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
121	C₂×D₁₇	—	2	512	no	—
122	C₂×D₁₇	—	2	16	no	—
123	C₂×D₁₇	—	2	16	no	—
124	C₂×D₁₇	—	2	448	no	—
125	C₂×D₁₇	C₂³×D₄	1	32	yes	complemented
126	C₄×C₁₇	—	2	8	no	—
127	Dic₁₇	—	2	16	no	—
128	C₂⁶	—	17	2	no	—
129	C₂⁶	—	17	28	no	—
130	C₂⁶	—	34	32	no	—
131	C₂⁶	—	34	16	no	—
132	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
133	C₂⁶	C₂×D₁₇	1	1	yes	characteristic, complemented
134	C₂³×D₄	—	2	16	no	—
135	C₂³×D₄	—	17	56	no	—
136	C₂³×D₄	—	34	32	no	—
137	C₂³×D₄	C₂×D₁₇	1	28	yes	complemented
138	C₂⁴×C₄	—	17	2	no	—
139	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic, complemented
140	C₂×#120	—	17	448	no	—
141	C₂²×C₂²⋊C₄	—	17	28	no	—
142	C₂²×C₂²⋊C₄	—	17	56	no	—
143	C₂×C₁₇	—	2	16	no	—
144	C₂×C₁₇	—	4	8	no	—
145	C₂×C₁₇	C₂²×#120	1	28	yes	complemented
146	C₂×C₁₇	C₂⁴×D₄	1	2	yes	—
147	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
148	D₁₇	—	2	32	no	—
149	C₂⁵	—	2	16	no	—
150	C₂⁵	—	2	16	no	—
151	C₂⁵	—	2	8	no	—
152	C₂⁵	—	17	32	no	—
153	C₂⁵	—	17	28	no	—
154	C₂⁵	—	17	112	no	—
155	C₂⁵	—	34	448	no	—
156	C₂⁵	—	34	512	no	—
157	C₂⁵	—	34	16	no	—
158	C₂⁵	—	34	16	no	—
159	C₂⁵	—	34	224	no	—
160	C₂⁵	C₂²⋊D₁₇	1	16	yes	complemented
161	C₂⁵	C₂²×D₁₇	1	14	yes	—
162	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
163	C₂⁵	C₂²×D₁₇	1	14	yes	complemented
164	C₂²×D₄	—	2	224	no	—
165	C₂²×D₄	—	17	224	no	—
166	C₂²×D₄	—	34	448	no	—
167	C₂²×D₄	C₂²×D₁₇	1	112	yes	complemented
168	C₂³×C₄	—	2	8	no	—
169	C₂³×C₄	—	17	28	no	—
170	C₂³×C₄	—	34	16	no	—
171	C₂³×C₄	C₂²×D₁₇	1	14	yes	complemented
172	#120	—	17	512	no	—
173	C₂×C₂²⋊C₄	—	17	112	no	—
174	C₂×C₂²⋊C₄	—	17	224	no	—
175	C₁₇	C₂³×#120	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	2	112	no	—
177	C₂⁴	—	2	128	no	—
178	C₂⁴	—	2	112	no	—
179	C₂⁴	—	2	56	no	—
180	C₂⁴	—	4	64	no	—
181	C₂⁴	—	17	224	no	—
182	C₂⁴	—	17	56	no	—
183	C₂⁴	—	17	64	no	—
184	C₂⁴	—	34	896	no	—
185	C₂⁴	—	34	3584	no	—
186	C₂⁴	—	34	112	no	—
187	C₂⁴	—	34	112	no	—
188	C₂⁴	—	34	448	no	—
189	C₂⁴	—	34	256	no	—
190	C₂⁴	D₄×D₁₇	1	16	yes	—
191	C₂⁴	C₂×C₂²⋊D₁₇	1	8	yes	—
192	C₂⁴	C₂×C₂²⋊D₁₇	1	112	yes	complemented
193	C₂⁴	C₂³×D₁₇	1	28	yes	—
194	C₂⁴	C₂³×D₁₇	1	7	yes	—
195	C₂⁴	C₂³×D₁₇	1	28	yes	complemented
196	C₂×D₄	—	2	448	no	—
197	C₂×D₄	—	17	128	no	—
198	C₂×D₄	—	34	896	no	—
199	C₂×D₄	C₂³×D₁₇	1	64	yes	complemented
200	C₂²×C₄	—	2	56	no	—
201	C₂²×C₄	—	17	56	no	—
202	C₂²×C₄	—	34	112	no	—
203	C₂²×C₄	C₂³×D₁₇	1	28	yes	complemented
204	C₂²⋊C₄	—	17	64	no	—
205	C₂²⋊C₄	—	17	128	no	—
206	C₂³	—	2	448	no	—
207	C₂³	—	2	112	no	—
208	C₂³	—	2	112	no	—
209	C₂³	—	2	56	no	—
210	C₂³	—	4	224	no	—
211	C₂³	—	17	224	no	—
212	C₂³	—	17	16	no	—
213	C₂³	—	34	256	no	—
214	C₂³	—	34	3584	no	—
215	C₂³	—	34	896	no	—
216	C₂³	—	34	112	no	—
217	C₂³	—	34	112	no	—
218	C₂³	—	34	128	no	—
219	C₂³	#10587623	1	64	yes	direct product factor
220	C₂³	C₂×D₄×D₁₇	1	56	yes	—
221	C₂³	C₂²×C₂²⋊D₁₇	1	28	yes	—
222	C₂³	C₂²×C₂²⋊D₁₇	1	112	yes	complemented
223	C₂³	C₂⁴×D₁₇	1	8	yes	—
224	C₂³	C₂⁴×D₁₇	1	7	yes	—
225	C₂³	C₂⁴×D₁₇	1	8	yes	complemented
226	D₄	—	2	128	no	—
227	D₄	—	34	256	no	—
228	C₂×C₄	—	2	56	no	—
229	C₂×C₄	—	17	16	no	—
230	C₂×C₄	—	34	112	no	—
231	C₂×C₄	C₂⁴×D₁₇	1	8	yes	complemented
232	C₂²	—	2	224	no	—
233	C₂²	—	2	16	no	—
234	C₂²	—	2	16	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	4	112	no	—
237	C₂²	—	17	32	no	—
238	C₂²	—	34	512	no	—
239	C₂²	—	34	448	no	—
240	C₂²	—	34	16	no	—
241	C₂²	—	34	16	no	—
242	C₂²	C₂×#10587623	1	112	yes	direct product factor
243	C₂²	C₂²×D₄×D₁₇	1	28	yes	—
244	C₂²	C₂³×C₂²⋊D₁₇	1	14	yes	—
245	C₂²	C₂³×C₂²⋊D₁₇	1	16	yes	complemented
246	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	34	16	no	—
249	C₂	—	2	16	no	—
250	C₂	—	4	8	no	—
251	C₂	—	34	32	no	—
252	C₂	C₂²×#10587623	1	28	yes	direct product factor
253	C₂	C₂³×D₄×D₁₇	1	2	yes	—
254	C₂	C₂⁴×C₂²⋊D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂⁴×D₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
hypercenter	C₂⁴×D₄	D₁₇
hypocenter	C₁₇	C₂³×#120
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₁₇	C₂	semidirect
2. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
3. C₂²×#10587623	C₂	direct (N × H)
4. C₂³×C₂²⋊Dic₁₇	C₂	semidirect
5. C₂³×#63227	C₂	semidirect
6. C₂⁴×D₄×C₁₇	C₂	semidirect
10. C₂⁵×D₁₇	C₂²	semidirect
12. C₂³×C₂²⋊D₁₇	C₂²	semidirect
13. C₂⁴×Dic₁₇	C₂²	semidirect
14. C₂×#10587623	C₂²	direct (N × H)
15. C₂²×C₂²⋊Dic₁₇	C₂²	semidirect
16. C₂²×#63227	C₂²	semidirect
18. C₂⁶×C₁₇	C₂²	semidirect
20. C₂³×D₄×C₁₇	C₂²	semidirect
21. C₂⁴×C₄×C₁₇	C₂²	semidirect
25. C₂⁵×C₁₇	D₄	semidirect
28. C₂⁵×C₁₇	C₂³	semidirect
34. C₂⁴×D₁₇	D₄	semidirect
36. C₂⁴×D₁₇	C₂³	semidirect
38. C₂²×D₄×C₁₇	C₂³	semidirect
40. C₂³×C₄×C₁₇	C₂³	semidirect
42. C₂²×C₂²⋊D₁₇	C₂³	semidirect
44. C₂³×Dic₁₇	C₂³	semidirect
45. C₂×C₂²⋊Dic₁₇	C₂³	semidirect
46. #10587623	C₂³	direct (N × H)
47. C₂×#63227	C₂³	semidirect
55. C₂⁴×C₁₇	C₂×D₄	semidirect
58. C₂⁴×C₁₇	C₂⁴	semidirect
65. C₂³×D₁₇	C₂×D₄	semidirect
67. C₂³×D₁₇	C₂⁴	semidirect
69. C₂×D₄×C₁₇	C₂⁴	semidirect
71. C₂²×C₄×C₁₇	C₂⁴	semidirect
73. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
75. C₂²×Dic₁₇	C₂⁴	semidirect
76. C₂²⋊Dic₁₇	C₂⁴	semidirect
77. #63227	C₂⁴	semidirect
84. C₂³×C₁₇	#120	semidirect
87. C₂³×C₁₇	C₂²×D₄	semidirect
90. C₂³×C₁₇	C₂⁵	semidirect
97. C₂²×D₁₇	C₂²×D₄	semidirect
101. C₂×C₄×C₁₇	C₂⁵	semidirect
104. C₂×Dic₁₇	C₂⁵	semidirect
107. C₂⁴×D₄	D₁₇	semidirect
116. C₂²×C₁₇	C₂×#120	semidirect
119. C₂²×C₁₇	C₂³×D₄	semidirect
125. C₂×D₁₇	C₂³×D₄	semidirect
133. C₂⁶	C₂×D₁₇	semidirect
137. C₂³×D₄	C₂×D₁₇	semidirect
139. C₂⁴×C₄	C₂×D₁₇	semidirect
145. C₂×C₁₇	C₂²×#120	semidirect
160. C₂⁵	C₂²⋊D₁₇	semidirect
163. C₂⁵	C₂²×D₁₇	semidirect
167. C₂²×D₄	C₂²×D₁₇	semidirect
171. C₂³×C₄	C₂²×D₁₇	semidirect
175. C₁₇	C₂³×#120	semidirect
192. C₂⁴	C₂×C₂²⋊D₁₇	semidirect
195. C₂⁴	C₂³×D₁₇	semidirect
199. C₂×D₄	C₂³×D₁₇	semidirect
203. C₂²×C₄	C₂³×D₁₇	semidirect
219. C₂³	#10587623	direct (N × H)
222. C₂³	C₂²×C₂²⋊D₁₇	semidirect
225. C₂³	C₂⁴×D₁₇	semidirect
231. C₂×C₄	C₂⁴×D₁₇	semidirect
242. C₂²	C₂×#10587623	direct (N × H)
245. C₂²	C₂³×C₂²⋊D₁₇	semidirect
252. C₂	C₂²×#10587623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×D₁₇	2
C₂⁴×C₂²⋊D₁₇	2
C₂²×#10587623	2
C₂³×C₂²⋊Dic₁₇	2
C₂³×#63227	2
C₂⁴×D₄×C₁₇	2
C₂³×#120	17

Subgroups — C23×#10587623 #17940954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587623 #17940954