Subgroups — C₂³×#10587624 #17940958

Order 4352.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₁₇	C₂	1	2	yes	maximal, complemented
2	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#10587624	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂²⋊Dic₁₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
8	C₂³×C₂²⋊D₁₇	—	2	32	no	—
9	C₂³×C₂²⋊D₁₇	C₂²	1	56	yes	complemented
10	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
12	C₂³×C₄×D₁₇	—	2	16	no	—
13	C₂³×C₄×D₁₇	C₂²	1	28	yes	complemented
14	C₂×#10587624	C₂²	1	448	yes	direct product factor
15	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	complemented
16	C₂²×C₂²⋊Dic₁₇	C₂²	1	56	yes	complemented
17	C₂⁶×C₁₇	C₂²	1	2	yes	—
18	C₂³×D₄×C₁₇	—	2	16	no	—
19	C₂³×D₄×C₁₇	C₂²	1	28	yes	complemented
20	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
21	C₂⁵×C₁₇	—	2	16	no	—
22	C₂⁵×C₁₇	—	2	16	no	—
23	C₂⁵×C₁₇	—	2	16	no	—
24	C₂⁵×C₁₇	C₂³	1	28	yes	—
25	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
26	C₂⁴×D₁₇	—	2	8	no	—
27	C₂⁴×D₁₇	—	2	8	no	—
28	C₂⁴×D₁₇	D₄	1	16	yes	complemented
29	C₂⁴×D₁₇	C₂³	1	14	yes	—
30	C₂²×D₄×C₁₇	—	2	224	no	—
31	C₂²×D₄×C₁₇	C₂³	1	112	yes	complemented
32	C₂³×C₄×C₁₇	D₄	1	16	yes	complemented
33	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
34	C₂²×C₂²⋊D₁₇	—	2	448	no	—
35	C₂²×C₂²⋊D₁₇	C₂³	1	224	yes	complemented
36	C₂³×Dic₁₇	—	2	8	no	—
37	C₂³×Dic₁₇	—	2	16	no	—
38	C₂³×Dic₁₇	C₂³	1	14	yes	—
39	C₂³×Dic₁₇	C₂³	1	28	yes	—
40	C₂²×C₄×D₁₇	—	2	224	no	—
41	C₂²×C₄×D₁₇	C₂³	1	112	yes	complemented
42	C₂×C₂²⋊Dic₁₇	C₂³	1	224	yes	complemented
43	#10587624	C₂³	1	512	yes	direct product factor
44	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
45	C₂⁴×C₁₇	—	2	112	no	—
46	C₂⁴×C₁₇	—	2	112	no	—
47	C₂⁴×C₁₇	—	2	112	no	—
48	C₂⁴×C₁₇	—	4	128	no	—
49	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
50	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
51	C₂⁴×C₁₇	#41	1	8	yes	—
52	C₂⁴×C₁₇	C₂⁴	1	56	yes	—
53	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
54	C₂³×D₁₇	—	2	128	no	—
55	C₂³×D₁₇	—	2	56	no	—
56	C₂³×D₁₇	—	2	56	no	—
57	C₂³×D₁₇	C₂×D₄	1	112	yes	complemented
58	C₂³×D₁₇	C₂⁴	1	28	yes	—
59	C₂×D₄×C₁₇	—	2	448	no	—
60	C₂×D₄×C₁₇	C₂⁴	1	64	yes	complemented
61	C₂²×C₄×C₁₇	C₂×D₄	1	112	yes	complemented
62	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
63	C₂×C₂²⋊D₁₇	—	2	896	no	—
64	C₂×C₂²⋊D₁₇	C₂⁴	1	128	yes	complemented
65	C₂²×Dic₁₇	—	2	56	no	—
66	C₂²×Dic₁₇	—	2	112	no	—
67	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
68	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
69	C₂×C₄×D₁₇	—	2	448	no	—
70	C₂×C₄×D₁₇	C₂⁴	1	64	yes	complemented
71	C₂²⋊Dic₁₇	C₂⁴	1	128	yes	complemented
72	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
73	C₂³×#121	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	112	no	—
75	C₂³×C₁₇	—	2	112	no	—
76	C₂³×C₁₇	—	2	112	no	—
77	C₂³×C₁₇	—	4	448	no	—
78	C₂³×C₁₇	#121	1	64	yes	complemented
79	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
80	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
81	C₂³×C₁₇	C₂×#41	1	28	yes	—
82	C₂³×C₁₇	C₂⁵	1	16	yes	—
83	C₂³×C₁₇	C₂⁵	1	7	yes	—
84	C₂²×D₁₇	—	2	448	no	—
85	C₂²×D₁₇	—	2	56	no	—
86	C₂²×D₁₇	—	2	56	no	—
87	C₂²×D₁₇	C₂²×D₄	1	112	yes	complemented
88	C₂²×D₁₇	C₂⁵	1	8	yes	—
89	D₄×C₁₇	—	2	128	no	—
90	C₂×C₄×C₁₇	C₂²×D₄	1	112	yes	complemented
91	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
92	C₂²⋊D₁₇	—	2	256	no	—
93	C₂×Dic₁₇	—	2	56	no	—
94	C₂×Dic₁₇	—	2	112	no	—
95	C₂×Dic₁₇	C₂⁵	1	8	yes	—
96	C₂×Dic₁₇	C₂⁵	1	16	yes	—
97	C₄×D₁₇	—	2	128	no	—
98	C₂⁴×D₄	—	17	2	no	—
99	C₂⁴×D₄	D₁₇	1	1	yes	characteristic, complemented
100	C₂⁵×C₄	—	17	1	no	—
101	C₂²×#121	—	17	56	no	—
102	C₂³×C₄⋊C₄	—	17	1	no	—
103	C₂³×C₂²⋊C₄	—	17	2	no	—
104	C₂²×C₁₇	—	2	16	no	—
105	C₂²×C₁₇	—	2	16	no	—
106	C₂²×C₁₇	—	2	16	no	—
107	C₂²×C₁₇	—	4	224	no	—
108	C₂²×C₁₇	C₂×#121	1	112	yes	complemented
109	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
110	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
111	C₂²×C₁₇	C₂²×#41	1	14	yes	—
112	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
113	C₂×D₁₇	—	2	8	no	—
114	C₂×D₁₇	—	2	8	no	—
115	C₂×D₁₇	—	2	224	no	—
116	C₂×D₁₇	C₂³×D₄	1	16	yes	complemented
117	C₄×C₁₇	C₂³×D₄	1	16	yes	complemented
118	Dic₁₇	—	2	8	no	—
119	Dic₁₇	—	2	16	no	—
120	C₂⁶	—	17	1	no	—
121	C₂⁶	C₂×D₁₇	1	2	yes	—
122	C₂³×D₄	—	2	16	no	—
123	C₂³×D₄	—	17	56	no	—
124	C₂³×D₄	—	34	32	no	—
125	C₂³×D₄	C₂×D₁₇	1	28	yes	complemented
126	C₂⁴×C₄	—	17	1	no	—
127	C₂⁴×C₄	—	17	2	no	—
128	C₂⁴×C₄	—	17	28	no	—
129	C₂⁴×C₄	—	34	16	no	—
130	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
131	C₂×#121	—	17	448	no	—
132	C₂²×C₄⋊C₄	—	17	28	no	—
133	C₂²×C₂²⋊C₄	—	17	56	no	—
134	C₂×C₁₇	—	4	16	no	—
135	C₂×C₁₇	C₂²×#121	1	28	yes	complemented
136	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
137	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
138	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
139	D₁₇	—	2	16	no	—
140	C₂⁵	—	2	16	no	—
141	C₂⁵	—	2	16	no	—
142	C₂⁵	—	2	16	no	—
143	C₂⁵	—	17	16	no	—
144	C₂⁵	—	17	14	no	—
145	C₂⁵	—	34	8	no	—
146	C₂⁵	—	34	8	no	—
147	C₂⁵	C₂²×D₁₇	1	28	yes	—
148	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
149	C₂²×D₄	—	2	224	no	—
150	C₂²×D₄	—	17	224	no	—
151	C₂²×D₄	—	34	448	no	—
152	C₂²×D₄	C₂²×D₁₇	1	112	yes	complemented
153	C₂³×C₄	—	17	14	no	—
154	C₂³×C₄	—	17	28	no	—
155	C₂³×C₄	—	17	112	no	—
156	C₂³×C₄	—	34	16	no	—
157	C₂³×C₄	—	34	8	no	—
158	C₂³×C₄	—	34	224	no	—
159	C₂³×C₄	C₂²⋊D₁₇	1	16	yes	complemented
160	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
161	#121	—	17	512	no	—
162	C₂×C₄⋊C₄	—	17	112	no	—
163	C₂×C₂²⋊C₄	—	17	224	no	—
164	C₁₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
165	C₂⁴	—	2	112	no	—
166	C₂⁴	—	2	112	no	—
167	C₂⁴	—	2	112	no	—
168	C₂⁴	—	4	128	no	—
169	C₂⁴	—	17	112	no	—
170	C₂⁴	—	17	28	no	—
171	C₂⁴	—	34	56	no	—
172	C₂⁴	—	34	56	no	—
173	C₂⁴	—	34	128	no	—
174	C₂⁴	D₄×D₁₇	1	8	yes	—
175	C₂⁴	#63254	1	8	yes	—
176	C₂⁴	C₂×C₂²⋊D₁₇	1	8	yes	—
177	C₂⁴	C₂³×D₁₇	1	56	yes	—
178	C₂⁴	C₂³×D₁₇	1	7	yes	—
179	C₂×D₄	—	2	448	no	—
180	C₂×D₄	—	17	128	no	—
181	C₂×D₄	—	34	896	no	—
182	C₂×D₄	C₂³×D₁₇	1	64	yes	complemented
183	C₂²×C₄	—	17	28	no	—
184	C₂²×C₄	—	17	56	no	—
185	C₂²×C₄	—	17	64	no	—
186	C₂²×C₄	—	34	112	no	—
187	C₂²×C₄	—	34	56	no	—
188	C₂²×C₄	—	34	448	no	—
189	C₂²×C₄	C₂×C₂²⋊D₁₇	1	112	yes	complemented
190	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
191	C₄⋊C₄	—	17	64	no	—
192	C₂²⋊C₄	—	17	128	no	—
193	C₂³	—	2	112	no	—
194	C₂³	—	2	112	no	—
195	C₂³	—	2	112	no	—
196	C₂³	—	4	448	no	—
197	C₂³	—	17	112	no	—
198	C₂³	—	17	8	no	—
199	C₂³	—	34	448	no	—
200	C₂³	—	34	56	no	—
201	C₂³	—	34	56	no	—
202	C₂³	#10587624	1	64	yes	direct product factor
203	C₂³	C₂×D₄×D₁₇	1	28	yes	—
204	C₂³	C₂×#63254	1	28	yes	—
205	C₂³	C₂²×C₂²⋊D₁₇	1	28	yes	—
206	C₂³	C₂⁴×D₁₇	1	16	yes	—
207	C₂³	C₂⁴×D₁₇	1	7	yes	—
208	D₄	—	2	128	no	—
209	D₄	—	34	256	no	—
210	C₂×C₄	—	17	8	no	—
211	C₂×C₄	—	17	16	no	—
212	C₂×C₄	—	34	112	no	—
213	C₂×C₄	—	34	56	no	—
214	C₂×C₄	—	34	128	no	—
215	C₂×C₄	C₂²×C₂²⋊D₁₇	1	112	yes	complemented
216	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
217	C₂²	—	2	16	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	4	224	no	—
221	C₂²	—	17	16	no	—
222	C₂²	—	34	224	no	—
223	C₂²	—	34	8	no	—
224	C₂²	—	34	8	no	—
225	C₂²	C₂×#10587624	1	112	yes	direct product factor
226	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
227	C₂²	C₂³×C₂²⋊D₁₇	1	14	yes	—
228	C₂²	C₂²×#63254	1	14	yes	—
229	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
230	C₄	—	34	16	no	—
231	C₄	—	34	8	no	—
232	C₄	C₂³×C₂²⋊D₁₇	1	16	yes	complemented
233	C₂	—	4	16	no	—
234	C₂	—	34	16	no	—
235	C₂	C₂²×#10587624	1	28	yes	direct product factor
236	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
237	C₂	C₂⁴×C₂²⋊D₁₇	1	1	yes	characteristic
238	C₂	C₂³×#63254	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×#121
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. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
2. C₂⁴×C₄×D₁₇	C₂	semidirect
3. C₂²×#10587624	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₇	C₂	semidirect
5. C₂³×C₂²⋊Dic₁₇	C₂	semidirect
6. C₂⁴×D₄×C₁₇	C₂	semidirect
9. C₂³×C₂²⋊D₁₇	C₂²	semidirect
13. C₂³×C₄×D₁₇	C₂²	semidirect
14. C₂×#10587624	C₂²	direct (N × H)
15. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
16. C₂²×C₂²⋊Dic₁₇	C₂²	semidirect
19. C₂³×D₄×C₁₇	C₂²	semidirect
28. C₂⁴×D₁₇	D₄	semidirect
31. C₂²×D₄×C₁₇	C₂³	semidirect
32. C₂³×C₄×C₁₇	D₄	semidirect
35. C₂²×C₂²⋊D₁₇	C₂³	semidirect
41. C₂²×C₄×D₁₇	C₂³	semidirect
42. C₂×C₂²⋊Dic₁₇	C₂³	semidirect
43. #10587624	C₂³	direct (N × H)
44. C₂×C₄⋊Dic₁₇	C₂³	semidirect
57. C₂³×D₁₇	C₂×D₄	semidirect
60. C₂×D₄×C₁₇	C₂⁴	semidirect
61. C₂²×C₄×C₁₇	C₂×D₄	semidirect
64. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
70. C₂×C₄×D₁₇	C₂⁴	semidirect
71. C₂²⋊Dic₁₇	C₂⁴	semidirect
72. C₄⋊Dic₁₇	C₂⁴	semidirect
78. C₂³×C₁₇	#121	semidirect
87. C₂²×D₁₇	C₂²×D₄	semidirect
90. C₂×C₄×C₁₇	C₂²×D₄	semidirect
99. C₂⁴×D₄	D₁₇	semidirect
108. C₂²×C₁₇	C₂×#121	semidirect
116. C₂×D₁₇	C₂³×D₄	semidirect
117. C₄×C₁₇	C₂³×D₄	semidirect
125. C₂³×D₄	C₂×D₁₇	semidirect
135. C₂×C₁₇	C₂²×#121	semidirect
152. C₂²×D₄	C₂²×D₁₇	semidirect
159. C₂³×C₄	C₂²⋊D₁₇	semidirect
164. C₁₇	C₂³×#121	semidirect
182. C₂×D₄	C₂³×D₁₇	semidirect
189. C₂²×C₄	C₂×C₂²⋊D₁₇	semidirect
202. C₂³	#10587624	direct (N × H)
215. C₂×C₄	C₂²×C₂²⋊D₁₇	semidirect
225. C₂²	C₂×#10587624	direct (N × H)
232. C₄	C₂³×C₂²⋊D₁₇	semidirect
235. C₂	C₂²×#10587624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₁₇	2
C₂⁴×C₄×D₁₇	2
C₂²×#10587624	2
C₂³×C₄⋊Dic₁₇	2
C₂³×C₂²⋊Dic₁₇	2
C₂⁴×D₄×C₁₇	2
C₂³×#121	17

Subgroups — C23×#10587624 #17940958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587624 #17940958