Subgroups — C₂³×#10587568 #17940956

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×D₆₈	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#10587568	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#63227	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×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	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₆₈	—	2	32	no	—
13	C₂³×D₆₈	C₂²	1	56	yes	complemented
14	C₂³×C₂²⋊D₁₇	—	2	16	no	—
15	C₂³×C₂²⋊D₁₇	C₂²	1	28	yes	complemented
16	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×#10587568	C₂²	1	448	yes	direct product factor
18	C₂²×#63227	C₂²	1	56	yes	complemented
19	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	complemented
21	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	28	yes	complemented
22	C₂⁵×C₁₇	—	2	16	no	—
23	C₂⁵×C₁₇	D₄	1	16	yes	complemented
24	C₂⁵×C₁₇	C₂³	1	14	yes	—
25	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
26	C₂⁴×D₁₇	—	2	512	no	—
27	C₂⁴×D₁₇	—	2	16	no	—
28	C₂⁴×D₁₇	—	2	16	no	—
29	C₂⁴×D₁₇	—	2	224	no	—
30	C₂⁴×D₁₇	—	2	448	no	—
31	C₂⁴×D₁₇	—	2	8	no	—
32	C₂⁴×D₁₇	—	2	16	no	—
33	C₂⁴×D₁₇	D₄	1	32	yes	complemented
34	C₂⁴×D₁₇	C₂³	1	28	yes	—
35	C₂⁴×D₁₇	C₂³	1	112	yes	complemented
36	C₂⁴×D₁₇	C₂³	1	14	yes	complemented
37	C₂³×C₄×C₁₇	—	2	16	no	—
38	C₂³×C₄×C₁₇	C₂³	1	28	yes	complemented
39	C₂²×C₂²⋊D₁₇	—	2	224	no	—
40	C₂²×C₂²⋊D₁₇	C₂³	1	112	yes	complemented
41	C₂³×Dic₁₇	—	2	8	no	—
42	C₂³×Dic₁₇	C₂³	1	14	yes	complemented
43	C₂²×D₆₈	—	2	448	no	—
44	C₂²×D₆₈	C₂³	1	224	yes	complemented
45	C₂×C₂²⋊C₄×C₁₇	C₂³	1	112	yes	complemented
46	C₂×#63227	C₂³	1	224	yes	complemented
47	#10587568	C₂³	1	512	yes	direct product factor
48	C₂⁴×C₁₇	—	2	128	no	—
49	C₂⁴×C₁₇	—	2	112	no	—
50	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
51	C₂⁴×C₁₇	C₂×D₄	1	16	yes	—
52	C₂⁴×C₁₇	C₂×D₄	1	112	yes	complemented
53	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
54	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
55	C₂³×D₁₇	—	2	3584	no	—
56	C₂³×D₁₇	—	2	256	no	—
57	C₂³×D₁₇	—	2	112	no	—
58	C₂³×D₁₇	—	2	112	no	—
59	C₂³×D₁₇	—	2	448	no	—
60	C₂³×D₁₇	—	2	896	no	—
61	C₂³×D₁₇	—	2	112	no	—
62	C₂³×D₁₇	—	2	56	no	—
63	C₂³×D₁₇	—	4	64	no	—
64	C₂³×D₁₇	C₂×D₄	1	224	yes	complemented
65	C₂³×D₁₇	C₂⁴	1	56	yes	—
66	C₂³×D₁₇	C₂⁴	1	64	yes	complemented
67	C₂³×D₁₇	C₂⁴	1	28	yes	complemented
68	C₂²×C₄×C₁₇	—	2	112	no	—
69	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	complemented
70	C₂×C₂²⋊D₁₇	—	2	448	no	—
71	C₂×C₂²⋊D₁₇	C₂⁴	1	64	yes	complemented
72	C₂²×Dic₁₇	—	2	56	no	—
73	C₂²×Dic₁₇	C₂⁴	1	28	yes	complemented
74	C₂×D₆₈	—	2	896	no	—
75	C₂×D₆₈	C₂⁴	1	128	yes	complemented
76	C₂²⋊C₄×C₁₇	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	#120	1	64	yes	complemented
82	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
83	C₂³×C₁₇	C₂²×D₄	1	56	yes	—
84	C₂³×C₁₇	C₂²×D₄	1	112	yes	complemented
85	C₂³×C₁₇	C₂⁵	1	8	yes	—
86	C₂³×C₁₇	C₂⁵	1	7	yes	—
87	C₂²×D₁₇	—	2	3584	no	—
88	C₂²×D₁₇	—	2	896	no	—
89	C₂²×D₁₇	—	2	112	no	—
90	C₂²×D₁₇	—	2	112	no	—
91	C₂²×D₁₇	—	2	128	no	—
92	C₂²×D₁₇	—	2	256	no	—
93	C₂²×D₁₇	—	2	112	no	—
94	C₂²×D₁₇	—	2	56	no	—
95	C₂²×D₁₇	—	4	224	no	—
96	C₂²×D₁₇	C₂²×D₄	1	224	yes	complemented
97	C₂²×D₁₇	C₂⁵	1	16	yes	—
98	C₂²×D₁₇	C₂⁵	1	8	yes	complemented
99	C₂×C₄×C₁₇	—	2	112	no	—
100	C₂×C₄×C₁₇	C₂⁵	1	16	yes	complemented
101	C₂²⋊D₁₇	—	2	128	no	—
102	C₂×Dic₁₇	—	2	56	no	—
103	C₂×Dic₁₇	C₂⁵	1	8	yes	complemented
104	D₆₈	—	2	256	no	—
105	C₂⁷	—	17	1	no	—
106	C₂⁴×D₄	—	17	1	no	—
107	C₂⁴×D₄	—	17	2	no	—
108	C₂²×#120	—	17	56	no	—
109	C₂³×C₂²⋊C₄	—	17	2	no	—
110	C₂³×C₂²⋊C₄	D₁₇	1	1	yes	characteristic, complemented
111	C₂²×C₁₇	—	2	16	no	—
112	C₂²×C₁₇	—	2	224	no	—
113	C₂²×C₁₇	C₂×#120	1	112	yes	complemented
114	C₂²×C₁₇	C₂³×D₄	1	28	yes	—
115	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
116	C₂²×C₁₇	C₂³×D₄	1	16	yes	complemented
117	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
118	C₂×D₁₇	—	2	512	no	—
119	C₂×D₁₇	—	2	16	no	—
120	C₂×D₁₇	—	2	16	no	—
121	C₂×D₁₇	—	2	448	no	—
122	C₂×D₁₇	—	2	16	no	—
123	C₂×D₁₇	—	2	8	no	—
124	C₂×D₁₇	—	4	112	no	—
125	C₂×D₁₇	C₂³×D₄	1	32	yes	complemented
126	C₄×C₁₇	—	2	16	no	—
127	Dic₁₇	—	2	8	no	—
128	C₂⁶	—	17	2	no	—
129	C₂⁶	—	17	28	no	—
130	C₂⁶	—	17	1	no	—
131	C₂⁶	—	34	32	no	—
132	C₂⁶	—	34	16	no	—
133	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
134	C₂³×D₄	—	17	28	no	—
135	C₂³×D₄	—	17	56	no	—
136	C₂³×D₄	—	34	32	no	—
137	C₂³×D₄	—	34	16	no	—
138	C₂⁴×C₄	—	17	1	no	—
139	C₂⁴×C₄	C₂×D₁₇	1	2	yes	complemented
140	C₂×#120	—	17	448	no	—
141	C₂²×C₂²⋊C₄	—	17	56	no	—
142	C₂²×C₂²⋊C₄	C₂×D₁₇	1	28	yes	complemented
143	C₂×C₁₇	—	2	16	no	—
144	C₂×C₁₇	C₂²×#120	1	28	yes	complemented
145	C₂×C₁₇	C₂⁴×D₄	1	2	yes	—
146	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
147	D₁₇	—	2	32	no	—
148	D₁₇	—	4	8	no	—
149	C₂⁵	—	2	16	no	—
150	C₂⁵	—	17	32	no	—
151	C₂⁵	—	17	28	no	—
152	C₂⁵	—	17	112	no	—
153	C₂⁵	—	17	14	no	—
154	C₂⁵	—	34	448	no	—
155	C₂⁵	—	34	512	no	—
156	C₂⁵	—	34	16	no	—
157	C₂⁵	—	34	224	no	—
158	C₂⁵	—	34	16	no	—
159	C₂⁵	—	34	8	no	—
160	C₂⁵	—	34	16	no	—
161	C₂⁵	D₆₈	1	16	yes	complemented
162	C₂⁵	C₂²×D₁₇	1	14	yes	—
163	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
164	C₂²×D₄	—	17	112	no	—
165	C₂²×D₄	—	17	224	no	—
166	C₂²×D₄	—	34	448	no	—
167	C₂²×D₄	—	34	224	no	—
168	C₂³×C₄	—	2	16	no	—
169	C₂³×C₄	—	17	14	no	—
170	C₂³×C₄	—	34	8	no	—
171	C₂³×C₄	C₂²×D₁₇	1	28	yes	complemented
172	#120	—	17	512	no	—
173	C₂×C₂²⋊C₄	—	17	224	no	—
174	C₂×C₂²⋊C₄	C₂²×D₁₇	1	112	yes	complemented
175	C₁₇	C₂³×#120	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	2	112	no	—
177	C₂⁴	—	2	128	no	—
178	C₂⁴	—	17	224	no	—
179	C₂⁴	—	17	56	no	—
180	C₂⁴	—	17	64	no	—
181	C₂⁴	—	17	28	no	—
182	C₂⁴	—	34	896	no	—
183	C₂⁴	—	34	3584	no	—
184	C₂⁴	—	34	112	no	—
185	C₂⁴	—	34	448	no	—
186	C₂⁴	—	34	112	no	—
187	C₂⁴	—	34	256	no	—
188	C₂⁴	—	34	56	no	—
189	C₂⁴	—	34	112	no	—
190	C₂⁴	—	68	64	no	—
191	C₂⁴	C₂×D₆₈	1	8	yes	—
192	C₂⁴	C₂×D₆₈	1	112	yes	complemented
193	C₂⁴	D₄×D₁₇	1	16	yes	—
194	C₂⁴	C₂³×D₁₇	1	28	yes	—
195	C₂⁴	C₂³×D₁₇	1	7	yes	—
196	C₂×D₄	—	17	64	no	—
197	C₂×D₄	—	17	128	no	—
198	C₂×D₄	—	34	896	no	—
199	C₂×D₄	—	34	448	no	—
200	C₂²×C₄	—	2	112	no	—
201	C₂²×C₄	—	17	28	no	—
202	C₂²×C₄	—	34	56	no	—
203	C₂²×C₄	C₂³×D₁₇	1	56	yes	complemented
204	C₂²⋊C₄	—	17	128	no	—
205	C₂²⋊C₄	C₂³×D₁₇	1	64	yes	complemented
206	C₂³	—	2	448	no	—
207	C₂³	—	2	112	no	—
208	C₂³	—	17	224	no	—
209	C₂³	—	17	16	no	—
210	C₂³	—	17	8	no	—
211	C₂³	—	34	256	no	—
212	C₂³	—	34	3584	no	—
213	C₂³	—	34	896	no	—
214	C₂³	—	34	112	no	—
215	C₂³	—	34	112	no	—
216	C₂³	—	34	128	no	—
217	C₂³	—	34	112	no	—
218	C₂³	—	34	56	no	—
219	C₂³	—	68	224	no	—
220	C₂³	#10587568	1	64	yes	direct product factor
221	C₂³	C₂²×D₆₈	1	28	yes	—
222	C₂³	C₂²×D₆₈	1	112	yes	complemented
223	C₂³	C₂×D₄×D₁₇	1	56	yes	—
224	C₂³	C₂⁴×D₁₇	1	8	yes	—
225	C₂³	C₂⁴×D₁₇	1	7	yes	—
226	D₄	—	34	256	no	—
227	D₄	—	34	128	no	—
228	C₂×C₄	—	2	112	no	—
229	C₂×C₄	—	17	8	no	—
230	C₂×C₄	—	34	56	no	—
231	C₂×C₄	C₂⁴×D₁₇	1	16	yes	complemented
232	C₂²	—	2	224	no	—
233	C₂²	—	2	16	no	—
234	C₂²	—	17	32	no	—
235	C₂²	—	34	512	no	—
236	C₂²	—	34	448	no	—
237	C₂²	—	34	16	no	—
238	C₂²	—	34	16	no	—
239	C₂²	—	34	16	no	—
240	C₂²	—	34	8	no	—
241	C₂²	—	68	112	no	—
242	C₂²	C₂×#10587568	1	112	yes	direct product factor
243	C₂²	C₂²×D₄×D₁₇	1	28	yes	—
244	C₂²	C₂³×D₆₈	1	14	yes	—
245	C₂²	C₂³×D₆₈	1	16	yes	complemented
246	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
247	C₄	—	2	16	no	—
248	C₄	—	34	8	no	—
249	C₂	—	2	16	no	—
250	C₂	—	34	32	no	—
251	C₂	—	68	8	no	—
252	C₂	C₂²×#10587568	1	28	yes	direct product factor
253	C₂	C₂³×D₄×D₁₇	1	2	yes	—
254	C₂	C₂⁴×D₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₂²⋊C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₁₇
hypocenter	C₁₇	C₂³×#120
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₁₇	C₂	semidirect
2. C₂⁴×D₆₈	C₂	semidirect
3. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
4. C₂²×#10587568	C₂	direct (N × H)
5. C₂³×#63227	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₁₇	C₂	semidirect
10. C₂⁵×D₁₇	C₂²	semidirect
11. C₂⁵×D₁₇	C₂²	semidirect
13. C₂³×D₆₈	C₂²	semidirect
15. C₂³×C₂²⋊D₁₇	C₂²	semidirect
16. C₂⁴×Dic₁₇	C₂²	semidirect
17. C₂×#10587568	C₂²	direct (N × H)
18. C₂²×#63227	C₂²	semidirect
20. C₂⁴×C₄×C₁₇	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₁₇	C₂²	semidirect
23. C₂⁵×C₁₇	D₄	semidirect
33. C₂⁴×D₁₇	D₄	semidirect
35. C₂⁴×D₁₇	C₂³	semidirect
36. C₂⁴×D₁₇	C₂³	semidirect
38. C₂³×C₄×C₁₇	C₂³	semidirect
40. C₂²×C₂²⋊D₁₇	C₂³	semidirect
42. C₂³×Dic₁₇	C₂³	semidirect
44. C₂²×D₆₈	C₂³	semidirect
45. C₂×C₂²⋊C₄×C₁₇	C₂³	semidirect
46. C₂×#63227	C₂³	semidirect
47. #10587568	C₂³	direct (N × H)
52. C₂⁴×C₁₇	C₂×D₄	semidirect
64. C₂³×D₁₇	C₂×D₄	semidirect
66. C₂³×D₁₇	C₂⁴	semidirect
67. C₂³×D₁₇	C₂⁴	semidirect
69. C₂²×C₄×C₁₇	C₂⁴	semidirect
71. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
73. C₂²×Dic₁₇	C₂⁴	semidirect
75. C₂×D₆₈	C₂⁴	semidirect
76. C₂²⋊C₄×C₁₇	C₂⁴	semidirect
77. #63227	C₂⁴	semidirect
81. C₂³×C₁₇	#120	semidirect
84. C₂³×C₁₇	C₂²×D₄	semidirect
96. C₂²×D₁₇	C₂²×D₄	semidirect
98. C₂²×D₁₇	C₂⁵	semidirect
100. C₂×C₄×C₁₇	C₂⁵	semidirect
103. C₂×Dic₁₇	C₂⁵	semidirect
110. C₂³×C₂²⋊C₄	D₁₇	semidirect
113. C₂²×C₁₇	C₂×#120	semidirect
116. C₂²×C₁₇	C₂³×D₄	semidirect
125. C₂×D₁₇	C₂³×D₄	semidirect
139. C₂⁴×C₄	C₂×D₁₇	semidirect
142. C₂²×C₂²⋊C₄	C₂×D₁₇	semidirect
144. C₂×C₁₇	C₂²×#120	semidirect
161. C₂⁵	D₆₈	semidirect
171. C₂³×C₄	C₂²×D₁₇	semidirect
174. C₂×C₂²⋊C₄	C₂²×D₁₇	semidirect
175. C₁₇	C₂³×#120	semidirect
192. C₂⁴	C₂×D₆₈	semidirect
203. C₂²×C₄	C₂³×D₁₇	semidirect
205. C₂²⋊C₄	C₂³×D₁₇	semidirect
220. C₂³	#10587568	direct (N × H)
222. C₂³	C₂²×D₆₈	semidirect
231. C₂×C₄	C₂⁴×D₁₇	semidirect
242. C₂²	C₂×#10587568	direct (N × H)
245. C₂²	C₂³×D₆₈	semidirect
252. C₂	C₂²×#10587568	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×D₆₈	2
C₂⁴×C₂²⋊D₁₇	2
C₂²×#10587568	2
C₂³×#63227	2
C₂³×C₂²⋊C₄×C₁₇	2
C₂³×#120	17

Subgroups — C23×#10587568 #17940956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587568 #17940956