Subgroups — C₂³×#14524088 #31769717

Order 17152.

65726 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524088	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11761628	C₂	1	2	yes	maximal, complemented
3	C₂³×#11761627	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₂²⋊D₆₇	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×D₆₇	C₂	1	4	yes	maximal, complemented
6	C₂⁴×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₇	C₂²	1	4	yes	complemented
8	C₂×#14524088	C₂²	1	1792	yes	direct product factor
9	C₂²×#11761628	C₂²	1	112	yes	complemented
10	C₂²×#11761627	C₂²	1	224	yes	complemented
11	C₂³×Dic₁₃₄	C₂²	1	2	yes	complemented
12	C₂³×D₂₆₈	C₂²	1	2	yes	complemented
13	C₂³×C₂²⋊D₆₇	C₂²	1	8	yes	complemented
14	C₂³×C₂²⋊D₆₇	C₂²	1	4	yes	complemented
15	C₂³×C₂²⋊D₆₇	C₂²	1	224	yes	complemented
16	C₂²×D₄×D₆₇	C₂²	1	224	yes	complemented
17	C₂⁴×Dic₆₇	C₂²	1	4	yes	complemented
18	C₂³×C₄×D₆₇	C₂²	1	4	yes	complemented
19	C₂⁶×C₆₇	C₂²	1	2	yes	complemented
20	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
21	C₂³×D₄×C₆₇	C₂²	1	56	yes	complemented
22	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂⁴×D₆₇	—	4	32	no	—
24	C₂⁴×D₆₇	C₂³	1	4	yes	—
25	C₂⁴×D₆₇	C₂³	1	112	yes	complemented
26	#14524088	C₂³	1	4096	yes	direct product factor
27	C₂×#11761628	C₂³	1	896	yes	complemented
28	C₂×#11761627	C₂³	1	1792	yes	complemented
29	C₂²×Dic₁₃₄	C₂³	1	56	yes	complemented
30	C₂²×D₂₆₈	C₂³	1	56	yes	complemented
31	C₂²×C₂²⋊D₆₇	C₂³	1	224	yes	complemented
32	C₂²×C₂²⋊D₆₇	C₂³	1	112	yes	complemented
33	C₂²×C₂²⋊D₆₇	C₂³	1	1792	yes	complemented
34	C₂×D₄×D₆₇	C₂³	1	1792	yes	complemented
35	C₂³×Dic₆₇	C₂³	1	4	yes	—
36	C₂³×Dic₆₇	C₂³	1	112	yes	complemented
37	C₂²×C₄×D₆₇	C₂³	1	112	yes	complemented
38	C₂⁵×C₆₇	—	4	16	no	—
39	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
40	C₂⁵×C₆₇	C₂³	1	4	yes	—
41	C₂⁵×C₆₇	C₂³	1	56	yes	complemented
42	C₂²×D₄×C₆₇	C₂³	1	112	yes	complemented
43	C₂²×D₄×C₆₇	C₂³	1	448	yes	complemented
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	28	yes	complemented
46	C₂³×D₆₇	—	2	32	no	—
47	C₂³×D₆₇	—	4	448	no	—
48	C₂³×D₆₇	C₂⁴	1	56	yes	—
49	C₂³×D₆₇	C₂⁴	1	448	yes	complemented
50	#11761628	C₂⁴	1	1024	yes	complemented
51	#11761627	C₂⁴	1	2048	yes	complemented
52	C₂×Dic₁₃₄	C₂⁴	1	224	yes	complemented
53	C₂×D₂₆₈	C₂⁴	1	224	yes	complemented
54	C₂×C₂²⋊D₆₇	C₂⁴	1	896	yes	complemented
55	C₂×C₂²⋊D₆₇	C₂⁴	1	448	yes	complemented
56	C₂×C₂²⋊D₆₇	C₂⁴	1	2048	yes	complemented
57	D₄×D₆₇	C₂⁴	1	2048	yes	complemented
58	C₂²×Dic₆₇	C₂⁴	1	56	yes	—
59	C₂²×Dic₆₇	C₂⁴	1	448	yes	complemented
60	C₂×C₄×D₆₇	C₂⁴	1	448	yes	complemented
61	C₂⁴×C₆₇	—	2	8	no	—
62	C₂⁴×C₆₇	—	2	32	no	—
63	C₂⁴×C₆₇	—	4	224	no	—
64	C₂⁴×C₆₇	C₂⁴	1	14	yes	—
65	C₂⁴×C₆₇	C₂⁴	1	56	yes	—
66	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂⁴×C₆₇	C₂⁴	1	224	yes	complemented
68	C₂×D₄×C₆₇	C₂⁴	1	448	yes	complemented
69	C₂×D₄×C₆₇	C₂⁴	1	512	yes	complemented
70	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	112	yes	complemented
72	C₂³×C₆₇	—	2	56	no	—
73	C₂³×C₆₇	—	2	224	no	—
74	C₂³×C₆₇	—	4	448	no	—
75	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	complemented
76	C₂³×C₆₇	C₂⁵	1	28	yes	—
77	C₂³×C₆₇	C₂⁵	1	112	yes	—
78	C₂³×C₆₇	C₂⁵	1	7	yes	—
79	C₂³×C₆₇	C₂⁵	1	128	yes	complemented
80	C₂²×D₆₇	—	2	224	no	—
81	C₂²×D₆₇	—	4	896	no	—
82	C₂²×D₆₇	C₂⁵	1	112	yes	—
83	C₂²×D₆₇	C₂⁵	1	256	yes	complemented
84	D₄×C₆₇	C₂⁵	1	256	yes	complemented
85	C₂×C₄×C₆₇	C₂⁵	1	56	yes	—
86	C₂×C₄×C₆₇	C₂⁵	1	64	yes	complemented
87	C₂²⋊D₆₇	C₂⁵	1	512	yes	complemented
88	C₂²⋊D₆₇	C₂⁵	1	256	yes	complemented
89	C₂×Dic₆₇	C₂⁵	1	112	yes	—
90	C₂×Dic₆₇	C₂⁵	1	256	yes	complemented
91	D₂₆₈	C₂⁵	1	128	yes	complemented
92	C₄×D₆₇	C₂⁵	1	256	yes	complemented
93	Dic₁₃₄	C₂⁵	1	128	yes	complemented
94	C₂²×C₆₇	—	2	224	no	—
95	C₂²×C₆₇	—	2	56	no	—
96	C₂²×C₆₇	—	4	128	no	—
97	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	28	yes	complemented
98	C₂²×C₆₇	C₂⁶	1	32	yes	—
99	C₂²×C₆₇	C₂⁶	1	8	yes	—
100	C₂²×C₆₇	C₂⁶	1	7	yes	—
101	C₂×D₆₇	—	2	224	no	—
102	C₂×D₆₇	—	4	256	no	—
103	C₂×D₆₇	C₂⁶	1	32	yes	—
104	C₄×C₆₇	C₂⁶	1	16	yes	—
105	Dic₆₇	C₂⁶	1	32	yes	—
106	C₂³×H(5,𝔽₂)	—	67	1	no	maximal, Sylow
107	C₂×C₆₇	—	2	32	no	—
108	C₂×C₆₇	—	2	8	no	—
109	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	14	yes	complemented
110	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
111	D₆₇	—	2	32	no	—
112	C₂²×H(5,𝔽₂)	—	67	112	no	—
113	C₂³×#41	—	67	4	no	—
114	C₂³×#41	—	67	2	no	—
115	C₂⁴×D₄	—	67	4	no	—
116	C₂⁴×D₄	—	67	4	no	—
117	C₂⁴×D₄	D₆₇	1	1	yes	characteristic, complemented
118	C₆₇	C₂³×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
119	C₂⁶	—	67	4	no	—
120	C₂⁶	C₂×D₆₇	1	2	yes	complemented
121	C₂×H(5,𝔽₂)	—	67	1792	no	—
122	C₂²×#41	—	67	224	no	—
123	C₂²×#41	—	67	112	no	—
124	C₂³×Q₈	—	67	2	no	—
125	C₂³×D₄	—	67	8	no	—
126	C₂³×D₄	—	67	4	no	—
127	C₂³×D₄	—	67	224	no	—
128	C₂³×D₄	—	67	224	no	—
129	C₂³×D₄	—	67	2	no	—
130	C₂³×D₄	C₂×D₆₇	1	4	yes	complemented
131	C₂³×D₄	C₂×D₆₇	1	56	yes	complemented
132	C₂⁴×C₄	—	67	4	no	—
133	C₂⁴×C₄	—	67	4	no	—
134	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
135	C₂⁵	—	4	16	no	—
136	C₂⁵	—	268	32	no	—
137	C₂⁵	—	67	112	no	—
138	C₂⁵	—	67	4	no	—
139	C₂⁵	C₂²×D₆₇	1	4	yes	—
140	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
141	C₂⁵	C₂²×D₆₇	1	56	yes	complemented
142	H(5,𝔽₂)	—	67	4096	no	—
143	C₂×#41	—	67	1792	no	—
144	C₂×#41	—	67	896	no	—
145	C₂²×Q₈	—	67	56	no	—
146	C₂²×D₄	—	67	224	no	—
147	C₂²×D₄	—	67	112	no	—
148	C₂²×D₄	—	67	1792	no	—
149	C₂²×D₄	—	67	1792	no	—
150	C₂²×D₄	—	67	56	no	—
151	C₂²×D₄	C₂²×D₆₇	1	112	yes	complemented
152	C₂²×D₄	C₂²×D₆₇	1	448	yes	complemented
153	C₂³×C₄	—	67	112	no	—
154	C₂³×C₄	—	67	4	no	—
155	C₂³×C₄	—	67	112	no	—
156	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
157	C₂³×C₄	C₂²×D₆₇	1	28	yes	complemented
158	C₂⁴	—	2	8	no	—
159	C₂⁴	—	2	32	no	—
160	C₂⁴	—	4	224	no	—
161	C₂⁴	—	268	448	no	—
162	C₂⁴	—	67	448	no	—
163	C₂⁴	—	67	56	no	—
164	C₂⁴	—	134	32	no	—
165	C₂⁴	C₂³×D₆₇	1	56	yes	—
166	C₂⁴	C₂³×D₆₇	1	14	yes	—
167	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
168	C₂⁴	C₂³×D₆₇	1	224	yes	complemented
169	#41	—	67	2048	no	—
170	#41	—	67	1024	no	—
171	C₂×Q₈	—	67	224	no	—
172	C₂×D₄	—	67	896	no	—
173	C₂×D₄	—	67	448	no	—
174	C₂×D₄	—	67	2048	no	—
175	C₂×D₄	—	67	2048	no	—
176	C₂×D₄	—	67	224	no	—
177	C₂×D₄	C₂³×D₆₇	1	448	yes	complemented
178	C₂×D₄	C₂³×D₆₇	1	512	yes	complemented
179	C₂²×C₄	—	67	56	no	—
180	C₂²×C₄	—	67	448	no	—
181	C₂²×C₄	—	67	448	no	—
182	C₂²×C₄	C₂³×D₆₇	1	28	yes	—
183	C₂²×C₄	C₂³×D₆₇	1	112	yes	complemented
184	C₂³	—	2	56	no	—
185	C₂³	—	2	224	no	—
186	C₂³	—	4	448	no	—
187	C₂³	—	268	896	no	—
188	C₂³	—	67	256	no	—
189	C₂³	—	67	112	no	—
190	C₂³	—	134	224	no	—
191	C₂³	#14524088	1	8	yes	direct product factor
192	C₂³	C₂⁴×D₆₇	1	112	yes	—
193	C₂³	C₂⁴×D₆₇	1	28	yes	—
194	C₂³	C₂⁴×D₆₇	1	7	yes	—
195	C₂³	C₂⁴×D₆₇	1	128	yes	complemented
196	Q₈	—	67	128	no	—
197	D₄	—	67	512	no	—
198	D₄	—	67	256	no	—
199	D₄	—	67	128	no	—
200	D₄	C₂⁴×D₆₇	1	256	yes	complemented
201	C₂×C₄	—	67	112	no	—
202	C₂×C₄	—	67	256	no	—
203	C₂×C₄	—	67	256	no	—
204	C₂×C₄	C₂⁴×D₆₇	1	56	yes	—
205	C₂×C₄	C₂⁴×D₆₇	1	64	yes	complemented
206	C₂²	—	2	56	no	—
207	C₂²	—	2	224	no	—
208	C₂²	—	4	128	no	—
209	C₂²	—	268	256	no	—
210	C₂²	—	67	32	no	—
211	C₂²	—	134	224	no	—
212	C₂²	C₂×#14524088	1	28	yes	direct product factor
213	C₂²	C₂⁵×D₆₇	1	32	yes	—
214	C₂²	C₂⁵×D₆₇	1	8	yes	—
215	C₂²	C₂⁵×D₆₇	1	7	yes	—
216	C₄	—	67	32	no	—
217	C₄	C₂⁵×D₆₇	1	16	yes	—
218	C₂	—	2	8	no	—
219	C₂	—	2	32	no	—
220	C₂	—	134	32	no	—
221	C₂	C₂²×#14524088	1	14	yes	direct product factor
222	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂⁴×D₄×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂⁴×D₄	D₆₇
hypocenter	C₆₇	C₂³×H(5,𝔽₂)
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14524088	C₂	direct (N × H)
2. C₂³×#11761628	C₂	semidirect
3. C₂³×#11761627	C₂	semidirect
4. C₂⁴×C₂²⋊D₆₇	C₂	semidirect
5. C₂³×D₄×D₆₇	C₂	semidirect
6. C₂⁴×D₄×C₆₇	C₂	semidirect
7. C₂⁵×D₆₇	C₂²	semidirect
8. C₂×#14524088	C₂²	direct (N × H)
9. C₂²×#11761628	C₂²	semidirect
10. C₂²×#11761627	C₂²	semidirect
11. C₂³×Dic₁₃₄	C₂²	semidirect
12. C₂³×D₂₆₈	C₂²	semidirect
13. C₂³×C₂²⋊D₆₇	C₂²	semidirect
14. C₂³×C₂²⋊D₆₇	C₂²	semidirect
15. C₂³×C₂²⋊D₆₇	C₂²	semidirect
16. C₂²×D₄×D₆₇	C₂²	semidirect
17. C₂⁴×Dic₆₇	C₂²	semidirect
18. C₂³×C₄×D₆₇	C₂²	semidirect
19. C₂⁶×C₆₇	C₂²	semidirect
20. C₂³×D₄×C₆₇	C₂²	semidirect
21. C₂³×D₄×C₆₇	C₂²	semidirect
22. C₂⁴×C₄×C₆₇	C₂²	semidirect
25. C₂⁴×D₆₇	C₂³	semidirect
26. #14524088	C₂³	direct (N × H)
27. C₂×#11761628	C₂³	semidirect
28. C₂×#11761627	C₂³	semidirect
29. C₂²×Dic₁₃₄	C₂³	semidirect
30. C₂²×D₂₆₈	C₂³	semidirect
31. C₂²×C₂²⋊D₆₇	C₂³	semidirect
32. C₂²×C₂²⋊D₆₇	C₂³	semidirect
33. C₂²×C₂²⋊D₆₇	C₂³	semidirect
34. C₂×D₄×D₆₇	C₂³	semidirect
36. C₂³×Dic₆₇	C₂³	semidirect
37. C₂²×C₄×D₆₇	C₂³	semidirect
41. C₂⁵×C₆₇	C₂³	semidirect
42. C₂²×D₄×C₆₇	C₂³	semidirect
43. C₂²×D₄×C₆₇	C₂³	semidirect
45. C₂³×C₄×C₆₇	C₂³	semidirect
49. C₂³×D₆₇	C₂⁴	semidirect
50. #11761628	C₂⁴	semidirect
51. #11761627	C₂⁴	semidirect
52. C₂×Dic₁₃₄	C₂⁴	semidirect
53. C₂×D₂₆₈	C₂⁴	semidirect
54. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
55. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
56. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
57. D₄×D₆₇	C₂⁴	semidirect
59. C₂²×Dic₆₇	C₂⁴	semidirect
60. C₂×C₄×D₆₇	C₂⁴	semidirect
67. C₂⁴×C₆₇	C₂⁴	semidirect
68. C₂×D₄×C₆₇	C₂⁴	semidirect
69. C₂×D₄×C₆₇	C₂⁴	semidirect
71. C₂²×C₄×C₆₇	C₂⁴	semidirect
75. C₂³×C₆₇	H(5,𝔽₂)	semidirect
79. C₂³×C₆₇	C₂⁵	semidirect
83. C₂²×D₆₇	C₂⁵	semidirect
84. D₄×C₆₇	C₂⁵	semidirect
86. C₂×C₄×C₆₇	C₂⁵	semidirect
87. C₂²⋊D₆₇	C₂⁵	semidirect
88. C₂²⋊D₆₇	C₂⁵	semidirect
90. C₂×Dic₆₇	C₂⁵	semidirect
91. D₂₆₈	C₂⁵	semidirect
92. C₄×D₆₇	C₂⁵	semidirect
93. Dic₁₃₄	C₂⁵	semidirect
97. C₂²×C₆₇	C₂×H(5,𝔽₂)	semidirect
109. C₂×C₆₇	C₂²×H(5,𝔽₂)	semidirect
117. C₂⁴×D₄	D₆₇	semidirect
118. C₆₇	C₂³×H(5,𝔽₂)	semidirect
120. C₂⁶	C₂×D₆₇	semidirect
130. C₂³×D₄	C₂×D₆₇	semidirect
131. C₂³×D₄	C₂×D₆₇	semidirect
134. C₂⁴×C₄	C₂×D₆₇	semidirect
141. C₂⁵	C₂²×D₆₇	semidirect
151. C₂²×D₄	C₂²×D₆₇	semidirect
152. C₂²×D₄	C₂²×D₆₇	semidirect
157. C₂³×C₄	C₂²×D₆₇	semidirect
168. C₂⁴	C₂³×D₆₇	semidirect
177. C₂×D₄	C₂³×D₆₇	semidirect
178. C₂×D₄	C₂³×D₆₇	semidirect
183. C₂²×C₄	C₂³×D₆₇	semidirect
191. C₂³	#14524088	direct (N × H)
195. C₂³	C₂⁴×D₆₇	semidirect
200. D₄	C₂⁴×D₆₇	semidirect
205. C₂×C₄	C₂⁴×D₆₇	semidirect
212. C₂²	C₂×#14524088	direct (N × H)
221. C₂	C₂²×#14524088	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×#14524088	2
C₂³×#11761628	2
C₂³×#11761627	2
C₂⁴×C₂²⋊D₆₇	2
C₂³×D₄×D₆₇	2
C₂⁴×D₄×C₆₇	2
C₂³×H(5,𝔽₂)	67

Subgroups — C23×#14524088 #31769717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14524088 #31769717