Subgroups — C₂³×#14524011 #31762461

Order 17152.

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×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#14524011	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×(C₂×D₆₇)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂⁴×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₇	—	2	32	no	—
8	C₂⁵×D₆₇	—	2	16	no	—
9	C₂⁵×D₆₇	C₂²	1	2	yes	—
10	C₂⁵×D₆₇	C₂²	1	28	yes	complemented
11	C₂³×C₂²⋊D₆₇	—	2	32	no	—
12	C₂³×C₂²⋊D₆₇	C₂²	1	56	yes	complemented
13	C₂⁴×Dic₆₇	C₂²	1	2	yes	complemented
14	C₂×#14524011	C₂²	1	448	yes	direct product factor
15	C₂²×C₂²⋊Dic₆₇	C₂²	1	28	yes	complemented
16	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	56	yes	complemented
17	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
18	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄×C₆₇	—	2	16	no	—
20	C₂³×D₄×C₆₇	C₂²	1	28	yes	complemented
21	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂⁴×D₆₇	—	2	512	no	—
23	C₂⁴×D₆₇	—	2	448	no	—
24	C₂⁴×D₆₇	—	2	16	no	—
25	C₂⁴×D₆₇	—	2	16	no	—
26	C₂⁴×D₆₇	—	2	224	no	—
27	C₂⁴×D₆₇	D₄	1	32	yes	complemented
28	C₂⁴×D₆₇	C₂³	1	28	yes	—
29	C₂⁴×D₆₇	C₂³	1	112	yes	complemented
30	C₂²×C₂²⋊D₆₇	—	2	448	no	—
31	C₂²×C₂²⋊D₆₇	C₂³	1	224	yes	complemented
32	C₂³×Dic₆₇	—	2	16	no	—
33	C₂³×Dic₆₇	C₂³	1	28	yes	complemented
34	#14524011	C₂³	1	512	yes	direct product factor
35	C₂×C₂²⋊Dic₆₇	C₂³	1	112	yes	complemented
36	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	224	yes	complemented
37	C₂⁵×C₆₇	—	2	16	no	—
38	C₂⁵×C₆₇	—	2	8	no	—
39	C₂⁵×C₆₇	—	2	16	no	—
40	C₂⁵×C₆₇	D₄	1	16	yes	complemented
41	C₂⁵×C₆₇	C₂³	1	14	yes	—
42	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
43	C₂⁵×C₆₇	C₂³	1	14	yes	complemented
44	C₂²×D₄×C₆₇	—	2	224	no	—
45	C₂²×D₄×C₆₇	C₂³	1	112	yes	complemented
46	C₂³×C₄×C₆₇	—	2	8	no	—
47	C₂³×C₄×C₆₇	C₂³	1	14	yes	complemented
48	C₂³×D₆₇	—	2	3584	no	—
49	C₂³×D₆₇	—	2	256	no	—
50	C₂³×D₆₇	—	2	112	no	—
51	C₂³×D₆₇	—	2	112	no	—
52	C₂³×D₆₇	—	2	448	no	—
53	C₂³×D₆₇	—	2	896	no	—
54	C₂³×D₆₇	C₂×D₄	1	224	yes	complemented
55	C₂³×D₆₇	C₂⁴	1	56	yes	—
56	C₂³×D₆₇	C₂⁴	1	64	yes	complemented
57	C₂×C₂²⋊D₆₇	—	2	896	no	—
58	C₂×C₂²⋊D₆₇	C₂⁴	1	128	yes	complemented
59	C₂²×Dic₆₇	—	2	112	no	—
60	C₂²×Dic₆₇	C₂⁴	1	56	yes	complemented
61	C₂²⋊Dic₆₇	C₂⁴	1	64	yes	complemented
62	(C₂×D₆₇)⋊C₄	C₂⁴	1	128	yes	complemented
63	C₂⁴×C₆₇	—	2	128	no	—
64	C₂⁴×C₆₇	—	2	112	no	—
65	C₂⁴×C₆₇	—	2	56	no	—
66	C₂⁴×C₆₇	—	2	112	no	—
67	C₂⁴×C₆₇	—	4	64	no	—
68	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
69	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
70	C₂⁴×C₆₇	C₂×D₄	1	112	yes	complemented
71	C₂⁴×C₆₇	C₂⁴	1	28	yes	—
72	C₂⁴×C₆₇	C₂⁴	1	7	yes	—
73	C₂⁴×C₆₇	C₂⁴	1	28	yes	complemented
74	C₂×D₄×C₆₇	—	2	448	no	—
75	C₂×D₄×C₆₇	C₂⁴	1	64	yes	complemented
76	C₂²×C₄×C₆₇	—	2	56	no	—
77	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	complemented
78	C₂³×C₆₇	—	2	448	no	—
79	C₂³×C₆₇	—	2	112	no	—
80	C₂³×C₆₇	—	2	56	no	—
81	C₂³×C₆₇	—	2	112	no	—
82	C₂³×C₆₇	—	4	224	no	—
83	C₂³×C₆₇	#120	1	64	yes	complemented
84	C₂³×C₆₇	C₂²×D₄	1	56	yes	—
85	C₂³×C₆₇	C₂²×D₄	1	28	yes	—
86	C₂³×C₆₇	C₂²×D₄	1	112	yes	complemented
87	C₂³×C₆₇	C₂⁵	1	8	yes	—
88	C₂³×C₆₇	C₂⁵	1	7	yes	—
89	C₂³×C₆₇	C₂⁵	1	8	yes	complemented
90	C₂²×D₆₇	—	2	3584	no	—
91	C₂²×D₆₇	—	2	896	no	—
92	C₂²×D₆₇	—	2	112	no	—
93	C₂²×D₆₇	—	2	112	no	—
94	C₂²×D₆₇	—	2	128	no	—
95	C₂²×D₆₇	—	2	256	no	—
96	C₂²×D₆₇	C₂²×D₄	1	224	yes	complemented
97	C₂²×D₆₇	C₂⁵	1	16	yes	—
98	D₄×C₆₇	—	2	128	no	—
99	C₂×C₄×C₆₇	—	2	56	no	—
100	C₂×C₄×C₆₇	C₂⁵	1	8	yes	complemented
101	C₂²⋊D₆₇	—	2	256	no	—
102	C₂×Dic₆₇	—	2	112	no	—
103	C₂×Dic₆₇	C₂⁵	1	16	yes	complemented
104	C₂²×C₆₇	—	2	16	no	—
105	C₂²×C₆₇	—	2	224	no	—
106	C₂²×C₆₇	—	2	16	no	—
107	C₂²×C₆₇	—	2	8	no	—
108	C₂²×C₆₇	—	4	112	no	—
109	C₂²×C₆₇	C₂×#120	1	112	yes	complemented
110	C₂²×C₆₇	C₂³×D₄	1	28	yes	—
111	C₂²×C₆₇	C₂³×D₄	1	14	yes	—
112	C₂²×C₆₇	C₂³×D₄	1	16	yes	complemented
113	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
114	C₂×D₆₇	—	2	512	no	—
115	C₂×D₆₇	—	2	16	no	—
116	C₂×D₆₇	—	2	16	no	—
117	C₂×D₆₇	—	2	448	no	—
118	C₂×D₆₇	C₂³×D₄	1	32	yes	complemented
119	C₄×C₆₇	—	2	8	no	—
120	Dic₆₇	—	2	16	no	—
121	C₂³×#120	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	16	no	—
123	C₂×C₆₇	—	4	8	no	—
124	C₂×C₆₇	C₂²×#120	1	28	yes	complemented
125	C₂×C₆₇	C₂⁴×D₄	1	2	yes	—
126	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
127	D₆₇	—	2	32	no	—
128	C₂⁷	—	67	1	no	—
129	C₂⁴×D₄	—	67	2	no	—
130	C₂⁴×D₄	D₆₇	1	1	yes	characteristic, complemented
131	C₂²×#120	—	67	56	no	—
132	C₂³×C₂²⋊C₄	—	67	1	no	—
133	C₂³×C₂²⋊C₄	—	67	2	no	—
134	C₆₇	C₂³×#120	1	1	yes	characteristic, complemented, Sylow
135	C₂⁶	—	67	2	no	—
136	C₂⁶	—	67	28	no	—
137	C₂⁶	—	134	32	no	—
138	C₂⁶	—	134	16	no	—
139	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
140	C₂⁶	C₂×D₆₇	1	1	yes	characteristic, complemented
141	C₂³×D₄	—	2	16	no	—
142	C₂³×D₄	—	67	56	no	—
143	C₂³×D₄	—	134	32	no	—
144	C₂³×D₄	C₂×D₆₇	1	28	yes	complemented
145	C₂⁴×C₄	—	67	2	no	—
146	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
147	C₂×#120	—	67	448	no	—
148	C₂²×C₂²⋊C₄	—	67	28	no	—
149	C₂²×C₂²⋊C₄	—	67	56	no	—
150	C₂⁵	—	2	16	no	—
151	C₂⁵	—	2	16	no	—
152	C₂⁵	—	2	8	no	—
153	C₂⁵	—	67	32	no	—
154	C₂⁵	—	67	28	no	—
155	C₂⁵	—	67	112	no	—
156	C₂⁵	—	134	448	no	—
157	C₂⁵	—	134	512	no	—
158	C₂⁵	—	134	16	no	—
159	C₂⁵	—	134	16	no	—
160	C₂⁵	—	134	224	no	—
161	C₂⁵	C₂²⋊D₆₇	1	16	yes	complemented
162	C₂⁵	C₂²×D₆₇	1	14	yes	—
163	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
164	C₂⁵	C₂²×D₆₇	1	14	yes	complemented
165	C₂²×D₄	—	2	224	no	—
166	C₂²×D₄	—	67	224	no	—
167	C₂²×D₄	—	134	448	no	—
168	C₂²×D₄	C₂²×D₆₇	1	112	yes	complemented
169	C₂³×C₄	—	2	8	no	—
170	C₂³×C₄	—	67	28	no	—
171	C₂³×C₄	—	134	16	no	—
172	C₂³×C₄	C₂²×D₆₇	1	14	yes	complemented
173	#120	—	67	512	no	—
174	C₂×C₂²⋊C₄	—	67	112	no	—
175	C₂×C₂²⋊C₄	—	67	224	no	—
176	C₂⁴	—	2	128	no	—
177	C₂⁴	—	2	112	no	—
178	C₂⁴	—	2	112	no	—
179	C₂⁴	—	2	56	no	—
180	C₂⁴	—	4	64	no	—
181	C₂⁴	—	67	224	no	—
182	C₂⁴	—	67	56	no	—
183	C₂⁴	—	67	64	no	—
184	C₂⁴	—	134	896	no	—
185	C₂⁴	—	134	3584	no	—
186	C₂⁴	—	134	256	no	—
187	C₂⁴	—	134	112	no	—
188	C₂⁴	—	134	112	no	—
189	C₂⁴	—	134	448	no	—
190	C₂⁴	D₄×D₆₇	1	16	yes	—
191	C₂⁴	C₂×C₂²⋊D₆₇	1	8	yes	—
192	C₂⁴	C₂×C₂²⋊D₆₇	1	112	yes	complemented
193	C₂⁴	C₂³×D₆₇	1	28	yes	—
194	C₂⁴	C₂³×D₆₇	1	7	yes	—
195	C₂⁴	C₂³×D₆₇	1	28	yes	complemented
196	C₂×D₄	—	2	448	no	—
197	C₂×D₄	—	67	128	no	—
198	C₂×D₄	—	134	896	no	—
199	C₂×D₄	C₂³×D₆₇	1	64	yes	complemented
200	C₂²×C₄	—	2	56	no	—
201	C₂²×C₄	—	67	56	no	—
202	C₂²×C₄	—	134	112	no	—
203	C₂²×C₄	C₂³×D₆₇	1	28	yes	complemented
204	C₂²⋊C₄	—	67	64	no	—
205	C₂²⋊C₄	—	67	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	—	67	224	no	—
212	C₂³	—	67	16	no	—
213	C₂³	—	134	256	no	—
214	C₂³	—	134	3584	no	—
215	C₂³	—	134	896	no	—
216	C₂³	—	134	112	no	—
217	C₂³	—	134	112	no	—
218	C₂³	—	134	128	no	—
219	C₂³	#14524011	1	64	yes	direct product factor
220	C₂³	C₂×D₄×D₆₇	1	56	yes	—
221	C₂³	C₂²×C₂²⋊D₆₇	1	28	yes	—
222	C₂³	C₂²×C₂²⋊D₆₇	1	112	yes	complemented
223	C₂³	C₂⁴×D₆₇	1	8	yes	—
224	C₂³	C₂⁴×D₆₇	1	7	yes	—
225	C₂³	C₂⁴×D₆₇	1	8	yes	complemented
226	D₄	—	2	128	no	—
227	D₄	—	134	256	no	—
228	C₂×C₄	—	2	56	no	—
229	C₂×C₄	—	67	16	no	—
230	C₂×C₄	—	134	112	no	—
231	C₂×C₄	C₂⁴×D₆₇	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	—	67	32	no	—
238	C₂²	—	134	512	no	—
239	C₂²	—	134	448	no	—
240	C₂²	—	134	16	no	—
241	C₂²	—	134	16	no	—
242	C₂²	C₂×#14524011	1	112	yes	direct product factor
243	C₂²	C₂²×D₄×D₆₇	1	28	yes	—
244	C₂²	C₂³×C₂²⋊D₆₇	1	14	yes	—
245	C₂²	C₂³×C₂²⋊D₆₇	1	16	yes	complemented
246	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	134	16	no	—
249	C₂	—	2	16	no	—
250	C₂	—	4	8	no	—
251	C₂	—	134	32	no	—
252	C₂	C₂²×#14524011	1	28	yes	direct product factor
253	C₂	C₂³×D₄×D₆₇	1	2	yes	—
254	C₂	C₂⁴×C₂²⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂⁴×D₄×C₆₇	C₂
socle	C₂⁵×C₆₇	C₂³
hypercenter	C₂⁴×D₄	D₆₇
hypocenter	C₆₇	C₂³×#120
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₆₇	C₂	semidirect
2. C₂⁴×C₂²⋊D₆₇	C₂	semidirect
3. C₂²×#14524011	C₂	direct (N × H)
4. C₂³×C₂²⋊Dic₆₇	C₂	semidirect
5. C₂³×(C₂×D₆₇)⋊C₄	C₂	semidirect
6. C₂⁴×D₄×C₆₇	C₂	semidirect
10. C₂⁵×D₆₇	C₂²	semidirect
12. C₂³×C₂²⋊D₆₇	C₂²	semidirect
13. C₂⁴×Dic₆₇	C₂²	semidirect
14. C₂×#14524011	C₂²	direct (N × H)
15. C₂²×C₂²⋊Dic₆₇	C₂²	semidirect
16. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
18. C₂⁶×C₆₇	C₂²	semidirect
20. C₂³×D₄×C₆₇	C₂²	semidirect
21. C₂⁴×C₄×C₆₇	C₂²	semidirect
27. C₂⁴×D₆₇	D₄	semidirect
29. C₂⁴×D₆₇	C₂³	semidirect
31. C₂²×C₂²⋊D₆₇	C₂³	semidirect
33. C₂³×Dic₆₇	C₂³	semidirect
34. #14524011	C₂³	direct (N × H)
35. C₂×C₂²⋊Dic₆₇	C₂³	semidirect
36. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
40. C₂⁵×C₆₇	D₄	semidirect
43. C₂⁵×C₆₇	C₂³	semidirect
45. C₂²×D₄×C₆₇	C₂³	semidirect
47. C₂³×C₄×C₆₇	C₂³	semidirect
54. C₂³×D₆₇	C₂×D₄	semidirect
56. C₂³×D₆₇	C₂⁴	semidirect
58. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
60. C₂²×Dic₆₇	C₂⁴	semidirect
61. C₂²⋊Dic₆₇	C₂⁴	semidirect
62. (C₂×D₆₇)⋊C₄	C₂⁴	semidirect
70. C₂⁴×C₆₇	C₂×D₄	semidirect
73. C₂⁴×C₆₇	C₂⁴	semidirect
75. C₂×D₄×C₆₇	C₂⁴	semidirect
77. C₂²×C₄×C₆₇	C₂⁴	semidirect
83. C₂³×C₆₇	#120	semidirect
86. C₂³×C₆₇	C₂²×D₄	semidirect
89. C₂³×C₆₇	C₂⁵	semidirect
96. C₂²×D₆₇	C₂²×D₄	semidirect
100. C₂×C₄×C₆₇	C₂⁵	semidirect
103. C₂×Dic₆₇	C₂⁵	semidirect
109. C₂²×C₆₇	C₂×#120	semidirect
112. C₂²×C₆₇	C₂³×D₄	semidirect
118. C₂×D₆₇	C₂³×D₄	semidirect
124. C₂×C₆₇	C₂²×#120	semidirect
130. C₂⁴×D₄	D₆₇	semidirect
134. C₆₇	C₂³×#120	semidirect
140. C₂⁶	C₂×D₆₇	semidirect
144. C₂³×D₄	C₂×D₆₇	semidirect
146. C₂⁴×C₄	C₂×D₆₇	semidirect
161. C₂⁵	C₂²⋊D₆₇	semidirect
164. C₂⁵	C₂²×D₆₇	semidirect
168. C₂²×D₄	C₂²×D₆₇	semidirect
172. C₂³×C₄	C₂²×D₆₇	semidirect
192. C₂⁴	C₂×C₂²⋊D₆₇	semidirect
195. C₂⁴	C₂³×D₆₇	semidirect
199. C₂×D₄	C₂³×D₆₇	semidirect
203. C₂²×C₄	C₂³×D₆₇	semidirect
219. C₂³	#14524011	direct (N × H)
222. C₂³	C₂²×C₂²⋊D₆₇	semidirect
225. C₂³	C₂⁴×D₆₇	semidirect
231. C₂×C₄	C₂⁴×D₆₇	semidirect
242. C₂²	C₂×#14524011	direct (N × H)
245. C₂²	C₂³×C₂²⋊D₆₇	semidirect
252. C₂	C₂²×#14524011	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×D₆₇	2
C₂⁴×C₂²⋊D₆₇	2
C₂²×#14524011	2
C₂³×C₂²⋊Dic₆₇	2
C₂³×(C₂×D₆₇)⋊C₄	2
C₂⁴×D₄×C₆₇	2
C₂³×#120	67

Subgroups — C23×#14524011 #31762461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14524011 #31762461