Subgroups — C₂×#577×C₆₇ #22558013

Order 8576.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#577×C₆₇	C₂	1	16	yes	maximal, direct product factor
4	C₂×#137×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×AGL(1,ℤ/8)×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂×#135×C₆₇	C₂	1	3	yes	maximal, complemented
7	C₂²×SD₁₆×C₆₇	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	2₋¹⁺⁴×C₆₇	—	2	4	no	—
10	2₋¹⁺⁴×C₆₇	C₂²	1	8	yes	complemented
11	H(5,𝔽₂)×C₆₇	—	2	4	no	—
12	H(5,𝔽₂)×C₆₇	C₂²	1	8	yes	complemented
13	C₂×#41×C₆₇	—	2	1	no	—
14	C₂×#41×C₆₇	—	2	3	no	—
15	C₂×#41×C₆₇	C₂²	1	3	yes	—
16	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#41×C₆₇	C₂²	1	3	yes	complemented
18	C₂²×Q₈×C₆₇	—	2	1	no	—
19	C₂²×Q₈×C₆₇	C₂²	1	3	yes	complemented
20	C₂²×D₄×C₆₇	—	2	3	no	—
21	C₂²×D₄×C₆₇	C₂²	1	3	yes	—
22	#137×C₆₇	C₂²	1	24	yes	complemented
23	AGL(1,ℤ/8)×C₆₇	C₂²	1	24	yes	complemented
24	#135×C₆₇	C₂²	1	24	yes	complemented
25	C₂×Q₁₆×C₆₇	C₂²	1	3	yes	complemented
26	C₂×SD₁₆×C₆₇	C₂²	1	3	yes	complemented
27	C₂×SD₁₆×C₆₇	C₂²	1	6	yes	complemented
28	C₂×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×SD₁₆×C₆₇	C₂²	1	24	yes	complemented
30	C₂×D₈×C₆₇	C₂²	1	3	yes	complemented
31	#131×C₆₇	C₂²	1	8	yes	complemented
32	C₂×M₁₆×C₆₇	C₂²	1	3	yes	complemented
33	C₂²×C₈×C₆₇	C₂²	1	3	yes	complemented
34	C₂⁴×C₆₇	—	2	3	no	—
35	#41×C₆₇	—	2	8	no	—
36	#41×C₆₇	—	2	24	no	—
37	#41×C₆₇	—	2	6	no	—
38	#41×C₆₇	—	2	6	no	—
39	#41×C₆₇	D₄	1	4	yes	—
40	#41×C₆₇	C₂³	1	12	yes	—
41	#41×C₆₇	C₂³	1	4	yes	—
42	#41×C₆₇	C₂³	1	12	yes	complemented
43	C₂×Q₈×C₆₇	—	2	8	no	—
44	C₂×Q₈×C₆₇	—	2	3	no	—
45	C₂×Q₈×C₆₇	—	2	6	no	—
46	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₆₇	C₂³	1	3	yes	—
48	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₆₇	C₂³	1	12	yes	complemented
50	C₂×D₄×C₆₇	—	2	3	no	—
51	C₂×D₄×C₆₇	—	2	3	no	—
52	C₂×D₄×C₆₇	—	2	24	no	—
53	C₂×D₄×C₆₇	—	2	3	no	—
54	C₂×D₄×C₆₇	—	2	6	no	—
55	C₂×D₄×C₆₇	D₄	1	3	yes	—
56	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₆₇	C₂³	1	3	yes	—
58	C₂×D₄×C₆₇	C₂³	1	12	yes	—
59	C₂²×C₄×C₆₇	—	2	3	no	—
60	C₂²×C₄×C₆₇	—	2	3	no	—
61	C₂²×C₄×C₆₇	—	2	3	no	—
62	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
63	Q₁₆×C₆₇	C₂³	1	12	yes	complemented
64	SD₁₆×C₆₇	C₂³	1	12	yes	complemented
65	SD₁₆×C₆₇	C₂³	1	24	yes	complemented
66	SD₁₆×C₆₇	C₂³	1	4	yes	complemented
67	D₈×C₆₇	C₂³	1	12	yes	complemented
68	M₁₆×C₆₇	C₂³	1	12	yes	complemented
69	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂³	1	3	yes	—
71	C₂×C₈×C₆₇	C₂³	1	12	yes	complemented
72	C₂³×C₆₇	—	2	1	no	—
73	C₂³×C₆₇	—	2	3	no	—
74	C₂³×C₆₇	—	2	12	no	—
75	C₂³×C₆₇	—	8	6	no	—
76	C₂³×C₆₇	C₂×D₄	1	3	yes	—
77	Q₈×C₆₇	—	2	3	no	—
78	Q₈×C₆₇	—	2	12	no	—
79	Q₈×C₆₇	—	2	1	no	—
80	Q₈×C₆₇	C₂×D₄	1	4	yes	—
81	Q₈×C₆₇	C₂⁴	1	6	yes	—
82	Q₈×C₆₇	C₂⁴	1	2	yes	—
83	D₄×C₆₇	—	2	12	no	—
84	D₄×C₆₇	—	2	12	no	—
85	D₄×C₆₇	—	2	12	no	—
86	D₄×C₆₇	—	2	3	no	—
87	D₄×C₆₇	—	2	1	no	—
88	D₄×C₆₇	C₂×D₄	1	12	yes	—
89	D₄×C₆₇	C₂⁴	1	2	yes	—
90	D₄×C₆₇	C₂⁴	1	6	yes	—
91	C₂×C₄×C₆₇	—	2	12	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	3	no	—
94	C₂×C₄×C₆₇	—	2	12	no	—
95	C₂×C₄×C₆₇	—	2	12	no	—
96	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
97	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
98	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
99	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₈×C₆₇	C₂⁴	1	2	yes	—
101	C₈×C₆₇	C₂⁴	1	6	yes	—
102	C₂²×C₆₇	—	2	6	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	6	no	—
105	C₂²×C₆₇	—	4	2	no	—
106	C₂²×C₆₇	—	4	6	no	—
107	C₂²×C₆₇	—	8	12	no	—
108	C₂²×C₆₇	C₂²×D₄	1	6	yes	—
109	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	6	no	—
112	C₄×C₆₇	C₂²×D₄	1	6	yes	—
113	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂⁵	1	1	yes	characteristic
115	C₂×C₆₇	—	2	6	no	—
116	C₂×C₆₇	—	4	2	no	—
117	C₂×C₆₇	—	4	6	no	—
118	C₂×C₆₇	#577	1	2	yes	direct product factor
119	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
120	C₂×#577	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₆₇	C₂×#577	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
123	C₂×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
124	#577	C₂×C₆₇	1	16	yes	direct product factor
125	C₂×#137	C₂×C₆₇	1	3	yes	complemented
126	C₂×AGL(1,ℤ/8)	C₂×C₆₇	1	3	yes	complemented
127	C₂×#135	C₂×C₆₇	1	3	yes	complemented
128	C₂²×SD₁₆	C₂×C₆₇	1	3	yes	complemented
129	C₂×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	2	4	no	—
131	2₋¹⁺⁴	C₂²×C₆₇	1	8	yes	complemented
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	C₂²×C₆₇	1	8	yes	complemented
134	C₂×#41	—	2	1	no	—
135	C₂×#41	—	2	3	no	—
136	C₂×#41	C₂²×C₆₇	1	3	yes	—
137	C₂×#41	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×#41	C₂²×C₆₇	1	3	yes	complemented
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×Q₈	C₂²×C₆₇	1	3	yes	complemented
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	C₂²×C₆₇	1	3	yes	—
143	#137	C₂²×C₆₇	1	24	yes	complemented
144	AGL(1,ℤ/8)	C₂²×C₆₇	1	24	yes	complemented
145	#135	C₂²×C₆₇	1	24	yes	complemented
146	C₂×Q₁₆	C₂²×C₆₇	1	3	yes	complemented
147	C₂×SD₁₆	C₂²×C₆₇	1	3	yes	complemented
148	C₂×SD₁₆	C₂²×C₆₇	1	6	yes	complemented
149	C₂×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
150	C₂×SD₁₆	C₂²×C₆₇	1	24	yes	complemented
151	C₂×D₈	C₂²×C₆₇	1	3	yes	complemented
152	#131	C₂²×C₆₇	1	8	yes	complemented
153	C₂×M₁₆	C₂²×C₆₇	1	3	yes	complemented
154	C₂²×C₈	C₂²×C₆₇	1	3	yes	complemented
155	C₂⁴	—	2	3	no	—
156	#41	—	2	8	no	—
157	#41	—	2	24	no	—
158	#41	—	2	6	no	—
159	#41	—	2	6	no	—
160	#41	D₄×C₆₇	1	4	yes	—
161	#41	C₂³×C₆₇	1	12	yes	—
162	#41	C₂³×C₆₇	1	4	yes	—
163	#41	C₂³×C₆₇	1	12	yes	complemented
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	3	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	D₄×C₆₇	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₆₇	1	3	yes	—
169	C₂×Q₈	C₂³×C₆₇	1	1	yes	characteristic
170	C₂×Q₈	C₂³×C₆₇	1	12	yes	complemented
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	24	no	—
174	C₂×D₄	—	2	3	no	—
175	C₂×D₄	—	2	6	no	—
176	C₂×D₄	D₄×C₆₇	1	3	yes	—
177	C₂×D₄	C₂³×C₆₇	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₆₇	1	3	yes	—
179	C₂×D₄	C₂³×C₆₇	1	12	yes	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	2	3	no	—
182	C₂²×C₄	—	2	3	no	—
183	C₂²×C₄	C₂³×C₆₇	1	3	yes	—
184	Q₁₆	C₂³×C₆₇	1	12	yes	complemented
185	SD₁₆	C₂³×C₆₇	1	12	yes	complemented
186	SD₁₆	C₂³×C₆₇	1	24	yes	complemented
187	SD₁₆	C₂³×C₆₇	1	4	yes	complemented
188	D₈	C₂³×C₆₇	1	12	yes	complemented
189	M₁₆	C₂³×C₆₇	1	12	yes	complemented
190	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
191	C₂×C₈	C₂³×C₆₇	1	3	yes	—
192	C₂×C₈	C₂³×C₆₇	1	12	yes	complemented
193	C₂³	—	2	1	no	—
194	C₂³	—	2	3	no	—
195	C₂³	—	2	12	no	—
196	C₂³	—	8	6	no	—
197	C₂³	C₂×D₄×C₆₇	1	3	yes	—
198	Q₈	—	2	3	no	—
199	Q₈	—	2	12	no	—
200	Q₈	—	2	1	no	—
201	Q₈	C₂×D₄×C₆₇	1	4	yes	—
202	Q₈	C₂⁴×C₆₇	1	6	yes	—
203	Q₈	C₂⁴×C₆₇	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	12	no	—
207	D₄	—	2	3	no	—
208	D₄	—	2	1	no	—
209	D₄	C₂×D₄×C₆₇	1	12	yes	—
210	D₄	C₂⁴×C₆₇	1	2	yes	—
211	D₄	C₂⁴×C₆₇	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	3	no	—
214	C₂×C₄	—	2	12	no	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	C₂×D₄×C₆₇	1	3	yes	—
218	C₂×C₄	C₂×D₄×C₆₇	1	6	yes	—
219	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₆₇	1	6	yes	—
221	C₈	C₂⁴×C₆₇	1	2	yes	—
222	C₈	C₂⁴×C₆₇	1	6	yes	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	6	no	—
226	C₂²	—	4	6	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	8	12	no	—
229	C₂²	C₂²×D₄×C₆₇	1	6	yes	—
230	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	6	no	—
233	C₄	C₂²×D₄×C₆₇	1	6	yes	—
234	C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
235	C₄	C₂⁵×C₆₇	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	4	2	no	—
239	C₂	#577×C₆₇	1	2	yes	direct product factor
240	C₂	C₂³×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂×H(5,𝔽₂)×C₆₇	C₂	semidirect
3. #577×C₆₇	C₂	direct (N × H)
4. C₂×#137×C₆₇	C₂	semidirect
5. C₂×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
6. C₂×#135×C₆₇	C₂	semidirect
7. C₂²×SD₁₆×C₆₇	C₂	semidirect
8. C₂×#131×C₆₇	C₂	semidirect
10. 2₋¹⁺⁴×C₆₇	C₂²	semidirect
12. H(5,𝔽₂)×C₆₇	C₂²	semidirect
17. C₂×#41×C₆₇	C₂²	semidirect
19. C₂²×Q₈×C₆₇	C₂²	semidirect
22. #137×C₆₇	C₂²	semidirect
23. AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
24. #135×C₆₇	C₂²	semidirect
25. C₂×Q₁₆×C₆₇	C₂²	semidirect
26. C₂×SD₁₆×C₆₇	C₂²	semidirect
27. C₂×SD₁₆×C₆₇	C₂²	semidirect
28. C₂×SD₁₆×C₆₇	C₂²	semidirect
29. C₂×SD₁₆×C₆₇	C₂²	semidirect
30. C₂×D₈×C₆₇	C₂²	semidirect
31. #131×C₆₇	C₂²	semidirect
32. C₂×M₁₆×C₆₇	C₂²	semidirect
33. C₂²×C₈×C₆₇	C₂²	semidirect
42. #41×C₆₇	C₂³	semidirect
49. C₂×Q₈×C₆₇	C₂³	semidirect
63. Q₁₆×C₆₇	C₂³	semidirect
64. SD₁₆×C₆₇	C₂³	semidirect
65. SD₁₆×C₆₇	C₂³	semidirect
66. SD₁₆×C₆₇	C₂³	semidirect
67. D₈×C₆₇	C₂³	semidirect
68. M₁₆×C₆₇	C₂³	semidirect
71. C₂×C₈×C₆₇	C₂³	semidirect
118. C₂×C₆₇	#577	direct (N × H)
120. C₂×#577	C₆₇	direct (N × H)
121. C₆₇	C₂×#577	direct (N × H)
122. C₂×2₋¹⁺⁴	C₂×C₆₇	semidirect
123. C₂×H(5,𝔽₂)	C₂×C₆₇	semidirect
124. #577	C₂×C₆₇	direct (N × H)
125. C₂×#137	C₂×C₆₇	semidirect
126. C₂×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
127. C₂×#135	C₂×C₆₇	semidirect
128. C₂²×SD₁₆	C₂×C₆₇	semidirect
129. C₂×#131	C₂×C₆₇	semidirect
131. 2₋¹⁺⁴	C₂²×C₆₇	semidirect
133. H(5,𝔽₂)	C₂²×C₆₇	semidirect
138. C₂×#41	C₂²×C₆₇	semidirect
140. C₂²×Q₈	C₂²×C₆₇	semidirect
143. #137	C₂²×C₆₇	semidirect
144. AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
145. #135	C₂²×C₆₇	semidirect
146. C₂×Q₁₆	C₂²×C₆₇	semidirect
147. C₂×SD₁₆	C₂²×C₆₇	semidirect
148. C₂×SD₁₆	C₂²×C₆₇	semidirect
149. C₂×SD₁₆	C₂²×C₆₇	semidirect
150. C₂×SD₁₆	C₂²×C₆₇	semidirect
151. C₂×D₈	C₂²×C₆₇	semidirect
152. #131	C₂²×C₆₇	semidirect
153. C₂×M₁₆	C₂²×C₆₇	semidirect
154. C₂²×C₈	C₂²×C₆₇	semidirect
163. #41	C₂³×C₆₇	semidirect
170. C₂×Q₈	C₂³×C₆₇	semidirect
184. Q₁₆	C₂³×C₆₇	semidirect
185. SD₁₆	C₂³×C₆₇	semidirect
186. SD₁₆	C₂³×C₆₇	semidirect
187. SD₁₆	C₂³×C₆₇	semidirect
188. D₈	C₂³×C₆₇	semidirect
189. M₁₆	C₂³×C₆₇	semidirect
192. C₂×C₈	C₂³×C₆₇	semidirect
239. C₂	#577×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₆₇	2
C₂×H(5,𝔽₂)×C₆₇	2
#577×C₆₇	2
C₂×#137×C₆₇	2
C₂×AGL(1,ℤ/8)×C₆₇	2
C₂×#135×C₆₇	2
C₂²×SD₁₆×C₆₇	2
C₂×#131×C₆₇	2
C₂×#577	67

Subgroups — C2×#577×C67 #22558013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#577×C₆₇ #22558013