Subgroups — #22574003

Order 8576.

1540 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	C₂×#14524089	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864057	C₂	1	4	yes	maximal, complemented
3	#16864034	C₂	1	12	yes	maximal, complemented
4	#16863970	C₂	1	3	yes	maximal, complemented
5	#16863960	C₂	1	3	yes	maximal, complemented
6	#16863942	C₂	1	3	yes	maximal, complemented
7	C₂×#14524032	C₂	1	3	yes	maximal, complemented
8	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#14524089	—	2	8	no	—
11	C₂×#11761629	C₂²	1	2	yes	—
12	C₂×#11761628	C₂²	1	3	yes	—
13	C₂²×D₂₆₈	C₂²	1	3	yes	—
14	C₂×D₄×D₆₇	C₂²	1	6	yes	—
15	#14524041	C₂²	1	3	yes	complemented
16	#14524039	C₂²	1	3	yes	complemented
17	#14524032	C₂²	1	12	yes	complemented
18	#14524032	C₂²	1	3	yes	complemented
19	#14524032	C₂²	1	6	yes	complemented
20	#14524030	C₂²	1	1	yes	characteristic, complemented
21	#14524026	C₂²	1	12	yes	complemented
22	#14524019	C₂²	1	12	yes	complemented
23	#14524017	C₂²	1	12	yes	complemented
24	#14524015	C₂²	1	12	yes	complemented
25	#14524011	C₂²	1	12	yes	complemented
26	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
27	D₄×Dic₆₇	C₂²	1	6	yes	complemented
28	C₄×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
29	#14524002	C₂²	1	3	yes	complemented
30	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	12	yes	complemented
31	C₂×C₄×Dic₆₇	C₂²	1	3	yes	complemented
32	2₋¹⁺⁴×C₆₇	—	2	8	no	—
33	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#41×C₆₇	C₂²	1	6	yes	complemented
35	C₂×#41×C₆₇	C₂²	1	3	yes	complemented
36	C₂²×Q₈×C₆₇	C₂²	1	2	yes	complemented
37	C₂²×Q₈×C₆₇	C₂²	1	3	yes	complemented
38	C₂³×D₆₇	C₂³	1	6	yes	—
39	#11761629	—	2	8	no	—
40	#11761628	—	2	12	no	—
41	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
42	C₂×D₂₆₈	—	2	12	no	—
43	C₂×D₂₆₈	C₂³	1	6	yes	—
44	C₂×D₂₆₈	C₂³	1	3	yes	—
45	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
46	C₂×C₂²⋊D₆₇	C₂³	1	12	yes	—
47	D₄×D₆₇	—	2	24	no	—
48	C₂²×Dic₆₇	C₂³	1	6	yes	—
49	C₂×C₄×D₆₇	C₂³	1	6	yes	—
50	C₄⋊Dic₆₇	C₂³	1	6	yes	—
51	Dic₆₇⋊C₄	C₂³	1	12	yes	—
52	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
53	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
54	(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
55	(C₂×D₆₇)⋊C₄	C₂³	1	24	yes	—
56	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
57	C₄×Dic₆₇	C₂³	1	3	yes	—
58	C₄×Dic₆₇	C₂³	1	6	yes	—
59	#41×C₆₇	—	2	24	no	—
60	#41×C₆₇	—	2	12	no	—
61	#41×C₆₇	D₄	1	8	yes	complemented
62	C₂×Q₈×C₆₇	—	2	8	no	—
63	C₂×Q₈×C₆₇	—	2	12	no	—
64	C₂×Q₈×C₆₇	C₂³	1	6	yes	—
65	C₂×Q₈×C₆₇	C₂³	1	3	yes	—
66	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₆₇	C₂³	1	6	yes	—
68	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₆₇	C₂³	1	3	yes	—
70	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
71	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
72	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
73	C₂³×C₆₇	C₂⁴	1	2	yes	—
74	C₂³×C₆₇	C₂⁴	1	3	yes	—
75	C₂²×D₆₇	—	2	12	no	—
76	C₂²×D₆₇	—	4	12	no	—
77	C₂²×D₆₇	C₂⁴	1	6	yes	—
78	Q₈×C₆₇	—	2	6	no	—
79	Q₈×C₆₇	—	2	12	no	—
80	Q₈×C₆₇	C₂×D₄	1	4	yes	—
81	D₄×C₆₇	—	2	12	no	—
82	D₄×C₆₇	—	2	2	no	—
83	D₄×C₆₇	C₂×D₄	1	12	yes	—
84	C₂×C₄×C₆₇	—	2	12	no	—
85	C₂×C₄×C₆₇	—	2	12	no	—
86	C₂×C₄×C₆₇	C₂×D₄	1	12	yes	—
87	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
88	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
90	C₂²⋊D₆₇	—	2	12	no	—
91	C₂²⋊D₆₇	—	4	12	no	—
92	C₂×Dic₆₇	—	4	6	no	—
93	C₂×Dic₆₇	C₂⁴	1	2	yes	—
94	C₂×Dic₆₇	C₂⁴	1	6	yes	—
95	C₂×Dic₆₇	C₂⁴	1	2	yes	—
96	D₂₆₈	—	2	12	no	—
97	D₂₆₈	—	2	6	no	—
98	C₄×D₆₇	—	2	12	no	—
99	Dic₁₃₄	—	2	2	no	—
100	C₂²×C₆₇	—	2	6	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	4	no	—
103	C₂²×C₆₇	C₂²×D₄	1	6	yes	—
104	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
105	C₂×D₆₇	—	2	6	no	—
106	C₂×D₆₇	—	2	12	no	—
107	C₂×D₆₇	—	8	12	no	—
108	C₄×C₆₇	—	2	6	no	—
109	C₄×C₆₇	C₂²×D₄	1	2	yes	—
110	C₄×C₆₇	C₂²×D₄	1	6	yes	—
111	Dic₆₇	—	2	2	no	—
112	Dic₆₇	—	2	2	no	—
113	Dic₆₇	—	2	6	no	—
114	C₂×C₆₇	—	2	6	no	—
115	C₂×C₆₇	—	4	2	no	—
116	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₆₇	#585	1	2	yes	—
118	D₆₇	—	4	6	no	—
119	#3500	—	67	1	no	maximal, Sylow
120	C₆₇	#3500	1	1	yes	characteristic, complemented, Sylow
121	C₂×2₋¹⁺⁴	D₆₇	1	1	yes	characteristic, complemented
122	C₂×H(5,𝔽₂)	—	67	1	no	—
123	#548	—	67	4	no	—
124	#546	—	67	12	no	—
125	#536	—	67	3	no	—
126	#535	—	67	3	no	—
127	#532	—	67	3	no	—
128	C₂×#124	—	67	3	no	—
129	C₄×#41	—	67	1	no	—
130	2₋¹⁺⁴	—	2	8	no	—
131	H(5,𝔽₂)	—	134	8	no	—
132	C₂×#41	—	67	2	no	—
133	C₂×#41	—	67	3	no	—
134	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
135	C₂×#41	C₂×D₆₇	1	3	yes	complemented
136	C₂×#41	C₂×D₆₇	1	6	yes	complemented
137	C₂²×Q₈	C₂×D₆₇	1	3	yes	complemented
138	C₂²×Q₈	C₂×D₆₇	1	2	yes	complemented
139	C₂²×D₄	—	67	6	no	—
140	C₂²×D₄	—	67	3	no	—
141	C₄⋊Q₈	—	67	3	no	—
142	#127	—	67	3	no	—
143	#124	—	67	3	no	—
144	#124	—	67	6	no	—
145	#124	—	67	12	no	—
146	#124	—	67	1	no	—
147	#123	—	67	12	no	—
148	C₂²⋊Q₈	—	67	12	no	—
149	#121	—	67	12	no	—
150	#121	—	67	12	no	—
151	#120	—	67	12	no	—
152	C₄×Q₈	—	67	2	no	—
153	C₄×D₄	—	67	6	no	—
154	C₄×D₄	—	67	12	no	—
155	#117	—	67	3	no	—
156	C₂×C₂²⋊C₄	—	67	12	no	—
157	C₂×C₄²	—	67	3	no	—
158	C₂⁴	—	67	6	no	—
159	#41	—	2	24	no	—
160	#41	—	2	12	no	—
161	#41	—	134	8	no	—
162	#41	—	134	12	no	—
163	#41	C₂²⋊D₆₇	1	8	yes	complemented
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	12	no	—
166	C₂×Q₈	—	67	1	no	—
167	C₂×Q₈	C₂²×D₆₇	1	6	yes	—
168	C₂×Q₈	C₂²×D₆₇	1	3	yes	—
169	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×D₄	—	67	12	no	—
171	C₂×D₄	—	67	6	no	—
172	C₂×D₄	—	67	6	no	—
173	C₂×D₄	—	67	3	no	—
174	C₂×D₄	—	134	24	no	—
175	C₂×D₄	—	134	12	no	—
176	C₂×D₄	C₂²×D₆₇	1	6	yes	—
177	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×D₄	C₂²×D₆₇	1	3	yes	—
179	C₂²×C₄	—	67	6	no	—
180	C₂²×C₄	—	67	6	no	—
181	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
182	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
183	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
184	C₄⋊C₄	—	67	6	no	—
185	C₄⋊C₄	—	67	12	no	—
186	C₂²⋊C₄	—	67	6	no	—
187	C₂²⋊C₄	—	67	4	no	—
188	C₂²⋊C₄	—	67	12	no	—
189	C₂²⋊C₄	—	67	24	no	—
190	C₄²	—	67	1	no	—
191	C₄²	—	67	3	no	—
192	C₄²	—	67	6	no	—
193	C₂³	—	268	12	no	—
194	C₂³	—	67	6	no	—
195	C₂³	—	134	12	no	—
196	C₂³	C₂³×D₆₇	1	2	yes	—
197	C₂³	C₂³×D₆₇	1	3	yes	—
198	Q₈	—	2	12	no	—
199	Q₈	—	2	6	no	—
200	Q₈	—	134	2	no	—
201	Q₈	C₂×C₂²⋊D₆₇	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	2	no	—
204	D₄	—	268	12	no	—
205	D₄	—	134	12	no	—
206	D₄	—	134	12	no	—
207	D₄	—	134	6	no	—
208	D₄	C₂×C₂²⋊D₆₇	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	268	6	no	—
212	C₂×C₄	—	67	2	no	—
213	C₂×C₄	—	67	6	no	—
214	C₂×C₄	—	67	2	no	—
215	C₂×C₄	—	134	12	no	—
216	C₂×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
217	C₂×C₄	C₂³×D₆₇	1	6	yes	—
218	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₆₇	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	536	12	no	—
224	C₂²	—	134	6	no	—
225	C₂²	—	134	12	no	—
226	C₂²	C₂²×C₂²⋊D₆₇	1	6	yes	—
227	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
228	C₄	—	2	6	no	—
229	C₄	—	134	2	no	—
230	C₄	—	134	2	no	—
231	C₄	—	134	6	no	—
232	C₄	C₂²×C₂²⋊D₆₇	1	6	yes	—
233	C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	2	no	—
236	C₂	—	268	6	no	—
237	C₂	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂	#16864233	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	C₂×2₋¹⁺⁴×C₆₇	C₂
hypercenter	C₂×2₋¹⁺⁴	D₆₇
hypocenter	C₆₇	#3500
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#14524089	C₂	semidirect
2. #16864057	C₂	semidirect
3. #16864034	C₂	semidirect
4. #16863970	C₂	semidirect
5. #16863960	C₂	semidirect
6. #16863942	C₂	semidirect
7. C₂×#14524032	C₂	semidirect
8. #41×Dic₆₇	C₂	semidirect
9. C₂×2₋¹⁺⁴×C₆₇	C₂	semidirect
15. #14524041	C₂²	semidirect
16. #14524039	C₂²	semidirect
17. #14524032	C₂²	semidirect
18. #14524032	C₂²	semidirect
19. #14524032	C₂²	semidirect
20. #14524030	C₂²	semidirect
21. #14524026	C₂²	semidirect
22. #14524019	C₂²	semidirect
23. #14524017	C₂²	semidirect
24. #14524015	C₂²	semidirect
25. #14524011	C₂²	semidirect
26. Q₈×Dic₆₇	C₂²	semidirect
27. D₄×Dic₆₇	C₂²	semidirect
28. C₄×C₂²⋊D₆₇	C₂²	semidirect
29. #14524002	C₂²	semidirect
30. C₂×(C₂×D₆₇)⋊C₄	C₂²	semidirect
31. C₂×C₄×Dic₆₇	C₂²	semidirect
34. C₂×#41×C₆₇	C₂²	semidirect
35. C₂×#41×C₆₇	C₂²	semidirect
36. C₂²×Q₈×C₆₇	C₂²	semidirect
37. C₂²×Q₈×C₆₇	C₂²	semidirect
61. #41×C₆₇	D₄	semidirect
120. C₆₇	#3500	semidirect
121. C₂×2₋¹⁺⁴	D₆₇	semidirect
135. C₂×#41	C₂×D₆₇	semidirect
136. C₂×#41	C₂×D₆₇	semidirect
137. C₂²×Q₈	C₂×D₆₇	semidirect
138. C₂²×Q₈	C₂×D₆₇	semidirect
163. #41	C₂²⋊D₆₇	semidirect