Subgroups — #22573983

Order 8576.

1748 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#16864065	C₂	1	4	yes	maximal, complemented
3	D₄×C₂²⋊D₆₇	C₂	1	12	yes	maximal, complemented
4	#16863963	C₂	1	3	yes	maximal, complemented
5	#16863960	C₂	1	3	yes	maximal, complemented
6	#16863942	C₂	1	3	yes	maximal, complemented
7	C₂×#14524039	C₂	1	3	yes	maximal, complemented
8	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#14524089	—	2	8	no	—
11	C₂×#11761629	C₂²	1	2	yes	complemented
12	C₂×#11761628	C₂²	1	3	yes	complemented
13	C₂²×D₂₆₈	C₂²	1	3	yes	complemented
14	C₂²×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
15	C₂×D₄×D₆₇	C₂²	1	6	yes	complemented
16	#14524041	C₂²	1	1	yes	characteristic, complemented
17	#14524039	C₂²	1	12	yes	complemented
18	#14524039	C₂²	1	3	yes	complemented
19	#14524039	C₂²	1	6	yes	complemented
20	#14524032	C₂²	1	3	yes	complemented
21	#14524030	C₂²	1	3	yes	complemented
22	#14524017	C₂²	1	12	yes	complemented
23	#14524015	C₂²	1	12	yes	complemented
24	#14524014	C₂²	1	12	yes	complemented
25	#14524012	C₂²	1	12	yes	complemented
26	#14524011	C₂²	1	12	yes	complemented
27	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
28	D₄×Dic₆₇	C₂²	1	6	yes	complemented
29	C₄×C₂²⋊D₆₇	C₂²	1	12	yes	complemented
30	#14524002	C₂²	1	3	yes	complemented
31	C₂×C₄×Dic₆₇	C₂²	1	3	yes	complemented
32	H(5,𝔽₂)×C₆₇	—	2	8	no	—
33	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#41×C₆₇	C₂²	1	2	yes	complemented
35	C₂×#41×C₆₇	C₂²	1	3	yes	complemented
36	C₂²×D₄×C₆₇	C₂²	1	6	yes	complemented
37	C₂²×D₄×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	—	4	24	no	—
46	C₂×C₂²⋊D₆₇	C₂³	1	12	yes	—
47	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
48	C₂×C₂²⋊D₆₇	C₂³	1	24	yes	—
49	D₄×D₆₇	—	2	24	no	—
50	C₂²×Dic₆₇	C₂³	1	6	yes	—
51	C₂×C₄×D₆₇	C₂³	1	6	yes	—
52	C₄⋊Dic₆₇	C₂³	1	6	yes	—
53	Dic₆₇⋊C₄	C₂³	1	4	yes	—
54	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
55	C₂²⋊Dic₆₇	C₂³	1	12	yes	—
56	(C₂×D₆₇)⋊C₄	C₂³	1	12	yes	—
57	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₄×Dic₆₇	C₂³	1	3	yes	—
59	C₄×Dic₆₇	C₂³	1	6	yes	—
60	C₂⁴×C₆₇	C₂³	1	6	yes	—
61	#41×C₆₇	—	2	8	no	—
62	#41×C₆₇	—	2	12	no	—
63	#41×C₆₇	D₄	1	8	yes	complemented
64	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₆₇	—	2	24	no	—
67	C₂×D₄×C₆₇	—	2	12	no	—
68	C₂×D₄×C₆₇	C₂³	1	6	yes	—
69	C₂×D₄×C₆₇	C₂³	1	6	yes	—
70	C₂×D₄×C₆₇	C₂³	1	3	yes	—
71	C₂×D₄×C₆₇	C₂³	1	3	yes	—
72	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
73	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
74	C₂³×C₆₇	—	2	12	no	—
75	C₂³×C₆₇	—	4	12	no	—
76	C₂³×C₆₇	C₂⁴	1	6	yes	—
77	C₂³×C₆₇	C₂⁴	1	3	yes	—
78	C₂²×D₆₇	—	2	12	no	—
79	C₂²×D₆₇	—	4	12	no	—
80	C₂²×D₆₇	C₂⁴	1	6	yes	—
81	Q₈×C₆₇	—	2	2	no	—
82	Q₈×C₆₇	C₂×D₄	1	4	yes	complemented
83	D₄×C₆₇	—	2	12	no	—
84	D₄×C₆₇	—	2	6	no	—
85	D₄×C₆₇	—	2	12	no	—
86	D₄×C₆₇	C₂×D₄	1	12	yes	complemented
87	C₂×C₄×C₆₇	—	2	12	no	—
88	C₂×C₄×C₆₇	C₂×D₄	1	12	yes	complemented
89	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
90	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
92	C₂²⋊D₆₇	—	2	12	no	—
93	C₂²⋊D₆₇	—	2	24	no	—
94	C₂²⋊D₆₇	—	4	24	no	—
95	C₂×Dic₆₇	—	2	12	no	—
96	C₂×Dic₆₇	C₂⁴	1	6	yes	—
97	C₂×Dic₆₇	C₂⁴	1	2	yes	—
98	C₂×Dic₆₇	C₂⁴	1	2	yes	—
99	D₂₆₈	—	2	12	no	—
100	D₂₆₈	—	2	6	no	—
101	C₄×D₆₇	—	2	12	no	—
102	Dic₁₃₄	—	2	2	no	—
103	C₂²×C₆₇	—	2	6	no	—
104	C₂²×C₆₇	—	2	6	no	—
105	C₂²×C₆₇	—	2	12	no	—
106	C₂²×C₆₇	—	8	12	no	—
107	C₂²×C₆₇	C₂²×D₄	1	6	yes	—
108	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
109	C₂×D₆₇	—	2	6	no	—
110	C₂×D₆₇	—	2	12	no	—
111	C₂×D₆₇	—	8	12	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	C₂²×D₄	1	2	yes	—
114	C₄×C₆₇	C₂²×D₄	1	6	yes	—
115	Dic₆₇	—	2	2	no	—
116	Dic₆₇	—	2	6	no	—
117	Dic₆₇	H(5,𝔽₂)	1	4	yes	complemented
118	C₂×C₆₇	—	2	6	no	—
119	C₂×C₆₇	—	4	6	no	—
120	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
121	C₂×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₆₇	—	4	6	no	—
123	#3498	—	67	1	no	maximal, Sylow
124	C₆₇	#3498	1	1	yes	characteristic, complemented, Sylow
125	C₂×H(5,𝔽₂)	—	67	1	no	—
126	C₂×H(5,𝔽₂)	D₆₇	1	1	yes	characteristic, complemented
127	#550	—	67	4	no	—
128	D₄²	—	67	12	no	—
129	#535	—	67	3	no	—
130	#535	—	67	3	no	—
131	#532	—	67	3	no	—
132	C₂×#127	—	67	3	no	—
133	C₄×#41	—	67	1	no	—
134	H(5,𝔽₂)	—	2	8	no	—
135	H(5,𝔽₂)	—	134	8	no	—
136	C₂×#41	—	67	2	no	—
137	C₂×#41	—	67	3	no	—
138	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
139	C₂×#41	C₂×D₆₇	1	3	yes	complemented
140	C₂×#41	C₂×D₆₇	1	2	yes	complemented
141	C₂²×D₄	—	67	12	no	—
142	C₂²×D₄	—	67	6	no	—
143	C₂²×D₄	—	67	3	no	—
144	C₂²×D₄	C₂×D₆₇	1	3	yes	complemented
145	C₂²×D₄	C₂×D₆₇	1	6	yes	complemented
146	C₄⋊Q₈	—	67	1	no	—
147	#127	—	67	3	no	—
148	#127	—	67	6	no	—
149	#127	—	67	12	no	—
150	#124	—	67	3	no	—
151	#124	—	67	3	no	—
152	#121	—	67	12	no	—
153	#121	—	67	12	no	—
154	#121	—	67	12	no	—
155	#120	—	67	12	no	—
156	#120	—	67	12	no	—
157	C₄×Q₈	—	67	2	no	—
158	C₄×D₄	—	67	6	no	—
159	C₄×D₄	—	67	12	no	—
160	#117	—	67	3	no	—
161	C₂×C₄²	—	67	3	no	—
162	C₂⁴	—	67	6	no	—
163	C₂⁴	C₂²×D₆₇	1	6	yes	—
164	#41	—	2	8	no	—
165	#41	—	2	12	no	—
166	#41	—	134	8	no	—
167	#41	—	134	12	no	—
168	#41	C₂²⋊D₆₇	1	8	yes	complemented
169	C₂×Q₈	—	67	1	no	—
170	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	268	24	no	—
175	C₂×D₄	—	67	24	no	—
176	C₂×D₄	—	67	12	no	—
177	C₂×D₄	—	67	6	no	—
178	C₂×D₄	—	67	6	no	—
179	C₂×D₄	—	67	3	no	—
180	C₂×D₄	—	134	24	no	—
181	C₂×D₄	—	134	12	no	—
182	C₂×D₄	C₂²×D₆₇	1	6	yes	—
183	C₂×D₄	C₂²×D₆₇	1	6	yes	—
184	C₂×D₄	C₂²×D₆₇	1	3	yes	—
185	C₂×D₄	C₂²×D₆₇	1	3	yes	—
186	C₂²×C₄	—	67	6	no	—
187	C₂²×C₄	—	67	6	no	—
188	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
189	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
190	C₄⋊C₄	—	67	6	no	—
191	C₄⋊C₄	—	67	4	no	—
192	C₂²⋊C₄	—	67	6	no	—
193	C₂²⋊C₄	—	67	12	no	—
194	C₂²⋊C₄	—	67	12	no	—
195	C₄²	—	67	1	no	—
196	C₄²	—	67	3	no	—
197	C₄²	—	67	6	no	—
198	C₂³	—	2	12	no	—
199	C₂³	—	4	12	no	—
200	C₂³	—	268	12	no	—
201	C₂³	—	67	6	no	—
202	C₂³	—	134	12	no	—
203	C₂³	C₂³×D₆₇	1	6	yes	—
204	C₂³	C₂³×D₆₇	1	3	yes	—
205	Q₈	—	2	2	no	—
206	Q₈	—	134	2	no	—
207	Q₈	C₂×C₂²⋊D₆₇	1	4	yes	complemented
208	D₄	—	2	12	no	—
209	D₄	—	2	12	no	—
210	D₄	—	2	6	no	—
211	D₄	—	268	24	no	—
212	D₄	—	134	24	no	—
213	D₄	—	134	12	no	—
214	D₄	—	134	12	no	—
215	D₄	—	134	6	no	—
216	D₄	C₂×C₂²⋊D₆₇	1	12	yes	complemented
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	67	2	no	—
219	C₂×C₄	—	67	6	no	—
220	C₂×C₄	—	67	2	no	—
221	C₂×C₄	—	134	12	no	—
222	C₂×C₄	—	134	12	no	—
223	C₂×C₄	C₂×C₂²⋊D₆₇	1	12	yes	complemented
224	C₂×C₄	C₂³×D₆₇	1	2	yes	—
225	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₆₇	1	3	yes	—
227	C₂²	—	2	6	no	—
228	C₂²	—	2	6	no	—
229	C₂²	—	2	12	no	—
230	C₂²	—	8	12	no	—
231	C₂²	—	536	12	no	—
232	C₂²	—	134	6	no	—
233	C₂²	—	134	12	no	—
234	C₂²	C₂²×C₂²⋊D₆₇	1	6	yes	—
235	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	67	4	no	—
238	C₄	—	134	2	no	—
239	C₄	—	134	6	no	—
240	C₄	C₂²×C₂²⋊D₆₇	1	6	yes	—
241	C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
242	C₂	—	2	6	no	—
243	C₂	—	4	6	no	—
244	C₂	—	268	6	no	—
245	C₂	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
246	C₂	H(5,𝔽₂)×D₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₆₇	C₂
hypercenter	C₂×H(5,𝔽₂)	D₆₇
hypocenter	C₆₇	#3498
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#14524089	C₂	semidirect
2. #16864065	C₂	semidirect
3. D₄×C₂²⋊D₆₇	C₂	semidirect
4. #16863963	C₂	semidirect
5. #16863960	C₂	semidirect
6. #16863942	C₂	semidirect
7. C₂×#14524039	C₂	semidirect
8. #41×Dic₆₇	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₆₇	C₂	semidirect
11. C₂×#11761629	C₂²	semidirect
12. C₂×#11761628	C₂²	semidirect
13. C₂²×D₂₆₈	C₂²	semidirect
14. C₂²×C₂²⋊D₆₇	C₂²	semidirect
15. C₂×D₄×D₆₇	C₂²	semidirect
16. #14524041	C₂²	semidirect
17. #14524039	C₂²	semidirect
18. #14524039	C₂²	semidirect
19. #14524039	C₂²	semidirect
20. #14524032	C₂²	semidirect
21. #14524030	C₂²	semidirect
22. #14524017	C₂²	semidirect
23. #14524015	C₂²	semidirect
24. #14524014	C₂²	semidirect
25. #14524012	C₂²	semidirect
26. #14524011	C₂²	semidirect
27. Q₈×Dic₆₇	C₂²	semidirect
28. D₄×Dic₆₇	C₂²	semidirect
29. C₄×C₂²⋊D₆₇	C₂²	semidirect
30. #14524002	C₂²	semidirect
31. C₂×C₄×Dic₆₇	C₂²	semidirect
34. C₂×#41×C₆₇	C₂²	semidirect
35. C₂×#41×C₆₇	C₂²	semidirect
36. C₂²×D₄×C₆₇	C₂²	semidirect
37. C₂²×D₄×C₆₇	C₂²	semidirect
63. #41×C₆₇	D₄	semidirect
82. Q₈×C₆₇	C₂×D₄	semidirect
86. D₄×C₆₇	C₂×D₄	semidirect
88. C₂×C₄×C₆₇	C₂×D₄	semidirect
117. Dic₆₇	H(5,𝔽₂)	semidirect
124. C₆₇	#3498	semidirect
126. C₂×H(5,𝔽₂)	D₆₇	semidirect
139. C₂×#41	C₂×D₆₇	semidirect
140. C₂×#41	C₂×D₆₇	semidirect
144. C₂²×D₄	C₂×D₆₇	semidirect
145. C₂²×D₄	C₂×D₆₇	semidirect
168. #41	C₂²⋊D₆₇	semidirect
207. Q₈	C₂×C₂²⋊D₆₇	semidirect
216. D₄	C₂×C₂²⋊D₆₇	semidirect
223. C₂×C₄	C₂×C₂²⋊D₆₇	semidirect