Subgroups — C₂×#16864074 #22573708

Order 8576.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864074	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14524036	C₂	1	2	yes	maximal, complemented
3	C₂×#14524030	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524025	C₂	1	2	yes	maximal, complemented
5	C₂×#14524021	C₂	1	2	yes	maximal, complemented
6	C₂×#14524011	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×C₂²⋊D₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#14524001	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₆₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₆₇	C₂²	1	2	yes	—
14	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
16	#14524036	C₂²	1	16	yes	complemented
17	#14524030	C₂²	1	8	yes	complemented
18	#14524025	C₂²	1	16	yes	complemented
19	#14524021	C₂²	1	16	yes	complemented
20	#14524011	C₂²	1	8	yes	—
21	C₄×C₂²⋊D₆₇	C₂²	1	16	yes	complemented
22	#14524001	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	complemented
24	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	complemented
25	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	complemented
26	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₆₇	C₂²	1	2	yes	complemented
29	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
30	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
31	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
32	C₂²⋊C₄×D₆₇	C₂²	1	16	yes	complemented
33	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×Dic₆₇	C₂²	1	2	yes	complemented
35	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#123×C₆₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₆₇	—	2	8	no	—
42	C₂³×D₆₇	—	4	2	no	—
43	C₂³×D₆₇	C₂³	1	4	yes	—
44	C₂³×D₆₇	C₂³	1	2	yes	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	C₂³	1	4	yes	complemented
47	C₂×C₂²⋊D₆₇	—	2	8	no	—
48	C₂×C₂²⋊D₆₇	C₂³	1	8	yes	—
49	C₂²×Dic₆₇	C₂³	1	2	yes	—
50	C₂²×Dic₆₇	C₂³	1	2	yes	—
51	C₂²×Dic₆₇	C₂³	1	2	yes	—
52	C₂×C₄×D₆₇	—	2	8	no	—
53	C₂×C₄×D₆₇	—	4	4	no	—
54	C₂×C₄×D₆₇	C₂³	1	8	yes	—
55	C₂×C₄×D₆₇	C₂³	1	8	yes	—
56	C₄⋊Dic₆₇	C₂³	1	8	yes	complemented
57	Dic₆₇⋊C₄	C₂³	1	8	yes	complemented
58	Dic₆₇⋊C₄	C₂³	1	8	yes	complemented
59	C₂²⋊Dic₆₇	C₂³	1	8	yes	—
60	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
61	C₂²⋊Dic₆₇	C₂³	1	8	yes	complemented
62	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
63	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
64	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
65	C₄×Dic₆₇	C₂³	1	4	yes	complemented
66	C₄×Dic₆₇	C₂³	1	8	yes	complemented
67	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₆₇	—	4	2	no	—
70	C₂×D₄×C₆₇	C₂³	1	4	yes	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂³	1	4	yes	complemented
76	C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
77	C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	complemented
79	C₂³×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	C₂⁴	1	2	yes	—
84	C₂³×C₆₇	C₂⁴	1	2	yes	—
85	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂²×D₆₇	—	2	4	no	—
87	C₂²×D₆₇	—	2	4	no	—
88	C₂²×D₆₇	—	2	16	no	—
89	C₂²×D₆₇	—	4	4	no	—
90	C₂²×D₆₇	—	4	16	no	—
91	C₂²×D₆₇	#41	1	8	yes	—
92	C₂²×D₆₇	C₂⁴	1	4	yes	—
93	D₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
99	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
100	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
101	C₂²⋊D₆₇	—	2	16	no	—
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	4	no	—
104	C₂×Dic₆₇	—	2	4	no	—
105	C₂×Dic₆₇	C₂⁴	1	4	yes	—
106	C₂×Dic₆₇	C₂⁴	1	4	yes	—
107	C₂×Dic₆₇	C₂⁴	1	4	yes	—
108	C₄×D₆₇	—	2	16	no	—
109	C₄×D₆₇	—	4	8	no	—
110	Dic₁₃₄	—	4	4	no	—
111	C₂²×C₆₇	—	2	4	no	—
112	C₂²×C₆₇	—	2	4	no	—
113	C₂²×C₆₇	—	2	2	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	4	4	no	—
116	C₂²×C₆₇	—	4	4	no	—
117	C₂²×C₆₇	C₂×#41	1	4	yes	—
118	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
120	C₂×D₆₇	—	2	16	no	—
121	C₂×D₆₇	—	2	4	no	—
122	C₂×D₆₇	—	2	4	no	—
123	C₂×D₆₇	—	4	16	no	—
124	C₂×D₆₇	C₂×#41	1	8	yes	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	4	no	—
128	Dic₆₇	—	2	4	no	—
129	Dic₆₇	—	2	4	no	—
130	Dic₆₇	—	2	4	no	—
131	C₂×C₆₇	—	4	2	no	—
132	C₂×C₆₇	—	4	2	no	—
133	C₂×C₆₇	#551	1	4	yes	complemented
134	C₂×C₆₇	C₂²×#41	1	2	yes	—
135	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	D₆₇	—	2	8	no	—
137	C₂×#551	—	67	1	no	maximal, Sylow
138	C₆₇	C₂×#551	1	1	yes	characteristic, complemented, Sylow
139	#551	—	67	16	no	—
140	C₂×#126	—	67	2	no	—
141	C₂×#124	—	67	1	no	—
142	C₂×#123	—	67	2	no	—
143	C₂×#123	D₆₇	1	1	yes	characteristic, complemented
144	C₂×C₂²⋊Q₈	—	67	2	no	—
145	C₂×#120	—	67	1	no	—
146	C₂×C₄×D₄	—	67	2	no	—
147	C₂×#117	—	67	2	no	—
148	C₂²×C₂²⋊C₄	—	67	2	no	—
149	C₂⁵	—	67	1	no	—
150	C₂²×Q₈	—	67	1	no	—
151	C₂²×D₄	—	67	2	no	—
152	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
153	C₂³×C₄	—	67	2	no	—
154	C₂³×C₄	—	67	2	no	—
155	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
156	#126	—	67	16	no	—
157	#124	—	67	8	no	—
158	#123	—	67	16	no	—
159	#123	C₂×D₆₇	1	8	yes	complemented
160	C₂²⋊Q₈	—	67	16	no	—
161	#120	—	67	8	no	—
162	C₄×D₄	—	67	16	no	—
163	#117	—	67	16	no	—
164	C₂×C₄⋊C₄	—	67	2	no	—
165	C₂×C₄⋊C₄	—	67	2	no	—
166	C₂×C₄⋊C₄	—	67	2	no	—
167	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	complemented
168	C₂×C₂²⋊C₄	—	67	2	no	—
169	C₂×C₂²⋊C₄	—	67	2	no	—
170	C₂×C₂²⋊C₄	—	67	1	no	—
171	C₂×C₂²⋊C₄	—	67	2	no	—
172	C₂×C₂²⋊C₄	—	67	2	no	—
173	C₂×C₂²⋊C₄	—	67	2	no	—
174	C₂×C₂²⋊C₄	—	67	16	no	—
175	C₂×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic, complemented
177	C₂×C₄²	—	67	1	no	—
178	C₂×C₄²	—	67	2	no	—
179	C₂⁴	—	268	2	no	—
180	C₂⁴	—	67	4	no	—
181	C₂⁴	—	67	2	no	—
182	C₂⁴	—	134	8	no	—
183	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
184	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
185	C₂×Q₈	—	268	2	no	—
186	C₂×Q₈	—	67	4	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	67	8	no	—
189	C₂×D₄	—	134	8	no	—
190	C₂×D₄	C₂²×D₆₇	1	4	yes	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	268	4	no	—
193	C₂²×C₄	—	67	8	no	—
194	C₂²×C₄	—	67	2	no	—
195	C₂²×C₄	—	67	2	no	—
196	C₂²×C₄	—	67	2	no	—
197	C₂²×C₄	—	67	8	no	—
198	C₂²×C₄	—	134	8	no	—
199	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
201	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
202	C₂²×C₄	C₂²×D₆₇	1	4	yes	complemented
203	C₄⋊C₄	—	67	8	no	—
204	C₄⋊C₄	—	67	8	no	—
205	C₄⋊C₄	—	67	8	no	—
206	C₄⋊C₄	C₂²×D₆₇	1	8	yes	complemented
207	C₂²⋊C₄	—	67	8	no	—
208	C₂²⋊C₄	—	67	8	no	—
209	C₂²⋊C₄	—	67	4	no	—
210	C₂²⋊C₄	—	67	8	no	—
211	C₂²⋊C₄	—	67	8	no	—
212	C₂²⋊C₄	—	67	8	no	—
213	C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	complemented
215	C₄²	—	67	4	no	—
216	C₄²	—	67	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	268	4	no	—
222	C₂³	—	268	16	no	—
223	C₂³	—	67	8	no	—
224	C₂³	—	67	4	no	—
225	C₂³	—	134	16	no	—
226	C₂³	—	134	4	no	—
227	C₂³	—	134	4	no	—
228	C₂³	C₂³×D₆₇	1	2	yes	—
229	C₂³	C₂³×D₆₇	1	2	yes	—
230	C₂³	C₂³×D₆₇	1	1	yes	characteristic
231	Q₈	—	268	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	134	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	268	8	no	—
239	C₂×C₄	—	67	4	no	—
240	C₂×C₄	—	67	4	no	—
241	C₂×C₄	—	67	4	no	—
242	C₂×C₄	—	134	16	no	—
243	C₂×C₄	—	134	4	no	—
244	C₂×C₄	—	134	4	no	—
245	C₂×C₄	—	134	4	no	—
246	C₂×C₄	C₂³×D₆₇	1	2	yes	—
247	C₂×C₄	C₂³×D₆₇	1	4	yes	—
248	C₂×C₄	C₂³×D₆₇	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	268	16	no	—
256	C₂²	—	67	8	no	—
257	C₂²	—	134	16	no	—
258	C₂²	—	134	4	no	—
259	C₂²	—	134	4	no	—
260	C₂²	#41×D₆₇	1	4	yes	—
261	C₂²	#14524088	1	2	yes	—
262	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	134	4	no	—
267	C₄	—	134	4	no	—
268	C₄	—	134	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	134	8	no	—
272	C₂	#16864074	1	4	yes	direct product factor
273	C₂	C₂×#41×D₆₇	1	2	yes	—
274	C₂	C₂×#14524088	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₂×#123×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×#123	D₆₇
hypocenter	C₆₇	C₂×#551
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #16864074	C₂	direct (N × H)
2. C₂×#14524036	C₂	semidirect
3. C₂×#14524030	C₂	semidirect
4. C₂×#14524025	C₂	semidirect
5. C₂×#14524021	C₂	semidirect
7. C₂×C₄×C₂²⋊D₆₇	C₂	semidirect
8. C₂×#14524001	C₂	semidirect
9. C₂×C₂²⋊C₄×D₆₇	C₂	semidirect
10. C₂×#123×C₆₇	C₂	semidirect
12. C₂²×Dic₁₃₄	C₂²	semidirect
16. #14524036	C₂²	semidirect
17. #14524030	C₂²	semidirect
18. #14524025	C₂²	semidirect
19. #14524021	C₂²	semidirect
21. C₄×C₂²⋊D₆₇	C₂²	semidirect
22. #14524001	C₂²	semidirect
23. C₂×C₄⋊Dic₆₇	C₂²	semidirect
24. C₂×Dic₆₇⋊C₄	C₂²	semidirect
25. C₂×Dic₆₇⋊C₄	C₂²	semidirect
28. C₂×C₂²⋊Dic₆₇	C₂²	semidirect
32. C₂²⋊C₄×D₆₇	C₂²	semidirect
33. C₂×C₄×Dic₆₇	C₂²	semidirect
34. C₂×C₄×Dic₆₇	C₂²	semidirect
36. C₂³×C₄×C₆₇	C₂²	semidirect
37. #123×C₆₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
46. C₂×Dic₁₃₄	C₂³	semidirect
56. C₄⋊Dic₆₇	C₂³	semidirect
57. Dic₆₇⋊C₄	C₂³	semidirect
58. Dic₆₇⋊C₄	C₂³	semidirect
61. C₂²⋊Dic₆₇	C₂³	semidirect
65. C₄×Dic₆₇	C₂³	semidirect
66. C₄×Dic₆₇	C₂³	semidirect
75. C₂²×C₄×C₆₇	C₂³	semidirect
76. C₄⋊C₄×C₆₇	C₂³	semidirect
78. C₂²⋊C₄×C₆₇	C₂³	semidirect
133. C₂×C₆₇	#551	semidirect
138. C₆₇	C₂×#551	semidirect
143. C₂×#123	D₆₇	semidirect
155. C₂³×C₄	C₂×D₆₇	semidirect
159. #123	C₂×D₆₇	semidirect
167. C₂×C₄⋊C₄	C₂×D₆₇	semidirect
176. C₂×C₂²⋊C₄	C₂×D₆₇	semidirect
202. C₂²×C₄	C₂²×D₆₇	semidirect
206. C₄⋊C₄	C₂²×D₆₇	semidirect
214. C₂²⋊C₄	C₂²×D₆₇	semidirect
272. C₂	#16864074	direct (N × H)

Subgroups — C2×#16864074 #22573708

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16864074 #22573708