Subgroups — #22569243

Order 8576.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#16863475	C₂	1	2	yes	maximal, complemented
3	#16863466	C₂	1	4	yes	maximal, complemented
4	C₈×D₂₆₈	C₂	1	2	yes	maximal, complemented
5	#16863453	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863358	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₈×D₆₇	C₂	1	2	yes	maximal, complemented
8	#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₂₆₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
11	C₂×#11761611	C₂²	1	4	yes	complemented
12	C₂×C₈×D₆₇	C₂²	1	4	yes	complemented
13	C₄×D₂₆₈	C₄	1	2	yes	—
14	C₄×D₂₆₈	C₄	1	1	yes	characteristic
15	C₄×D₂₆₈	C₄	1	1	yes	characteristic
16	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
17	C₄×D₂₆₈	C₂²	1	2	yes	—
18	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
20	C₂×(C₂×D₆₇)⋊C₄	C₄	1	2	yes	—
21	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
22	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
23	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	complemented
24	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	complemented
25	#14523954	C₂²	1	2	yes	complemented
26	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
28	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
31	C₂³×D₆₇	C₂×C₄	1	2	yes	—
32	C₂×D₂₆₈	—	2	4	no	—
33	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
34	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
35	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₆₇	—	4	4	no	—
38	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₆₇	C₂³	1	2	yes	—
41	C₂×C₄×D₆₇	C₂³	1	2	yes	—
42	#11761611	—	4	4	no	—
43	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
44	C₈×D₆₇	—	4	4	no	—
45	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
48	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
49	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂×C₈×C₆₇	C₂³	1	2	yes	—
57	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₆₇	—	2	4	no	—
63	C₂²×D₆₇	—	2	4	no	—
64	C₂²×D₆₇	—	4	2	no	—
65	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₆₇	—	4	2	no	—
77	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
79	D₂₆₈	—	2	4	no	—
80	D₂₆₈	—	2	2	no	—
81	D₂₆₈	M₁₆	1	4	yes	complemented
82	C₄×D₆₇	—	2	4	no	—
83	C₄×D₆₇	—	2	4	no	—
84	C₄×D₆₇	—	4	2	no	—
85	C₄×D₆₇	—	4	2	no	—
86	C₈×C₆₇	—	2	2	no	—
87	C₈×C₆₇	—	2	2	no	—
88	C₆₇⋊C₈	—	2	4	no	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	#131	1	2	yes	—
92	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₆₇	—	2	2	no	—
94	C₂×D₆₇	—	2	2	no	—
95	C₂×D₆₇	—	2	2	no	—
96	C₂×D₆₇	—	2	2	no	—
97	C₂×D₆₇	—	2	2	no	—
98	C₂×D₆₇	—	4	8	no	—
99	C₂×D₆₇	C₂×M₁₆	1	4	yes	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
105	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
106	Dic₆₇	—	2	2	no	—
107	Dic₆₇	—	2	2	no	—
108	C₂×C₆₇	—	2	2	no	—
109	C₂×C₆₇	#518	1	1	yes	characteristic
110	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
112	D₆₇	—	2	4	no	—
113	D₆₇	—	4	2	no	—
114	#2974	—	67	1	no	maximal, Sylow
115	C₆₇	#2974	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄×D₄	—	67	1	no	—
117	#436	—	67	2	no	—
118	#435	—	67	4	no	—
119	C₈×D₄	—	67	2	no	—
120	#432	—	67	1	no	—
121	#431	D₆₇	1	1	yes	characteristic, complemented
122	#407	—	67	2	no	—
123	C₂×C₂²⋊C₈	—	67	2	no	—
124	C₂²×D₄	—	67	1	no	—
125	C₂³×C₄	—	67	2	no	—
126	C₂×M₁₆	—	67	4	no	—
127	C₂²×C₈	—	67	4	no	—
128	C₄×D₄	—	67	1	no	—
129	C₄×D₄	—	67	2	no	—
130	C₄×D₄	—	67	1	no	—
131	C₄×D₄	—	67	1	no	—
132	C₄×D₄	—	67	2	no	—
133	C₄×D₄	—	67	1	no	—
134	C₂×C₄⋊C₄	—	67	1	no	—
135	C₂×C₂²⋊C₄	—	67	2	no	—
136	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
137	C₄⋊C₈	—	67	2	no	—
138	C₄⋊C₈	C₂×D₆₇	1	2	yes	complemented
139	C₂²⋊C₈	—	67	2	no	—
140	C₂²⋊C₈	—	67	4	no	—
141	C₂²⋊C₈	—	67	4	no	—
142	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
143	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
144	#97	—	67	2	no	—
145	C₄×C₈	C₂×D₆₇	1	2	yes	complemented
146	C₂⁴	—	67	2	no	—
147	C₂×D₄	—	67	1	no	—
148	C₂×D₄	—	67	2	no	—
149	C₂×D₄	—	67	1	no	—
150	C₂×D₄	—	134	4	no	—
151	C₂²×C₄	—	268	4	no	—
152	C₂²×C₄	—	67	2	no	—
153	C₂²×C₄	—	67	2	no	—
154	C₂²×C₄	—	67	2	no	—
155	C₂²×C₄	—	67	2	no	—
156	C₂²×C₄	—	67	2	no	—
157	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
160	M₁₆	—	268	4	no	—
161	C₂×C₈	—	268	4	no	—
162	C₂×C₈	—	67	4	no	—
163	C₂×C₈	C₂²×D₆₇	1	2	yes	—
164	C₂×C₈	C₂²×D₆₇	1	2	yes	—
165	C₄⋊C₄	—	67	2	no	—
166	C₄⋊C₄	—	67	1	no	—
167	C₄⋊C₄	—	67	1	no	—
168	C₂²⋊C₄	—	67	2	no	—
169	C₂²⋊C₄	—	67	2	no	—
170	C₂²⋊C₄	—	67	2	no	—
171	C₂²⋊C₄	—	67	2	no	—
172	C₄²	C₄×D₆₇	1	1	yes	characteristic
173	C₄²	C₄×D₆₇	1	1	yes	characteristic
174	C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	C₂³	—	268	2	no	—
177	C₂³	—	67	2	no	—
178	C₂³	—	67	2	no	—
179	C₂³	—	134	4	no	—
180	C₂³	—	134	4	no	—
181	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
182	D₄	—	67	4	no	—
183	D₄	—	134	4	no	—
184	D₄	—	134	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	268	2	no	—
189	C₂×C₄	—	268	2	no	—
190	C₂×C₄	—	268	2	no	—
191	C₂×C₄	—	67	2	no	—
192	C₂×C₄	—	67	2	no	—
193	C₂×C₄	—	134	4	no	—
194	C₂×C₄	—	134	4	no	—
195	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	134	4	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	268	8	no	—
207	C₂²	—	67	4	no	—
208	C₂²	—	134	2	no	—
209	C₂²	—	134	2	no	—
210	C₂²	—	134	2	no	—
211	C₂²	—	134	2	no	—
212	C₂²	—	134	2	no	—
213	C₂²	#14524049	1	2	yes	—
214	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	134	2	no	—
220	C₄	—	134	2	no	—
221	C₄	M₁₆×D₆₇	1	2	yes	—
222	C₄	#14524089	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	268	2	no	—
225	C₂	—	134	4	no	—
226	C₂	#16863870	1	1	yes	characteristic
227	C₂	C₂×M₁₆×D₆₇	1	1	yes	characteristic
228	C₂	C₂×#14524049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#431×C₆₇	C₂
hypercenter	#431	D₆₇
hypocenter	C₆₇	#2974
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16863475	C₂	semidirect
3. #16863466	C₂	semidirect
4. C₈×D₂₆₈	C₂	semidirect
5. #16863453	C₂	semidirect
6. #16863358	C₂	semidirect
7. C₂²⋊C₈×D₆₇	C₂	semidirect
8. #431×C₆₇	C₂	semidirect
11. C₂×#11761611	C₂²	semidirect
12. C₂×C₈×D₆₇	C₂²	semidirect
21. C₄⋊(C₆₇⋊C₈)	C₂²	semidirect
23. (C₂×D₆₇)⋊C₈	C₂²	semidirect
24. (C₂×D₆₇)⋊C₈	C₂²	semidirect
25. #14523954	C₂²	semidirect
27. C₄⋊C₈×C₆₇	C₂²	semidirect
30. C₄×C₈×C₆₇	C₂²	semidirect
81. D₂₆₈	M₁₆	semidirect
115. C₆₇	#2974	semidirect
121. #431	D₆₇	semidirect
138. C₄⋊C₈	C₂×D₆₇	semidirect
145. C₄×C₈	C₂×D₆₇	semidirect