Subgroups — #22568901

Order 8576.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863903	C₂	1	1	yes	maximal, characteristic
3	#16863465	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₆₇	C₂	1	4	yes	maximal, complemented
5	#16863424	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863363	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863359	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863351	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₆₇	C₂²	1	1	yes	characteristic
11	C₂×#11761612	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₆₇	—	2	4	no	—
14	C₂×C₈×D₆₇	C₂²	1	2	yes	—
15	C₂×C₈×D₆₇	C₂²	1	2	yes	—
16	#14524024	C₄	1	2	yes	—
17	#14524019	C₄	1	2	yes	—
18	#14524014	C₄	1	2	yes	—
19	#14524011	C₄	1	2	yes	—
20	C₄×C₂²⋊D₆₇	C₂²	1	4	yes	—
21	#14524002	C₂²	1	1	yes	characteristic
22	Dic₆₇⋊C₈	C₂²	1	4	yes	complemented
23	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
24	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	—
25	#14523953	C₂²	1	2	yes	complemented
26	C₈×Dic₆₇	C₂²	1	2	yes	complemented
27	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
28	#131×C₆₇	—	2	4	no	—
29	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₆₇	C₂²	1	2	yes	complemented
31	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂²	1	2	yes	complemented
33	C₂³×D₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₆₇	C₂×C₄	1	4	yes	—
35	C₂²×Dic₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₆₇	—	2	2	no	—
37	C₂×C₄×D₆₇	—	2	2	no	—
38	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₆₇	C₂³	1	2	yes	—
40	#11761612	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₈×D₆₇	—	2	4	no	—
45	C₈×D₆₇	—	2	4	no	—
46	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
47	Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
50	(C₂×D₆₇)⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₆₇	C₂³	1	2	yes	—
52	#41×C₆₇	—	2	2	no	—
53	#41×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	2	no	—
60	M₁₆×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	D₄	1	4	yes	complemented
63	C₂×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₆₇	—	2	2	no	—
69	C₂²×D₆₇	—	2	4	no	—
70	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
71	Q₈×C₆₇	—	2	2	no	—
72	D₄×C₆₇	—	2	4	no	—
73	D₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₆₇	#41	1	2	yes	—
80	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₆₇	—	2	8	no	—
82	C₂×Dic₆₇	—	2	2	no	—
83	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
84	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
85	C₄×D₆₇	—	2	4	no	—
86	C₄×D₆₇	—	2	2	no	—
87	C₄×D₆₇	—	2	2	no	—
88	C₈×C₆₇	—	2	2	no	—
89	C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₆₇⋊C₈	—	2	2	no	—
92	C₆₇⋊C₈	—	2	2	no	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	—	2	2	no	—
95	C₂²×C₆₇	—	2	4	no	—
96	C₂²×C₆₇	C₄×D₄	1	2	yes	—
97	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₆₇	—	2	8	no	—
99	C₂×D₆₇	—	2	2	no	—
100	C₂×D₆₇	—	2	2	no	—
101	C₂×D₆₇	#131	1	4	yes	complemented
102	C₄×C₆₇	—	2	2	no	—
103	C₄×C₆₇	C₄×D₄	1	2	yes	—
104	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
106	Dic₆₇	—	2	2	no	—
107	Dic₆₇	—	2	4	no	—
108	C₂×C₆₇	—	2	2	no	—
109	C₂×C₆₇	—	4	2	no	—
110	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#131	1	2	yes	—
112	D₆₇	—	2	4	no	—
113	#2929	—	67	1	no	maximal, Sylow
114	C₆₇	#2929	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	D₆₇	1	1	yes	characteristic, complemented
116	C₂³×C₈	—	67	1	no	—
117	#525	—	67	1	no	—
118	#435	—	67	4	no	—
119	C₈×D₄	—	67	4	no	—
120	#424	—	67	1	no	—
121	#408	—	67	1	no	—
122	#407	—	67	1	no	—
123	#405	—	67	1	no	—
124	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
125	C₂³×C₄	—	67	1	no	—
126	#131	—	2	4	no	—
127	C₂×M₁₆	—	67	1	no	—
128	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
129	C₂×M₁₆	C₂×D₆₇	1	2	yes	complemented
130	C₂²×C₈	—	67	1	no	—
131	C₂²×C₈	—	67	2	no	—
132	C₂²×C₈	—	67	2	no	—
133	C₂²×C₈	—	134	4	no	—
134	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₆₇	1	2	yes	complemented
136	#123	—	67	2	no	—
137	C₂²⋊Q₈	—	67	2	no	—
138	#121	—	67	2	no	—
139	#120	—	67	2	no	—
140	C₄×D₄	—	67	4	no	—
141	#117	—	67	1	no	—
142	C₄⋊C₈	—	67	4	no	—
143	C₂²⋊C₈	—	67	4	no	—
144	C₂²⋊C₈	—	67	4	no	—
145	#97	—	67	2	no	—
146	C₄×C₈	—	67	2	no	—
147	C₂⁴	—	67	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₆₇	1	1	yes	characteristic
151	C₂×D₄	—	67	4	no	—
152	C₂×D₄	C₄×D₆₇	1	2	yes	—
153	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
154	C₂²×C₄	—	67	1	no	—
155	C₂²×C₄	—	67	2	no	—
156	C₂²×C₄	—	67	2	no	—
157	C₂²×C₄	—	134	2	no	—
158	C₂²×C₄	—	134	2	no	—
159	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	134	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	67	2	no	—
166	C₂×C₈	—	67	2	no	—
167	C₂×C₈	—	134	2	no	—
168	C₂×C₈	—	134	4	no	—
169	C₂×C₈	—	134	4	no	—
170	C₂×C₈	C₂²⋊D₆₇	1	4	yes	complemented
171	C₂×C₈	C₂²×D₆₇	1	2	yes	—
172	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₄⋊C₄	—	67	2	no	—
175	C₄⋊C₄	—	67	4	no	—
176	C₂²⋊C₄	—	67	2	no	—
177	C₂²⋊C₄	—	67	4	no	—
178	C₂²⋊C₄	—	67	4	no	—
179	C₄²	—	67	2	no	—
180	C₂³	—	67	2	no	—
181	C₂³	—	134	4	no	—
182	C₂³	—	134	2	no	—
183	C₂³	C₂×C₄×D₆₇	1	2	yes	—
184	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	134	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	67	4	no	—
192	C₂×C₄	—	67	2	no	—
193	C₂×C₄	—	134	2	no	—
194	C₂×C₄	—	134	4	no	—
195	C₂×C₄	—	134	2	no	—
196	C₂×C₄	—	134	2	no	—
197	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
200	C₂×C₄	#11761628	1	2	yes	—
201	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	134	2	no	—
204	C₈	—	134	2	no	—
205	C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
206	C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	67	4	no	—
211	C₂²	—	134	8	no	—
212	C₂²	—	134	2	no	—
213	C₂²	—	134	2	no	—
214	C₂²	C₄×C₂²⋊D₆₇	1	2	yes	—
215	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	134	2	no	—
218	C₄	—	134	4	no	—
219	C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₄	C₂×#11761628	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	134	4	no	—
225	C₂	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂	#131×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#131×C₆₇	C₂
hypercenter	C₂×#131	D₆₇
hypocenter	C₆₇	#2929
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₆₇	C₂	semidirect
3. #16863465	C₂	semidirect
4. C₈×C₂²⋊D₆₇	C₂	semidirect
5. #16863424	C₂	semidirect
6. #16863363	C₂	semidirect
7. #16863359	C₂	semidirect
8. #16863351	C₂	semidirect
9. C₂×#131×C₆₇	C₂	semidirect
22. Dic₆₇⋊C₈	C₂²	semidirect
23. C₂²⋊(C₆₇⋊C₈)	C₂²	semidirect
25. #14523953	C₂²	semidirect
26. C₈×Dic₆₇	C₂²	semidirect
30. C₂×M₁₆×C₆₇	C₂²	semidirect
32. C₂²×C₈×C₆₇	C₂²	semidirect
62. C₂×C₈×C₆₇	D₄	semidirect
101. C₂×D₆₇	#131	semidirect
114. C₆₇	#2929	semidirect
115. C₂×#131	D₆₇	semidirect
129. C₂×M₁₆	C₂×D₆₇	semidirect
135. C₂²×C₈	C₂×D₆₇	semidirect
170. C₂×C₈	C₂²⋊D₆₇	semidirect