Subgroups — #22568955

Order 8576.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524051	C₂	1	2	yes	maximal, complemented
2	#16863946	C₂	1	1	yes	maximal, characteristic
3	#16863475	C₂	1	2	yes	maximal, complemented
4	#16863469	C₂	1	4	yes	maximal, complemented
5	C₈×D₂₆₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863364	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11761628	C₂²	1	2	yes	—
10	#14524051	—	2	8	no	—
11	C₂×#11761611	C₂²	1	4	yes	—
12	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×D₆₇	C₂²	1	4	yes	—
14	#14524043	C₄	1	1	yes	characteristic
15	#14524040	C₄	1	1	yes	characteristic
16	#14524031	C₄	1	2	yes	—
17	#14524017	C₄	1	2	yes	—
18	#14524017	C₄	1	2	yes	—
19	C₄×D₂₆₈	C₂²	1	2	yes	—
20	C₄×D₂₆₈	C₂²	1	2	yes	—
21	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	—
25	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	—
26	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#97×C₆₇	C₂²	1	2	yes	—
30	C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	#11761628	—	2	4	no	—
32	#11761628	—	2	4	no	—
33	C₂×Dic₁₃₄	C₂×C₄	1	2	yes	—
34	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
35	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
36	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊D₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₆₇	C₂³	1	4	yes	—
39	#11761611	—	2	8	no	—
40	C₂×C₆₇⋊C₈	—	2	4	no	—
41	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₈×D₆₇	—	2	8	no	—
44	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
46	(C₂×D₆₇)⋊C₄	C₂×C₄	1	4	yes	—
47	(C₂×D₆₇)⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	D₄	1	4	yes	complemented
53	C₂×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₆₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	4	1	no	—
63	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₆₇	#41	1	2	yes	—
70	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₆₇	—	2	8	no	—
72	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
73	D₂₆₈	—	2	4	no	—
74	D₂₆₈	—	2	4	no	—
75	D₂₆₈	—	4	2	no	—
76	C₄×D₆₇	—	2	4	no	—
77	C₄×D₆₇	—	2	4	no	—
78	Dic₁₃₄	—	2	4	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	C₂×D₄	1	4	yes	—
81	C₆₇⋊C₈	—	2	2	no	—
82	C₆₇⋊C₈	—	2	2	no	—
83	C₂²×C₆₇	—	2	1	no	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	C₄×D₄	1	2	yes	—
86	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₆₇	—	2	4	no	—
88	C₂×D₆₇	—	2	4	no	—
89	C₂×D₆₇	—	2	4	no	—
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	—	2	1	no	—
93	C₄×C₆₇	C₄×D₄	1	2	yes	—
94	C₄×C₆₇	#131	1	2	yes	complemented
95	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
97	Dic₆₇	—	2	4	no	—
98	C₂×C₆₇	—	2	2	no	—
99	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
101	C₂×C₆₇	#568	1	1	yes	characteristic
102	D₆₇	—	4	4	no	—
103	#2935	—	67	1	no	maximal, Sylow
104	C₆₇	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	67	2	no	—
106	#532	—	67	1	no	—
107	#436	—	67	2	no	—
108	#435	—	67	4	no	—
109	C₈×D₄	—	67	2	no	—
110	C₂×C₄⋊C₈	—	67	1	no	—
111	#408	—	67	2	no	—
112	C₄×M₁₆	D₆₇	1	1	yes	characteristic, complemented
113	C₂×#41	—	67	2	no	—
114	#131	—	134	8	no	—
115	C₂×M₁₆	—	67	4	no	—
116	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
118	C₂²×C₈	—	67	2	no	—
119	C₂²×C₈	—	67	4	no	—
120	C₄⋊Q₈	—	67	1	no	—
121	#127	—	67	1	no	—
122	#124	—	67	2	no	—
123	#121	—	67	2	no	—
124	#121	—	67	2	no	—
125	C₄×D₄	—	67	2	no	—
126	C₄×D₄	—	67	2	no	—
127	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
128	C₄⋊C₈	—	67	1	no	—
129	C₄⋊C₈	—	67	2	no	—
130	C₄⋊C₈	—	67	1	no	—
131	C₂²⋊C₈	—	67	4	no	—
132	C₂²⋊C₈	—	67	4	no	—
133	#97	C₂×D₆₇	1	2	yes	—
134	C₄×C₈	C₂×D₆₇	1	2	yes	—
135	#41	—	134	4	no	—
136	#41	—	134	4	no	—
137	C₂×Q₈	—	67	2	no	—
138	C₂×D₄	—	67	4	no	—
139	C₂×D₄	—	67	2	no	—
140	C₂×D₄	—	67	2	no	—
141	C₂×D₄	—	67	2	no	—
142	C₂²×C₄	—	67	4	no	—
143	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	134	8	no	—
148	M₁₆	D₂₆₈	1	4	yes	complemented
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	—	67	2	no	—
151	C₂×C₈	—	134	4	no	—
152	C₂×C₈	—	134	8	no	—
153	C₂×C₈	C₂²×D₆₇	1	2	yes	—
154	C₂×C₈	C₂²×D₆₇	1	2	yes	—
155	C₄⋊C₄	—	67	2	no	—
156	C₄⋊C₄	—	67	2	no	—
157	C₂²⋊C₄	—	67	4	no	—
158	C₂²⋊C₄	—	67	4	no	—
159	C₄²	C₄×D₆₇	1	1	yes	characteristic
160	C₄²	C₄×D₆₇	1	1	yes	characteristic
161	C₄²	C₂²×D₆₇	1	1	yes	characteristic
162	C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	C₂³	—	67	4	no	—
164	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
165	Q₈	—	134	4	no	—
166	D₄	—	268	2	no	—
167	D₄	—	134	8	no	—
168	D₄	—	134	4	no	—
169	D₄	—	134	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	67	4	no	—
173	C₂×C₄	—	134	4	no	—
174	C₂×C₄	—	134	4	no	—
175	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
180	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
181	C₂×C₄	#11761628	1	2	yes	—
182	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	134	2	no	—
185	C₈	—	134	2	no	—
186	C₈	C₂×D₂₆₈	1	4	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	134	4	no	—
190	C₂²	—	134	4	no	—
191	C₂²	—	134	4	no	—
192	C₂²	C₄×D₂₆₈	1	2	yes	—
193	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	134	4	no	—
198	C₄	C₄×D₂₆₈	1	2	yes	—
199	C₄	#14524051	1	2	yes	complemented
200	C₄	C₂²×D₂₆₈	1	1	yes	characteristic
201	C₄	C₂×#11761628	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	268	4	no	—
204	C₂	C₂×C₄×D₂₆₈	1	1	yes	characteristic
205	C₂	C₂×#14524051	1	1	yes	characteristic
206	C₂	#16864162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₄×M₁₆×C₆₇	C₂
hypercenter	C₄×M₁₆	D₆₇
hypocenter	C₆₇	#2935
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14524051	C₂	semidirect
3. #16863475	C₂	semidirect
4. #16863469	C₂	semidirect
5. C₈×D₂₆₈	C₂	semidirect
6. C₂×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
7. #16863364	C₂	semidirect
8. C₄×M₁₆×C₆₇	C₂	semidirect
52. M₁₆×C₆₇	D₄	semidirect
94. C₄×C₆₇	#131	semidirect
104. C₆₇	#2935	semidirect
112. C₄×M₁₆	D₆₇	semidirect
148. M₁₆	D₂₆₈	semidirect
199. C₄	#14524051	semidirect