Subgroups — #22568944

Order 8576.

544 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×M₁₆×D₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#16863473	C₂	1	2	yes	maximal, complemented
4	#16863466	C₂	1	4	yes	maximal, complemented
5	#16863461	C₂	1	2	yes	maximal, complemented
6	#16863359	C₂	1	2	yes	maximal, complemented
7	C₄×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
8	#423×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×#11761612	C₂²	1	1	yes	characteristic
12	C₂×#11761612	C₂²	1	1	yes	characteristic
13	C₂×#11761611	C₂²	1	4	yes	complemented
14	M₁₆×D₆₇	—	2	8	no	—
15	C₂×C₈×D₆₇	C₂²	1	4	yes	complemented
16	C₄×D₂₆₈	C₄	1	2	yes	—
17	C₄×D₂₆₈	C₄	1	1	yes	characteristic
18	C₄×D₂₆₈	C₄	1	1	yes	characteristic
19	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
20	C₄×D₂₆₈	C₂²	1	2	yes	—
21	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
23	C₂×(C₂×D₆₇)⋊C₄	C₄	1	2	yes	—
24	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	complemented
25	(C₂×D₆₇)⋊C₈	C₂²	1	4	yes	complemented
26	#14523954	C₂²	1	2	yes	complemented
27	C₄×C₆₇⋊C₈	C₂²	1	2	yes	complemented
28	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
31	C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
32	C₂³×D₆₇	C₂×C₄	1	2	yes	—
33	C₂×D₂₆₈	—	2	4	no	—
34	C₂×D₂₆₈	C₂×C₄	1	2	yes	—
35	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
36	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₆₇	—	2	4	no	—
39	C₂×C₄×D₆₇	—	2	4	no	—
40	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₆₇	C₂³	1	2	yes	—
43	C₂×C₄×D₆₇	C₂³	1	2	yes	—
44	#11761612	—	2	2	no	—
45	#11761612	D₄	1	4	yes	complemented
46	#11761611	—	2	8	no	—
47	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
49	C₈×D₆₇	—	2	8	no	—
50	C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
53	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	M₁₆×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	C₂³	1	4	yes	—
62	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₆₇	—	2	4	no	—
68	C₂²×D₆₇	—	2	4	no	—
69	C₂²×D₆₇	—	4	2	no	—
70	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	4	1	no	—
74	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₆₇	#41	1	2	yes	—
81	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₆₇	—	2	4	no	—
83	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
85	D₂₆₈	—	2	4	no	—
86	D₂₆₈	—	2	2	no	—
87	D₂₆₈	M₁₆	1	4	yes	complemented
88	C₄×D₆₇	—	2	4	no	—
89	C₄×D₆₇	—	2	4	no	—
90	C₄×D₆₇	—	2	2	no	—
91	C₄×D₆₇	—	2	2	no	—
92	C₄×D₆₇	—	2	2	no	—
93	C₄×D₆₇	—	2	2	no	—
94	C₈×C₆₇	—	2	4	no	—
95	C₆₇⋊C₈	—	2	2	no	—
96	C₆₇⋊C₈	C₂×D₄	1	4	yes	complemented
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	C₄×D₄	1	2	yes	—
100	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₆₇	—	2	2	no	—
102	C₂×D₆₇	—	2	2	no	—
103	C₂×D₆₇	—	2	2	no	—
104	C₂×D₆₇	—	2	2	no	—
105	C₂×D₆₇	—	2	2	no	—
106	C₂×D₆₇	—	4	8	no	—
107	C₂×D₆₇	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₄×C₆₇	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	Dic₆₇	—	2	2	no	—
116	Dic₆₇	—	2	2	no	—
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₆₇	#568	1	1	yes	characteristic
121	D₆₇	—	2	4	no	—
122	D₆₇	—	4	2	no	—
123	D₄×M₁₆	—	67	1	no	maximal, Sylow
124	C₆₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	67	2	no	—
126	C₂×C₄×D₄	—	67	1	no	—
127	#436	—	67	2	no	—
128	#435	—	67	4	no	—
129	C₈×D₄	—	67	2	no	—
130	#423	D₆₇	1	1	yes	characteristic, complemented
131	#407	—	67	2	no	—
132	C₄×M₁₆	—	67	1	no	—
133	C₂²×D₄	—	67	1	no	—
134	C₂³×C₄	—	67	2	no	—
135	C₂×M₁₆	—	67	1	no	—
136	C₂×M₁₆	—	67	1	no	—
137	C₂×M₁₆	—	67	4	no	—
138	C₂×M₁₆	—	134	8	no	—
139	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
140	C₂²×C₈	—	67	4	no	—
141	C₄×D₄	—	67	1	no	—
142	C₄×D₄	—	67	2	no	—
143	C₄×D₄	—	67	1	no	—
144	C₄×D₄	—	67	1	no	—
145	C₄×D₄	—	67	2	no	—
146	C₄×D₄	—	67	1	no	—
147	C₂×C₄⋊C₄	—	67	1	no	—
148	C₂×C₂²⋊C₄	—	67	2	no	—
149	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂×D₆₇	1	2	yes	complemented
151	C₄⋊C₈	C₂×D₆₇	1	2	yes	complemented
152	C₂²⋊C₈	—	67	4	no	—
153	C₂²⋊C₈	—	67	4	no	—
154	#97	—	67	2	no	—
155	C₄×C₈	—	67	2	no	—
156	C₂⁴	—	67	2	no	—
157	C₂×D₄	—	67	1	no	—
158	C₂×D₄	—	67	2	no	—
159	C₂×D₄	—	67	1	no	—
160	C₂×D₄	—	134	4	no	—
161	C₂²×C₄	—	67	2	no	—
162	C₂²×C₄	—	67	2	no	—
163	C₂²×C₄	—	67	2	no	—
164	C₂²×C₄	—	67	2	no	—
165	C₂²×C₄	—	67	2	no	—
166	C₂²×C₄	—	134	4	no	—
167	C₂²×C₄	—	134	4	no	—
168	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	67	4	no	—
173	M₁₆	—	134	8	no	—
174	M₁₆	—	134	2	no	—
175	C₂×C₈	—	67	2	no	—
176	C₂×C₈	—	67	2	no	—
177	C₂×C₈	—	134	8	no	—
178	C₂×C₈	C₂²×D₆₇	1	4	yes	—
179	C₄⋊C₄	—	67	2	no	—
180	C₄⋊C₄	—	67	1	no	—
181	C₄⋊C₄	—	67	1	no	—
182	C₂²⋊C₄	—	67	2	no	—
183	C₂²⋊C₄	—	67	2	no	—
184	C₂²⋊C₄	—	67	2	no	—
185	C₂²⋊C₄	—	67	2	no	—
186	C₄²	C₄×D₆₇	1	1	yes	characteristic
187	C₄²	C₄×D₆₇	1	1	yes	characteristic
188	C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	C₄²	C₂²×D₆₇	1	1	yes	characteristic
190	C₂³	—	268	2	no	—
191	C₂³	—	67	2	no	—
192	C₂³	—	67	2	no	—
193	C₂³	—	134	4	no	—
194	C₂³	—	134	4	no	—
195	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
196	D₄	—	67	4	no	—
197	D₄	—	134	4	no	—
198	D₄	—	134	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	67	2	no	—
202	C₂×C₄	—	67	2	no	—
203	C₂×C₄	—	134	4	no	—
204	C₂×C₄	—	134	4	no	—
205	C₂×C₄	—	134	4	no	—
206	C₂×C₄	—	134	2	no	—
207	C₂×C₄	—	134	2	no	—
208	C₂×C₄	—	134	2	no	—
209	C₂×C₄	—	134	2	no	—
210	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₄	D₄×D₆₇	1	2	yes	—
216	C₂×C₄	#11761629	1	2	yes	—
217	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	67	4	no	—
220	C₈	—	134	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	268	8	no	—
224	C₂²	—	67	4	no	—
225	C₂²	—	134	2	no	—
226	C₂²	—	134	2	no	—
227	C₂²	—	134	2	no	—
228	C₂²	—	134	2	no	—
229	C₂²	—	134	2	no	—
230	C₂²	#14524005	1	2	yes	—
231	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	134	2	no	—
236	C₄	—	134	2	no	—
237	C₄	#14524005	1	2	yes	—
238	C₄	M₁₆×D₆₇	1	2	yes	complemented
239	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
240	C₄	C₂×#11761629	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	268	2	no	—
243	C₂	—	134	4	no	—
244	C₂	C₂×#14524005	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₆₇	1	1	yes	characteristic
246	C₂	#16864162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#423×C₆₇	C₂
hypercenter	#423	D₆₇
hypocenter	C₆₇	D₄×M₁₆
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₆₇	C₂	semidirect
3. #16863473	C₂	semidirect
4. #16863466	C₂	semidirect
5. #16863461	C₂	semidirect
6. #16863359	C₂	semidirect
7. C₄×#11761612	C₂	semidirect
8. #423×C₆₇	C₂	semidirect
13. C₂×#11761611	C₂²	semidirect
15. C₂×C₈×D₆₇	C₂²	semidirect
24. (C₂×D₆₇)⋊C₈	C₂²	semidirect
25. (C₂×D₆₇)⋊C₈	C₂²	semidirect
26. #14523954	C₂²	semidirect
27. C₄×C₆₇⋊C₈	C₂²	semidirect
30. C₄⋊C₈×C₆₇	C₂²	semidirect
31. C₄⋊C₈×C₆₇	C₂²	semidirect
45. #11761612	D₄	semidirect
87. D₂₆₈	M₁₆	semidirect
96. C₆₇⋊C₈	C₂×D₄	semidirect
107. C₂×D₆₇	C₂×M₁₆	semidirect
112. C₄×C₆₇	C₂×M₁₆	semidirect
124. C₆₇	D₄×M₁₆	semidirect
130. #423	D₆₇	semidirect
150. C₄⋊C₈	C₂×D₆₇	semidirect
151. C₄⋊C₈	C₂×D₆₇	semidirect
238. C₄	M₁₆×D₆₇	semidirect