Subgroups — #22569072

Order 8576.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#16863872	C₂	1	2	yes	maximal
3	#16863501	C₂	1	2	yes	maximal, complemented
4	#16863500	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863490	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863433	C₂	1	1	yes	maximal, characteristic
9	#16863403	C₂	1	2	yes	maximal
10	#16863350	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₆₇	C₂²	1	2	yes	—
13	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₂×#11761618	C₂²	1	2	yes	—
15	Q₁₆×D₆₇	C₄	1	8	yes	complemented
16	C₂×#11761611	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₆₇	C₂²	1	2	yes	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₄×Dic₁₃₄	C₂²	1	2	yes	—
21	#14524008	—	2	2	no	—
22	#14524008	C₂²	1	2	yes	—
23	#14524001	—	2	2	no	—
24	#14524001	C₂²	1	2	yes	—
25	#14524000	—	2	2	no	—
26	#14524000	C₂²	1	1	yes	characteristic
27	#14523977	C₂²	1	1	yes	characteristic
28	#14523974	C₂²	1	2	yes	—
29	#14523968	C₂²	1	2	yes	—
30	#14523967	C₂²	1	2	yes	—
31	#14523966	C₂²	1	2	yes	—
32	#14523954	C₂²	1	1	yes	characteristic
33	#14523953	C₂²	1	1	yes	characteristic
34	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
37	#107×C₆₇	C₂²	1	1	yes	characteristic
38	#103×C₆₇	C₂²	1	2	yes	—
39	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×Dic₁₃₄	—	2	2	no	—
41	C₂×Dic₁₃₄	C₂³	1	2	yes	—
42	Q₈×D₆₇	—	2	4	no	—
43	Q₈×D₆₇	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₆₇	—	2	2	no	—
45	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
46	Dic₂₆₈	C₂×C₄	1	4	yes	—
47	#11761618	C₂×C₄	1	8	yes	—
48	#11761611	—	2	2	no	—
49	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₆₇	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₆₇	—	2	2	no	—
53	C₄⋊Dic₆₇	C₂³	1	2	yes	—
54	Dic₆₇⋊C₄	—	2	2	no	—
55	Dic₆₇⋊C₄	—	2	2	no	—
56	Dic₆₇⋊C₄	D₄	1	1	yes	characteristic
57	Dic₆₇⋊C₄	D₄	1	1	yes	characteristic
58	(C₂×D₆₇)⋊C₄	—	2	2	no	—
59	(C₂×D₆₇)⋊C₄	D₄	1	1	yes	characteristic
60	(C₂×D₆₇)⋊C₄	D₄	1	1	yes	characteristic
61	C₄×Dic₆₇	—	2	2	no	—
62	C₄×Dic₆₇	—	2	2	no	—
63	C₄×Dic₆₇	—	2	2	no	—
64	C₄×Dic₆₇	—	2	2	no	—
65	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
67	Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	2	2	no	—
76	Q₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₆₇	—	2	2	no	—
83	C₂×Dic₆₇	—	2	2	no	—
84	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄×D₆₇	—	2	8	no	—
88	C₄×D₆₇	C₂²×C₄	1	2	yes	—
89	C₄×D₆₇	#41	1	2	yes	—
90	Dic₁₃₄	—	2	2	no	—
91	Dic₁₃₄	—	2	8	no	—
92	Dic₁₃₄	C₂²×C₄	1	4	yes	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₆₇⋊C₈	—	2	1	no	—
96	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₆₇	—	2	1	no	—
99	C₂×D₆₇	—	2	1	no	—
100	C₂×D₆₇	C₄×D₄	1	2	yes	—
101	C₄×C₆₇	—	2	4	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	4	2	no	—
105	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
107	Dic₆₇	—	2	4	no	—
108	Dic₆₇	—	2	1	no	—
109	Dic₆₇	—	2	1	no	—
110	Dic₆₇	—	4	2	no	—
111	Dic₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₆₇	#578	1	1	yes	characteristic
114	C₂×C₆₇	#578	1	1	yes	characteristic
115	D₆₇	—	2	2	no	—
116	#2950	—	67	1	no	maximal, Sylow
117	C₆₇	#2950	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	67	1	no	—
119	#519	—	67	2	no	—
120	#441	—	67	2	no	—
121	#441	—	67	2	no	—
122	C₄×Q₁₆	—	67	1	no	—
123	C₄×Q₁₆	—	67	1	no	—
124	C₄×Q₁₆	—	67	1	no	—
125	C₄×Q₁₆	D₆₇	1	1	yes	characteristic, complemented
126	#427	—	67	1	no	—
127	#419	—	67	2	no	—
128	#405	—	67	1	no	—
129	C₂²×Q₈	—	67	2	no	—
130	C₂×Q₁₆	—	67	2	no	—
131	C₂×Q₁₆	—	67	8	no	—
132	C₂×Q₁₆	—	67	1	no	—
133	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	67	1	no	—
135	C₂²×C₈	—	67	1	no	—
136	C₄×Q₈	—	67	2	no	—
137	C₄×Q₈	—	67	2	no	—
138	C₄×Q₈	—	67	2	no	—
139	C₄×Q₈	—	134	2	no	—
140	C₄×Q₈	—	134	2	no	—
141	C₄×Q₈	C₂×D₆₇	1	2	yes	—
142	#117	—	67	1	no	—
143	#117	—	67	2	no	—
144	#117	—	134	2	no	—
145	#117	—	134	2	no	—
146	#107	—	67	1	no	—
147	#107	C₂×D₆₇	1	1	yes	characteristic
148	#106	—	67	2	no	—
149	#103	—	67	2	no	—
150	#103	—	67	2	no	—
151	#103	—	67	2	no	—
152	#103	C₂×D₆₇	1	2	yes	—
153	#97	—	67	1	no	—
154	#97	—	67	1	no	—
155	C₄×C₈	—	67	1	no	—
156	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂×Q₈	—	67	8	no	—
158	C₂×Q₈	—	67	2	no	—
159	C₂×Q₈	—	134	2	no	—
160	C₂×Q₈	—	134	4	no	—
161	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
162	C₂²×C₄	—	67	1	no	—
163	C₂²×C₄	—	134	2	no	—
164	Q₁₆	—	67	8	no	—
165	Q₁₆	—	67	4	no	—
166	Q₁₆	C₄×D₆₇	1	4	yes	—
167	M₁₆	—	134	2	no	—
168	C₂×C₈	—	67	1	no	—
169	C₂×C₈	—	67	4	no	—
170	C₂×C₈	—	67	1	no	—
171	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	67	1	no	—
175	C₄⋊C₄	—	67	1	no	—
176	C₄⋊C₄	—	67	2	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
181	C₂²⋊C₄	—	67	1	no	—
182	C₂²⋊C₄	—	67	1	no	—
183	C₂²⋊C₄	—	134	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	67	1	no	—
186	C₄²	—	134	2	no	—
187	C₄²	—	134	2	no	—
188	C₄²	—	134	2	no	—
189	C₄²	—	134	2	no	—
190	C₄²	C₂²×D₆₇	1	1	yes	characteristic
191	C₂³	—	67	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	67	4	no	—
194	Q₈	—	134	8	no	—
195	Q₈	—	134	2	no	—
196	Q₈	C₂×C₄×D₆₇	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	67	1	no	—
200	C₂×C₄	—	67	2	no	—
201	C₂×C₄	—	67	2	no	—
202	C₂×C₄	—	67	1	no	—
203	C₂×C₄	—	67	1	no	—
204	C₂×C₄	—	134	8	no	—
205	C₂×C₄	—	134	2	no	—
206	C₂×C₄	—	134	2	no	—
207	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
208	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	67	2	no	—
212	C₈	—	134	1	no	—
213	C₈	C₂×C₄×D₆₇	1	2	yes	—
214	C₂²	—	67	2	no	—
215	C₂²	—	134	1	no	—
216	C₂²	—	134	1	no	—
217	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	268	2	no	—
223	C₄	—	67	2	no	—
224	C₄	—	134	4	no	—
225	C₄	—	134	1	no	—
226	C₄	—	134	1	no	—
227	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
228	C₄	#41×D₆₇	1	1	yes	characteristic
229	C₂	—	134	2	no	—
230	C₂	C₄×D₄×D₆₇	1	1	yes	characteristic
231	C₂	#16864209	1	1	yes	characteristic
232	C₂	#16864210	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#2950
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #16863501	C₂	semidirect
4. #16863500	C₂	semidirect
5. C₄×Dic₂₆₈	C₂	semidirect
6. Q₁₆×Dic₆₇	C₂	semidirect
7. #16863490	C₂	semidirect
11. C₄×Q₁₆×C₆₇	C₂	semidirect
15. Q₁₆×D₆₇	C₄	semidirect
117. C₆₇	#2950	semidirect
125. C₄×Q₁₆	D₆₇	semidirect