Subgroups — Q₈×C₁₆×D₆₇ #31009310

Order 17152.

582 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	C₈×Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₃₄	C₂	1	3	yes	maximal, complemented
4	#22562111	C₂	1	3	yes	maximal, complemented
5	C₄⋊C₁₆×D₆₇	C₂	1	3	yes	maximal
6	C₄×C₁₆×D₆₇	C₂	1	3	yes	maximal
7	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₆₇	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₆₇	C₂²	1	3	yes	—
10	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
11	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₄	C₂²	1	3	yes	—
13	#16863520	C₄	1	3	yes	—
14	C₄⋊C₈×D₆₇	C₄	1	3	yes	—
15	C₄×C₈×D₆₇	C₂²	1	3	yes	—
16	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	3	yes	—
17	Dic₆₇⋊C₁₆	C₂²	1	6	yes	—
18	C₄×C₆₇⋊C₁₆	C₂²	1	3	yes	—
19	C₁₆×Dic₆₇	C₂²	1	3	yes	—
20	C₁₆×Dic₆₇	C₂²	1	3	yes	—
21	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₆₇	C₂²	1	3	yes	—
23	C₄×C₁₆×C₆₇	C₂²	1	3	yes	—
24	C₂×Q₈×D₆₇	C₈	1	1	yes	characteristic
25	C₂×C₈×D₆₇	C₂×C₄	1	3	yes	—
26	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
27	Q₈×Dic₆₇	C₈	1	1	yes	characteristic
28	C₄×Dic₁₃₄	C₂×C₄	1	3	yes	—
29	#14524008	C₈	1	3	yes	—
30	C₄⋊C₄×D₆₇	C₈	1	3	yes	—
31	C₄²×D₆₇	C₂×C₄	1	3	yes	—
32	C₂×C₆₇⋊C₁₆	C₂³	1	3	yes	—
33	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
34	C₁₆×D₆₇	—	2	6	no	—
35	C₁₆×D₆₇	Q₈	1	4	yes	direct product factor
36	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	3	yes	—
37	Dic₆₇⋊C₈	C₂×C₄	1	6	yes	—
38	C₄×C₆₇⋊C₈	C₂×C₄	1	3	yes	—
39	C₈×Dic₆₇	C₂×C₄	1	3	yes	—
40	C₈×Dic₆₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
43	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
45	C₄×C₈×C₆₇	C₂³	1	3	yes	—
46	C₂×Dic₁₃₄	C₂×C₈	1	3	yes	—
47	Q₈×D₆₇	C₁₆	1	8	yes	direct product factor
48	C₂×C₄×D₆₇	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₆₇	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₆₇⋊C₈	C₂²×C₄	1	3	yes	—
52	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
53	C₈×D₆₇	—	2	6	no	—
54	C₈×D₆₇	C₂×Q₈	1	2	yes	—
55	C₈×D₆₇	#41	1	2	yes	—
56	C₄⋊Dic₆₇	C₂×C₈	1	3	yes	—
57	Dic₆₇⋊C₄	C₂×C₈	1	6	yes	—
58	C₄×Dic₆₇	C₂×C₈	1	3	yes	—
59	C₄×Dic₆₇	C₂×C₈	1	3	yes	—
60	C₄×Dic₆₇	C₂²×C₄	1	3	yes	—
61	C₆₇⋊C₁₆	—	2	3	no	—
62	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	complemented
63	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
65	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	C₂×C₈	1	3	yes	—
67	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
68	C₁₆×C₆₇	—	2	3	no	—
69	C₁₆×C₆₇	C₂×Q₈	1	2	yes	complemented
70	C₂²×D₆₇	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₆₇	C₂×C₁₆	1	4	yes	complemented
72	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₆₇	C₂²×C₈	1	3	yes	—
76	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×Dic₆₇	C₂²×C₈	1	3	yes	—
78	C₄×D₆₇	—	2	6	no	—
79	C₄×D₆₇	C₂×C₁₆	1	12	yes	—
80	C₄×D₆₇	C₄×Q₈	1	2	yes	—
81	C₄×D₆₇	#131	1	2	yes	—
82	Dic₁₃₄	C₂×C₁₆	1	12	yes	complemented
83	C₈×C₆₇	—	2	3	no	—
84	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
86	C₆₇⋊C₈	—	2	3	no	—
87	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
88	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
89	C₂×D₆₇	C₈×Q₈	1	2	yes	—
90	C₂×D₆₇	C₂²×C₁₆	1	2	yes	—
91	C₂×D₆₇	#504	1	2	yes	—
92	C₄×C₆₇	—	2	3	no	—
93	C₄×C₆₇	C₂²×C₁₆	1	6	yes	—
94	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
96	Dic₆₇	—	2	3	no	—
97	Dic₆₇	C₈×Q₈	1	2	yes	—
98	Dic₆₇	C₂²×C₁₆	1	6	yes	—
99	C₂×Q₈×C₁₆	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
101	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
103	D₆₇	Q₈×C₁₆	1	4	yes	direct product factor
104	C₂×C₈×Q₈	—	67	1	no	—
105	Q₈×C₁₆	—	67	1	no	—
106	Q₈×C₁₆	—	67	3	no	—
107	Q₈×C₁₆	—	67	3	no	—
108	Q₈×C₁₆	D₆₇	1	1	yes	characteristic, direct product factor
109	C₂×C₄⋊C₁₆	—	67	3	no	—
110	C₂×C₄×C₁₆	—	67	3	no	—
111	C₆₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	—	67	1	no	—
113	C₂²×C₁₆	—	67	1	no	—
114	C₂²×C₁₆	—	67	3	no	—
115	C₈×Q₈	—	67	1	no	—
116	C₈×Q₈	—	67	3	no	—
117	C₈×Q₈	—	67	3	no	—
118	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	67	3	no	—
120	C₂×C₄×C₈	—	67	3	no	—
121	C₄⋊C₁₆	—	67	6	no	—
122	C₄⋊C₁₆	—	67	3	no	—
123	C₄⋊C₁₆	C₂×D₆₇	1	3	yes	—
124	C₄×C₁₆	—	67	3	no	—
125	C₄×C₁₆	—	67	3	no	—
126	C₄×C₁₆	—	67	3	no	—
127	C₄×C₁₆	C₂×D₆₇	1	3	yes	—
128	C₂²×Q₈	—	67	1	no	—
129	C₂²×C₈	—	67	1	no	—
130	C₂²×C₈	—	67	3	no	—
131	C₄×Q₈	—	67	1	no	—
132	C₄×Q₈	—	67	3	no	—
133	C₄×Q₈	—	67	3	no	—
134	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	67	3	no	—
136	C₂×C₄²	—	67	3	no	—
137	C₂×C₁₆	—	67	1	no	—
138	C₂×C₁₆	—	67	4	no	—
139	C₂×C₁₆	—	67	3	no	—
140	C₂×C₁₆	—	134	6	no	—
141	C₂×C₁₆	C₂²×D₆₇	1	3	yes	—
142	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
143	C₄⋊C₈	—	67	6	no	—
144	C₄⋊C₈	—	67	3	no	—
145	C₄⋊C₈	C₄×D₆₇	1	3	yes	—
146	C₄×C₈	—	67	3	no	—
147	C₄×C₈	—	67	3	no	—
148	C₄×C₈	—	67	3	no	—
149	C₄×C₈	C₂²×D₆₇	1	3	yes	—
150	C₂×Q₈	—	67	8	no	—
151	C₂×Q₈	—	67	3	no	—
152	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
153	C₂²×C₄	—	67	3	no	—
154	C₂²×C₄	—	67	1	no	—
155	C₂²×C₄	—	67	3	no	—
156	C₂×C₈	—	67	1	no	—
157	C₂×C₈	—	67	2	no	—
158	C₂×C₈	—	67	2	no	—
159	C₂×C₈	—	67	3	no	—
160	C₂×C₈	—	134	6	no	—
161	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
162	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
163	C₄⋊C₄	—	67	6	no	—
164	C₄⋊C₄	—	67	3	no	—
165	C₄⋊C₄	C₈×D₆₇	1	3	yes	—
166	C₄²	—	67	3	no	—
167	C₄²	—	67	3	no	—
168	C₄²	—	67	3	no	—
169	C₄²	C₂×C₄×D₆₇	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	67	2	no	—
172	C₁₆	—	134	3	no	—
173	C₁₆	Q₈×D₆₇	1	2	yes	direct product factor
174	C₂³	—	67	1	no	—
175	Q₈	—	67	12	no	—
176	Q₈	C₁₆×D₆₇	1	4	yes	direct product factor
177	C₂×C₄	—	67	1	no	—
178	C₂×C₄	—	67	3	no	—
179	C₂×C₄	—	67	12	no	—
180	C₂×C₄	—	67	2	no	—
181	C₂×C₄	—	67	2	no	—
182	C₂×C₄	—	67	3	no	—
183	C₂×C₄	—	134	6	no	—
184	C₂×C₄	C₂×C₈×D₆₇	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₆₇	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	67	2	no	—
189	C₈	—	134	3	no	—
190	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
191	C₈	#41×D₆₇	1	1	yes	characteristic
192	C₂²	—	67	2	no	—
193	C₂²	—	67	2	no	—
194	C₂²	—	67	2	no	—
195	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	67	2	no	—
198	C₄	—	67	6	no	—
199	C₄	—	134	3	no	—
200	C₄	C₂×C₁₆×D₆₇	1	6	yes	—
201	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
202	C₄	#131×D₆₇	1	1	yes	characteristic
203	C₂	—	67	4	no	—
204	C₂	C₈×Q₈×D₆₇	1	1	yes	characteristic
205	C₂	C₂²×C₁₆×D₆₇	1	1	yes	characteristic
206	C₂	#504×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	Q₈×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂²×D₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×C₆₇⋊C₁₆	C₂	semidirect
3. C₁₆×Dic₁₃₄	C₂	semidirect
4. #22562111	C₂	semidirect
7. Q₈×C₁₆×C₆₇	C₂	semidirect
35. C₁₆×D₆₇	Q₈	direct (N × H)
47. Q₈×D₆₇	C₁₆	direct (N × H)
62. C₆₇⋊C₁₆	C₂×Q₈	semidirect
69. C₁₆×C₆₇	C₂×Q₈	semidirect
71. Q₈×C₆₇	C₂×C₁₆	semidirect
82. Dic₁₃₄	C₂×C₁₆	semidirect
103. D₆₇	Q₈×C₁₆	direct (N × H)
108. Q₈×C₁₆	D₆₇	direct (N × H)
111. C₆₇	C₂×Q₈×C₁₆	semidirect
173. C₁₆	Q₈×D₆₇	direct (N × H)
176. Q₈	C₁₆×D₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈×Q₈×D₆₇	2
Q₈×C₆₇⋊C₁₆	2
C₁₆×Dic₁₃₄	2
#22562111	2
C₄⋊C₁₆×D₆₇	2
C₄×C₁₆×D₆₇	2
Q₈×C₁₆×C₆₇	2
C₂×Q₈×C₁₆	67

Subgroups — Q8×C16×D67 #31009310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₆×D₆₇ #31009310