Subgroups — #41×C₆₇⋊C₁₆ #31009350

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×C₆₇⋊C₁₆	C₂	1	6	yes	maximal, complemented
3	#22562008	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	6	yes	—
7	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	6	yes	—
8	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	6	yes	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₆₇⋊C₁₆	C₂²	1	6	yes	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	3	yes	—
12	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₆₇	C₄	1	3	yes	—
16	C₈×D₄×C₆₇	C₂²	1	3	yes	—
17	#431×C₆₇	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
19	C₂×C₆₇⋊C₁₆	—	2	12	no	—
20	C₂×C₆₇⋊C₁₆	C₂³	1	6	yes	—
21	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
22	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
25	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₆₇	C₈	1	3	yes	—
28	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
29	#117×C₆₇	C₈	1	3	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₆₇	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₆₇	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₆₇	C₂³	1	3	yes	—
37	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₆₇⋊C₁₆	—	2	6	no	—
39	C₆₇⋊C₁₆	#41	1	4	yes	direct product factor
40	#41×C₆₇	C₁₆	1	8	yes	complemented
41	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₆₇	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₆₇	—	2	6	no	—
47	C₂×C₈×C₆₇	—	2	6	no	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₆₇	C₂×C₈	1	3	yes	—
53	C₄⋊C₄×C₆₇	C₂×C₈	1	3	yes	—
54	C₂²⋊C₄×C₆₇	C₂×C₈	1	6	yes	—
55	C₄²×C₆₇	C₂×C₈	1	3	yes	—
56	C₄²×C₆₇	C₂×C₈	1	3	yes	—
57	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
60	Q₈×C₆₇	C₂×C₁₆	1	4	yes	complemented
61	D₄×C₆₇	C₂×C₁₆	1	12	yes	complemented
62	C₂×C₄×C₆₇	—	2	3	no	—
63	C₂×C₄×C₆₇	—	2	3	no	—
64	C₂×C₄×C₆₇	—	2	6	no	—
65	C₂×C₄×C₆₇	C₂×C₁₆	1	12	yes	complemented
66	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₆₇	—	2	3	no	—
73	C₈×C₆₇	—	2	3	no	—
74	C₈×C₆₇	#131	1	2	yes	—
75	C₈×C₆₇	C₂×#41	1	2	yes	—
76	C₂²×C₆₇	—	2	3	no	—
77	C₂²×C₆₇	—	2	3	no	—
78	C₂²×C₆₇	C₂²×C₁₆	1	6	yes	—
79	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₆₇	—	2	3	no	—
81	C₄×C₆₇	—	2	3	no	—
82	C₄×C₆₇	C₂²×C₁₆	1	2	yes	—
83	C₄×C₆₇	C₂²×C₁₆	1	6	yes	—
84	C₄×C₆₇	#504	1	2	yes	—
85	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
86	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
87	C₁₆×#41	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	6	no	—
89	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
90	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
91	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
92	C₈×#41	D₆₇	1	1	yes	characteristic
93	Q₈×C₁₆	—	67	2	no	—
94	D₄×C₁₆	—	67	6	no	—
95	#2162	—	67	3	no	—
96	C₂×C₄×C₁₆	—	67	3	no	—
97	C₆₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
98	C₄×#41	Dic₆₇	1	1	yes	characteristic
99	C₂²×C₁₆	—	67	6	no	—
100	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
101	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
102	C₈×D₄	Dic₆₇	1	3	yes	—
103	C₈×D₄	C₂×D₆₇	1	3	yes	—
104	#431	Dic₆₇	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
106	C₄⋊C₁₆	—	67	6	no	—
107	C₂²⋊C₁₆	—	67	6	no	—
108	C₄×C₁₆	—	67	6	no	—
109	C₄×C₁₆	—	67	1	no	—
110	C₄×C₁₆	—	67	3	no	—
111	C₂×#41	C₆₇⋊C₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
113	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
114	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
116	C₄×D₄	C₆₇⋊C₈	1	3	yes	—
117	C₄×D₄	C₂×Dic₆₇	1	3	yes	—
118	#117	C₆₇⋊C₈	1	3	yes	—
119	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
120	C₂×C₁₆	—	67	2	no	—
121	C₂×C₁₆	—	67	6	no	—
122	C₂×C₁₆	—	134	12	no	—
123	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₆₇	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₆₇	1	6	yes	—
126	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
127	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
128	C₄×C₈	C₂²×D₆₇	1	3	yes	—
129	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
130	#41	C₆₇⋊C₁₆	1	8	yes	direct product factor
131	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
132	C₂×D₄	C₂×C₆₇⋊C₈	1	3	yes	—
133	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
134	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
139	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
140	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
142	C₄⋊C₄	C₂×C₆₇⋊C₈	1	3	yes	—
143	C₄⋊C₄	C₂×C₆₇⋊C₈	1	3	yes	—
144	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	6	yes	—
145	C₄²	C₂×C₆₇⋊C₈	1	3	yes	—
146	C₄²	C₂×C₆₇⋊C₈	1	3	yes	—
147	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
148	C₄²	C₂²×Dic₆₇	1	3	yes	—
149	C₁₆	—	67	4	no	—
150	C₁₆	—	134	6	no	—
151	C₂³	C₂²×C₆₇⋊C₈	1	3	yes	—
152	Q₈	C₂×C₆₇⋊C₁₆	1	4	yes	complemented
153	D₄	C₂×C₆₇⋊C₁₆	1	12	yes	complemented
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	C₂×C₆₇⋊C₁₆	1	12	yes	complemented
158	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
159	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
160	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
162	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#14524050	1	2	yes	—
167	C₈	#41×D₆₇	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×C₆₇⋊C₁₆	1	6	yes	—
171	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×C₆₇⋊C₁₆	1	2	yes	—
175	C₄	C₂²×C₆₇⋊C₁₆	1	6	yes	—
176	C₄	#16863813	1	2	yes	—
177	C₄	#41×Dic₆₇	1	1	yes	characteristic
178	C₄	C₂×#14524050	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
181	C₂	C₂³×C₆₇⋊C₁₆	1	1	yes	characteristic
182	C₂	C₂×#16863813	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂×#41×C₆₇	C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	C₁₆×#41
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₆₇⋊C₁₆	C₂	semidirect
2. D₄×C₆₇⋊C₁₆	C₂	semidirect
3. #22562008	C₂	semidirect
4. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
39. C₆₇⋊C₁₆	#41	direct (N × H)
40. #41×C₆₇	C₁₆	semidirect
60. Q₈×C₆₇	C₂×C₁₆	semidirect
61. D₄×C₆₇	C₂×C₁₆	semidirect
65. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
97. C₆₇	C₁₆×#41	semidirect
130. #41	C₆₇⋊C₁₆	direct (N × H)
152. Q₈	C₂×C₆₇⋊C₁₆	semidirect
153. D₄	C₂×C₆₇⋊C₁₆	semidirect
157. C₂×C₄	C₂×C₆₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₆₇⋊C₁₆	2
D₄×C₆₇⋊C₁₆	2
#22562008	2
C₂×C₄×C₆₇⋊C₁₆	2
C₈×#41×C₆₇	2
C₁₆×#41	67

Subgroups — #41×C67⋊C16 #31009350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×C₆₇⋊C₁₆ #31009350