Subgroups — C₂³×Q₁₆×D₇₁ #32911232

Order 18176.

11166 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₇₁	C₂	1	2	yes	maximal
2	C₂³×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11765855	C₂	1	2	yes	maximal, complemented
4	C₂²×Q₁₆×D₇₁	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Dic₁₄₂	—	2	2	no	—
8	C₂³×Dic₁₄₂	C₂²	1	2	yes	—
9	C₂²×Q₈×D₇₁	—	2	28	no	—
10	C₂²×Q₈×D₇₁	C₂²	1	56	yes	—
11	C₂³×C₄×D₇₁	—	2	2	no	—
12	C₂³×C₄×D₇₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₈₄	C₂²	1	28	yes	complemented
14	C₂²×#11765855	C₂²	1	56	yes	complemented
15	C₂×Q₁₆×D₇₁	C₂²	1	448	yes	direct product factor
16	C₂³×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₇₁	C₂²	1	28	yes	—
18	C₂³×Q₈×C₇₁	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₇₁	C₂²	1	28	yes	complemented
20	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂⁴×D₇₁	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₄₂	—	2	56	no	—
23	C₂²×Dic₁₄₂	—	2	14	no	—
24	C₂²×Dic₁₄₂	C₂³	1	28	yes	—
25	C₂×Q₈×D₇₁	—	2	336	no	—
26	C₂×Q₈×D₇₁	C₂³	1	224	yes	—
27	C₂³×Dic₇₁	—	2	2	no	—
28	C₂³×Dic₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₁	—	2	56	no	—
30	C₂²×C₄×D₇₁	D₄	1	14	yes	—
31	C₂²×C₄×D₇₁	C₂³	1	14	yes	—
32	C₂×Dic₂₈₄	C₂³	1	112	yes	complemented
33	C₂×#11765855	C₂³	1	224	yes	complemented
34	Q₁₆×D₇₁	C₂³	1	512	yes	direct product factor
35	C₂²×C₇₁⋊C₈	C₂³	1	14	yes	—
36	C₂×C₈×D₇₁	C₂³	1	112	yes	—
37	C₂²×Q₈×C₇₁	—	2	14	no	—
38	C₂²×Q₈×C₇₁	C₂³	1	28	yes	—
39	C₂³×C₄×C₇₁	—	2	2	no	—
40	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
41	C₂×Q₁₆×C₇₁	C₂³	1	112	yes	complemented
42	C₂²×C₈×C₇₁	C₂³	1	14	yes	—
43	C₂³×D₇₁	Q₁₆	1	16	yes	direct product factor
44	C₂³×D₇₁	C₂×D₄	1	14	yes	—
45	C₂×Dic₁₄₂	—	2	84	no	—
46	C₂×Dic₁₄₂	—	2	224	no	—
47	C₂×Dic₁₄₂	C₂⁴	1	56	yes	—
48	Q₈×D₇₁	—	2	448	no	—
49	Q₈×D₇₁	C₂⁴	1	128	yes	—
50	C₂²×Dic₇₁	—	2	28	no	—
51	C₂²×Dic₇₁	C₂×D₄	1	14	yes	—
52	C₂×C₄×D₇₁	—	2	224	no	—
53	C₂×C₄×D₇₁	C₂×D₄	1	84	yes	—
54	C₂×C₄×D₇₁	C₂⁴	1	28	yes	—
55	Dic₂₈₄	C₂⁴	1	64	yes	complemented
56	#11765855	C₂⁴	1	128	yes	complemented
57	C₂×C₇₁⋊C₈	C₂⁴	1	28	yes	—
58	C₈×D₇₁	C₂⁴	1	64	yes	—
59	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₇₁	—	2	84	no	—
61	C₂×Q₈×C₇₁	C₂⁴	1	56	yes	—
62	C₂²×C₄×C₇₁	—	2	28	no	—
63	C₂²×C₄×C₇₁	C₂×D₄	1	7	yes	—
64	C₂²×C₄×C₇₁	C₂⁴	1	7	yes	—
65	Q₁₆×C₇₁	C₂⁴	1	64	yes	complemented
66	C₂×C₈×C₇₁	C₂⁴	1	28	yes	—
67	C₂³×C₇₁	C₂×Q₁₆	1	8	yes	complemented
68	C₂³×C₇₁	C₂²×D₄	1	7	yes	—
69	C₂²×D₇₁	C₂×Q₁₆	1	112	yes	direct product factor
70	C₂²×D₇₁	C₂²×D₄	1	28	yes	—
71	Q₈×C₇₁	—	2	56	no	—
72	Q₈×C₇₁	C₂⁵	1	16	yes	—
73	C₂×C₄×C₇₁	—	2	56	no	—
74	C₂×C₄×C₇₁	C₂²×D₄	1	21	yes	—
75	C₂×C₄×C₇₁	C₂⁵	1	7	yes	—
76	C₂×Dic₇₁	—	2	56	no	—
77	C₂×Dic₇₁	C₂²×D₄	1	28	yes	—
78	C₄×D₇₁	—	2	128	no	—
79	C₄×D₇₁	C₂²×D₄	1	56	yes	—
80	C₄×D₇₁	C₂⁵	1	8	yes	—
81	Dic₁₄₂	—	2	56	no	—
82	Dic₁₄₂	—	2	128	no	—
83	Dic₁₄₂	C₂⁵	1	16	yes	—
84	C₈×C₇₁	C₂⁵	1	8	yes	—
85	C₇₁⋊C₈	C₂⁵	1	8	yes	—
86	C₂²×C₇₁	C₂²×Q₁₆	1	28	yes	complemented
87	C₂²×C₇₁	C₂³×D₄	1	7	yes	—
88	C₂×D₇₁	C₂²×Q₁₆	1	112	yes	direct product factor
89	C₂×D₇₁	C₂³×D₄	1	8	yes	—
90	C₄×C₇₁	—	2	16	no	—
91	C₄×C₇₁	C₂³×D₄	1	7	yes	—
92	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
93	Dic₇₁	—	2	16	no	—
94	Dic₇₁	C₂³×D₄	1	8	yes	—
95	C₂⁴×Q₁₆	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	C₂³×Q₁₆	1	14	yes	complemented
97	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
98	D₇₁	C₂³×Q₁₆	1	16	yes	direct product factor
99	C₂⁴×Q₈	—	71	2	no	—
100	C₂³×Q₁₆	—	71	2	no	—
101	C₂³×Q₁₆	—	71	56	no	—
102	C₂³×Q₁₆	—	71	1	no	—
103	C₂³×Q₁₆	D₇₁	1	1	yes	characteristic, direct product factor
104	C₂⁴×C₈	—	71	1	no	—
105	C₇₁	C₂⁴×Q₁₆	1	1	yes	characteristic, complemented, Sylow
106	C₂³×Q₈	—	71	56	no	—
107	C₂³×Q₈	—	71	2	no	—
108	C₂³×Q₈	—	142	2	no	—
109	C₂³×Q₈	—	142	28	no	—
110	C₂³×Q₈	C₂×D₇₁	1	2	yes	—
111	C₂⁴×C₄	—	71	1	no	—
112	C₂⁴×C₄	—	142	2	no	—
113	C₂²×Q₁₆	—	71	56	no	—
114	C₂²×Q₁₆	—	71	448	no	—
115	C₂²×Q₁₆	—	71	28	no	—
116	C₂²×Q₁₆	C₂×D₇₁	1	28	yes	direct product factor
117	C₂³×C₈	—	71	1	no	—
118	C₂³×C₈	—	71	28	no	—
119	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
120	C₂⁵	—	71	1	no	—
121	C₂²×Q₈	—	2	14	no	—
122	C₂²×Q₈	—	71	224	no	—
123	C₂²×Q₈	—	71	28	no	—
124	C₂²×Q₈	—	142	56	no	—
125	C₂²×Q₈	—	142	336	no	—
126	C₂²×Q₈	—	142	14	no	—
127	C₂²×Q₈	C₂²×D₇₁	1	28	yes	—
128	C₂³×C₄	—	2	2	no	—
129	C₂³×C₄	—	71	1	no	—
130	C₂³×C₄	—	71	14	no	—
131	C₂³×C₄	—	71	14	no	—
132	C₂³×C₄	—	142	56	no	—
133	C₂³×C₄	—	142	2	no	—
134	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
135	C₂×Q₁₆	—	71	224	no	—
136	C₂×Q₁₆	—	71	512	no	—
137	C₂×Q₁₆	—	71	112	no	—
138	C₂×Q₁₆	C₂²×D₇₁	1	112	yes	direct product factor
139	C₂²×C₈	—	71	14	no	—
140	C₂²×C₈	—	71	112	no	—
141	C₂²×C₈	C₂²×D₇₁	1	14	yes	—
142	C₂⁴	—	71	14	no	—
143	C₂⁴	—	71	16	no	—
144	C₂⁴	D₄×D₇₁	1	1	yes	characteristic
145	C₂×Q₈	—	2	84	no	—
146	C₂×Q₈	—	71	128	no	—
147	C₂×Q₈	—	71	56	no	—
148	C₂×Q₈	—	142	224	no	—
149	C₂×Q₈	—	142	448	no	—
150	C₂×Q₈	—	142	84	no	—
151	C₂×Q₈	C₂³×D₇₁	1	56	yes	—
152	C₂²×C₄	—	2	28	no	—
153	C₂²×C₄	—	71	14	no	—
154	C₂²×C₄	—	71	84	no	—
155	C₂²×C₄	—	71	28	no	—
156	C₂²×C₄	—	142	224	no	—
157	C₂²×C₄	—	142	28	no	—
158	C₂²×C₄	D₄×D₇₁	1	7	yes	—
159	C₂²×C₄	C₂³×D₇₁	1	7	yes	—
160	Q₁₆	—	71	128	no	—
161	Q₁₆	—	71	64	no	—
162	Q₁₆	C₂³×D₇₁	1	64	yes	direct product factor
163	C₂×C₈	—	71	28	no	—
164	C₂×C₈	—	71	64	no	—
165	C₂×C₈	C₂³×D₇₁	1	28	yes	—
166	C₂³	—	71	28	no	—
167	C₂³	—	71	112	no	—
168	C₂³	Q₁₆×D₇₁	1	8	yes	direct product factor
169	C₂³	C₂×D₄×D₇₁	1	7	yes	—
170	Q₈	—	2	56	no	—
171	Q₈	—	71	16	no	—
172	Q₈	—	142	128	no	—
173	Q₈	—	142	56	no	—
174	Q₈	C₂⁴×D₇₁	1	16	yes	—
175	C₂×C₄	—	2	56	no	—
176	C₂×C₄	—	71	28	no	—
177	C₂×C₄	—	71	56	no	—
178	C₂×C₄	—	71	8	no	—
179	C₂×C₄	—	142	128	no	—
180	C₂×C₄	—	142	56	no	—
181	C₂×C₄	C₂×D₄×D₇₁	1	21	yes	—
182	C₂×C₄	C₂⁴×D₇₁	1	7	yes	—
183	C₈	—	71	8	no	—
184	C₈	C₂⁴×D₇₁	1	8	yes	—
185	C₂²	—	71	112	no	—
186	C₂²	—	71	8	no	—
187	C₂²	C₂×Q₁₆×D₇₁	1	28	yes	direct product factor
188	C₂²	C₂²×D₄×D₇₁	1	7	yes	—
189	C₄	—	2	16	no	—
190	C₄	—	71	8	no	—
191	C₄	—	142	16	no	—
192	C₄	C₂²×D₄×D₇₁	1	7	yes	—
193	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
194	C₂	—	71	16	no	—
195	C₂	C₂²×Q₁₆×D₇₁	1	14	yes	direct product factor
196	C₂	C₂³×D₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₁
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂³×Q₁₆×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂×D₄
prime radical	C₂⁴×D₇₁	D₄
hypercenter	C₂³×Q₁₆	D₇₁
hypocenter	C₇₁	C₂⁴×Q₁₆
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₂₈₄	C₂	semidirect
3. C₂³×#11765855	C₂	semidirect
4. C₂²×Q₁₆×D₇₁	C₂	direct (N × H)
6. C₂³×Q₁₆×C₇₁	C₂	semidirect
13. C₂²×Dic₂₈₄	C₂²	semidirect
14. C₂²×#11765855	C₂²	semidirect
15. C₂×Q₁₆×D₇₁	C₂²	direct (N × H)
19. C₂²×Q₁₆×C₇₁	C₂²	semidirect
32. C₂×Dic₂₈₄	C₂³	semidirect
33. C₂×#11765855	C₂³	semidirect
34. Q₁₆×D₇₁	C₂³	direct (N × H)
41. C₂×Q₁₆×C₇₁	C₂³	semidirect
43. C₂³×D₇₁	Q₁₆	direct (N × H)
55. Dic₂₈₄	C₂⁴	semidirect
56. #11765855	C₂⁴	semidirect
65. Q₁₆×C₇₁	C₂⁴	semidirect
67. C₂³×C₇₁	C₂×Q₁₆	semidirect
69. C₂²×D₇₁	C₂×Q₁₆	direct (N × H)
86. C₂²×C₇₁	C₂²×Q₁₆	semidirect
88. C₂×D₇₁	C₂²×Q₁₆	direct (N × H)
96. C₂×C₇₁	C₂³×Q₁₆	semidirect
98. D₇₁	C₂³×Q₁₆	direct (N × H)
103. C₂³×Q₁₆	D₇₁	direct (N × H)
105. C₇₁	C₂⁴×Q₁₆	semidirect
116. C₂²×Q₁₆	C₂×D₇₁	direct (N × H)
138. C₂×Q₁₆	C₂²×D₇₁	direct (N × H)
162. Q₁₆	C₂³×D₇₁	direct (N × H)
168. C₂³	Q₁₆×D₇₁	direct (N × H)
187. C₂²	C₂×Q₁₆×D₇₁	direct (N × H)
195. C₂	C₂²×Q₁₆×D₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₇₁	2
C₂³×Dic₂₈₄	2
C₂³×#11765855	2
C₂²×Q₁₆×D₇₁	2
C₂³×C₈×D₇₁	2
C₂³×Q₁₆×C₇₁	2
C₂⁴×Q₁₆	71

Subgroups — C23×Q16×D71 #32911232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₁₆×D₇₁ #32911232