Subgroups — C₂×D₁₆×Dic₇₁ #32150713

Order 18176.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₇₁	C₂	1	2	yes	maximal, complemented
2	D₁₆×Dic₇₁	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985825	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17985800	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×Dic₇₁	—	2	2	no	—
8	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	D₈×Dic₇₁	—	2	4	no	—
10	D₈×Dic₇₁	C₂²	1	8	yes	complemented
11	C₂×#14556062	C₂²	1	2	yes	—
12	C₂×#14556049	—	2	2	no	—
13	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
14	#17985825	C₂²	1	4	yes	complemented
15	#17985800	C₂²	1	8	yes	complemented
16	C₁₆×Dic₇₁	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
18	C₂×D₁₆×C₇₁	C₄	1	8	yes	complemented
19	C₂×D₁₆×C₇₁	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂³×Dic₇₁	—	4	2	no	—
22	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
23	D₄×Dic₇₁	—	2	8	no	—
24	D₄×Dic₇₁	—	4	4	no	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₂²⋊Dic₇₁	—	4	2	no	—
27	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
30	#14556062	—	2	2	no	—
31	#14556062	C₂³	1	4	yes	—
32	#14556049	—	2	8	no	—
33	C₈×Dic₇₁	D₄	1	2	yes	—
34	C₈×Dic₇₁	C₂³	1	2	yes	—
35	C₂²×D₄×C₇₁	—	2	2	no	—
36	C₂×D₈×C₇₁	—	2	2	no	—
37	C₂×D₈×C₇₁	—	2	4	no	—
38	C₂×D₈×C₇₁	C₂×C₄	1	8	yes	complemented
39	C₂×D₈×C₇₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	D₁₆×C₇₁	C₂×C₄	1	16	yes	complemented
42	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
44	C₂²×Dic₇₁	—	4	2	no	—
45	C₂²×Dic₇₁	—	4	8	no	—
46	C₂²×Dic₇₁	—	8	8	no	—
47	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
48	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
49	C₂×C₇₁⋊C₈	—	2	2	no	—
50	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
51	C₄⋊Dic₇₁	—	2	4	no	—
52	C₄⋊Dic₇₁	—	4	2	no	—
53	C₂²⋊Dic₇₁	—	4	8	no	—
54	C₄×Dic₇₁	D₈	1	2	yes	—
55	C₄×Dic₇₁	C₂×D₄	1	2	yes	—
56	C₇₁⋊C₁₆	—	2	2	no	—
57	C₂⁴×C₇₁	—	4	2	no	—
58	C₂×D₄×C₇₁	—	2	8	no	—
59	C₂×D₄×C₇₁	—	2	4	no	—
60	C₂×D₄×C₇₁	—	4	2	no	—
61	C₂×D₄×C₇₁	—	4	4	no	—
62	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
63	D₈×C₇₁	—	2	8	no	—
64	D₈×C₇₁	—	2	4	no	—
65	D₈×C₇₁	C₂²×C₄	1	8	yes	complemented
66	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	#41	1	2	yes	—
69	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
70	C₁₆×C₇₁	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₇₁	—	4	4	no	—
72	C₂³×C₇₁	—	4	8	no	—
73	C₂³×C₇₁	—	8	4	no	—
74	C₂³×C₇₁	—	8	4	no	—
75	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
76	D₄×C₇₁	—	2	8	no	—
77	D₄×C₇₁	—	4	4	no	—
78	D₄×C₇₁	—	4	8	no	—
79	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#135	1	2	yes	—
82	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₇₁	—	2	2	no	—
84	C₂×Dic₇₁	—	4	4	no	—
85	C₂×Dic₇₁	—	8	4	no	—
86	C₂×Dic₇₁	—	8	16	no	—
87	C₂×Dic₇₁	D₁₆	1	4	yes	direct product factor
88	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
89	C₂×Dic₇₁	C₂×D₈	1	2	yes	—
90	C₈×C₇₁	C₄×D₄	1	2	yes	—
91	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
93	C₇₁⋊C₈	—	2	2	no	—
94	C₂²×C₇₁	—	4	8	no	—
95	C₂²×C₇₁	—	8	4	no	—
96	C₂²×C₇₁	—	8	4	no	—
97	C₂²×C₇₁	—	8	16	no	—
98	C₂²×C₇₁	C₄×D₈	1	2	yes	—
99	C₂²×C₇₁	C₂×D₁₆	1	2	yes	—
100	C₂²×C₇₁	#508	1	2	yes	—
101	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₇₁	C₄×D₈	1	2	yes	—
103	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
104	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
105	Dic₇₁	—	2	2	no	—
106	Dic₇₁	—	8	4	no	—
107	Dic₇₁	C₂×D₁₆	1	4	yes	direct product factor
108	C₂×C₄×D₁₆	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	—	8	8	no	—
110	C₂×C₇₁	C₄×D₁₆	1	4	yes	complemented
111	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
112	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
114	C₂²×D₁₆	D₇₁	1	1	yes	characteristic
115	C₂×C₄×D₈	—	71	2	no	—
116	C₄×D₁₆	—	71	8	no	—
117	C₂×#366	—	71	1	no	—
118	C₂×#357	—	71	2	no	—
119	C₂×C₄×C₁₆	—	71	1	no	—
120	C₇₁	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₈	C₂×D₇₁	1	2	yes	—
122	C₂×C₄×D₄	—	142	2	no	—
123	C₂×D₁₆	Dic₇₁	1	8	yes	direct product factor
124	C₂×D₁₆	C₂×D₇₁	1	4	yes	—
125	C₂²×C₁₆	—	71	1	no	—
126	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
127	C₄×D₈	—	71	8	no	—
128	C₄×D₈	—	142	4	no	—
129	C₂×#107	—	71	2	no	—
130	C₂×#102	—	142	2	no	—
131	C₂×C₄×C₈	—	71	1	no	—
132	#366	—	71	4	no	—
133	#357	—	71	8	no	—
134	C₄×C₁₆	—	71	4	no	—
135	C₂²×D₄	—	2	2	no	—
136	C₂³×C₄	—	284	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	C₂×Dic₇₁	1	8	yes	complemented
140	C₂×D₈	C₂²×D₇₁	1	4	yes	—
141	C₂²×C₈	—	71	1	no	—
142	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
143	C₄×D₄	—	284	4	no	—
144	C₄×D₄	—	142	8	no	—
145	C₂×C₄⋊C₄	—	142	2	no	—
146	C₂×C₂²⋊C₄	—	284	2	no	—
147	C₂×C₄²	—	71	1	no	—
148	D₁₆	C₂×Dic₇₁	1	16	yes	direct product factor
149	C₂×C₁₆	—	71	2	no	—
150	C₂×C₁₆	—	142	2	no	—
151	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
153	#107	—	71	4	no	—
154	#107	—	142	2	no	—
155	#102	—	142	8	no	—
156	C₄×C₈	—	71	2	no	—
157	C₄×C₈	—	71	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	284	2	no	—
164	C₂²×C₄	—	284	8	no	—
165	C₂²×C₄	—	568	8	no	—
166	C₂²×C₄	—	71	1	no	—
167	C₂²×C₄	—	71	1	no	—
168	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
169	D₈	—	2	4	no	—
170	D₈	—	2	8	no	—
171	D₈	C₂²×Dic₇₁	1	8	yes	complemented
172	C₂×C₈	—	71	2	no	—
173	C₂×C₈	—	142	2	no	—
174	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
175	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
176	C₂×C₈	#11765864	1	2	yes	—
177	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
178	C₄⋊C₄	—	284	2	no	—
179	C₄⋊C₄	—	142	4	no	—
180	C₂²⋊C₄	—	284	8	no	—
181	C₄²	—	71	2	no	—
182	C₄²	—	71	2	no	—
183	C₁₆	—	142	2	no	—
184	C₁₆	C₂²×Dic₇₁	1	4	yes	complemented
185	C₂³	—	4	8	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	4	no	—
189	C₂³	D₈×D₇₁	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	D₄	—	4	8	no	—
192	D₄	—	4	4	no	—
193	C₂×C₄	—	284	4	no	—
194	C₂×C₄	—	568	4	no	—
195	C₂×C₄	—	568	16	no	—
196	C₂×C₄	—	71	4	no	—
197	C₂×C₄	—	71	2	no	—
198	C₂×C₄	—	71	2	no	—
199	C₂×C₄	—	142	2	no	—
200	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
201	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
202	C₂×C₄	#14556147	1	2	yes	—
203	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
204	C₈	—	142	2	no	—
205	C₈	D₄×Dic₇₁	1	2	yes	—
206	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
207	C₈	C₂×#11765864	1	1	yes	characteristic
208	C₂²	—	4	8	no	—
209	C₂²	—	8	4	no	—
210	C₂²	—	8	16	no	—
211	C₂²	—	8	4	no	—
212	C₂²	D₈×Dic₇₁	1	2	yes	—
213	C₂²	D₁₆×D₇₁	1	2	yes	—
214	C₂²	#17986428	1	2	yes	—
215	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
216	C₄	—	568	4	no	—
217	C₄	—	71	4	no	—
218	C₄	—	142	2	no	—
219	C₄	D₈×Dic₇₁	1	2	yes	—
220	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
221	C₄	C₂×#14556147	1	1	yes	characteristic
222	C₂	—	8	8	no	—
223	C₂	D₁₆×Dic₇₁	1	4	yes	direct product factor
224	C₂	C₂×D₈×Dic₇₁	1	1	yes	characteristic
225	C₂	C₂×D₁₆×D₇₁	1	1	yes	characteristic
226	C₂	C₂×#17986428	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting, prime radical	C₂²×D₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂²×D₁₆	D₇₁
hypocenter	C₇₁	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×Dic₇₁	C₂	semidirect
2. D₁₆×Dic₇₁	C₂	direct (N × H)
3. C₂×#17985825	C₂	semidirect
4. C₂×#17985800	C₂	semidirect
5. C₂×C₁₆×Dic₇₁	C₂	semidirect
10. D₈×Dic₇₁	C₂²	semidirect
14. #17985825	C₂²	semidirect
15. #17985800	C₂²	semidirect
16. C₁₆×Dic₇₁	C₂²	semidirect
18. C₂×D₁₆×C₇₁	C₄	semidirect
38. C₂×D₈×C₇₁	C₂×C₄	semidirect
41. D₁₆×C₇₁	C₂×C₄	semidirect
42. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
65. D₈×C₇₁	C₂²×C₄	semidirect
70. C₁₆×C₇₁	C₂²×C₄	semidirect
87. C₂×Dic₇₁	D₁₆	direct (N × H)
107. Dic₇₁	C₂×D₁₆	direct (N × H)
110. C₂×C₇₁	C₄×D₁₆	semidirect
120. C₇₁	C₂×C₄×D₁₆	semidirect
123. C₂×D₁₆	Dic₇₁	direct (N × H)
139. C₂×D₈	C₂×Dic₇₁	semidirect
148. D₁₆	C₂×Dic₇₁	direct (N × H)
151. C₂×C₁₆	C₂×Dic₇₁	semidirect
171. D₈	C₂²×Dic₇₁	semidirect
184. C₁₆	C₂²×Dic₇₁	semidirect
223. C₂	D₁₆×Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₈×Dic₇₁	2
D₁₆×Dic₇₁	2
C₂×#17985825	2
C₂×#17985800	2
C₂×C₁₆×Dic₇₁	2
C₂²×D₁₆×C₇₁	2
C₂×C₄×D₁₆	71

Subgroups — C2×D16×Dic71 #32150713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×Dic₇₁ #32150713