Subgroups — C₂×C₁₆×D₂₈₄ #32150603

Order 18176.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₇₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic
3	C₁₆×D₂₈₄	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₇₁)⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₇₁	C₂²	1	2	yes	—
8	C₂×C₄×D₂₈₄	C₄	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₁₆×D₇₁	—	2	8	no	—
11	C₂×C₁₆×D₇₁	C₂²	1	8	yes	complemented
12	C₈×D₂₈₄	C₄	1	4	yes	—
13	C₈×D₂₈₄	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₇₁)⋊C₈	C₄	1	2	yes	—
16	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	complemented
17	(C₂×D₇₁)⋊C₁₆	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₇₁	C₂²	1	4	yes	complemented
22	C₂²×D₂₈₄	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₇₁	—	2	4	no	—
26	C₂×C₈×D₇₁	—	2	4	no	—
27	C₂×C₈×D₇₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₇₁	C₂³	1	4	yes	—
29	C₄×D₂₈₄	C₈	1	4	yes	—
30	C₄×D₂₈₄	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
32	C₂×(C₂×D₇₁)⋊C₄	C₈	1	2	yes	—
33	C₂×C₇₁⋊C₁₆	—	2	4	no	—
34	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
35	C₁₆×D₇₁	—	2	16	no	—
36	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
37	(C₂×D₇₁)⋊C₈	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₁	—	2	2	no	—
42	C₂×C₁₆×C₇₁	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
44	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
45	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₁	C₂³	1	2	yes	—
47	C₂³×D₇₁	C₂×C₈	1	2	yes	—
48	C₂×D₂₈₄	C₁₆	1	8	yes	direct product factor
49	C₂×D₂₈₄	C₂×C₈	1	4	yes	—
50	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₇₁	—	2	4	no	—
52	C₂×C₄×D₇₁	—	2	4	no	—
53	C₂×C₄×D₇₁	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₇₁	C₂²×C₄	1	4	yes	—
55	C₂×C₇₁⋊C₈	—	2	4	no	—
56	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
57	C₈×D₇₁	—	2	8	no	—
58	C₈×D₇₁	—	2	4	no	—
59	C₈×D₇₁	—	2	4	no	—
60	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
61	(C₂×D₇₁)⋊C₄	C₂×C₈	1	8	yes	—
62	C₇₁⋊C₁₆	—	2	4	no	—
63	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	—	2	2	no	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₁	#41	1	2	yes	—
71	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×C₈	1	2	yes	—
73	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₇₁	—	2	2	no	—
75	C₁₆×C₇₁	C₂×D₄	1	4	yes	complemented
76	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₇₁	—	2	4	no	—
78	C₂²×D₇₁	—	2	4	no	—
79	C₂²×D₇₁	C₂×C₁₆	1	8	yes	complemented
80	C₂²×D₇₁	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	C₂×C₁₆	1	4	yes	complemented
83	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₇₁	#131	1	2	yes	—
88	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₇₁	—	2	4	no	—
90	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
91	D₂₈₄	C₂×C₁₆	1	16	yes	direct product factor
92	C₄×D₇₁	—	2	4	no	—
93	C₄×D₇₁	—	2	4	no	—
94	C₄×D₇₁	—	2	8	no	—
95	C₈×C₇₁	—	2	2	no	—
96	C₈×C₇₁	C₄×D₄	1	2	yes	—
97	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
99	C₇₁⋊C₈	—	2	4	no	—
100	C₂²×C₇₁	C₈×D₄	1	2	yes	—
101	C₂²×C₇₁	C₂²×C₁₆	1	2	yes	—
102	C₂²×C₇₁	#504	1	2	yes	—
103	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
104	C₂×D₇₁	—	2	4	no	—
105	C₂×D₇₁	—	2	4	no	—
106	C₂×D₇₁	—	2	16	no	—
107	C₂×D₇₁	C₂²×C₁₆	1	8	yes	complemented
108	C₄×C₇₁	—	2	2	no	—
109	C₄×C₇₁	C₈×D₄	1	2	yes	—
110	C₄×C₇₁	C₂²×C₁₆	1	4	yes	complemented
111	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
113	Dic₇₁	—	2	4	no	—
114	C₂×D₄×C₁₆	—	71	1	no	maximal, Sylow
115	C₂×C₇₁	D₄×C₁₆	1	4	yes	complemented
116	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₇₁	C₂³×C₁₆	1	1	yes	characteristic
118	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
119	D₇₁	—	2	8	no	—
120	C₂³×C₁₆	—	71	2	no	—
121	C₂×C₈×D₄	—	71	1	no	—
122	D₄×C₁₆	—	71	8	no	—
123	C₂×C₄⋊C₁₆	—	71	1	no	—
124	C₂×C₂²⋊C₁₆	—	71	2	no	—
125	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
126	C₇₁	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₈	—	71	2	no	—
128	C₂×C₄×D₄	—	71	1	no	—
129	C₂²×C₁₆	—	71	2	no	—
130	C₂²×C₁₆	—	71	8	no	—
131	C₂²×C₁₆	—	142	8	no	—
132	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
133	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
134	C₈×D₄	—	71	4	no	—
135	C₈×D₄	—	71	4	no	—
136	C₂×C₄⋊C₈	—	71	1	no	—
137	C₂×C₂²⋊C₈	—	71	2	no	—
138	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
139	C₄⋊C₁₆	—	71	4	no	—
140	C₂²⋊C₁₆	—	71	8	no	—
141	C₄×C₁₆	C₂×D₇₁	1	4	yes	complemented
142	C₂²×D₄	—	71	1	no	—
143	C₂³×C₄	—	71	2	no	—
144	C₂²×C₈	—	71	2	no	—
145	C₂²×C₈	—	71	4	no	—
146	C₂²×C₈	—	71	4	no	—
147	C₂²×C₈	—	142	4	no	—
148	C₂²×C₈	—	142	4	no	—
149	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
151	C₄×D₄	—	71	4	no	—
152	C₄×D₄	—	71	4	no	—
153	C₂×C₄⋊C₄	—	71	1	no	—
154	C₂×C₂²⋊C₄	—	71	2	no	—
155	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	71	4	no	—
158	C₂×C₁₆	—	142	4	no	—
159	C₂×C₁₆	—	142	16	no	—
160	C₂×C₁₆	D₂₈₄	1	4	yes	direct product factor
161	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
162	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
163	C₄⋊C₈	—	71	4	no	—
164	C₂²⋊C₈	—	71	8	no	—
165	C₄×C₈	C₄×D₇₁	1	2	yes	—
166	C₄×C₈	C₂²×D₇₁	1	2	yes	—
167	C₂⁴	—	71	2	no	—
168	C₂×D₄	—	71	8	no	—
169	C₂×D₄	—	71	4	no	—
170	C₂²×C₄	—	71	2	no	—
171	C₂²×C₄	—	71	4	no	—
172	C₂²×C₄	—	71	4	no	—
173	C₂²×C₄	—	142	4	no	—
174	C₂²×C₄	—	142	4	no	—
175	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
176	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	71	4	no	—
180	C₂×C₈	—	142	4	no	—
181	C₂×C₈	—	142	4	no	—
182	C₂×C₈	—	142	4	no	—
183	C₂×C₈	—	142	8	no	—
184	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
187	C₂×C₈	#11765865	1	2	yes	—
188	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
189	C₄⋊C₄	—	71	4	no	—
190	C₂²⋊C₄	—	71	8	no	—
191	C₄²	C₈×D₇₁	1	2	yes	—
192	C₄²	C₂×C₄×D₇₁	1	2	yes	—
193	C₁₆	—	2	2	no	—
194	C₁₆	—	142	4	no	—
195	C₁₆	C₂×D₂₈₄	1	4	yes	direct product factor
196	C₂³	—	71	8	no	—
197	C₂³	—	71	4	no	—
198	C₂³	—	142	4	no	—
199	C₂³	—	142	4	no	—
200	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
201	D₄	—	71	16	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	71	4	no	—
204	C₂×C₄	—	142	4	no	—
205	C₂×C₄	—	142	8	no	—
206	C₂×C₄	—	142	4	no	—
207	C₂×C₄	—	142	4	no	—
208	C₂×C₄	C₁₆×D₇₁	1	4	yes	complemented
209	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
213	C₂×C₄	#14556134	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	142	4	no	—
217	C₈	C₄×D₂₈₄	1	2	yes	—
218	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
219	C₈	C₂×#11765865	1	1	yes	characteristic
220	C₂²	—	71	8	no	—
221	C₂²	—	142	4	no	—
222	C₂²	—	142	16	no	—
223	C₂²	—	142	4	no	—
224	C₂²	C₈×D₂₈₄	1	2	yes	—
225	C₂²	C₂×C₁₆×D₇₁	1	2	yes	—
226	C₂²	#17986415	1	2	yes	—
227	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	142	4	no	—
230	C₄	C₈×D₂₈₄	1	2	yes	—
231	C₄	C₂×C₁₆×D₇₁	1	4	yes	complemented
232	C₄	C₂×C₄×D₂₈₄	1	1	yes	characteristic
233	C₄	C₂×#14556134	1	1	yes	characteristic
234	C₂	—	142	8	no	—
235	C₂	C₁₆×D₂₈₄	1	4	yes	direct product factor
236	C₂	C₂×C₈×D₂₈₄	1	1	yes	characteristic
237	C₂	C₂²×C₁₆×D₇₁	1	1	yes	characteristic
238	C₂	C₂×#17986415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₈
prime radical	C₂²×D₂₈₄	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₇₁	C₂	semidirect
3. C₁₆×D₂₈₄	C₂	direct (N × H)
4. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
5. C₂×(C₂×D₇₁)⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₁₆×C₇₁	C₂	semidirect
11. C₂×C₁₆×D₇₁	C₂²	semidirect
16. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
17. (C₂×D₇₁)⋊C₁₆	C₂²	semidirect
21. C₄×C₁₆×C₇₁	C₂²	semidirect
42. C₂×C₁₆×C₇₁	D₄	semidirect
48. C₂×D₂₈₄	C₁₆	direct (N × H)
75. C₁₆×C₇₁	C₂×D₄	semidirect
79. C₂²×D₇₁	C₂×C₁₆	semidirect
82. C₂×C₄×C₇₁	C₂×C₁₆	semidirect
91. D₂₈₄	C₂×C₁₆	direct (N × H)
107. C₂×D₇₁	C₂²×C₁₆	semidirect
110. C₄×C₇₁	C₂²×C₁₆	semidirect
115. C₂×C₇₁	D₄×C₁₆	semidirect
125. C₂×C₄×C₁₆	D₇₁	semidirect
126. C₇₁	C₂×D₄×C₁₆	semidirect
141. C₄×C₁₆	C₂×D₇₁	semidirect
160. C₂×C₁₆	D₂₈₄	direct (N × H)
195. C₁₆	C₂×D₂₈₄	direct (N × H)
208. C₂×C₄	C₁₆×D₇₁	semidirect
231. C₄	C₂×C₁₆×D₇₁	semidirect
235. C₂	C₁₆×D₂₈₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₇₁	2
C₂×C₈×D₂₈₄	2
C₁₆×D₂₈₄	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×(C₂×D₇₁)⋊C₁₆	2
C₂×C₄×C₁₆×C₇₁	2
C₂×D₄×C₁₆	71

Subgroups — C2×C16×D284 #32150603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×D₂₈₄ #32150603