Subgroups — C₂×C₇×D₄×C₁₆ #13210645

Order 1792.

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	C₂³×C₇×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×D₄×C₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	—	2	8	no	—
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
12	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇×C₁₆	C₂²	1	8	yes	complemented
14	C₇×C₈×D₄	C₄	1	4	yes	—
15	C₇×C₈×D₄	C₂²	1	4	yes	—
16	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₇×C₂²⋊C₁₆	C₂²	1	8	yes	complemented
21	C₄×C₇×C₁₆	C₂²	1	4	yes	complemented
22	C₂×D₄×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₇×D₄	C₈	1	1	yes	characteristic
24	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	—	2	4	no	—
26	C₂²×C₇×C₈	—	2	4	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂³	1	4	yes	—
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₄×C₇×D₄	C₈	1	4	yes	—
33	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
34	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₂×C₇×C₁₆	—	2	2	no	—
39	C₂×C₇×C₁₆	—	2	16	no	—
40	C₂×C₇×C₁₆	D₄	1	4	yes	direct product factor
41	C₂×C₇×C₁₆	C₂³	1	4	yes	—
42	C₂×C₇×C₁₆	C₂³	1	2	yes	—
43	C₂×C₇×C₁₆	C₂³	1	2	yes	—
44	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	C₂³×C₁₆	C₂×C₇	1	2	yes	complemented
49	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
50	D₄×C₁₆	C₂×C₇	1	8	yes	direct product factor
51	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂×C₂²⋊C₁₆	C₂×C₇	1	2	yes	complemented
53	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂⁴×C₇	C₂×C₈	1	2	yes	—
55	C₂×C₇×D₄	C₁₆	1	8	yes	direct product factor
56	C₂×C₇×D₄	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	4	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	2	8	no	—
68	C₂×C₇×C₈	—	2	4	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₇×C₈	#41	1	2	yes	—
75	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
76	C₇×C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₇×C₂²⋊C₄	C₂×C₈	1	8	yes	—
78	C₄²×C₇	C₂×C₈	1	2	yes	—
79	C₄²×C₇	C₂²×C₄	1	2	yes	—
80	C₇×C₁₆	—	2	4	no	—
81	C₇×C₁₆	—	2	2	no	—
82	C₇×C₁₆	C₂×D₄	1	4	yes	direct product factor
83	C₂³×C₈	C₂²×C₇	1	2	yes	—
84	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
85	C₂²×C₁₆	—	2	8	no	—
86	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
87	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
88	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
89	C₂²×C₁₆	C₂²×C₇	1	8	yes	complemented
90	C₈×D₄	C₄×C₇	1	4	yes	—
91	C₈×D₄	C₂²×C₇	1	4	yes	—
92	C₂×C₄⋊C₈	C₄×C₇	1	1	yes	characteristic
93	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
94	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
95	C₄⋊C₁₆	C₂²×C₇	1	4	yes	complemented
96	C₂²⋊C₁₆	C₂²×C₇	1	8	yes	complemented
97	C₄×C₁₆	C₂²×C₇	1	4	yes	complemented
98	C₂³×C₇	—	2	4	no	—
99	C₂³×C₇	—	2	4	no	—
100	C₂³×C₇	C₂×C₁₆	1	8	yes	complemented
101	C₂³×C₇	C₂²×C₈	1	4	yes	—
102	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
103	C₇×D₄	C₂×C₁₆	1	16	yes	direct product factor
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	2	8	no	—
109	C₂×C₄×C₇	C₂×C₁₆	1	4	yes	complemented
110	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₇	#131	1	2	yes	—
116	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
117	C₇×C₈	—	2	4	no	—
118	C₇×C₈	—	2	2	no	—
119	C₇×C₈	C₄×D₄	1	2	yes	—
120	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
121	C₇×C₈	C₂×#41	1	1	yes	characteristic
122	C₂²×D₄	C₇×C₈	1	1	yes	characteristic
123	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
128	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₇	1	4	yes	—
130	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
131	C₄×D₄	C₇×C₈	1	4	yes	—
132	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
133	C₂×C₄⋊C₄	C₇×C₈	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	C₇×C₈	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	16	no	—
139	C₂×C₁₆	C₇×D₄	1	4	yes	direct product factor
140	C₂×C₁₆	C₂³×C₇	1	2	yes	—
141	C₂×C₁₆	C₂³×C₇	1	4	yes	—
142	C₂×C₁₆	C₂³×C₇	1	2	yes	—
143	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
144	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
145	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
146	C₄×C₈	C₂³×C₇	1	2	yes	—
147	C₂²×C₇	—	2	4	no	—
148	C₂²×C₇	—	2	4	no	—
149	C₂²×C₇	—	2	16	no	—
150	C₂²×C₇	C₈×D₄	1	2	yes	—
151	C₂²×C₇	C₂²×C₁₆	1	2	yes	—
152	C₂²×C₇	C₂²×C₁₆	1	8	yes	complemented
153	C₂²×C₇	#504	1	2	yes	—
154	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	—	2	4	no	—
157	C₄×C₇	C₈×D₄	1	2	yes	—
158	C₄×C₇	C₂²×C₁₆	1	4	yes	complemented
159	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₇	C₂×#131	1	1	yes	characteristic
161	C₂⁴	C₂×C₇×C₈	1	2	yes	—
162	C₂×D₄	C₇×C₁₆	1	8	yes	direct product factor
163	C₂×D₄	C₂×C₇×C₈	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
168	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
169	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
181	C₂×C₈	C₇×#41	1	2	yes	—
182	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂×C₇×C₈	1	4	yes	—
184	C₂²⋊C₄	C₂×C₇×C₈	1	8	yes	—
185	C₄²	C₂×C₇×C₈	1	2	yes	—
186	C₄²	C₂²×C₄×C₇	1	2	yes	—
187	C₁₆	—	2	4	no	—
188	C₁₆	—	2	2	no	—
189	C₁₆	C₂×C₇×D₄	1	4	yes	direct product factor
190	C₂×C₇	—	2	8	no	—
191	C₂×C₇	D₄×C₁₆	1	4	yes	direct product factor
192	C₂×C₇	C₂×C₈×D₄	1	1	yes	characteristic
193	C₂×C₇	C₂³×C₁₆	1	1	yes	characteristic
194	C₂×C₇	C₂×#504	1	1	yes	characteristic
195	C₂³	—	2	4	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂×C₇×C₁₆	1	8	yes	complemented
198	C₂³	C₂²×C₇×C₈	1	4	yes	—
199	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
200	D₄	C₂×C₇×C₁₆	1	16	yes	direct product factor
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	C₂×C₇×C₁₆	1	4	yes	complemented
207	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
208	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
209	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
210	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
211	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
212	C₂×C₄	C₇×#131	1	2	yes	—
213	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	2	2	no	—
216	C₈	C₄×C₇×D₄	1	2	yes	—
217	C₈	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₈	C₂×C₇×#41	1	1	yes	characteristic
219	C₇	C₂×D₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	4	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	4	no	—
223	C₂²	C₇×C₈×D₄	1	2	yes	—
224	C₂²	C₂²×C₇×C₁₆	1	2	yes	—
225	C₂²	C₂²×C₇×C₁₆	1	8	yes	complemented
226	C₂²	C₇×#504	1	2	yes	—
227	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₇×C₈×D₄	1	2	yes	—
231	C₄	C₂²×C₇×C₁₆	1	4	yes	complemented
232	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
233	C₄	C₂×C₇×#131	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	C₇×D₄×C₁₆	1	4	yes	direct product factor
236	C₂	C₂×C₇×C₈×D₄	1	1	yes	characteristic
237	C₂	C₂³×C₇×C₁₆	1	1	yes	characteristic
238	C₂	C₂×C₇×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇×C₁₆	C₂²
commutator	C₂	C₂³×C₇×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁴×C₇
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂²×C₇×D₄	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×C₁₆	C₂	semidirect
3. C₇×D₄×C₁₆	C₂	direct (N × H)
4. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₇×C₂²⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₇×C₁₆	C₂	semidirect
13. C₂²×C₇×C₁₆	C₂²	semidirect
19. C₇×C₄⋊C₁₆	C₂²	semidirect
20. C₇×C₂²⋊C₁₆	C₂²	semidirect
21. C₄×C₇×C₁₆	C₂²	semidirect
22. C₂×D₄×C₁₆	C₇	direct (N × H)
40. C₂×C₇×C₁₆	D₄	direct (N × H)
48. C₂³×C₁₆	C₂×C₇	semidirect
50. D₄×C₁₆	C₂×C₇	direct (N × H)
51. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
52. C₂×C₂²⋊C₁₆	C₂×C₇	semidirect
53. C₂×C₄×C₁₆	C₂×C₇	semidirect
55. C₂×C₇×D₄	C₁₆	direct (N × H)
82. C₇×C₁₆	C₂×D₄	direct (N × H)
89. C₂²×C₁₆	C₂²×C₇	semidirect
95. C₄⋊C₁₆	C₂²×C₇	semidirect
96. C₂²⋊C₁₆	C₂²×C₇	semidirect
97. C₄×C₁₆	C₂²×C₇	semidirect
100. C₂³×C₇	C₂×C₁₆	semidirect
103. C₇×D₄	C₂×C₁₆	direct (N × H)
109. C₂×C₄×C₇	C₂×C₁₆	semidirect
139. C₂×C₁₆	C₇×D₄	direct (N × H)
152. C₂²×C₇	C₂²×C₁₆	semidirect
158. C₄×C₇	C₂²×C₁₆	semidirect
162. C₂×D₄	C₇×C₁₆	direct (N × H)
189. C₁₆	C₂×C₇×D₄	direct (N × H)
191. C₂×C₇	D₄×C₁₆	direct (N × H)
197. C₂³	C₂×C₇×C₁₆	semidirect
200. D₄	C₂×C₇×C₁₆	direct (N × H)
206. C₂×C₄	C₂×C₇×C₁₆	semidirect
219. C₇	C₂×D₄×C₁₆	direct (N × H)
225. C₂²	C₂²×C₇×C₁₆	semidirect
231. C₄	C₂²×C₇×C₁₆	semidirect
235. C₂	C₇×D₄×C₁₆	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×C₇×C₁₆	2
C₂×C₇×C₈×D₄	2
C₇×D₄×C₁₆	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×C₂²⋊C₁₆	2
C₂×C₄×C₇×C₁₆	2
C₂×D₄×C₁₆	7

Subgroups — C2×C7×D4×C16 #13210645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×D₄×C₁₆ #13210645