Subgroups — C₂×D₃×C₇×M₁₆ #13061459

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₇×M₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
6	C₂×D₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×M₁₆	—	3	1	no	maximal, Hall
8	C₂³×D₃×C₇	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×Dic₃	C₄	1	1	yes	characteristic
11	C₂×C₄×D₃×C₇	C₄	1	4	yes	—
12	C₂×C₄×D₃×C₇	C₄	1	1	yes	characteristic
13	C₂×C₄×D₃×C₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃×C₇	C₂²	1	4	yes	—
17	C₃×C₇×M₁₆	C₂²	1	4	yes	complemented
18	C₂×C₃×C₇×C₈	C₂²	1	2	yes	complemented
19	C₇×#208	C₂²	1	4	yes	complemented
20	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
21	C₇×ΓL(1,𝔽₂₅)	C₂²	1	8	yes	complemented
22	D₃×C₇×C₈	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₇	—	3	1	no	—
24	C₂×C₇×M₁₆	—	3	1	no	—
25	C₂×C₇×M₁₆	—	3	8	no	—
26	C₂×C₇×M₁₆	—	3	2	no	—
27	C₂×C₇×M₁₆	D₃	1	1	yes	characteristic, direct product factor
28	C₂²×C₇×C₈	—	3	2	no	—
29	C₂×D₃×M₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₃×C₇	—	2	4	no	—
32	C₂²×D₃×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₃×C₇	C₂×C₄	1	4	yes	—
34	C₂²×D₃×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×Dic₃	C₂×C₄	1	4	yes	—
40	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₄×D₃×C₇	C₂×C₄	1	2	yes	—
43	C₄×D₃×C₇	C₂×C₄	1	4	yes	—
44	C₄×D₃×C₇	C₂×C₄	1	2	yes	—
45	C₄×D₃×C₇	C₂×C₄	1	4	yes	—
46	C₄×D₃×C₇	C₂³	1	2	yes	—
47	C₄×D₃×C₇	C₂³	1	2	yes	—
48	C₃×C₇×C₈	C₂³	1	4	yes	complemented
49	C₇×C₃⋊C₈	C₂³	1	4	yes	complemented
50	C₂⁴×C₇	—	3	1	no	—
51	C₂²×C₄×C₇	—	3	1	no	—
52	C₂²×C₄×C₇	—	3	1	no	—
53	C₂²×C₄×C₇	—	3	4	no	—
54	C₂²×C₄×C₇	—	3	1	no	—
55	C₂²×C₄×C₇	—	3	4	no	—
56	C₂²×C₄×C₇	—	3	1	no	—
57	C₂²×C₄×C₇	—	3	1	no	—
58	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
59	C₇×M₁₆	—	3	4	no	—
60	C₇×M₁₆	—	3	8	no	—
61	C₇×M₁₆	C₂×D₃	1	4	yes	direct product factor
62	C₂×C₇×C₈	—	3	2	no	—
63	C₂×C₇×C₈	—	3	8	no	—
64	C₂×C₇×C₈	C₂×D₃	1	2	yes	complemented
65	C₂²×C₄×D₃	C₂×C₇	1	1	yes	characteristic
66	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
67	C₂×#208	C₂×C₇	1	1	yes	characteristic, complemented
68	D₃×M₁₆	C₂×C₇	1	8	yes	direct product factor
69	C₂×ΓL(1,𝔽₂₅)	C₂×C₇	1	2	yes	complemented
70	C₂×D₃×C₈	C₂×C₇	1	2	yes	complemented
71	C₂²×C₃×C₇	—	2	2	no	—
72	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×D₃×C₇	—	2	2	no	—
75	C₂×D₃×C₇	—	2	8	no	—
76	C₂×D₃×C₇	M₁₆	1	4	yes	direct product factor
77	C₂×D₃×C₇	C₂²×C₄	1	2	yes	—
78	C₂×D₃×C₇	C₂²×C₄	1	2	yes	—
79	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
80	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
82	C₇×Dic₃	C₂²×C₄	1	2	yes	—
83	C₇×Dic₃	C₂²×C₄	1	2	yes	—
84	C₂²×M₁₆	—	3	1	no	Sylow
85	C₂³×C₇	—	3	1	no	—
86	C₂³×C₇	—	3	4	no	—
87	C₂³×C₇	—	3	1	no	—
88	C₂³×C₇	—	6	4	no	—
89	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
90	C₂×C₄×C₇	—	3	4	no	—
91	C₂×C₄×C₇	—	3	1	no	—
92	C₂×C₄×C₇	—	3	1	no	—
93	C₂×C₄×C₇	—	3	4	no	—
94	C₂×C₄×C₇	—	3	4	no	—
95	C₂×C₄×C₇	—	3	2	no	—
96	C₂×C₄×C₇	—	3	2	no	—
97	C₂×C₄×C₇	—	3	2	no	—
98	C₂×C₄×C₇	—	3	2	no	—
99	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
101	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
102	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
103	C₇×C₈	—	3	4	no	—
104	C₇×C₈	C₂²×D₃	1	4	yes	complemented
105	C₂³×D₃	C₄×C₇	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
107	C₂²×Dic₃	C₄×C₇	1	1	yes	characteristic
108	C₂×C₄×D₃	C₄×C₇	1	4	yes	—
109	C₂×C₄×D₃	C₄×C₇	1	1	yes	characteristic
110	C₂×C₄×D₃	C₄×C₇	1	1	yes	characteristic
111	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄×D₃	C₂²×C₇	1	4	yes	—
114	C₃×M₁₆	C₂²×C₇	1	4	yes	complemented
115	C₂×C₃×C₈	C₂²×C₇	1	2	yes	complemented
116	#208	C₂²×C₇	1	4	yes	complemented
117	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	complemented
118	ΓL(1,𝔽₂₅)	C₂²×C₇	1	8	yes	complemented
119	D₃×C₈	C₂²×C₇	1	8	yes	complemented
120	C₂×C₃×C₇	—	2	2	no	—
121	C₂×C₃×C₇	C₂×M₁₆	1	2	yes	complemented
122	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
123	D₃×C₇	—	2	2	no	—
124	D₃×C₇	C₂×M₁₆	1	4	yes	direct product factor
125	C₂³×C₄	—	3	1	no	—
126	C₂×M₁₆	—	3	1	no	—
127	C₂×M₁₆	—	3	8	no	—
128	C₂×M₁₆	—	3	2	no	—
129	C₂×M₁₆	D₃×C₇	1	1	yes	characteristic, direct product factor
130	C₂²×C₈	—	3	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	3	4	no	—
133	C₂²×C₇	—	3	2	no	—
134	C₂²×C₇	—	3	2	no	—
135	C₂²×C₇	—	6	8	no	—
136	C₂²×C₇	—	6	2	no	—
137	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
138	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
139	C₄×C₇	—	3	2	no	—
140	C₄×C₇	—	3	2	no	—
141	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
142	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
143	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
144	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂²×D₃	—	2	4	no	—
146	C₂²×D₃	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂²×D₃	C₂×C₄×C₇	1	4	yes	—
148	C₂²×D₃	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
153	C₂×Dic₃	C₂×C₄×C₇	1	4	yes	—
154	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
155	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
156	C₄×D₃	C₂×C₄×C₇	1	2	yes	—
157	C₄×D₃	C₂×C₄×C₇	1	4	yes	—
158	C₄×D₃	C₂×C₄×C₇	1	2	yes	—
159	C₄×D₃	C₂×C₄×C₇	1	4	yes	—
160	C₄×D₃	C₂³×C₇	1	2	yes	—
161	C₄×D₃	C₂³×C₇	1	2	yes	—
162	C₃×C₈	C₂³×C₇	1	4	yes	complemented
163	C₃⋊C₈	C₂³×C₇	1	4	yes	complemented
164	C₃×C₇	C₂²×M₁₆	1	1	yes	characteristic, complemented, Hall
165	C₂⁴	—	3	1	no	—
166	C₂²×C₄	—	3	1	no	—
167	C₂²×C₄	—	3	1	no	—
168	C₂²×C₄	—	3	4	no	—
169	C₂²×C₄	—	3	1	no	—
170	C₂²×C₄	—	3	1	no	—
171	C₂²×C₄	—	3	4	no	—
172	C₂²×C₄	—	3	1	no	—
173	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
174	M₁₆	—	3	4	no	—
175	M₁₆	—	3	8	no	—
176	M₁₆	C₂×D₃×C₇	1	4	yes	direct product factor
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	3	8	no	—
179	C₂×C₈	C₂×D₃×C₇	1	2	yes	complemented
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	—	3	4	no	—
182	C₂×C₇	—	6	2	no	—
183	C₂×C₇	D₃×M₁₆	1	2	yes	direct product factor
184	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₂²×C₄×C₇	1	2	yes	—
187	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
188	C₂×D₃	—	2	2	no	—
189	C₂×D₃	—	2	8	no	—
190	C₂×D₃	C₇×M₁₆	1	4	yes	direct product factor
191	C₂×D₃	C₂²×C₄×C₇	1	2	yes	—
192	C₂×D₃	C₂²×C₄×C₇	1	2	yes	—
193	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
194	C₃×C₄	C₂²×C₄×C₇	1	2	yes	—
195	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
196	Dic₃	C₂²×C₄×C₇	1	2	yes	—
197	Dic₃	C₂²×C₄×C₇	1	2	yes	—
198	C₂³	—	3	1	no	—
199	C₂³	—	3	4	no	—
200	C₂³	—	3	1	no	—
201	C₂³	—	6	4	no	—
202	C₂³	C₄×D₃×C₇	1	1	yes	characteristic
203	C₂×C₄	—	3	4	no	—
204	C₂×C₄	—	3	1	no	—
205	C₂×C₄	—	3	1	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	3	4	no	—
208	C₂×C₄	—	3	4	no	—
209	C₂×C₄	—	3	2	no	—
210	C₂×C₄	—	3	2	no	—
211	C₂×C₄	—	3	2	no	—
212	C₂×C₄	C₄×D₃×C₇	1	2	yes	—
213	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
216	C₈	—	3	4	no	—
217	C₈	C₂²×D₃×C₇	1	4	yes	complemented
218	C₇	C₂×D₃×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	C₂×C₇×M₁₆	1	2	yes	complemented
221	C₂×C₃	C₂³×C₄×C₇	1	1	yes	characteristic
222	D₃	—	2	2	no	—
223	D₃	C₂×C₇×M₁₆	1	4	yes	direct product factor
224	C₂²	—	2	2	no	—
225	C₂²	—	3	4	no	—
226	C₂²	—	3	2	no	—
227	C₂²	—	3	2	no	—
228	C₂²	—	6	8	no	—
229	C₂²	—	6	2	no	—
230	C₂²	C₂×C₄×D₃×C₇	1	2	yes	—
231	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
232	C₄	—	3	2	no	—
233	C₄	—	3	2	no	—
234	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
235	C₄	C₂×C₄×D₃×C₇	1	2	yes	—
236	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
237	C₃	C₂²×C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	3	4	no	—
240	C₂	—	6	2	no	—
241	C₂	D₃×C₇×M₁₆	1	2	yes	direct product factor
242	C₂	C₂²×C₄×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂³×C₄×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×D₃×C₇	C₄
cosocle	C₃×C₄	C₂⁴×C₇
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₂²×C₇×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₂×C₇×#208	C₂	semidirect
4. D₃×C₇×M₁₆	C₂	direct (N × H)
5. C₂×C₇×ΓL(1,𝔽₂₅)	C₂	semidirect
6. C₂×D₃×C₇×C₈	C₂	semidirect
17. C₃×C₇×M₁₆	C₂²	semidirect
18. C₂×C₃×C₇×C₈	C₂²	semidirect
19. C₇×#208	C₂²	semidirect
20. C₂×C₇×C₃⋊C₈	C₂²	semidirect
21. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
22. D₃×C₇×C₈	C₂²	semidirect
27. C₂×C₇×M₁₆	D₃	direct (N × H)
29. C₂×D₃×M₁₆	C₇	direct (N × H)
48. C₃×C₇×C₈	C₂³	semidirect
49. C₇×C₃⋊C₈	C₂³	semidirect
61. C₇×M₁₆	C₂×D₃	direct (N × H)
64. C₂×C₇×C₈	C₂×D₃	semidirect
66. C₂×C₃×M₁₆	C₂×C₇	semidirect
67. C₂×#208	C₂×C₇	semidirect
68. D₃×M₁₆	C₂×C₇	direct (N × H)
69. C₂×ΓL(1,𝔽₂₅)	C₂×C₇	semidirect
70. C₂×D₃×C₈	C₂×C₇	semidirect
76. C₂×D₃×C₇	M₁₆	direct (N × H)
104. C₇×C₈	C₂²×D₃	semidirect
114. C₃×M₁₆	C₂²×C₇	semidirect
115. C₂×C₃×C₈	C₂²×C₇	semidirect
116. #208	C₂²×C₇	semidirect
117. C₂×C₃⋊C₈	C₂²×C₇	semidirect
118. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
119. D₃×C₈	C₂²×C₇	semidirect
121. C₂×C₃×C₇	C₂×M₁₆	semidirect
124. D₃×C₇	C₂×M₁₆	direct (N × H)
129. C₂×M₁₆	D₃×C₇	direct (N × H)
162. C₃×C₈	C₂³×C₇	semidirect
163. C₃⋊C₈	C₂³×C₇	semidirect
164. C₃×C₇	C₂²×M₁₆	semidirect
176. M₁₆	C₂×D₃×C₇	direct (N × H)
179. C₂×C₈	C₂×D₃×C₇	semidirect
183. C₂×C₇	D₃×M₁₆	direct (N × H)
190. C₂×D₃	C₇×M₁₆	direct (N × H)
217. C₈	C₂²×D₃×C₇	semidirect
218. C₇	C₂×D₃×M₁₆	direct (N × H)
220. C₂×C₃	C₂×C₇×M₁₆	semidirect
223. D₃	C₂×C₇×M₁₆	direct (N × H)
237. C₃	C₂²×C₇×M₁₆	semidirect
241. C₂	D₃×C₇×M₁₆	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×C₄×D₃×C₇	2
C₂×C₃×C₇×M₁₆	2
C₂×C₇×#208	2
D₃×C₇×M₁₆	2
C₂×C₇×ΓL(1,𝔽₂₅)	2
C₂×D₃×C₇×C₈	2
C₂²×C₇×M₁₆	3
C₂×D₃×M₁₆	7

Subgroups — C2×D3×C7×M16 #13061459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₇×M₁₆ #13061459