Subgroups — C₂²×C₄×C₇×M₁₆ #13237268

Order 1792.

3950 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×M₁₆	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇×M₁₆	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#97	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₇×C₈	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₇	C₄	1	2	yes	—
7	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₇×M₁₆	C₄	1	32	yes	direct product factor
9	C₂²×C₇×M₁₆	C₂²	1	24	yes	—
10	C₂³×C₇×C₈	C₂²	1	4	yes	—
11	C₂²×C₄²×C₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₇	C₄	1	12	yes	—
13	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₇	C₂²	1	12	yes	—
15	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
16	C₄×C₇×M₁₆	C₂²	1	64	yes	direct product factor
17	C₂×C₇×#97	C₂²	1	24	yes	complemented
18	C₂×C₄×C₇×C₈	C₂²	1	24	yes	complemented
19	C₂²×C₄×M₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	—	2	16	no	—
22	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₇	C₂×C₄	1	24	yes	—
24	C₂³×C₄×C₇	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₇	C₂³	1	6	yes	—
29	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₂×C₇×M₁₆	C₂×C₄	1	192	yes	direct product factor
31	C₂×C₇×M₁₆	C₂³	1	32	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	32	yes	complemented
33	C₂²×C₇×C₈	C₂³	1	24	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	48	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	24	yes	—
37	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
38	C₂×C₄²×C₇	C₂³	1	6	yes	—
39	C₂×C₄²×C₇	C₂³	1	16	yes	—
40	C₂×C₄²×C₇	C₂³	1	6	yes	—
41	C₇×#97	C₂³	1	32	yes	complemented
42	C₄×C₇×C₈	C₂³	1	32	yes	complemented
43	C₂³×M₁₆	C₂×C₇	1	2	yes	—
44	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic
45	C₂×C₄×M₁₆	C₂×C₇	1	24	yes	direct product factor
46	C₂²×#97	C₂×C₇	1	2	yes	complemented
47	C₂²×C₄×C₈	C₂×C₇	1	2	yes	complemented
48	C₂⁴×C₇	—	2	8	no	—
49	C₂⁴×C₇	C₄²	1	8	yes	—
50	C₂⁴×C₇	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	—	2	8	no	—
53	C₂²×C₄×C₇	—	2	96	no	—
54	C₂²×C₄×C₇	C₄²	1	48	yes	—
55	C₂²×C₄×C₇	C₄²	1	8	yes	—
56	C₂²×C₄×C₇	M₁₆	1	16	yes	direct product factor
57	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	32	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	48	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
67	C₇×M₁₆	C₂²×C₄	1	128	yes	direct product factor
68	C₂×C₇×C₈	C₂²×C₄	1	96	yes	complemented
69	C₂×C₇×C₈	C₂⁴	1	16	yes	—
70	C₄²×C₇	C₂²×C₄	1	12	yes	—
71	C₄²×C₇	C₂²×C₄	1	32	yes	—
72	C₄²×C₇	C₂²×C₄	1	12	yes	—
73	C₄²×C₇	C₂⁴	1	4	yes	—
74	C₄²×C₇	C₂⁴	1	4	yes	—
75	C₂⁴×C₄	C₄×C₇	1	2	yes	—
76	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
77	C₂²×M₁₆	C₄×C₇	1	32	yes	direct product factor
78	C₂²×M₁₆	C₂²×C₇	1	24	yes	—
79	C₂³×C₈	C₂²×C₇	1	4	yes	—
80	C₂²×C₄²	C₄×C₇	1	12	yes	—
81	C₂²×C₄²	C₄×C₇	1	2	yes	—
82	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
83	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
84	C₂²×C₄²	C₂²×C₇	1	12	yes	—
85	C₄×M₁₆	C₂²×C₇	1	64	yes	direct product factor
86	C₂×#97	C₂²×C₇	1	24	yes	complemented
87	C₂×C₄×C₈	C₂²×C₇	1	24	yes	complemented
88	C₂³×C₇	—	2	24	no	—
89	C₂³×C₇	—	2	32	no	—
90	C₂³×C₇	C₂×C₄²	1	24	yes	—
91	C₂³×C₇	C₂×C₄²	1	4	yes	—
92	C₂³×C₇	C₂×M₁₆	1	8	yes	—
93	C₂³×C₇	C₂³×C₄	1	4	yes	—
94	C₂³×C₇	C₂³×C₄	1	3	yes	—
95	C₂×C₄×C₇	—	2	24	no	—
96	C₂×C₄×C₇	—	2	64	no	—
97	C₂×C₄×C₇	C₂×C₄²	1	48	yes	—
98	C₂×C₄×C₇	C₂×C₄²	1	12	yes	—
99	C₂×C₄×C₇	C₂×C₄²	1	24	yes	—
100	C₂×C₄×C₇	C₂×M₁₆	1	48	yes	direct product factor
101	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
102	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
104	C₂×C₄×C₇	C₂³×C₄	1	16	yes	—
105	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
106	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
107	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
108	C₇×C₈	C₂³×C₄	1	32	yes	complemented
109	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
110	C₂³×C₄	—	2	16	no	—
111	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
112	C₂³×C₄	C₂×C₄×C₇	1	24	yes	—
113	C₂³×C₄	C₂×C₄×C₇	1	6	yes	—
114	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
115	C₂³×C₄	C₂×C₄×C₇	1	16	yes	—
116	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
117	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂³×C₄	C₂³×C₇	1	6	yes	—
119	C₂×M₁₆	C₂×C₄×C₇	1	192	yes	direct product factor
120	C₂×M₁₆	C₂³×C₇	1	32	yes	—
121	C₂²×C₈	C₂×C₄×C₇	1	32	yes	complemented
122	C₂²×C₈	C₂³×C₇	1	24	yes	—
123	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
124	C₂×C₄²	C₂×C₄×C₇	1	48	yes	—
125	C₂×C₄²	C₂×C₄×C₇	1	24	yes	—
126	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
127	C₂×C₄²	C₂³×C₇	1	6	yes	—
128	C₂×C₄²	C₂³×C₇	1	6	yes	—
129	C₂×C₄²	C₂³×C₇	1	16	yes	—
130	#97	C₂³×C₇	1	32	yes	complemented
131	C₄×C₈	C₂³×C₇	1	32	yes	complemented
132	C₂²×C₇	—	2	48	no	—
133	C₂²×C₇	—	2	8	no	—
134	C₂²×C₇	C₄×M₁₆	1	16	yes	direct product factor
135	C₂²×C₇	C₂²×C₄²	1	8	yes	—
136	C₂²×C₇	C₂²×C₄²	1	6	yes	—
137	C₂²×C₇	C₂²×M₁₆	1	12	yes	—
138	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
139	C₄×C₇	—	2	8	no	—
140	C₄×C₇	C₂²×C₄²	1	6	yes	—
141	C₄×C₇	C₂²×C₄²	1	8	yes	—
142	C₄×C₇	C₂²×M₁₆	1	16	yes	direct product factor
143	C₄×C₇	C₂⁴×C₄	1	2	yes	—
144	C₂⁴	—	2	8	no	—
145	C₂⁴	C₄²×C₇	1	8	yes	—
146	C₂⁴	C₂²×C₄×C₇	1	6	yes	—
147	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	96	no	—
150	C₂²×C₄	C₄²×C₇	1	48	yes	—
151	C₂²×C₄	C₄²×C₇	1	8	yes	—
152	C₂²×C₄	C₇×M₁₆	1	16	yes	direct product factor
153	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
154	C₂²×C₄	C₂²×C₄×C₇	1	32	yes	—
155	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
156	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
157	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
158	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
159	C₂²×C₄	C₂²×C₄×C₇	1	48	yes	—
160	C₂²×C₄	C₂²×C₄×C₇	1	12	yes	—
161	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
162	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
163	M₁₆	C₂²×C₄×C₇	1	128	yes	direct product factor
164	C₂×C₈	C₂²×C₄×C₇	1	96	yes	complemented
165	C₂×C₈	C₂⁴×C₇	1	16	yes	—
166	C₄²	C₂²×C₄×C₇	1	12	yes	—
167	C₄²	C₂²×C₄×C₇	1	32	yes	—
168	C₄²	C₂²×C₄×C₇	1	12	yes	—
169	C₄²	C₂⁴×C₇	1	4	yes	—
170	C₄²	C₂⁴×C₇	1	4	yes	—
171	C₂×C₇	—	2	8	no	—
172	C₂×C₇	C₂×C₄×M₁₆	1	12	yes	direct product factor
173	C₂×C₇	C₂³×C₄²	1	1	yes	characteristic
174	C₂×C₇	C₂³×M₁₆	1	2	yes	—
175	C₂³	—	2	32	no	—
176	C₂³	—	2	24	no	—
177	C₂³	C₂×C₄²×C₇	1	24	yes	—
178	C₂³	C₂×C₄²×C₇	1	4	yes	—
179	C₂³	C₂×C₇×M₁₆	1	8	yes	—
180	C₂³	C₂³×C₄×C₇	1	4	yes	—
181	C₂³	C₂³×C₄×C₇	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×C₄²×C₇	1	48	yes	—
185	C₂×C₄	C₂×C₄²×C₇	1	12	yes	—
186	C₂×C₄	C₂×C₄²×C₇	1	24	yes	—
187	C₂×C₄	C₂×C₇×M₁₆	1	48	yes	direct product factor
188	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
189	C₂×C₄	C₂³×C₄×C₇	1	3	yes	—
190	C₂×C₄	C₂³×C₄×C₇	1	4	yes	—
191	C₂×C₄	C₂³×C₄×C₇	1	12	yes	—
192	C₂×C₄	C₂³×C₄×C₇	1	16	yes	—
193	C₂×C₄	C₂³×C₄×C₇	1	8	yes	—
194	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
195	C₈	C₂³×C₄×C₇	1	32	yes	complemented
196	C₇	C₂²×C₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	—	2	8	no	—
198	C₂²	—	2	48	no	—
199	C₂²	C₄×C₇×M₁₆	1	16	yes	direct product factor
200	C₂²	C₂²×C₄²×C₇	1	8	yes	—
201	C₂²	C₂²×C₄²×C₇	1	6	yes	—
202	C₂²	C₂²×C₇×M₁₆	1	12	yes	—
203	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	C₂²×C₄²×C₇	1	6	yes	—
206	C₄	C₂²×C₄²×C₇	1	8	yes	—
207	C₄	C₂²×C₇×M₁₆	1	16	yes	direct product factor
208	C₄	C₂⁴×C₄×C₇	1	2	yes	—
209	C₂	—	2	8	no	—
210	C₂	C₂×C₄×C₇×M₁₆	1	12	yes	direct product factor
211	C₂	C₂³×C₄²×C₇	1	1	yes	characteristic
212	C₂	C₂³×C₇×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₇	C₂²
commutator	C₂	C₂³×C₄²×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂⁴×C₇	C₂²×C₄
prime radical	C₂⁵×C₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₇×M₁₆	C₂	direct (N × H)
4. C₂²×C₇×#97	C₂	semidirect
5. C₂²×C₄×C₇×C₈	C₂	semidirect
8. C₂²×C₇×M₁₆	C₄	direct (N × H)
16. C₄×C₇×M₁₆	C₂²	direct (N × H)
17. C₂×C₇×#97	C₂²	semidirect
18. C₂×C₄×C₇×C₈	C₂²	semidirect
19. C₂²×C₄×M₁₆	C₇	direct (N × H)
30. C₂×C₇×M₁₆	C₂×C₄	direct (N × H)
32. C₂²×C₇×C₈	C₂×C₄	semidirect
41. C₇×#97	C₂³	semidirect
42. C₄×C₇×C₈	C₂³	semidirect
45. C₂×C₄×M₁₆	C₂×C₇	direct (N × H)
46. C₂²×#97	C₂×C₇	semidirect
47. C₂²×C₄×C₈	C₂×C₇	semidirect
56. C₂²×C₄×C₇	M₁₆	direct (N × H)
67. C₇×M₁₆	C₂²×C₄	direct (N × H)
68. C₂×C₇×C₈	C₂²×C₄	semidirect
77. C₂²×M₁₆	C₄×C₇	direct (N × H)
85. C₄×M₁₆	C₂²×C₇	direct (N × H)
86. C₂×#97	C₂²×C₇	semidirect
87. C₂×C₄×C₈	C₂²×C₇	semidirect
100. C₂×C₄×C₇	C₂×M₁₆	direct (N × H)
108. C₇×C₈	C₂³×C₄	semidirect
119. C₂×M₁₆	C₂×C₄×C₇	direct (N × H)
121. C₂²×C₈	C₂×C₄×C₇	semidirect
130. #97	C₂³×C₇	semidirect
131. C₄×C₈	C₂³×C₇	semidirect
134. C₂²×C₇	C₄×M₁₆	direct (N × H)
142. C₄×C₇	C₂²×M₁₆	direct (N × H)
152. C₂²×C₄	C₇×M₁₆	direct (N × H)
163. M₁₆	C₂²×C₄×C₇	direct (N × H)
164. C₂×C₈	C₂²×C₄×C₇	semidirect
172. C₂×C₇	C₂×C₄×M₁₆	direct (N × H)
187. C₂×C₄	C₂×C₇×M₁₆	direct (N × H)
195. C₈	C₂³×C₄×C₇	semidirect
196. C₇	C₂²×C₄×M₁₆	direct (N × H)
199. C₂²	C₄×C₇×M₁₆	direct (N × H)
207. C₄	C₂²×C₇×M₁₆	direct (N × H)
210. C₂	C₂×C₄×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
C₂³×C₇×M₁₆	2
C₂³×C₄²×C₇	2
C₂×C₄×C₇×M₁₆	2
C₂²×C₇×#97	2
C₂²×C₄×C₇×C₈	2
C₂²×C₄×M₁₆	7

Subgroups — C22×C4×C7×M16 #13237268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₇×M₁₆ #13237268