Subgroups — C₂³×C₇×#123 #13239874

Order 1792.

15150 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#123	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₇×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₇	C₂²	1	1	yes	characteristic
8	C₂⁶×C₇	C₂²	1	1	yes	characteristic
9	C₂³×C₇×D₄	—	2	16	no	—
10	C₂³×C₇×D₄	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₇	—	2	16	no	—
12	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₇	C₂²	1	28	yes	complemented
16	C₂×C₇×#123	C₂²	1	448	yes	direct product factor
17	C₂²×C₇×C₄⋊C₄	C₂²	1	56	yes	complemented
18	C₂²×C₇×C₂²⋊C₄	C₂²	1	28	yes	complemented
19	C₂²×C₇×C₂²⋊C₄	C₂²	1	56	yes	complemented
20	C₂³×#123	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₇	—	2	16	no	—
22	C₂⁵×C₇	—	2	8	no	—
23	C₂⁵×C₇	—	2	16	no	—
24	C₂⁵×C₇	D₄	1	16	yes	—
25	C₂⁵×C₇	C₂³	1	14	yes	—
26	C₂⁵×C₇	C₂³	1	14	yes	—
27	C₂⁵×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₇×D₄	—	2	224	no	—
29	C₂²×C₇×D₄	C₂³	1	112	yes	—
30	C₂³×C₄×C₇	—	2	8	no	—
31	C₂³×C₄×C₇	—	2	16	no	—
32	C₂³×C₄×C₇	—	2	224	no	—
33	C₂³×C₄×C₇	—	2	16	no	—
34	C₂³×C₄×C₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₇	C₂³	1	28	yes	—
36	C₂³×C₄×C₇	C₂³	1	28	yes	—
37	C₂³×C₄×C₇	C₂³	1	112	yes	complemented
38	C₇×#123	C₂³	1	512	yes	direct product factor
39	C₂×C₇×C₄⋊C₄	C₂³	1	224	yes	complemented
40	C₂×C₇×C₂²⋊C₄	C₂³	1	112	yes	complemented
41	C₂×C₇×C₂²⋊C₄	C₂³	1	224	yes	complemented
42	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic
43	C₂⁵×C₄	C₂×C₇	1	1	yes	characteristic, complemented
44	C₂²×#123	C₂×C₇	1	56	yes	direct product factor
45	C₂³×C₄⋊C₄	C₂×C₇	1	2	yes	complemented
46	C₂³×C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
47	C₂³×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂⁴×C₇	—	2	128	no	—
49	C₂⁴×C₇	—	2	112	no	—
50	C₂⁴×C₇	—	2	56	no	—
51	C₂⁴×C₇	—	2	112	no	—
52	C₂⁴×C₇	—	4	64	no	—
53	C₂⁴×C₇	C₂×D₄	1	112	yes	—
54	C₂⁴×C₇	C₂×D₄	1	8	yes	—
55	C₂⁴×C₇	#41	1	16	yes	—
56	C₂⁴×C₇	C₂⁴	1	28	yes	—
57	C₂⁴×C₇	C₂⁴	1	28	yes	—
58	C₂⁴×C₇	C₂⁴	1	7	yes	—
59	C₂×C₇×D₄	—	2	448	no	—
60	C₂×C₇×D₄	C₂⁴	1	64	yes	—
61	C₂²×C₄×C₇	—	2	112	no	—
62	C₂²×C₄×C₇	—	2	448	no	—
63	C₂²×C₄×C₇	—	2	56	no	—
64	C₂²×C₄×C₇	—	2	112	no	—
65	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₇	C₂⁴	1	64	yes	complemented
69	C₇×C₄⋊C₄	C₂⁴	1	128	yes	complemented
70	C₇×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
71	C₇×C₂²⋊C₄	C₂⁴	1	128	yes	complemented
72	C₂⁶	C₂²×C₇	1	1	yes	characteristic
73	C₂⁶	C₂²×C₇	1	1	yes	characteristic
74	C₂³×D₄	—	2	16	no	—
75	C₂³×D₄	C₂²×C₇	1	28	yes	—
76	C₂⁴×C₄	—	2	16	no	—
77	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
78	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
79	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
80	C₂⁴×C₄	C₂²×C₇	1	28	yes	complemented
81	C₂×#123	C₂²×C₇	1	448	yes	direct product factor
82	C₂²×C₄⋊C₄	C₂²×C₇	1	56	yes	complemented
83	C₂²×C₂²⋊C₄	C₂²×C₇	1	28	yes	complemented
84	C₂²×C₂²⋊C₄	C₂²×C₇	1	56	yes	complemented
85	C₂³×C₇	—	2	448	no	—
86	C₂³×C₇	—	2	112	no	—
87	C₂³×C₇	—	2	56	no	—
88	C₂³×C₇	—	2	112	no	—
89	C₂³×C₇	—	4	224	no	—
90	C₂³×C₇	#123	1	64	yes	direct product factor
91	C₂³×C₇	C₂²×D₄	1	112	yes	—
92	C₂³×C₇	C₂²×D₄	1	28	yes	—
93	C₂³×C₇	C₂×#41	1	56	yes	—
94	C₂³×C₇	C₂⁵	1	8	yes	—
95	C₂³×C₇	C₂⁵	1	8	yes	—
96	C₂³×C₇	C₂⁵	1	7	yes	—
97	C₇×D₄	—	2	128	no	—
98	C₂×C₄×C₇	—	2	112	no	—
99	C₂×C₄×C₇	—	2	128	no	—
100	C₂×C₄×C₇	—	2	56	no	—
101	C₂×C₄×C₇	—	2	112	no	—
102	C₂×C₄×C₇	C₂⁵	1	16	yes	—
103	C₂×C₄×C₇	C₂⁵	1	16	yes	—
104	C₂×C₄×C₇	C₂⁵	1	8	yes	—
105	C₂⁵	—	2	16	no	—
106	C₂⁵	—	2	8	no	—
107	C₂⁵	—	2	16	no	—
108	C₂⁵	C₇×D₄	1	16	yes	—
109	C₂⁵	C₂³×C₇	1	14	yes	—
110	C₂⁵	C₂³×C₇	1	14	yes	—
111	C₂⁵	C₂³×C₇	1	1	yes	characteristic
112	C₂²×D₄	—	2	224	no	—
113	C₂²×D₄	C₂³×C₇	1	112	yes	—
114	C₂³×C₄	—	2	16	no	—
115	C₂³×C₄	—	2	224	no	—
116	C₂³×C₄	—	2	8	no	—
117	C₂³×C₄	—	2	16	no	—
118	C₂³×C₄	C₂³×C₇	1	28	yes	—
119	C₂³×C₄	C₂³×C₇	1	28	yes	—
120	C₂³×C₄	C₂³×C₇	1	14	yes	—
121	C₂³×C₄	C₂³×C₇	1	112	yes	complemented
122	#123	C₂³×C₇	1	512	yes	direct product factor
123	C₂×C₄⋊C₄	C₂³×C₇	1	224	yes	complemented
124	C₂×C₂²⋊C₄	C₂³×C₇	1	112	yes	complemented
125	C₂×C₂²⋊C₄	C₂³×C₇	1	224	yes	complemented
126	C₂²×C₇	—	2	16	no	—
127	C₂²×C₇	—	2	224	no	—
128	C₂²×C₇	—	2	8	no	—
129	C₂²×C₇	—	2	16	no	—
130	C₂²×C₇	—	4	112	no	—
131	C₂²×C₇	C₂×#123	1	112	yes	direct product factor
132	C₂²×C₇	C₂³×D₄	1	16	yes	—
133	C₂²×C₇	C₂³×D₄	1	14	yes	—
134	C₂²×C₇	C₂²×#41	1	28	yes	—
135	C₂²×C₇	C₂⁶	1	1	yes	characteristic
136	C₄×C₇	—	2	16	no	—
137	C₄×C₇	—	2	8	no	—
138	C₄×C₇	—	2	16	no	—
139	C₂⁴	—	2	128	no	—
140	C₂⁴	—	2	112	no	—
141	C₂⁴	—	2	56	no	—
142	C₂⁴	—	2	112	no	—
143	C₂⁴	—	4	64	no	—
144	C₂⁴	C₂×C₇×D₄	1	112	yes	—
145	C₂⁴	C₂×C₇×D₄	1	8	yes	—
146	C₂⁴	C₇×#41	1	16	yes	—
147	C₂⁴	C₂⁴×C₇	1	28	yes	—
148	C₂⁴	C₂⁴×C₇	1	28	yes	—
149	C₂⁴	C₂⁴×C₇	1	7	yes	—
150	C₂×D₄	—	2	448	no	—
151	C₂×D₄	C₂⁴×C₇	1	64	yes	—
152	C₂²×C₄	—	2	112	no	—
153	C₂²×C₄	—	2	448	no	—
154	C₂²×C₄	—	2	56	no	—
155	C₂²×C₄	—	2	112	no	—
156	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
157	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
158	C₂²×C₄	C₂⁴×C₇	1	28	yes	—
159	C₂²×C₄	C₂⁴×C₇	1	64	yes	complemented
160	C₄⋊C₄	C₂⁴×C₇	1	128	yes	complemented
161	C₂²⋊C₄	C₂⁴×C₇	1	64	yes	complemented
162	C₂²⋊C₄	C₂⁴×C₇	1	128	yes	complemented
163	C₂×C₇	—	2	16	no	—
164	C₂×C₇	—	4	8	no	—
165	C₂×C₇	C₂²×#123	1	28	yes	direct product factor
166	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
167	C₂×C₇	C₂³×#41	1	2	yes	—
168	C₂³	—	2	112	no	—
169	C₂³	—	2	448	no	—
170	C₂³	—	2	56	no	—
171	C₂³	—	2	112	no	—
172	C₂³	—	4	224	no	—
173	C₂³	C₇×#123	1	64	yes	direct product factor
174	C₂³	C₂²×C₇×D₄	1	112	yes	—
175	C₂³	C₂²×C₇×D₄	1	28	yes	—
176	C₂³	C₂×C₇×#41	1	56	yes	—
177	C₂³	C₂⁵×C₇	1	8	yes	—
178	C₂³	C₂⁵×C₇	1	8	yes	—
179	C₂³	C₂⁵×C₇	1	7	yes	—
180	D₄	—	2	128	no	—
181	C₂×C₄	—	2	112	no	—
182	C₂×C₄	—	2	128	no	—
183	C₂×C₄	—	2	56	no	—
184	C₂×C₄	—	2	112	no	—
185	C₂×C₄	C₂⁵×C₇	1	16	yes	—
186	C₂×C₄	C₂⁵×C₇	1	16	yes	—
187	C₂×C₄	C₂⁵×C₇	1	8	yes	—
188	C₇	C₂³×#123	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	2	16	no	—
190	C₂²	—	2	224	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×C₇×#123	1	112	yes	direct product factor
195	C₂²	C₂³×C₇×D₄	1	16	yes	—
196	C₂²	C₂³×C₇×D₄	1	14	yes	—
197	C₂²	C₂²×C₇×#41	1	28	yes	—
198	C₂²	C₂⁶×C₇	1	1	yes	characteristic
199	C₄	—	2	16	no	—
200	C₄	—	2	8	no	—
201	C₄	—	2	16	no	—
202	C₂	—	2	16	no	—
203	C₂	—	4	8	no	—
204	C₂	C₂²×C₇×#123	1	28	yes	direct product factor
205	C₂	C₂⁴×C₇×D₄	1	1	yes	characteristic
206	C₂	C₂³×C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
prime radical	C₂⁴×C₇×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁵×C₄×C₇	C₂	semidirect
3. C₂²×C₇×#123	C₂	direct (N × H)
4. C₂³×C₇×C₄⋊C₄	C₂	semidirect
5. C₂³×C₇×C₂²⋊C₄	C₂	semidirect
6. C₂³×C₇×C₂²⋊C₄	C₂	semidirect
15. C₂⁴×C₄×C₇	C₂²	semidirect
16. C₂×C₇×#123	C₂²	direct (N × H)
17. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
19. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
20. C₂³×#123	C₇	direct (N × H)
37. C₂³×C₄×C₇	C₂³	semidirect
38. C₇×#123	C₂³	direct (N × H)
39. C₂×C₇×C₄⋊C₄	C₂³	semidirect
40. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
41. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
43. C₂⁵×C₄	C₂×C₇	semidirect
44. C₂²×#123	C₂×C₇	direct (N × H)
45. C₂³×C₄⋊C₄	C₂×C₇	semidirect
46. C₂³×C₂²⋊C₄	C₂×C₇	semidirect
47. C₂³×C₂²⋊C₄	C₂×C₇	semidirect
68. C₂²×C₄×C₇	C₂⁴	semidirect
69. C₇×C₄⋊C₄	C₂⁴	semidirect
70. C₇×C₂²⋊C₄	C₂⁴	semidirect
71. C₇×C₂²⋊C₄	C₂⁴	semidirect
80. C₂⁴×C₄	C₂²×C₇	semidirect
81. C₂×#123	C₂²×C₇	direct (N × H)
82. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
83. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
84. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
90. C₂³×C₇	#123	direct (N × H)
121. C₂³×C₄	C₂³×C₇	semidirect
122. #123	C₂³×C₇	direct (N × H)
123. C₂×C₄⋊C₄	C₂³×C₇	semidirect
124. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
125. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
131. C₂²×C₇	C₂×#123	direct (N × H)
159. C₂²×C₄	C₂⁴×C₇	semidirect
160. C₄⋊C₄	C₂⁴×C₇	semidirect
161. C₂²⋊C₄	C₂⁴×C₇	semidirect
162. C₂²⋊C₄	C₂⁴×C₇	semidirect
165. C₂×C₇	C₂²×#123	direct (N × H)
173. C₂³	C₇×#123	direct (N × H)
188. C₇	C₂³×#123	direct (N × H)
194. C₂²	C₂×C₇×#123	direct (N × H)
204. C₂	C₂²×C₇×#123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×D₄	2
C₂⁵×C₄×C₇	2
C₂²×C₇×#123	2
C₂³×C₇×C₄⋊C₄	2
C₂³×C₇×C₂²⋊C₄	2
C₂³×#123	7

Subgroups — C23×C7×#123 #13239874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#123 #13239874