Subgroups — C₂³×C₇×#121 #13239872

Order 1792.

19998 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	C₂⁴×C₇×D₄	C₂	1	2	yes	maximal, complemented
2	C₂⁴×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁵×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×#121	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₇	C₂²	1	2	yes	—
8	C₂⁶×C₇	C₂²	1	1	yes	characteristic
9	C₂³×C₇×D₄	—	2	32	no	—
10	C₂³×C₇×D₄	—	2	16	no	—
11	C₂³×C₇×D₄	C₂²	1	56	yes	complemented
12	C₂³×C₇×D₄	C₂²	1	28	yes	complemented
13	C₂⁴×C₄×C₇	—	2	16	no	—
14	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₇	C₂²	1	28	yes	complemented
18	C₂×C₇×#121	C₂²	1	448	yes	direct product factor
19	C₂²×C₇×C₄⋊C₄	C₂²	1	28	yes	complemented
20	C₂²×C₇×C₂²⋊C₄	C₂²	1	56	yes	complemented
21	C₂³×#121	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂⁵×C₇	—	2	8	no	—
23	C₂⁵×C₇	—	2	8	no	—
24	C₂⁵×C₇	—	2	16	no	—
25	C₂⁵×C₇	—	2	16	no	—
26	C₂⁵×C₇	—	2	16	no	—
27	C₂⁵×C₇	D₄	1	16	yes	complemented
28	C₂⁵×C₇	C₂³	1	28	yes	—
29	C₂⁵×C₇	C₂³	1	14	yes	—
30	C₂⁵×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₇×D₄	—	2	448	no	—
32	C₂²×C₇×D₄	—	2	224	no	—
33	C₂²×C₇×D₄	C₂³	1	224	yes	complemented
34	C₂²×C₇×D₄	C₂³	1	112	yes	complemented
35	C₂³×C₄×C₇	—	2	16	no	—
36	C₂³×C₄×C₇	—	2	8	no	—
37	C₂³×C₄×C₇	—	2	224	no	—
38	C₂³×C₄×C₇	D₄	1	16	yes	complemented
39	C₂³×C₄×C₇	C₂³	1	28	yes	—
40	C₂³×C₄×C₇	C₂³	1	14	yes	—
41	C₂³×C₄×C₇	C₂³	1	14	yes	—
42	C₂³×C₄×C₇	C₂³	1	112	yes	complemented
43	C₇×#121	C₂³	1	512	yes	direct product factor
44	C₂×C₇×C₄⋊C₄	C₂³	1	112	yes	complemented
45	C₂×C₇×C₂²⋊C₄	C₂³	1	224	yes	complemented
46	C₂⁴×D₄	C₂×C₇	1	2	yes	complemented
47	C₂⁴×D₄	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂⁵×C₄	C₂×C₇	1	1	yes	characteristic, complemented
49	C₂²×#121	C₂×C₇	1	56	yes	direct product factor
50	C₂³×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂³×C₂²⋊C₄	C₂×C₇	1	2	yes	complemented
52	C₂⁴×C₇	—	2	128	no	—
53	C₂⁴×C₇	—	2	56	no	—
54	C₂⁴×C₇	—	2	56	no	—
55	C₂⁴×C₇	—	2	112	no	—
56	C₂⁴×C₇	—	2	112	no	—
57	C₂⁴×C₇	—	2	112	no	—
58	C₂⁴×C₇	—	4	128	no	—
59	C₂⁴×C₇	C₂×D₄	1	8	yes	—
60	C₂⁴×C₇	C₂×D₄	1	8	yes	—
61	C₂⁴×C₇	C₂×D₄	1	112	yes	complemented
62	C₂⁴×C₇	#41	1	8	yes	—
63	C₂⁴×C₇	C₂⁴	1	56	yes	—
64	C₂⁴×C₇	C₂⁴	1	28	yes	—
65	C₂⁴×C₇	C₂⁴	1	7	yes	—
66	C₂×C₇×D₄	—	2	896	no	—
67	C₂×C₇×D₄	—	2	448	no	—
68	C₂×C₇×D₄	C₂⁴	1	128	yes	complemented
69	C₂×C₇×D₄	C₂⁴	1	64	yes	complemented
70	C₂²×C₄×C₇	—	2	56	no	—
71	C₂²×C₄×C₇	—	2	112	no	—
72	C₂²×C₄×C₇	—	2	448	no	—
73	C₂²×C₄×C₇	C₂×D₄	1	112	yes	complemented
74	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
75	C₂²×C₄×C₇	C₂⁴	1	56	yes	—
76	C₂²×C₄×C₇	C₂⁴	1	28	yes	—
77	C₂²×C₄×C₇	C₂⁴	1	64	yes	complemented
78	C₇×C₄⋊C₄	C₂⁴	1	64	yes	complemented
79	C₇×C₂²⋊C₄	C₂⁴	1	128	yes	complemented
80	C₂⁶	C₂²×C₇	1	2	yes	—
81	C₂⁶	C₂²×C₇	1	1	yes	characteristic
82	C₂³×D₄	—	2	32	no	—
83	C₂³×D₄	—	2	16	no	—
84	C₂³×D₄	C₂²×C₇	1	56	yes	complemented
85	C₂³×D₄	C₂²×C₇	1	28	yes	complemented
86	C₂⁴×C₄	—	2	16	no	—
87	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
88	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
89	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂⁴×C₄	C₂²×C₇	1	28	yes	complemented
91	C₂×#121	C₂²×C₇	1	448	yes	direct product factor
92	C₂²×C₄⋊C₄	C₂²×C₇	1	28	yes	complemented
93	C₂²×C₂²⋊C₄	C₂²×C₇	1	56	yes	complemented
94	C₂³×C₇	—	2	448	no	—
95	C₂³×C₇	—	2	56	no	—
96	C₂³×C₇	—	2	56	no	—
97	C₂³×C₇	—	2	112	no	—
98	C₂³×C₇	—	2	112	no	—
99	C₂³×C₇	—	2	112	no	—
100	C₂³×C₇	—	4	448	no	—
101	C₂³×C₇	#121	1	64	yes	direct product factor
102	C₂³×C₇	C₂²×D₄	1	28	yes	—
103	C₂³×C₇	C₂²×D₄	1	28	yes	—
104	C₂³×C₇	C₂²×D₄	1	112	yes	complemented
105	C₂³×C₇	C₂×#41	1	28	yes	—
106	C₂³×C₇	C₂⁵	1	16	yes	—
107	C₂³×C₇	C₂⁵	1	8	yes	—
108	C₂³×C₇	C₂⁵	1	7	yes	—
109	C₇×D₄	—	2	256	no	—
110	C₇×D₄	—	2	128	no	—
111	C₂×C₄×C₇	—	2	56	no	—
112	C₂×C₄×C₇	—	2	112	no	—
113	C₂×C₄×C₇	—	2	128	no	—
114	C₂×C₄×C₇	C₂²×D₄	1	112	yes	complemented
115	C₂×C₄×C₇	C₂⁵	1	8	yes	—
116	C₂×C₄×C₇	C₂⁵	1	16	yes	—
117	C₂×C₄×C₇	C₂⁵	1	8	yes	—
118	C₂⁵	—	2	8	no	—
119	C₂⁵	—	2	8	no	—
120	C₂⁵	—	2	16	no	—
121	C₂⁵	—	2	16	no	—
122	C₂⁵	—	2	16	no	—
123	C₂⁵	C₇×D₄	1	16	yes	complemented
124	C₂⁵	C₂³×C₇	1	28	yes	—
125	C₂⁵	C₂³×C₇	1	14	yes	—
126	C₂⁵	C₂³×C₇	1	1	yes	characteristic
127	C₂²×D₄	—	2	448	no	—
128	C₂²×D₄	—	2	224	no	—
129	C₂²×D₄	C₂³×C₇	1	224	yes	complemented
130	C₂²×D₄	C₂³×C₇	1	112	yes	complemented
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	—	2	16	no	—
133	C₂³×C₄	—	2	224	no	—
134	C₂³×C₄	C₇×D₄	1	16	yes	complemented
135	C₂³×C₄	C₂³×C₇	1	14	yes	—
136	C₂³×C₄	C₂³×C₇	1	28	yes	—
137	C₂³×C₄	C₂³×C₇	1	14	yes	—
138	C₂³×C₄	C₂³×C₇	1	112	yes	complemented
139	#121	C₂³×C₇	1	512	yes	direct product factor
140	C₂×C₄⋊C₄	C₂³×C₇	1	112	yes	complemented
141	C₂×C₂²⋊C₄	C₂³×C₇	1	224	yes	complemented
142	C₂²×C₇	—	2	8	no	—
143	C₂²×C₇	—	2	8	no	—
144	C₂²×C₇	—	2	224	no	—
145	C₂²×C₇	—	2	16	no	—
146	C₂²×C₇	—	2	16	no	—
147	C₂²×C₇	—	2	16	no	—
148	C₂²×C₇	—	4	224	no	—
149	C₂²×C₇	C₂×#121	1	112	yes	direct product factor
150	C₂²×C₇	C₂³×D₄	1	14	yes	—
151	C₂²×C₇	C₂³×D₄	1	14	yes	—
152	C₂²×C₇	C₂³×D₄	1	16	yes	complemented
153	C₂²×C₇	C₂²×#41	1	14	yes	—
154	C₂²×C₇	C₂⁶	1	1	yes	characteristic
155	C₄×C₇	—	2	8	no	—
156	C₄×C₇	—	2	16	no	—
157	C₄×C₇	C₂³×D₄	1	16	yes	complemented
158	C₂⁴	—	2	128	no	—
159	C₂⁴	—	2	56	no	—
160	C₂⁴	—	2	56	no	—
161	C₂⁴	—	2	112	no	—
162	C₂⁴	—	2	112	no	—
163	C₂⁴	—	2	112	no	—
164	C₂⁴	—	4	128	no	—
165	C₂⁴	C₂×C₇×D₄	1	8	yes	—
166	C₂⁴	C₂×C₇×D₄	1	8	yes	—
167	C₂⁴	C₂×C₇×D₄	1	112	yes	complemented
168	C₂⁴	C₇×#41	1	8	yes	—
169	C₂⁴	C₂⁴×C₇	1	56	yes	—
170	C₂⁴	C₂⁴×C₇	1	28	yes	—
171	C₂⁴	C₂⁴×C₇	1	7	yes	—
172	C₂×D₄	—	2	896	no	—
173	C₂×D₄	—	2	448	no	—
174	C₂×D₄	C₂⁴×C₇	1	128	yes	complemented
175	C₂×D₄	C₂⁴×C₇	1	64	yes	complemented
176	C₂²×C₄	—	2	56	no	—
177	C₂²×C₄	—	2	112	no	—
178	C₂²×C₄	—	2	448	no	—
179	C₂²×C₄	C₂×C₇×D₄	1	112	yes	complemented
180	C₂²×C₄	C₂⁴×C₇	1	28	yes	—
181	C₂²×C₄	C₂⁴×C₇	1	56	yes	—
182	C₂²×C₄	C₂⁴×C₇	1	28	yes	—
183	C₂²×C₄	C₂⁴×C₇	1	64	yes	complemented
184	C₄⋊C₄	C₂⁴×C₇	1	64	yes	complemented
185	C₂²⋊C₄	C₂⁴×C₇	1	128	yes	complemented
186	C₂×C₇	—	2	16	no	—
187	C₂×C₇	—	4	16	no	—
188	C₂×C₇	C₂²×#121	1	28	yes	direct product factor
189	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
190	C₂×C₇	C₂⁴×D₄	1	1	yes	characteristic
191	C₂×C₇	C₂³×#41	1	1	yes	characteristic
192	C₂³	—	2	56	no	—
193	C₂³	—	2	56	no	—
194	C₂³	—	2	448	no	—
195	C₂³	—	2	112	no	—
196	C₂³	—	2	112	no	—
197	C₂³	—	2	112	no	—
198	C₂³	—	4	448	no	—
199	C₂³	C₇×#121	1	64	yes	direct product factor
200	C₂³	C₂²×C₇×D₄	1	28	yes	—
201	C₂³	C₂²×C₇×D₄	1	28	yes	—
202	C₂³	C₂²×C₇×D₄	1	112	yes	complemented
203	C₂³	C₂×C₇×#41	1	28	yes	—
204	C₂³	C₂⁵×C₇	1	16	yes	—
205	C₂³	C₂⁵×C₇	1	8	yes	—
206	C₂³	C₂⁵×C₇	1	7	yes	—
207	D₄	—	2	256	no	—
208	D₄	—	2	128	no	—
209	C₂×C₄	—	2	56	no	—
210	C₂×C₄	—	2	112	no	—
211	C₂×C₄	—	2	128	no	—
212	C₂×C₄	C₂²×C₇×D₄	1	112	yes	complemented
213	C₂×C₄	C₂⁵×C₇	1	8	yes	—
214	C₂×C₄	C₂⁵×C₇	1	16	yes	—
215	C₂×C₄	C₂⁵×C₇	1	8	yes	—
216	C₇	C₂³×#121	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	8	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	224	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	4	224	no	—
224	C₂²	C₂×C₇×#121	1	112	yes	direct product factor
225	C₂²	C₂³×C₇×D₄	1	14	yes	—
226	C₂²	C₂³×C₇×D₄	1	14	yes	—
227	C₂²	C₂³×C₇×D₄	1	16	yes	complemented
228	C₂²	C₂²×C₇×#41	1	14	yes	—
229	C₂²	C₂⁶×C₇	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	2	16	no	—
232	C₄	C₂³×C₇×D₄	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	4	16	no	—
235	C₂	C₂²×C₇×#121	1	28	yes	direct product factor
236	C₂	C₂⁴×C₇×D₄	1	1	yes	characteristic
237	C₂	C₂⁴×C₇×D₄	1	1	yes	characteristic
238	C₂	C₂³×C₇×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
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
1. C₂⁴×C₇×D₄	C₂	semidirect
2. C₂⁴×C₇×D₄	C₂	semidirect
3. C₂⁵×C₄×C₇	C₂	semidirect
4. C₂²×C₇×#121	C₂	direct (N × H)
5. C₂³×C₇×C₄⋊C₄	C₂	semidirect
6. C₂³×C₇×C₂²⋊C₄	C₂	semidirect
11. C₂³×C₇×D₄	C₂²	semidirect
12. C₂³×C₇×D₄	C₂²	semidirect
17. C₂⁴×C₄×C₇	C₂²	semidirect
18. C₂×C₇×#121	C₂²	direct (N × H)
19. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
20. C₂²×C₇×C₂²⋊C₄	C₂²	semidirect
21. C₂³×#121	C₇	direct (N × H)
27. C₂⁵×C₇	D₄	semidirect
33. C₂²×C₇×D₄	C₂³	semidirect
34. C₂²×C₇×D₄	C₂³	semidirect
38. C₂³×C₄×C₇	D₄	semidirect
42. C₂³×C₄×C₇	C₂³	semidirect
43. C₇×#121	C₂³	direct (N × H)
44. C₂×C₇×C₄⋊C₄	C₂³	semidirect
45. C₂×C₇×C₂²⋊C₄	C₂³	semidirect
46. C₂⁴×D₄	C₂×C₇	semidirect
47. C₂⁴×D₄	C₂×C₇	semidirect
48. C₂⁵×C₄	C₂×C₇	semidirect
49. C₂²×#121	C₂×C₇	direct (N × H)
50. C₂³×C₄⋊C₄	C₂×C₇	semidirect
51. C₂³×C₂²⋊C₄	C₂×C₇	semidirect
61. C₂⁴×C₇	C₂×D₄	semidirect
68. C₂×C₇×D₄	C₂⁴	semidirect
69. C₂×C₇×D₄	C₂⁴	semidirect
73. C₂²×C₄×C₇	C₂×D₄	semidirect
77. C₂²×C₄×C₇	C₂⁴	semidirect
78. C₇×C₄⋊C₄	C₂⁴	semidirect
79. C₇×C₂²⋊C₄	C₂⁴	semidirect
84. C₂³×D₄	C₂²×C₇	semidirect
85. C₂³×D₄	C₂²×C₇	semidirect
90. C₂⁴×C₄	C₂²×C₇	semidirect
91. C₂×#121	C₂²×C₇	direct (N × H)
92. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
93. C₂²×C₂²⋊C₄	C₂²×C₇	semidirect
101. C₂³×C₇	#121	direct (N × H)
104. C₂³×C₇	C₂²×D₄	semidirect
114. C₂×C₄×C₇	C₂²×D₄	semidirect
123. C₂⁵	C₇×D₄	semidirect
129. C₂²×D₄	C₂³×C₇	semidirect
130. C₂²×D₄	C₂³×C₇	semidirect
134. C₂³×C₄	C₇×D₄	semidirect
138. C₂³×C₄	C₂³×C₇	semidirect
139. #121	C₂³×C₇	direct (N × H)
140. C₂×C₄⋊C₄	C₂³×C₇	semidirect
141. C₂×C₂²⋊C₄	C₂³×C₇	semidirect
149. C₂²×C₇	C₂×#121	direct (N × H)
152. C₂²×C₇	C₂³×D₄	semidirect
157. C₄×C₇	C₂³×D₄	semidirect
167. C₂⁴	C₂×C₇×D₄	semidirect
174. C₂×D₄	C₂⁴×C₇	semidirect
175. C₂×D₄	C₂⁴×C₇	semidirect
179. C₂²×C₄	C₂×C₇×D₄	semidirect
183. C₂²×C₄	C₂⁴×C₇	semidirect
184. C₄⋊C₄	C₂⁴×C₇	semidirect
185. C₂²⋊C₄	C₂⁴×C₇	semidirect
188. C₂×C₇	C₂²×#121	direct (N × H)
199. C₂³	C₇×#121	direct (N × H)
202. C₂³	C₂²×C₇×D₄	semidirect
212. C₂×C₄	C₂²×C₇×D₄	semidirect
216. C₇	C₂³×#121	direct (N × H)
224. C₂²	C₂×C₇×#121	direct (N × H)
227. C₂²	C₂³×C₇×D₄	semidirect
232. C₄	C₂³×C₇×D₄	semidirect
235. C₂	C₂²×C₇×#121	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×#121	2
C₂³×C₇×C₄⋊C₄	2
C₂³×C₇×C₂²⋊C₄	2
C₂³×#121	7

Subgroups — C23×C7×#121 #13239872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#121 #13239872