Subgroups — C₃×C₇×#440 #13056951

Order 1344.

240 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₇×#102	C₂	1	1	yes	maximal, characteristic
7	C₃×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#440	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	—	2	1	no	—
12	C₃×C₇×SD₁₆	C₄	1	4	yes	complemented
13	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₇×C₄⋊C₄	—	2	1	no	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₃×C₇×C₂²⋊C₄	—	2	1	no	—
19	C₃×C₄²×C₇	—	2	1	no	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₂×C₇×SD₁₆	C₂×C₃	1	1	yes	characteristic
22	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
23	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
24	C₇×#107	C₂×C₃	1	1	yes	characteristic, complemented
25	C₇×#103	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×#102	C₂×C₃	1	1	yes	characteristic
27	C₇×#97	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#440	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂³×C₃×C₇	—	2	1	no	—
30	C₃×C₇×Q₈	—	2	1	no	—
31	C₃×C₇×Q₈	C₂×C₄	1	2	yes	complemented
32	C₃×C₇×D₄	—	2	1	no	—
33	C₃×C₇×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₇	—	2	1	no	—
35	C₂×C₃×C₄×C₇	—	2	1	no	—
36	C₂×C₃×C₄×C₇	—	2	1	no	—
37	C₂×C₃×C₄×C₇	—	4	2	no	—
38	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₃×C₇×C₈	—	2	1	no	—
42	C₃×C₇×C₈	C₂×C₄	1	2	yes	complemented
43	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	1	no	—
46	C₇×SD₁₆	C₃×C₄	1	4	yes	complemented
47	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₇×C₄⋊C₄	—	2	1	no	—
50	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₇×C₂²⋊C₄	—	2	1	no	—
53	C₄²×C₇	—	2	1	no	—
54	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×SD₁₆	C₂×C₇	1	1	yes	characteristic
56	C₃×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
57	C₃×C₄×D₄	C₂×C₇	1	1	yes	characteristic
58	C₃×#107	C₂×C₇	1	1	yes	characteristic, complemented
59	C₃×#103	C₂×C₇	1	1	yes	characteristic, complemented
60	C₃×#102	C₂×C₇	1	1	yes	characteristic
61	C₃×#97	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂²×C₃×C₇	—	2	2	no	—
63	C₂²×C₃×C₇	—	4	1	no	—
64	C₂²×C₃×C₇	—	4	1	no	—
65	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₇	—	2	2	no	—
67	C₃×C₄×C₇	—	2	1	no	—
68	C₃×C₄×C₇	—	2	1	no	—
69	C₃×C₄×C₇	—	4	1	no	—
70	C₃×C₄×C₇	—	4	1	no	—
71	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄×C₇	#41	1	1	yes	characteristic
73	#440	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₇	—	2	1	no	—
75	C₇×Q₈	—	2	1	no	—
76	C₇×Q₈	C₂×C₃×C₄	1	2	yes	complemented
77	C₇×D₄	—	2	1	no	—
78	C₇×D₄	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	1	no	—
81	C₂×C₄×C₇	—	2	1	no	—
82	C₂×C₄×C₇	—	4	2	no	—
83	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₇×C₈	—	2	1	no	—
87	C₇×C₈	C₂×C₃×C₄	1	2	yes	complemented
88	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
89	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₃×SD₁₆	C₄×C₇	1	4	yes	complemented
92	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	1	no	—
95	C₃×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
96	C₃×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
97	C₃×C₂²⋊C₄	—	2	1	no	—
98	C₃×C₄²	—	2	1	no	—
99	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₃×C₇	—	4	2	no	—
101	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₃×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₃×C₇	#137	1	1	yes	characteristic
104	C₂×SD₁₆	C₂×C₃×C₇	1	1	yes	characteristic
105	C₄×Q₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
106	C₄×D₄	C₂×C₃×C₇	1	1	yes	characteristic
107	#107	C₂×C₃×C₇	1	1	yes	characteristic, complemented
108	#103	C₂×C₃×C₇	1	1	yes	characteristic, complemented
109	#102	C₂×C₃×C₇	1	1	yes	characteristic
110	#97	C₂×C₃×C₇	1	1	yes	characteristic, complemented
111	C₂²×C₇	—	2	2	no	—
112	C₂²×C₇	—	4	1	no	—
113	C₂²×C₇	—	4	1	no	—
114	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
115	C₄×C₇	—	2	2	no	—
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	4	1	no	—
119	C₄×C₇	—	4	1	no	—
120	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
121	C₄×C₇	C₃×#41	1	1	yes	characteristic
122	C₂³×C₃	—	2	1	no	—
123	C₃×Q₈	—	2	1	no	—
124	C₃×Q₈	C₂×C₄×C₇	1	2	yes	complemented
125	C₃×D₄	—	2	1	no	—
126	C₃×D₄	C₂×C₄×C₇	1	2	yes	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
134	C₃×C₈	—	2	1	no	—
135	C₃×C₈	C₂×C₄×C₇	1	2	yes	complemented
136	C₃×C₇	#440	1	1	yes	characteristic, direct product factor, Hall
137	C₂×Q₈	C₂²×C₃×C₇	1	1	yes	characteristic
138	C₂×D₄	C₂²×C₃×C₇	1	1	yes	characteristic
139	C₂²×C₄	—	2	1	no	—
140	SD₁₆	C₃×C₄×C₇	1	4	yes	complemented
141	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	C₂²×C₃×C₇	1	1	yes	characteristic
145	C₄⋊C₄	C₂²×C₃×C₇	1	1	yes	characteristic
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
149	C₂×C₇	—	4	2	no	—
150	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
151	C₂×C₇	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₇	C₃×#137	1	1	yes	characteristic
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	—	4	1	no	—
155	C₂²×C₃	—	4	1	no	—
156	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	4	1	no	—
161	C₃×C₄	—	4	1	no	—
162	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
163	C₃×C₄	C₇×#41	1	1	yes	characteristic
164	C₂³	—	2	1	no	—
165	Q₈	—	2	1	no	—
166	Q₈	C₂×C₃×C₄×C₇	1	2	yes	complemented
167	D₄	—	2	1	no	—
168	D₄	C₂×C₃×C₄×C₇	1	2	yes	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
174	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	C₂×C₃×C₄×C₇	1	2	yes	complemented
178	C₇	C₃×#440	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×C₃	—	4	2	no	—
180	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
181	C₂×C₃	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₃	C₇×#137	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	1	no	—
186	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
193	C₄	C₃×C₇×#41	1	1	yes	characteristic
194	C₃	C₇×#440	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	4	2	no	—
196	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic
197	C₂	C₃×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₂	C₃×C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂×D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
prime radical	C₂×C₃×C₇×D₄	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ C₃×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×Q₈	C₂	semidirect
4. C₃×C₇×#107	C₂	semidirect
5. C₃×C₇×#103	C₂	semidirect
7. C₃×C₇×#97	C₂	semidirect
8. C₇×#440	C₃	direct (N × H)
12. C₃×C₇×SD₁₆	C₄	semidirect
22. C₄×C₇×Q₈	C₂×C₃	semidirect
24. C₇×#107	C₂×C₃	semidirect
25. C₇×#103	C₂×C₃	semidirect
27. C₇×#97	C₂×C₃	semidirect
28. C₃×#440	C₇	direct (N × H)
31. C₃×C₇×Q₈	C₂×C₄	semidirect
42. C₃×C₇×C₈	C₂×C₄	semidirect
46. C₇×SD₁₆	C₃×C₄	semidirect
56. C₃×C₄×Q₈	C₂×C₇	semidirect
58. C₃×#107	C₂×C₇	semidirect
59. C₃×#103	C₂×C₇	semidirect
61. C₃×#97	C₂×C₇	semidirect
73. #440	C₃×C₇	direct (N × H)
76. C₇×Q₈	C₂×C₃×C₄	semidirect
87. C₇×C₈	C₂×C₃×C₄	semidirect
91. C₃×SD₁₆	C₄×C₇	semidirect
105. C₄×Q₈	C₂×C₃×C₇	semidirect
107. #107	C₂×C₃×C₇	semidirect
108. #103	C₂×C₃×C₇	semidirect
110. #97	C₂×C₃×C₇	semidirect
124. C₃×Q₈	C₂×C₄×C₇	semidirect
135. C₃×C₈	C₂×C₄×C₇	semidirect
136. C₃×C₇	#440	direct (N × H)
140. SD₁₆	C₃×C₄×C₇	semidirect
166. Q₈	C₂×C₃×C₄×C₇	semidirect
177. C₈	C₂×C₃×C₄×C₇	semidirect
178. C₇	C₃×#440	direct (N × H)
194. C₃	C₇×#440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×SD₁₆	2
C₃×C₄×C₇×Q₈	2
C₃×C₄×C₇×D₄	2
C₃×C₇×#107	2
C₃×C₇×#103	2
C₃×C₇×#102	2
C₃×C₇×#97	2
C₇×#440	3
C₃×#440	7

Subgroups — C3×C7×#440 #13056951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#440 #13056951