Subgroups — C₂×#6470⋊C₃ #13057478

Order 1344.

532 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×#1157⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6470⋊C₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1131⋊C₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
7	C₂×#6470	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
9	C₂×(C₂²⋊D₇)⋊C₃	C₄	1	2	yes	—
10	#1157⋊C₃	C₄	1	4	yes	—
11	#1157⋊C₃	C₂²	1	4	yes	—
12	C₂×C₄⋊AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
13	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
14	C₂×ΓL(1,𝔽₈)⋊Q₈	C₄	1	1	yes	characteristic
15	M₁₆×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
16	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
17	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	4	yes	complemented
20	#1131⋊C₃	C₂²	1	8	yes	complemented
21	C₈×AGL(1,𝔽₇)	C₂²	1	8	yes	complemented
22	C₂×#1157	C₂×C₃	1	1	yes	characteristic
23	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#6470	C₂×C₃	1	8	yes	complemented
26	C₂×#1131	C₂×C₃	1	2	yes	complemented
27	C₂×C₈×D₇	C₂×C₃	1	2	yes	complemented
28	C₂×C₃×#131	—	7	1	no	maximal, Hall
29	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
30	C₂²×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
31	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
33	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
34	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
35	(C₂²⋊D₇)⋊C₃	C₂×C₄	1	8	yes	—
36	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	2	yes	—
37	C₄⋊AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
38	C₄×AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
39	C₄×AGL(1,𝔽₇)	C₂³	1	4	yes	—
40	ΓL(1,𝔽₈)⋊Q₈	C₂×C₄	1	4	yes	—
41	C₈×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
42	ΓL(1,𝔽₈)⋊C₈	C₂³	1	2	yes	—
43	ΓL(1,𝔽₈)⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₂²⋊D₇	C₃×C₄	1	2	yes	—
46	#1157	C₃×C₄	1	4	yes	—
47	#1157	C₂²×C₃	1	4	yes	—
48	C₂×D₂₈	C₃×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
50	C₂×Dic₁₄	C₃×C₄	1	1	yes	characteristic
51	C₇×M₁₆	C₂²×C₃	1	4	yes	complemented
52	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
53	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₇⋊C₈	C₂²×C₃	1	4	yes	complemented
56	#1131	C₂²×C₃	1	8	yes	complemented
57	C₈×D₇	C₂²×C₃	1	8	yes	complemented
58	C₂×C₃×#41	—	7	1	no	—
59	C₃×#131	—	7	8	no	—
60	C₂×C₃×M₁₆	—	7	2	no	—
61	C₂×C₃×M₁₆	—	7	1	no	—
62	C₂²×C₃×C₈	—	7	2	no	—
63	C₂²×C₃×C₈	—	7	1	no	—
64	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
65	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
66	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
67	C₂×AGL(1,𝔽₇)	—	2	4	no	—
68	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	4	yes	—
69	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
70	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
71	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
72	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	4	yes	—
73	C₂×#131	—	7	1	no	Sylow
74	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂²×D₇	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
79	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
80	C₂²⋊D₇	C₂×C₃×C₄	1	8	yes	—
81	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
82	D₂₈	C₂×C₃×C₄	1	4	yes	—
83	C₄×D₇	C₂×C₃×C₄	1	4	yes	—
84	C₄×D₇	C₂³×C₃	1	4	yes	—
85	Dic₁₄	C₂×C₃×C₄	1	4	yes	—
86	C₇×C₈	C₂³×C₃	1	4	yes	—
87	C₇⋊C₈	C₂³×C₃	1	2	yes	—
88	C₇⋊C₈	C₂³×C₃	1	2	yes	—
89	C₃×#41	—	7	4	no	—
90	C₃×#41	—	7	4	no	—
91	C₂×C₃×Q₈	—	7	1	no	—
92	C₂×C₃×D₄	—	7	1	no	—
93	C₂×C₃×D₄	—	7	2	no	—
94	C₂²×C₃×C₄	—	7	2	no	—
95	C₂²×C₃×C₄	—	7	1	no	—
96	C₃×M₁₆	—	7	8	no	—
97	C₃×M₁₆	—	7	4	no	—
98	C₂×C₃×C₈	—	7	1	no	—
99	C₂×C₃×C₈	—	7	1	no	—
100	C₂×C₃×C₈	—	7	8	no	—
101	C₂×C₃×C₈	—	7	4	no	—
102	C₂×C₃×C₈	—	7	2	no	—
103	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
104	C₂×ΓL(1,𝔽₈)	#131	1	2	yes	complemented
105	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
106	AGL(1,𝔽₇)	—	2	4	no	—
107	C₂×#41	—	7	1	no	—
108	#131	—	7	8	no	—
109	C₂×M₁₆	—	7	2	no	—
110	C₂×M₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
111	C₂²×C₈	—	7	1	no	—
112	C₂²×C₈	—	7	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
115	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×D₇	—	2	4	no	—
117	C₂×D₇	C₂²×C₃×C₄	1	4	yes	—
118	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
119	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
121	Dic₇	C₂²×C₃×C₄	1	4	yes	—
122	C₂³×C₃	—	7	2	no	—
123	C₂³×C₃	—	7	1	no	—
124	C₃×Q₈	—	7	4	no	—
125	C₃×D₄	—	7	4	no	—
126	C₃×D₄	—	7	8	no	—
127	C₂×C₃×C₄	—	7	4	no	—
128	C₂×C₃×C₄	—	7	4	no	—
129	C₂×C₃×C₄	—	7	2	no	—
130	C₂×C₃×C₄	—	7	1	no	—
131	C₂×C₃×C₄	—	7	2	no	—
132	C₂×C₃×C₄	—	7	2	no	—
133	C₂×C₃×C₄	—	7	1	no	—
134	C₃×C₈	—	7	2	no	—
135	C₃×C₈	—	7	2	no	—
136	C₃×C₈	—	7	4	no	—
137	ΓL(1,𝔽₈)	C₂×#131	1	1	yes	characteristic, complemented, Hall
138	#41	—	7	4	no	—
139	#41	—	7	4	no	—
140	C₂×Q₈	—	7	1	no	—
141	C₂×D₄	—	7	2	no	—
142	C₂×D₄	—	7	1	no	—
143	C₂²×C₄	—	7	2	no	—
144	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
145	M₁₆	—	7	8	no	—
146	M₁₆	C₂×AGL(1,𝔽₇)	1	4	yes	complemented
147	C₂×C₈	—	7	4	no	—
148	C₂×C₈	—	7	1	no	—
149	C₂×C₈	—	7	1	no	—
150	C₂×C₈	—	7	8	no	—
151	C₂×C₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	C₃×#131	1	2	yes	complemented
154	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
155	D₇	—	2	4	no	—
156	C₂²×C₃	—	7	4	no	—
157	C₂²×C₃	—	7	1	no	—
158	C₂²×C₃	—	7	2	no	—
159	C₂²×C₃	—	14	2	no	—
160	C₂²×C₃	—	14	4	no	—
161	C₃×C₄	—	7	4	no	—
162	C₃×C₄	—	7	1	no	—
163	C₃×C₄	—	7	2	no	—
164	C₃×C₄	—	7	1	no	—
165	C₂³	—	7	2	no	—
166	C₂³	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
167	Q₈	—	7	4	no	—
168	D₄	—	7	8	no	—
169	D₄	—	7	4	no	—
170	C₂×C₄	—	7	2	no	—
171	C₂×C₄	—	7	4	no	—
172	C₂×C₄	—	7	4	no	—
173	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
174	C₂×C₄	C₄×AGL(1,𝔽₇)	1	2	yes	—
175	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
176	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
177	C₈	—	7	2	no	—
178	C₈	—	7	2	no	—
179	C₈	C₂²×AGL(1,𝔽₇)	1	4	yes	—
180	C₇	C₂×C₃×#131	1	1	yes	characteristic, complemented, Sylow
181	C₂×C₃	—	7	1	no	—
182	C₂×C₃	—	7	2	no	—
183	C₂×C₃	—	14	2	no	—
184	C₂×C₃	—	14	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	7	4	no	—
187	C₂²	—	14	4	no	—
188	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
189	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
190	C₄	—	7	4	no	—
191	C₄	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
192	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
193	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
194	C₃	—	7	1	no	Sylow
195	C₂	—	2	2	no	—
196	C₂	—	14	4	no	—
197	C₂	#6470⋊C₃	1	2	yes	direct product factor
198	C₂	C₂²×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₂×C₇×M₁₆	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×C₄
prime radical	C₂×#1157⋊C₃	C₂
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	C₂×M₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×#131
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂²×ΓL(1,𝔽₈)⋊C₈	C₂	semidirect
4. #6470⋊C₃	C₂	direct (N × H)
5. C₂×#1131⋊C₃	C₂	semidirect
6. C₂×C₈×AGL(1,𝔽₇)	C₂	semidirect
7. C₂×#6470	C₃	semidirect
15. M₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
19. C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	semidirect
20. #1131⋊C₃	C₂²	semidirect
21. C₈×AGL(1,𝔽₇)	C₂²	semidirect
23. C₂×C₇×M₁₆	C₂×C₃	semidirect
24. C₂²×C₇⋊C₈	C₂×C₃	semidirect
25. #6470	C₂×C₃	semidirect
26. C₂×#1131	C₂×C₃	semidirect
27. C₂×C₈×D₇	C₂×C₃	semidirect
51. C₇×M₁₆	C₂²×C₃	semidirect
55. C₂×C₇⋊C₈	C₂²×C₃	semidirect
56. #1131	C₂²×C₃	semidirect
57. C₈×D₇	C₂²×C₃	semidirect
104. C₂×ΓL(1,𝔽₈)	#131	semidirect
110. C₂×M₁₆	AGL(1,𝔽₇)	semidirect
137. ΓL(1,𝔽₈)	C₂×#131	semidirect
146. M₁₆	C₂×AGL(1,𝔽₇)	semidirect
153. C₂×C₇	C₃×#131	semidirect
180. C₇	C₂×C₃×#131	semidirect
197. C₂	#6470⋊C₃	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×#1157⋊C₃	2
C₂×M₁₆×ΓL(1,𝔽₈)	2
C₂²×ΓL(1,𝔽₈)⋊C₈	2
#6470⋊C₃	2
C₂×#1131⋊C₃	2
C₂×C₈×AGL(1,𝔽₇)	2
C₂×#6470	3
C₂×C₃×#131	7

Subgroups — C2×#6470⋊C3 #13057478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6470⋊C₃ #13057478