Subgroups — C₂²×C₇×AGL(1,ℤ/8) #11744530

Order 896.

1720 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×AGL(1,ℤ/8)	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×D₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₇	—	2	1	no	—
8	C₂×C₇×#41	—	2	6	no	—
9	C₂×C₇×#41	C₂²	1	12	yes	complemented
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₇×D₄	—	2	1	no	—
12	C₂²×C₇×D₄	—	2	6	no	—
13	C₂²×C₇×D₄	—	2	1	no	—
14	C₂²×C₇×D₄	C₂²	1	2	yes	—
15	C₂²×C₇×D₄	C₂²	1	12	yes	complemented
16	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₇	—	2	1	no	—
18	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
19	C₇×AGL(1,ℤ/8)	C₂²	1	64	yes	direct product factor
20	C₂×C₇×SD₁₆	C₂²	1	24	yes	complemented
21	C₂×C₇×D₈	C₂²	1	24	yes	complemented
22	C₂×C₇×M₁₆	C₂²	1	12	yes	complemented
23	C₂²×C₇×C₈	C₂²	1	2	yes	complemented
24	C₂²×AGL(1,ℤ/8)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁴×C₇	—	2	12	no	—
26	C₂⁴×C₇	—	2	1	no	—
27	C₂⁴×C₇	—	2	2	no	—
28	C₂⁴×C₇	—	4	8	no	—
29	C₂⁴×C₇	D₄	1	1	yes	characteristic
30	C₇×#41	—	2	24	no	—
31	C₇×#41	C₂³	1	16	yes	complemented
32	C₂×C₇×Q₈	—	2	3	no	—
33	C₂×C₇×Q₈	C₂³	1	6	yes	complemented
34	C₂×C₇×D₄	—	2	12	no	—
35	C₂×C₇×D₄	—	2	6	no	—
36	C₂×C₇×D₄	—	2	12	no	—
37	C₂×C₇×D₄	—	2	24	no	—
38	C₂×C₇×D₄	—	2	3	no	—
39	C₂×C₇×D₄	C₂³	1	12	yes	—
40	C₂×C₇×D₄	C₂³	1	16	yes	complemented
41	C₂×C₇×D₄	C₂³	1	6	yes	complemented
42	C₂²×C₄×C₇	—	2	1	no	—
43	C₂²×C₄×C₇	—	2	12	no	—
44	C₂²×C₄×C₇	D₄	1	6	yes	—
45	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	6	yes	—
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×SD₁₆	C₂³	1	32	yes	complemented
49	C₇×D₈	C₂³	1	32	yes	complemented
50	C₇×M₁₆	C₂³	1	16	yes	complemented
51	C₂×C₇×C₈	C₂³	1	12	yes	complemented
52	C₂²×#41	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂³×D₄	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂×AGL(1,ℤ/8)	C₂×C₇	1	24	yes	direct product factor
55	C₂²×SD₁₆	C₂×C₇	1	2	yes	complemented
56	C₂²×D₈	C₂×C₇	1	2	yes	complemented
57	C₂²×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂³×C₇	—	2	4	no	—
59	C₂³×C₇	—	2	6	no	—
60	C₂³×C₇	—	2	12	no	—
61	C₂³×C₇	—	2	16	no	—
62	C₂³×C₇	—	4	48	no	—
63	C₂³×C₇	—	4	4	no	—
64	C₂³×C₇	—	4	8	no	—
65	C₂³×C₇	C₂×D₄	1	6	yes	—
66	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
67	C₇×Q₈	—	2	6	no	—
68	C₇×Q₈	C₂⁴	1	4	yes	complemented
69	C₇×D₄	—	2	16	no	—
70	C₇×D₄	—	2	12	no	—
71	C₇×D₄	—	2	16	no	—
72	C₇×D₄	—	2	6	no	—
73	C₇×D₄	C₂⁴	1	8	yes	—
74	C₇×D₄	C₂⁴	1	4	yes	complemented
75	C₂×C₄×C₇	—	2	6	no	—
76	C₂×C₄×C₇	—	2	16	no	—
77	C₂×C₄×C₇	C₂×D₄	1	6	yes	—
78	C₂×C₄×C₇	C₂×D₄	1	3	yes	—
79	C₂×C₄×C₇	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₇	C₂⁴	1	3	yes	—
81	C₂×C₄×C₇	C₂⁴	1	4	yes	—
82	C₇×C₈	C₂⁴	1	8	yes	complemented
83	C₂⁵	—	2	1	no	—
84	C₂×#41	—	2	6	no	—
85	C₂×#41	C₂²×C₇	1	12	yes	complemented
86	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
87	C₂²×D₄	—	2	1	no	—
88	C₂²×D₄	—	2	1	no	—
89	C₂²×D₄	—	2	6	no	—
90	C₂²×D₄	C₂²×C₇	1	2	yes	—
91	C₂²×D₄	C₂²×C₇	1	12	yes	complemented
92	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
93	C₂³×C₄	—	2	1	no	—
94	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
95	AGL(1,ℤ/8)	C₂²×C₇	1	64	yes	direct product factor
96	C₂×SD₁₆	C₂²×C₇	1	24	yes	complemented
97	C₂×D₈	C₂²×C₇	1	24	yes	complemented
98	C₂×M₁₆	C₂²×C₇	1	12	yes	complemented
99	C₂²×C₈	C₂²×C₇	1	2	yes	complemented
100	C₂²×C₇	—	2	12	no	—
101	C₂²×C₇	—	2	4	no	—
102	C₂²×C₇	—	2	8	no	—
103	C₂²×C₇	—	4	32	no	—
104	C₂²×C₇	—	4	12	no	—
105	C₂²×C₇	—	4	24	no	—
106	C₂²×C₇	AGL(1,ℤ/8)	1	4	yes	direct product factor
107	C₂²×C₇	C₂²×D₄	1	4	yes	—
108	C₂²×C₇	C₂²×D₄	1	3	yes	—
109	C₄×C₇	—	2	4	no	—
110	C₄×C₇	C₂²×D₄	1	4	yes	—
111	C₄×C₇	C₂²×D₄	1	3	yes	—
112	C₄×C₇	C₂⁵	1	1	yes	characteristic
113	C₂⁴	—	2	1	no	—
114	C₂⁴	—	2	2	no	—
115	C₂⁴	—	2	12	no	—
116	C₂⁴	—	4	8	no	—
117	C₂⁴	C₇×D₄	1	1	yes	characteristic
118	#41	—	2	24	no	—
119	#41	C₂³×C₇	1	16	yes	complemented
120	C₂×Q₈	—	2	3	no	—
121	C₂×Q₈	C₂³×C₇	1	6	yes	complemented
122	C₂×D₄	—	2	12	no	—
123	C₂×D₄	—	2	6	no	—
124	C₂×D₄	—	2	12	no	—
125	C₂×D₄	—	2	24	no	—
126	C₂×D₄	—	2	3	no	—
127	C₂×D₄	C₂³×C₇	1	12	yes	—
128	C₂×D₄	C₂³×C₇	1	16	yes	complemented
129	C₂×D₄	C₂³×C₇	1	6	yes	complemented
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	12	no	—
132	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₇×D₄	1	6	yes	—
134	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₇	1	6	yes	—
136	SD₁₆	C₂³×C₇	1	32	yes	complemented
137	D₈	C₂³×C₇	1	32	yes	complemented
138	M₁₆	C₂³×C₇	1	16	yes	complemented
139	C₂×C₈	C₂³×C₇	1	12	yes	complemented
140	C₂×C₇	—	2	4	no	—
141	C₂×C₇	—	4	4	no	—
142	C₂×C₇	—	4	8	no	—
143	C₂×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	direct product factor
144	C₂×C₇	C₂³×D₄	1	1	yes	characteristic
145	C₂³	—	2	4	no	—
146	C₂³	—	2	6	no	—
147	C₂³	—	2	12	no	—
148	C₂³	—	2	16	no	—
149	C₂³	—	4	48	no	—
150	C₂³	—	4	4	no	—
151	C₂³	—	4	8	no	—
152	C₂³	C₂×C₇×D₄	1	6	yes	—
153	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
154	Q₈	—	2	6	no	—
155	Q₈	C₂⁴×C₇	1	4	yes	complemented
156	D₄	—	2	16	no	—
157	D₄	—	2	12	no	—
158	D₄	—	2	16	no	—
159	D₄	—	2	6	no	—
160	D₄	C₂⁴×C₇	1	8	yes	—
161	D₄	C₂⁴×C₇	1	4	yes	complemented
162	C₂×C₄	—	2	6	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	C₂×C₇×D₄	1	6	yes	—
165	C₂×C₄	C₂×C₇×D₄	1	3	yes	—
166	C₂×C₄	C₂×C₇×D₄	1	12	yes	—
167	C₂×C₄	C₂⁴×C₇	1	3	yes	—
168	C₂×C₄	C₂⁴×C₇	1	4	yes	—
169	C₈	C₂⁴×C₇	1	8	yes	complemented
170	C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	—	2	12	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	4	32	no	—
175	C₂²	—	4	24	no	—
176	C₂²	—	4	12	no	—
177	C₂²	C₇×AGL(1,ℤ/8)	1	4	yes	direct product factor
178	C₂²	C₂²×C₇×D₄	1	4	yes	—
179	C₂²	C₂²×C₇×D₄	1	3	yes	—
180	C₄	—	2	4	no	—
181	C₄	C₂²×C₇×D₄	1	4	yes	—
182	C₄	C₂²×C₇×D₄	1	3	yes	—
183	C₄	C₂⁵×C₇	1	1	yes	characteristic
184	C₂	—	2	4	no	—
185	C₂	—	4	8	no	—
186	C₂	—	4	4	no	—
187	C₂	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	direct product factor
188	C₂	C₂³×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×#41	C₂	semidirect
2. C₂³×C₇×D₄	C₂	semidirect
3. C₂×C₇×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂²×C₇×SD₁₆	C₂	semidirect
5. C₂²×C₇×D₈	C₂	semidirect
6. C₂²×C₇×M₁₆	C₂	semidirect
9. C₂×C₇×#41	C₂²	semidirect
10. C₂²×C₇×Q₈	C₂²	semidirect
15. C₂²×C₇×D₄	C₂²	semidirect
16. C₂²×C₇×D₄	C₂²	semidirect
19. C₇×AGL(1,ℤ/8)	C₂²	direct (N × H)
20. C₂×C₇×SD₁₆	C₂²	semidirect
21. C₂×C₇×D₈	C₂²	semidirect
22. C₂×C₇×M₁₆	C₂²	semidirect
23. C₂²×C₇×C₈	C₂²	semidirect
24. C₂²×AGL(1,ℤ/8)	C₇	direct (N × H)
31. C₇×#41	C₂³	semidirect
33. C₂×C₇×Q₈	C₂³	semidirect
40. C₂×C₇×D₄	C₂³	semidirect
41. C₂×C₇×D₄	C₂³	semidirect
48. C₇×SD₁₆	C₂³	semidirect
49. C₇×D₈	C₂³	semidirect
50. C₇×M₁₆	C₂³	semidirect
51. C₂×C₇×C₈	C₂³	semidirect
52. C₂²×#41	C₂×C₇	semidirect
53. C₂³×D₄	C₂×C₇	semidirect
54. C₂×AGL(1,ℤ/8)	C₂×C₇	direct (N × H)
55. C₂²×SD₁₆	C₂×C₇	semidirect
56. C₂²×D₈	C₂×C₇	semidirect
57. C₂²×M₁₆	C₂×C₇	semidirect
68. C₇×Q₈	C₂⁴	semidirect
74. C₇×D₄	C₂⁴	semidirect
82. C₇×C₈	C₂⁴	semidirect
85. C₂×#41	C₂²×C₇	semidirect
86. C₂²×Q₈	C₂²×C₇	semidirect
91. C₂²×D₄	C₂²×C₇	semidirect
92. C₂²×D₄	C₂²×C₇	semidirect
95. AGL(1,ℤ/8)	C₂²×C₇	direct (N × H)
96. C₂×SD₁₆	C₂²×C₇	semidirect
97. C₂×D₈	C₂²×C₇	semidirect
98. C₂×M₁₆	C₂²×C₇	semidirect
99. C₂²×C₈	C₂²×C₇	semidirect
106. C₂²×C₇	AGL(1,ℤ/8)	direct (N × H)
119. #41	C₂³×C₇	semidirect
121. C₂×Q₈	C₂³×C₇	semidirect
128. C₂×D₄	C₂³×C₇	semidirect
129. C₂×D₄	C₂³×C₇	semidirect
136. SD₁₆	C₂³×C₇	semidirect
137. D₈	C₂³×C₇	semidirect
138. M₁₆	C₂³×C₇	semidirect
139. C₂×C₈	C₂³×C₇	semidirect
143. C₂×C₇	C₂×AGL(1,ℤ/8)	direct (N × H)
155. Q₈	C₂⁴×C₇	semidirect
161. D₄	C₂⁴×C₇	semidirect
169. C₈	C₂⁴×C₇	semidirect
170. C₇	C₂²×AGL(1,ℤ/8)	direct (N × H)
177. C₂²	C₇×AGL(1,ℤ/8)	direct (N × H)
187. C₂	C₂×C₇×AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#41	2
C₂³×C₇×D₄	2
C₂×C₇×AGL(1,ℤ/8)	2
C₂²×C₇×SD₁₆	2
C₂²×C₇×D₈	2
C₂²×C₇×M₁₆	2
C₂²×AGL(1,ℤ/8)	7

Subgroups — C22×C7×AGL(1,ℤ/8) #11744530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×AGL(1,ℤ/8) #11744530