Subgroups — C₂×C₄⋊C₄×AGL(1,𝔽₇) #13057236

Order 1344.

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

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
4. C₂×C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂	semidirect
5. C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂	semidirect
6. C₄⋊C₄×AGL(1,𝔽₇)	C₂	direct (N × H)
7. C₂×D₇×C₄⋊C₄	C₃	semidirect
14. C₂×C₄×AGL(1,𝔽₇)	C₄	semidirect
17. C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
18. C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	semidirect
19. (ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	semidirect
22. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
23. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
24. C₂×Dic₇⋊C₄	C₂×C₃	semidirect
25. D₇×C₄⋊C₄	C₂×C₃	semidirect
33. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	semidirect
37. C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	semidirect
41. C₄×AGL(1,𝔽₇)	C₂×C₄	semidirect
48. C₂×C₄×D₇	C₃×C₄	semidirect
51. C₇×C₄⋊C₄	C₂²×C₃	semidirect
52. C₄⋊Dic₇	C₂²×C₃	semidirect
53. Dic₇⋊C₄	C₂²×C₃	semidirect
64. C₂×AGL(1,𝔽₇)	C₄⋊C₄	direct (N × H)
69. C₄×ΓL(1,𝔽₈)	C₂²×C₄	semidirect
71. ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	semidirect
79. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
83. C₂×Dic₇	C₂×C₃×C₄	semidirect
87. C₄×D₇	C₂×C₃×C₄	semidirect
100. C₂×ΓL(1,𝔽₈)	C₂×C₄⋊C₄	semidirect
104. AGL(1,𝔽₇)	C₂×C₄⋊C₄	direct (N × H)
110. C₂×C₄⋊C₄	AGL(1,𝔽₇)	direct (N × H)
115. C₂×D₇	C₃×C₄⋊C₄	semidirect
120. C₄×C₇	C₂²×C₃×C₄	semidirect
122. Dic₇	C₂²×C₃×C₄	semidirect
138. ΓL(1,𝔽₈)	C₂²×C₄⋊C₄	semidirect
150. C₄⋊C₄	C₂×AGL(1,𝔽₇)	direct (N × H)
151. C₂×C₇	C₂×C₃×C₄⋊C₄	semidirect
155. D₇	C₂×C₃×C₄⋊C₄	semidirect
180. C₂×C₄	C₄×AGL(1,𝔽₇)	semidirect
183. C₇	C₂²×C₃×C₄⋊C₄	semidirect
200. C₄	C₂×C₄×AGL(1,𝔽₇)	semidirect
203. C₂	C₄⋊C₄×AGL(1,𝔽₇)	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×C₄×AGL(1,𝔽₇)	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₂×C₄⋊(ΓL(1,𝔽₈)⋊C₄)	2
C₂×(ΓL(1,𝔽₈)⋊C₄)⋊C₄	2
C₄⋊C₄×AGL(1,𝔽₇)	2
C₂×D₇×C₄⋊C₄	3
C₂²×C₃×C₄⋊C₄	7

Subgroups — C2×C4⋊C4×AGL(1,𝔽7) #13057236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×AGL(1,𝔽₇) #13057236