Subgroups — C₅×C₂²⋊C₄×AGL(1,𝔽₇) #72218139

Order 3360.

528 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	C₂³×C₅×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
3	C₅×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×(C₂²⋊Dic₇)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₅×D₇×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂³×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₂²×C₅×AGL(1,𝔽₇)	—	2	4	no	—
9	C₂²×C₅×AGL(1,𝔽₇)	C₄	1	4	yes	complemented
10	C₂²×C₅×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
11	C₂²×C₅×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
13	C₂×C₅×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
14	C₄×C₅×AGL(1,𝔽₇)	—	2	4	no	—
15	C₂²⋊C₄×AGL(1,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₅×D₇	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄×C₅×D₇	C₂×C₃	1	2	yes	complemented
18	C₅×C₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×C₂²⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
21	C₂×C₃×C₅×C₂²⋊C₄	—	7	1	no	maximal, Hall
22	C₂²×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
23	C₂²×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	complemented
24	C₂²×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
25	C₂×C₅×AGL(1,𝔽₇)	—	2	8	no	—
26	C₂×C₅×AGL(1,𝔽₇)	—	2	2	no	—
27	C₂×C₅×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
28	C₂×C₅×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	complemented
29	C₂×C₅×AGL(1,𝔽₇)	D₄	1	4	yes	—
30	C₄×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
31	C₅×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
32	C₂³×AGL(1,𝔽₇)	C₂×C₅	1	1	yes	characteristic
33	C₂×C₄×AGL(1,𝔽₇)	C₂×C₅	1	2	yes	complemented
34	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented
35	(C₂²⋊Dic₇)⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented
36	(C₂×AGL(1,𝔽₇))⋊C₄	C₂×C₅	1	2	yes	complemented
37	C₂³×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
38	C₂²×C₅×D₇	—	2	4	no	—
39	C₂²×C₅×D₇	C₃×C₄	1	4	yes	complemented
40	C₂²×C₅×D₇	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×D₇	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₄×C₅×C₇	C₂²×C₃	1	2	yes	complemented
43	C₂×C₅×Dic₇	C₂²×C₃	1	2	yes	complemented
44	C₄×C₅×D₇	—	2	4	no	—
45	C₂⁴×C₃×C₅	—	7	1	no	—
46	C₂²×C₃×C₄×C₅	—	7	2	no	—
47	C₃×C₅×C₂²⋊C₄	—	7	1	no	—
48	C₃×C₅×C₂²⋊C₄	—	7	2	no	—
49	C₃×C₅×C₂²⋊C₄	—	7	1	no	—
50	D₇×C₂²⋊C₄	C₃×C₅	1	1	yes	characteristic, complemented, Hall
51	C₂×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
52	C₂×C₅×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
54	C₅×AGL(1,𝔽₇)	—	2	2	no	—
55	C₅×AGL(1,𝔽₇)	C₂²⋊C₄	1	4	yes	direct product factor
56	C₂³×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
57	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
58	C₂²×AGL(1,𝔽₇)	C₄×C₅	1	4	yes	complemented
59	C₂²×AGL(1,𝔽₇)	C₂²×C₅	1	1	yes	characteristic
60	C₂²×AGL(1,𝔽₇)	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₄×ΓL(1,𝔽₈)	C₂²×C₅	1	2	yes	complemented
62	C₂×ΓL(1,𝔽₈)⋊C₄	C₂²×C₅	1	2	yes	complemented
63	C₄×AGL(1,𝔽₇)	—	2	4	no	—
64	C₂×C₅×C₂²⋊C₄	—	7	1	no	Hall
65	C₂²×C₅×C₇	—	2	2	no	—
66	C₂²×C₅×C₇	C₂×C₃×C₄	1	2	yes	complemented
67	C₂²×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
68	C₂×C₅×D₇	—	2	2	no	—
69	C₂×C₅×D₇	—	2	8	no	—
70	C₂×C₅×D₇	C₂×C₃×C₄	1	2	yes	—
71	C₂×C₅×D₇	C₂×C₃×C₄	1	2	yes	complemented
72	C₂×C₅×D₇	C₃×D₄	1	4	yes	—
73	C₄×C₅×C₇	—	2	2	no	—
74	C₅×Dic₇	—	2	2	no	—
75	C₂³×C₃×C₅	—	7	1	no	—
76	C₂³×C₃×C₅	—	7	1	no	—
77	C₂³×C₃×C₅	—	7	4	no	—
78	C₂³×C₃×C₅	—	7	1	no	—
79	C₂³×C₃×C₅	—	14	4	no	—
80	C₂×C₃×C₄×C₅	—	7	2	no	—
81	C₂×C₃×C₄×C₅	—	7	2	no	—
82	C₂×C₃×C₄×C₅	—	14	4	no	—
83	C₂³×D₇	C₂×C₃×C₅	1	1	yes	characteristic
84	C₂×C₄×D₇	C₂×C₃×C₅	1	2	yes	complemented
85	C₇×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂²⋊Dic₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	(C₂×D₇)⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
88	C₅×ΓL(1,𝔽₈)	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
89	C₂×C₃×C₂²⋊C₄	—	7	1	no	Hall
90	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂²×ΓL(1,𝔽₈)	C₂×C₄×C₅	1	2	yes	complemented
92	C₂²×ΓL(1,𝔽₈)	C₂³×C₅	1	1	yes	characteristic
93	C₂×AGL(1,𝔽₇)	—	2	2	no	—
94	C₂×AGL(1,𝔽₇)	—	2	8	no	—
95	C₂×AGL(1,𝔽₇)	C₂×C₄×C₅	1	2	yes	—
96	C₂×AGL(1,𝔽₇)	C₂×C₄×C₅	1	2	yes	complemented
97	C₂×AGL(1,𝔽₇)	C₅×D₄	1	4	yes	—
98	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
99	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
100	C₂⁴×C₅	—	7	1	no	—
101	C₂²×C₄×C₅	—	7	2	no	—
102	C₅×C₂²⋊C₄	—	7	1	no	—
103	C₅×C₂²⋊C₄	—	7	2	no	—
104	C₅×C₂²⋊C₄	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
105	C₂×C₅×C₇	—	2	2	no	—
106	C₂×C₅×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₂×C₅×C₇	C₂×C₃×D₄	1	2	yes	—
108	C₅×D₇	—	2	2	no	—
109	C₅×D₇	C₃×C₂²⋊C₄	1	4	yes	complemented
110	C₂²×C₃×C₅	—	7	4	no	—
111	C₂²×C₃×C₅	—	7	2	no	—
112	C₂²×C₃×C₅	—	7	2	no	—
113	C₂²×C₃×C₅	—	7	2	no	—
114	C₂²×C₃×C₅	—	7	1	no	—
115	C₂²×C₃×C₅	—	14	2	no	—
116	C₂²×C₃×C₅	—	14	8	no	—
117	C₂²×C₃×C₅	—	14	2	no	—
118	C₃×C₄×C₅	—	14	2	no	—
119	C₃×C₄×C₅	—	14	2	no	—
120	C₂³×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
121	C₂²×D₇	—	2	4	no	—
122	C₂²×D₇	C₃×C₄×C₅	1	4	yes	complemented
123	C₂²×D₇	C₂²×C₃×C₅	1	1	yes	characteristic
124	C₂²×D₇	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×C₃×C₅	1	2	yes	complemented
126	C₂×Dic₇	C₂²×C₃×C₅	1	2	yes	complemented
127	C₄×D₇	—	2	4	no	—
128	C₂⁴×C₃	—	7	1	no	—
129	C₂²×C₃×C₄	—	7	2	no	—
130	C₃×C₂²⋊C₄	—	7	2	no	—
131	C₃×C₂²⋊C₄	—	7	1	no	—
132	C₃×C₂²⋊C₄	—	7	1	no	—
133	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
134	C₂×ΓL(1,𝔽₈)	C₂²×C₄×C₅	1	1	yes	characteristic
135	C₂×ΓL(1,𝔽₈)	C₂×C₅×D₄	1	2	yes	—
136	AGL(1,𝔽₇)	—	2	2	no	—
137	AGL(1,𝔽₇)	C₅×C₂²⋊C₄	1	4	yes	direct product factor
138	C₂³×C₅	—	7	1	no	—
139	C₂³×C₅	—	7	4	no	—
140	C₂³×C₅	—	7	1	no	—
141	C₂³×C₅	—	14	4	no	—
142	C₂³×C₅	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
143	C₂×C₄×C₅	—	7	2	no	—
144	C₂×C₄×C₅	—	14	4	no	—
145	C₂×C₄×C₅	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
146	C₅×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
147	C₂×C₂²⋊C₄	—	7	1	no	Sylow
148	C₂×C₃×C₅	—	7	4	no	—
149	C₂×C₃×C₅	—	7	1	no	—
150	C₂×C₃×C₅	—	7	2	no	—
151	C₂×C₃×C₅	—	14	2	no	—
152	C₂×C₃×C₅	—	14	2	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	C₂×C₃×C₄×C₅	1	2	yes	complemented
155	C₂²×C₇	C₂³×C₃×C₅	1	1	yes	characteristic
156	C₂×D₇	—	2	2	no	—
157	C₂×D₇	—	2	8	no	—
158	C₂×D₇	C₂×C₃×C₄×C₅	1	2	yes	—
159	C₂×D₇	C₂×C₃×C₄×C₅	1	2	yes	complemented
160	C₂×D₇	C₃×C₅×D₄	1	4	yes	—
161	C₄×C₇	—	2	2	no	—
162	Dic₇	—	2	2	no	—
163	C₂³×C₃	—	7	1	no	—
164	C₂³×C₃	—	7	4	no	—
165	C₂³×C₃	—	7	1	no	—
166	C₂³×C₃	—	7	1	no	—
167	C₂³×C₃	—	14	4	no	—
168	C₂×C₃×C₄	—	7	2	no	—
169	C₂×C₃×C₄	—	7	2	no	—
170	C₂×C₃×C₄	—	14	4	no	—
171	ΓL(1,𝔽₈)	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	7	2	no	—
174	C₂²×C₅	—	7	4	no	—
175	C₂²×C₅	—	7	2	no	—
176	C₂²×C₅	—	14	8	no	—
177	C₂²×C₅	—	14	2	no	—
178	C₂²×C₅	C₄×AGL(1,𝔽₇)	1	2	yes	complemented
179	C₂²×C₅	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	14	2	no	—
182	C₂⁴	—	7	1	no	—
183	C₂²×C₄	—	7	2	no	—
184	C₂²⋊C₄	—	7	1	no	—
185	C₂²⋊C₄	—	7	2	no	—
186	C₂²⋊C₄	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
187	C₃×C₅	—	7	1	no	Hall
188	C₂×C₇	—	2	2	no	—
189	C₂×C₇	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
190	C₂×C₇	C₂×C₃×C₅×D₄	1	2	yes	—
191	D₇	—	2	2	no	—
192	D₇	C₃×C₅×C₂²⋊C₄	1	4	yes	complemented
193	C₂²×C₃	—	7	4	no	—
194	C₂²×C₃	—	7	2	no	—
195	C₂²×C₃	—	7	2	no	—
196	C₂²×C₃	—	7	2	no	—
197	C₂²×C₃	—	7	1	no	—
198	C₂²×C₃	—	14	2	no	—
199	C₂²×C₃	—	14	8	no	—
200	C₂²×C₃	—	14	2	no	—
201	C₃×C₄	—	14	2	no	—
202	C₃×C₄	—	14	2	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	—	7	4	no	—
205	C₂×C₅	—	14	2	no	—
206	C₂×C₅	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
207	C₂×C₅	D₄×AGL(1,𝔽₇)	1	2	yes	—
208	C₂³	—	7	1	no	—
209	C₂³	—	7	4	no	—
210	C₂³	—	7	1	no	—
211	C₂³	—	14	4	no	—
212	C₂³	C₂×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
213	C₂×C₄	—	7	2	no	—
214	C₂×C₄	—	14	4	no	—
215	C₂×C₄	C₂×C₅×AGL(1,𝔽₇)	1	2	yes	complemented
216	C₇	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
217	C₂×C₃	—	7	4	no	—
218	C₂×C₃	—	7	1	no	—
219	C₂×C₃	—	7	2	no	—
220	C₂×C₃	—	14	2	no	—
221	C₂×C₃	—	14	2	no	—
222	C₅	C₂²⋊C₄×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	2	no	—
224	C₂²	—	7	2	no	—
225	C₂²	—	7	4	no	—
226	C₂²	—	7	2	no	—
227	C₂²	—	14	8	no	—
228	C₂²	—	14	2	no	—
229	C₂²	C₄×C₅×AGL(1,𝔽₇)	1	2	yes	complemented
230	C₂²	C₂²×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	14	2	no	—
233	C₃	—	7	1	no	Sylow
234	C₂	—	2	2	no	—
235	C₂	—	7	4	no	—
236	C₂	—	14	2	no	—
237	C₂	C₂×C₄×C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
238	C₂	C₅×D₄×AGL(1,𝔽₇)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂²×C₃×C₄×C₅
Frattini	C₂²	C₂²×C₅×AGL(1,𝔽₇)
Fitting	C₅×C₇×C₂²⋊C₄	C₂×C₃
socle	C₂²×C₅×C₇	C₂³×C₃
prime radical	C₂³×C₅×AGL(1,𝔽₇)	C₂
cosocle	C₂²×C₇	C₂³×C₃×C₅
hypercenter	C₅×C₂²⋊C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×AGL(1,𝔽₇)	C₂	semidirect
3. C₅×C₂²⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
4. C₅×(C₂²⋊Dic₇)⋊C₃	C₂	semidirect
5. C₅×(C₂×AGL(1,𝔽₇))⋊C₄	C₂	semidirect
6. C₅×D₇×C₂²⋊C₄	C₃	semidirect
9. C₂²×C₅×AGL(1,𝔽₇)	C₄	semidirect
12. C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂²	semidirect
13. C₂×C₅×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
15. C₂²⋊C₄×AGL(1,𝔽₇)	C₅	direct (N × H)
17. C₂×C₄×C₅×D₇	C₂×C₃	semidirect
18. C₅×C₇×C₂²⋊C₄	C₂×C₃	semidirect
19. C₅×C₂²⋊Dic₇	C₂×C₃	semidirect
20. C₅×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
23. C₂²×C₅×ΓL(1,𝔽₈)	C₂×C₄	semidirect
28. C₂×C₅×AGL(1,𝔽₇)	C₂×C₄	semidirect
33. C₂×C₄×AGL(1,𝔽₇)	C₂×C₅	semidirect
34. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂×C₅	semidirect
35. (C₂²⋊Dic₇)⋊C₃	C₂×C₅	semidirect
36. (C₂×AGL(1,𝔽₇))⋊C₄	C₂×C₅	semidirect
39. C₂²×C₅×D₇	C₃×C₄	semidirect
42. C₂×C₄×C₅×C₇	C₂²×C₃	semidirect
43. C₂×C₅×Dic₇	C₂²×C₃	semidirect
50. D₇×C₂²⋊C₄	C₃×C₅	semidirect
55. C₅×AGL(1,𝔽₇)	C₂²⋊C₄	direct (N × H)
58. C₂²×AGL(1,𝔽₇)	C₄×C₅	semidirect
61. C₂×C₄×ΓL(1,𝔽₈)	C₂²×C₅	semidirect
62. C₂×ΓL(1,𝔽₈)⋊C₄	C₂²×C₅	semidirect
66. C₂²×C₅×C₇	C₂×C₃×C₄	semidirect
71. C₂×C₅×D₇	C₂×C₃×C₄	semidirect
84. C₂×C₄×D₇	C₂×C₃×C₅	semidirect
85. C₇×C₂²⋊C₄	C₂×C₃×C₅	semidirect
86. C₂²⋊Dic₇	C₂×C₃×C₅	semidirect
87. (C₂×D₇)⋊C₄	C₂×C₃×C₅	semidirect
88. C₅×ΓL(1,𝔽₈)	C₂×C₂²⋊C₄	semidirect
91. C₂²×ΓL(1,𝔽₈)	C₂×C₄×C₅	semidirect
96. C₂×AGL(1,𝔽₇)	C₂×C₄×C₅	semidirect
104. C₅×C₂²⋊C₄	AGL(1,𝔽₇)	direct (N × H)
109. C₅×D₇	C₃×C₂²⋊C₄	semidirect
122. C₂²×D₇	C₃×C₄×C₅	semidirect
125. C₂×C₄×C₇	C₂²×C₃×C₅	semidirect
126. C₂×Dic₇	C₂²×C₃×C₅	semidirect
137. AGL(1,𝔽₇)	C₅×C₂²⋊C₄	direct (N × H)
145. C₂×C₄×C₅	C₂×AGL(1,𝔽₇)	semidirect
146. C₅×C₇	C₂×C₃×C₂²⋊C₄	semidirect
154. C₂²×C₇	C₂×C₃×C₄×C₅	semidirect
159. C₂×D₇	C₂×C₃×C₄×C₅	semidirect
171. ΓL(1,𝔽₈)	C₂×C₅×C₂²⋊C₄	semidirect
178. C₂²×C₅	C₄×AGL(1,𝔽₇)	semidirect
186. C₂²⋊C₄	C₅×AGL(1,𝔽₇)	direct (N × H)
192. D₇	C₃×C₅×C₂²⋊C₄	semidirect
215. C₂×C₄	C₂×C₅×AGL(1,𝔽₇)	semidirect
216. C₇	C₂×C₃×C₅×C₂²⋊C₄	semidirect
222. C₅	C₂²⋊C₄×AGL(1,𝔽₇)	direct (N × H)
229. C₂²	C₄×C₅×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×AGL(1,𝔽₇)	2
C₂×C₄×C₅×AGL(1,𝔽₇)	2
C₅×C₂²⋊C₄×ΓL(1,𝔽₈)	2
C₅×(C₂²⋊Dic₇)⋊C₃	2
C₅×(C₂×AGL(1,𝔽₇))⋊C₄	2
C₅×D₇×C₂²⋊C₄	3
C₂²⋊C₄×AGL(1,𝔽₇)	5
C₂×C₃×C₅×C₂²⋊C₄	7

Subgroups — C5×C22⋊C4×AGL(1,𝔽7) #72218139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₂²⋊C₄×AGL(1,𝔽₇) #72218139