Subgroups — AGL(1,𝔽₇)×#498 #72229556

Order 2688.

848 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×AGL(1,𝔽₇)	C₂	1	2	yes	maximal
2	ΓL(1,𝔽₈)×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	#89851⋊C₃	C₂	1	2	yes	maximal, complemented
4	#106×AGL(1,𝔽₇)	C₂	1	4	yes	maximal
5	#89627⋊C₃	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
7	#89452⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	D₇×#498	C₃	1	1	yes	maximal, characteristic, complemented, Hall
9	C₂×Q₈×AGL(1,𝔽₇)	—	2	2	no	—
10	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂²	1	2	yes	—
11	#6507⋊C₃	—	2	2	no	—
12	C₂×C₈×AGL(1,𝔽₇)	C₂²	1	2	yes	—
13	C₄⋊C₄×AGL(1,𝔽₇)	—	2	2	no	—
14	C₄⋊C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
15	#6451⋊C₃	—	2	2	no	—
16	#6451⋊C₃	C₂²	1	4	yes	—
17	C₄²×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	#6432⋊C₃	C₂²	1	2	yes	—
19	ΓL(1,𝔽₈)×#106	C₂²	1	4	yes	—
20	C₄×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
21	#6391⋊C₃	C₂²	1	4	yes	—
22	C₈×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	—
23	#6382⋊C₃	C₂²	1	8	yes	—
24	C₄×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
25	D₇×C₄⋊Q₈	C₂×C₃	1	2	yes	—
26	C₇×#498	C₂×C₃	1	1	yes	characteristic, complemented
27	#89851	C₂×C₃	1	2	yes	complemented
28	D₇×#106	C₂×C₃	1	4	yes	—
29	#89627	C₂×C₃	1	4	yes	complemented
30	C₄×C₈×D₇	C₂×C₃	1	1	yes	characteristic
31	#89452	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#498	—	7	1	no	maximal, Hall
33	C₂×Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
34	Q₈×AGL(1,𝔽₇)	—	4	8	no	—
35	C₂×C₄×AGL(1,𝔽₇)	—	2	4	no	—
36	C₂×C₄×AGL(1,𝔽₇)	D₄	1	2	yes	—
37	C₂×C₄×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
38	C₂×ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
39	C₂×ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
40	C₂×C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₄⋊C₄×ΓL(1,𝔽₈)	—	2	2	no	—
42	C₄⋊C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
43	C₄²×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	—	2	2	no	—
45	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂³	1	4	yes	—
46	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	—	2	4	no	—
47	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	—	2	4	no	—
48	C₄×ΓL(1,𝔽₈)⋊C₄	D₄	1	2	yes	—
49	C₄×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
50	C₂×ΓL(1,𝔽₈)⋊C₈	C₂³	1	2	yes	—
51	C₈×AGL(1,𝔽₇)	Q₈	1	8	yes	complemented
52	C₂×Q₈×D₇	—	2	2	no	—
53	C₇×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	#6507	—	2	2	no	—
55	C₂×C₈×D₇	C₂²×C₃	1	2	yes	—
56	D₇×C₄⋊C₄	—	2	2	no	—
57	D₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
58	#6451	—	2	2	no	—
59	#6451	C₂²×C₃	1	4	yes	—
60	C₄²×D₇	C₂²×C₃	1	1	yes	characteristic
61	#6432	C₂²×C₃	1	2	yes	—
62	C₇×#106	C₂²×C₃	1	4	yes	—
63	C₄×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
64	#6391	C₂²×C₃	1	4	yes	—
65	C₈×Dic₇	C₂²×C₃	1	2	yes	—
66	#6382	C₂²×C₃	1	8	yes	—
67	C₄×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₄⋊Q₈	—	7	2	no	—
69	C₃×#498	—	7	2	no	—
70	C₃×#498	—	7	4	no	—
71	C₃×#498	—	7	1	no	—
72	C₃×#498	—	7	1	no	—
73	C₂×C₃×#106	—	7	4	no	—
74	C₂×C₃×C₄×C₈	—	7	1	no	—
75	C₂²×AGL(1,𝔽₇)	C₂×D₄	1	1	yes	characteristic
76	Q₈×ΓL(1,𝔽₈)	—	4	4	no	—
77	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
78	C₂×C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
79	C₂×C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
80	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
81	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
83	C₄×AGL(1,𝔽₇)	—	4	8	no	—
84	C₄×AGL(1,𝔽₇)	SD₁₆	1	8	yes	—
85	C₄×AGL(1,𝔽₇)	C₂×Q₈	1	4	yes	—
86	ΓL(1,𝔽₈)⋊Q₈	—	4	4	no	—
87	ΓL(1,𝔽₈)⋊Q₈	—	4	8	no	—
88	C₈×ΓL(1,𝔽₈)	C₂×Q₈	1	4	yes	complemented
89	ΓL(1,𝔽₈)⋊C₈	C₂×Q₈	1	4	yes	complemented
90	C₂×#498	—	7	1	no	Sylow
91	C₂×C₇×Q₈	—	2	2	no	—
92	Q₈×D₇	—	4	8	no	—
93	C₂×C₄×D₇	—	2	4	no	—
94	C₂×C₄×D₇	C₃×D₄	1	2	yes	—
95	C₂×C₄×D₇	C₂³×C₃	1	1	yes	characteristic
96	C₂×Dic₁₄	—	2	2	no	—
97	C₂×Dic₁₄	—	2	4	no	—
98	C₂×C₇×C₈	C₂³×C₃	1	2	yes	—
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	C₂³×C₃	1	4	yes	—
101	C₄²×C₇	C₂³×C₃	1	1	yes	characteristic
102	C₄⋊Dic₇	—	2	2	no	—
103	C₄⋊Dic₇	C₂³×C₃	1	4	yes	—
104	Dic₇⋊C₄	—	2	4	no	—
105	Dic₇⋊C₄	—	2	4	no	—
106	C₄×Dic₇	C₃×D₄	1	2	yes	—
107	C₄×Dic₇	C₂³×C₃	1	1	yes	characteristic
108	C₂×C₇⋊C₈	C₂³×C₃	1	2	yes	—
109	C₈×D₇	C₃×Q₈	1	8	yes	complemented
110	C₂²×C₃×Q₈	—	14	2	no	—
111	C₂²×C₃×C₈	—	7	2	no	—
112	C₃×C₄⋊Q₈	—	7	2	no	—
113	C₃×C₄⋊Q₈	—	7	4	no	—
114	C₃×C₄⋊Q₈	—	7	2	no	—
115	C₃×C₄⋊Q₈	—	14	2	no	—
116	C₃×C₄⋊Q₈	—	14	2	no	—
117	C₂×C₃×C₄⋊C₄	—	7	4	no	—
118	C₂×C₃×C₄⋊C₄	—	14	2	no	—
119	C₂×C₃×C₄²	—	7	1	no	—
120	C₃×#106	—	7	8	no	—
121	C₃×#106	—	7	4	no	—
122	C₃×#106	—	7	4	no	—
123	C₃×C₄×C₈	—	7	1	no	—
124	C₃×C₄×C₈	—	7	2	no	—
125	C₃×C₄×C₈	—	7	1	no	—
126	C₂²×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
127	C₂×AGL(1,𝔽₇)	C₄⋊Q₈	1	2	yes	—
128	C₂×AGL(1,𝔽₇)	C₂×SD₁₆	1	4	yes	—
129	C₄×ΓL(1,𝔽₈)	—	4	4	no	—
130	C₄×ΓL(1,𝔽₈)	C₂×SD₁₆	1	4	yes	—
131	C₄×ΓL(1,𝔽₈)	C₂²×Q₈	1	2	yes	—
132	ΓL(1,𝔽₈)⋊C₄	—	4	4	no	—
133	ΓL(1,𝔽₈)⋊C₄	C₄⋊Q₈	1	2	yes	—
134	ΓL(1,𝔽₈)⋊C₄	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄⋊Q₈	—	7	2	no	—
136	#498	—	7	2	no	—
137	#498	—	7	4	no	—
138	#498	—	7	1	no	—
139	#498	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
140	C₂×#106	—	7	4	no	—
141	C₂×C₄×C₈	—	7	1	no	—
142	C₂²×D₇	C₂×C₃×D₄	1	1	yes	characteristic
143	C₇×Q₈	—	4	4	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	C₂×C₃×D₄	1	2	yes	—
146	C₂×C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
147	C₂×Dic₇	—	2	4	no	—
148	C₂×Dic₇	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂×Dic₇	C₂×C₃×D₄	1	2	yes	—
150	C₄×D₇	—	4	8	no	—
151	C₄×D₇	C₃×SD₁₆	1	8	yes	—
152	C₄×D₇	C₂×C₃×Q₈	1	4	yes	—
153	Dic₁₄	—	4	4	no	—
154	Dic₁₄	—	4	8	no	—
155	C₇×C₈	C₂×C₃×Q₈	1	4	yes	complemented
156	C₇⋊C₈	C₂×C₃×Q₈	1	4	yes	complemented
157	C₂×C₃×Q₈	—	14	2	no	—
158	C₂×C₃×Q₈	—	14	4	no	—
159	C₂×C₃×Q₈	—	14	2	no	—
160	C₂×C₃×Q₈	—	28	8	no	—
161	C₂²×C₃×C₄	—	7	2	no	—
162	C₂²×C₃×C₄	—	7	1	no	—
163	C₂²×C₃×C₄	—	14	4	no	—
164	C₂×C₃×C₈	—	7	2	no	—
165	C₂×C₃×C₈	—	7	8	no	—
166	C₂×C₃×C₈	—	7	2	no	—
167	C₃×C₄⋊C₄	—	7	4	no	—
168	C₃×C₄⋊C₄	—	7	4	no	—
169	C₃×C₄⋊C₄	—	14	2	no	—
170	C₃×C₄⋊C₄	—	14	4	no	—
171	C₃×C₄⋊C₄	—	14	4	no	—
172	C₃×C₄⋊C₄	—	14	2	no	—
173	C₃×C₄²	—	7	1	no	—
174	C₃×C₄²	—	7	2	no	—
175	C₃×C₄²	—	7	1	no	—
176	C₂×ΓL(1,𝔽₈)	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₂×ΓL(1,𝔽₈)	C₂²×SD₁₆	1	2	yes	—
178	AGL(1,𝔽₇)	#498	1	4	yes	direct product factor
179	C₂²×Q₈	—	14	2	no	—
180	C₂²×C₈	—	7	2	no	—
181	C₄⋊Q₈	—	7	4	no	—
182	C₄⋊Q₈	—	7	2	no	—
183	C₄⋊Q₈	—	14	2	no	—
184	C₄⋊Q₈	—	14	2	no	—
185	C₄⋊Q₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
186	C₂×C₄⋊C₄	—	7	4	no	—
187	C₂×C₄⋊C₄	—	14	2	no	—
188	C₂×C₄²	—	7	1	no	—
189	#106	—	7	4	no	—
190	#106	—	7	8	no	—
191	#106	C₂×AGL(1,𝔽₇)	1	4	yes	—
192	C₄×C₈	—	7	2	no	—
193	C₄×C₈	—	7	1	no	—
194	C₄×C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
195	C₂²×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
196	C₂×D₇	C₃×C₄⋊Q₈	1	2	yes	—
197	C₂×D₇	C₂×C₃×SD₁₆	1	4	yes	—
198	C₄×C₇	—	4	4	no	—
199	C₄×C₇	C₂×C₃×SD₁₆	1	4	yes	—
200	C₄×C₇	C₂²×C₃×Q₈	1	2	yes	—
201	Dic₇	—	4	4	no	—
202	Dic₇	C₃×C₄⋊Q₈	1	2	yes	—
203	Dic₇	C₂×C₃×SD₁₆	1	4	yes	—
204	C₂³×C₃	—	7	1	no	—
205	C₃×Q₈	—	28	4	no	—
206	C₃×Q₈	—	28	8	no	—
207	C₃×Q₈	—	28	4	no	—
208	C₂×C₃×C₄	—	7	8	no	—
209	C₂×C₃×C₄	—	7	4	no	—
210	C₂×C₃×C₄	—	7	1	no	—
211	C₂×C₃×C₄	—	7	2	no	—
212	C₂×C₃×C₄	—	7	2	no	—
213	C₂×C₃×C₄	—	7	1	no	—
214	C₂×C₃×C₄	—	14	4	no	—
215	C₂×C₃×C₄	—	14	4	no	—
216	C₂×C₃×C₄	—	28	8	no	—
217	C₃×C₈	—	7	4	no	—
218	C₃×C₈	—	7	4	no	—
219	ΓL(1,𝔽₈)	C₂×#498	1	1	yes	characteristic, complemented, Hall
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	14	4	no	—
222	C₂×Q₈	—	14	2	no	—
223	C₂×Q₈	—	28	8	no	—
224	C₂²×C₄	—	7	2	no	—
225	C₂²×C₄	—	7	1	no	—
226	C₂²×C₄	—	14	4	no	—
227	C₂×C₈	—	7	2	no	—
228	C₂×C₈	—	7	8	no	—
229	C₂×C₈	C₂²×AGL(1,𝔽₇)	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	7	4	no	—
232	C₄⋊C₄	—	14	4	no	—
233	C₄⋊C₄	—	14	4	no	—
234	C₄⋊C₄	—	14	2	no	—
235	C₄⋊C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
236	C₄²	—	7	2	no	—
237	C₄²	—	7	1	no	—
238	C₄²	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
239	C₂×C₇	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
240	C₂×C₇	C₂²×C₃×SD₁₆	1	2	yes	—
241	D₇	C₃×#498	1	4	yes	complemented
242	C₂²×C₃	—	7	4	no	—
243	C₂²×C₃	—	7	2	no	—
244	C₂²×C₃	—	7	1	no	—
245	C₃×C₄	—	7	4	no	—
246	C₃×C₄	—	7	2	no	—
247	C₃×C₄	—	7	2	no	—
248	C₃×C₄	—	7	4	no	—
249	C₃×C₄	—	28	4	no	—
250	C₃×C₄	—	28	4	no	—
251	C₂³	—	7	1	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	28	8	no	—
254	Q₈	—	28	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	7	1	no	—
257	C₂×C₄	—	7	2	no	—
258	C₂×C₄	—	7	4	no	—
259	C₂×C₄	—	7	8	no	—
260	C₂×C₄	—	14	4	no	—
261	C₂×C₄	—	28	8	no	—
262	C₂×C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
263	C₂×C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
264	C₈	—	7	4	no	—
265	C₈	Q₈×AGL(1,𝔽₇)	1	4	yes	complemented
266	C₇	C₂×C₃×#498	1	1	yes	characteristic, complemented, Sylow
267	C₂×C₃	—	7	4	no	—
268	C₂×C₃	—	7	1	no	—
269	C₂×C₃	—	7	2	no	—
270	C₂²	—	7	2	no	—
271	C₂²	—	7	4	no	—
272	C₂²	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	7	2	no	—
275	C₄	—	7	4	no	—
276	C₄	—	28	4	no	—
277	C₄	SD₁₆×AGL(1,𝔽₇)	1	4	yes	—
278	C₄	C₂×Q₈×AGL(1,𝔽₇)	1	2	yes	—
279	C₃	—	7	1	no	Sylow
280	C₂	—	7	4	no	—
281	C₂	C₄⋊Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
282	C₂	C₂×SD₁₆×AGL(1,𝔽₇)	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#498	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×D₄
prime radical	C₂²×AGL(1,𝔽₇)	C₂×D₄
hypercenter	#498	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×#498
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×#498	C₂	semidirect
3. #89851⋊C₃	C₂	semidirect
5. #89627⋊C₃	C₂	semidirect
7. #89452⋊C₃	C₂	semidirect
8. D₇×#498	C₃	semidirect
26. C₇×#498	C₂×C₃	semidirect
27. #89851	C₂×C₃	semidirect
29. #89627	C₂×C₃	semidirect
31. #89452	C₂×C₃	semidirect
51. C₈×AGL(1,𝔽₇)	Q₈	semidirect
88. C₈×ΓL(1,𝔽₈)	C₂×Q₈	semidirect
89. ΓL(1,𝔽₈)⋊C₈	C₂×Q₈	semidirect
109. C₈×D₇	C₃×Q₈	semidirect
139. #498	AGL(1,𝔽₇)	direct (N × H)
155. C₇×C₈	C₂×C₃×Q₈	semidirect
156. C₇⋊C₈	C₂×C₃×Q₈	semidirect
178. AGL(1,𝔽₇)	#498	direct (N × H)
219. ΓL(1,𝔽₈)	C₂×#498	semidirect
241. D₇	C₃×#498	semidirect
265. C₈	Q₈×AGL(1,𝔽₇)	semidirect
266. C₇	C₂×C₃×#498	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×#498	2
#89851⋊C₃	2
#106×AGL(1,𝔽₇)	2
#89627⋊C₃	2
C₄×C₈×AGL(1,𝔽₇)	2
#89452⋊C₃	2
D₇×#498	3
C₂×C₃×#498	7

Subgroups — AGL(1,𝔽7)×#498 #72229556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₇)×#498 #72229556