Subgroups — #124×AGL(1,𝔽₇) #13057408

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
3	ΓL(1,𝔽₈)×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#6510⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#6499⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#6448⋊C₃	C₂	1	4	yes	maximal, complemented
7	C₂²⋊C₄×AGL(1,𝔽₇)	C₂	1	4	yes	maximal, complemented
8	#6438⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
10	D₇×#124	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂³×AGL(1,𝔽₇)	C₂²	1	2	yes	—
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×(C₂²⋊D₇)⋊C₃	C₂²	1	2	yes	—
15	Q₈×AGL(1,𝔽₇)	—	2	4	no	—
16	D₄×AGL(1,𝔽₇)	—	2	4	no	—
17	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
18	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
19	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	2	yes	—
20	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	complemented
23	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented
25	(C₂²⋊Dic₇)⋊C₃	C₂²	1	4	yes	complemented
26	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	complemented
27	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	complemented
28	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	2	yes	complemented
29	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×D₄×D₇	C₂×C₃	1	1	yes	characteristic
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6510	C₂×C₃	1	1	yes	characteristic, complemented
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	#6448	C₂×C₃	1	4	yes	complemented
36	D₇×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
37	#6438	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×C₃×#124	—	7	1	no	maximal, Hall
40	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
42	C₂²×AGL(1,𝔽₇)	—	2	8	no	—
43	C₂²×AGL(1,𝔽₇)	C₂³	1	2	yes	—
44	C₂²×AGL(1,𝔽₇)	C₂³	1	1	yes	characteristic
45	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
46	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
48	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	(C₂²⋊D₇)⋊C₃	—	2	4	no	—
51	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
52	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	2	yes	—
53	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	1	yes	characteristic
54	C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
55	C₄×AGL(1,𝔽₇)	—	2	4	no	—
56	C₄×AGL(1,𝔽₇)	—	2	4	no	—
57	C₄×AGL(1,𝔽₇)	D₄	1	4	yes	—
58	ΓL(1,𝔽₈)⋊Q₈	—	2	2	no	—
59	ΓL(1,𝔽₈)⋊Q₈	—	2	4	no	—
60	C₂³×D₇	C₂²×C₃	1	2	yes	—
61	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
62	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
64	Q₈×D₇	—	2	4	no	—
65	D₄×D₇	—	2	4	no	—
66	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
69	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×Dic₁₄	C₂²×C₃	1	2	yes	complemented
72	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
73	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	complemented
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
76	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	complemented
77	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₂²×C₃×Q₈	—	7	1	no	—
80	C₂²×C₃×D₄	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	—	7	4	no	—
83	C₃×#124	—	7	1	no	—
84	C₃×#124	—	7	1	no	—
85	C₃×#124	—	7	1	no	—
86	C₂×C₃×C₂²⋊C₄	—	7	4	no	—
87	C₂×C₃×C₄²	—	7	1	no	—
88	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
89	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
90	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
91	C₂×AGL(1,𝔽₇)	—	2	2	no	—
92	C₂×AGL(1,𝔽₇)	—	2	4	no	—
93	C₂×AGL(1,𝔽₇)	—	4	8	no	—
94	C₂×AGL(1,𝔽₇)	C₂×D₄	1	2	yes	—
95	C₂×AGL(1,𝔽₇)	#41	1	4	yes	—
96	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
97	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
98	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
99	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
100	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
101	ΓL(1,𝔽₈)⋊C₄	C₂×D₄	1	2	yes	—
102	C₂×#124	—	7	1	no	Sylow
103	C₂³×C₇	C₂³×C₃	1	2	yes	—
104	C₂²×D₇	—	2	4	no	—
105	C₂²×D₇	—	2	8	no	—
106	C₂²×D₇	C₂³×C₃	1	2	yes	—
107	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
108	C₇×Q₈	—	2	2	no	—
109	C₇×D₄	—	2	2	no	—
110	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
111	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
112	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
113	C₂²⋊D₇	—	2	4	no	—
114	C₂×Dic₇	C₂³×C₃	1	2	yes	—
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
117	D₂₈	—	2	2	no	—
118	C₄×D₇	—	2	4	no	—
119	C₄×D₇	—	2	4	no	—
120	C₄×D₇	C₃×D₄	1	4	yes	—
121	Dic₁₄	—	2	2	no	—
122	Dic₁₄	—	2	4	no	—
123	C₂⁴×C₃	—	7	2	no	—
124	C₂×C₃×Q₈	—	7	2	no	—
125	C₂×C₃×Q₈	—	7	1	no	—
126	C₂×C₃×Q₈	—	7	1	no	—
127	C₂×C₃×Q₈	—	14	4	no	—
128	C₂×C₃×D₄	—	7	1	no	—
129	C₂×C₃×D₄	—	7	2	no	—
130	C₂×C₃×D₄	—	7	1	no	—
131	C₂×C₃×D₄	—	14	4	no	—
132	C₂²×C₃×C₄	—	7	1	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₂²×C₃×C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	4	no	—
136	C₃×C₂²⋊C₄	—	7	4	no	—
137	C₃×C₂²⋊C₄	—	7	4	no	—
138	C₃×C₂²⋊C₄	—	7	4	no	—
139	C₃×C₄²	—	7	2	no	—
140	C₃×C₄²	—	7	1	no	—
141	C₃×C₄²	—	7	1	no	—
142	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
143	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
144	C₂×ΓL(1,𝔽₈)	C₂×#41	1	2	yes	—
145	AGL(1,𝔽₇)	—	4	2	no	—
146	AGL(1,𝔽₇)	#124	1	4	yes	direct product factor
147	C₂²×Q₈	—	7	1	no	—
148	C₂²×D₄	—	7	1	no	—
149	#124	—	7	1	no	—
150	#124	—	7	1	no	—
151	#124	—	7	4	no	—
152	#124	—	7	1	no	—
153	#124	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
154	C₂×C₂²⋊C₄	—	7	4	no	—
155	C₂×C₄²	—	7	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	4	no	—
158	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	4	no	—
161	C₂×D₇	—	4	8	no	—
162	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
163	C₂×D₇	C₃×#41	1	4	yes	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
167	Dic₇	—	2	2	no	—
168	Dic₇	—	2	2	no	—
169	Dic₇	C₂×C₃×D₄	1	2	yes	—
170	C₂³×C₃	—	7	1	no	—
171	C₂³×C₃	—	7	2	no	—
172	C₂³×C₃	—	7	2	no	—
173	C₂³×C₃	—	14	4	no	—
174	C₂³×C₃	—	14	8	no	—
175	C₃×Q₈	—	14	2	no	—
176	C₃×Q₈	—	14	4	no	—
177	C₃×Q₈	—	14	2	no	—
178	C₃×D₄	—	14	2	no	—
179	C₃×D₄	—	14	2	no	—
180	C₃×D₄	—	14	4	no	—
181	C₂×C₃×C₄	—	7	4	no	—
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	—	7	2	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	—	7	1	no	—
186	C₂×C₃×C₄	—	7	2	no	—
187	C₂×C₃×C₄	—	7	2	no	—
188	C₂×C₃×C₄	—	14	4	no	—
189	C₂×C₃×C₄	—	14	4	no	—
190	ΓL(1,𝔽₈)	C₂×#124	1	1	yes	characteristic, complemented, Hall
191	C₂⁴	—	7	2	no	—
192	C₂×Q₈	—	7	2	no	—
193	C₂×Q₈	—	7	1	no	—
194	C₂×Q₈	—	14	4	no	—
195	C₂×Q₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
196	C₂×D₄	—	7	2	no	—
197	C₂×D₄	—	7	1	no	—
198	C₂×D₄	—	14	4	no	—
199	C₂×D₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
200	C₂²×C₄	—	7	1	no	—
201	C₂²×C₄	—	7	2	no	—
202	C₂²×C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	4	no	—
204	C₂²⋊C₄	—	7	4	no	—
205	C₂²⋊C₄	—	7	4	no	—
206	C₂²⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	complemented
207	C₄²	—	7	2	no	—
208	C₄²	—	7	1	no	—
209	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
210	C₂×C₇	—	4	2	no	—
211	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₂×C₇	C₂×C₃×#41	1	2	yes	—
213	D₇	—	4	2	no	—
214	D₇	C₃×#124	1	4	yes	complemented
215	C₂²×C₃	—	7	2	no	—
216	C₂²×C₃	—	7	4	no	—
217	C₂²×C₃	—	7	1	no	—
218	C₂²×C₃	—	14	2	no	—
219	C₂²×C₃	—	14	4	no	—
220	C₂²×C₃	—	14	2	no	—
221	C₂²×C₃	—	14	4	no	—
222	C₂²×C₃	—	28	8	no	—
223	C₃×C₄	—	7	2	no	—
224	C₃×C₄	—	7	2	no	—
225	C₃×C₄	—	14	2	no	—
226	C₃×C₄	—	14	2	no	—
227	C₃×C₄	—	14	2	no	—
228	C₃×C₄	—	14	2	no	—
229	C₂³	—	7	1	no	—
230	C₂³	—	7	2	no	—
231	C₂³	—	14	4	no	—
232	C₂³	—	14	8	no	—
233	C₂³	C₂²×AGL(1,𝔽₇)	1	2	yes	—
234	Q₈	—	2	2	no	—
235	Q₈	—	14	4	no	—
236	Q₈	—	14	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	14	4	no	—
239	D₄	—	14	2	no	—
240	C₂×C₄	—	7	1	no	—
241	C₂×C₄	—	7	4	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	—	7	2	no	—
244	C₂×C₄	—	14	4	no	—
245	C₂×C₄	—	14	4	no	—
246	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
247	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
248	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
249	C₇	C₂×C₃×#124	1	1	yes	characteristic, complemented, Sylow
250	C₂×C₃	—	7	4	no	—
251	C₂×C₃	—	7	1	no	—
252	C₂×C₃	—	7	2	no	—
253	C₂×C₃	—	28	2	no	—
254	C₂×C₃	—	28	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	7	2	no	—
258	C₂²	—	7	4	no	—
259	C₂²	—	14	2	no	—
260	C₂²	—	14	4	no	—
261	C₂²	—	28	8	no	—
262	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	7	2	no	—
266	C₄	—	14	2	no	—
267	C₄	—	14	2	no	—
268	C₄	D₄×AGL(1,𝔽₇)	1	2	yes	—
269	C₃	—	7	1	no	Sylow
270	C₂	—	4	2	no	—
271	C₂	—	7	4	no	—
272	C₂	—	28	2	no	—
273	C₂	C₂×D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
274	C₂	#41×AGL(1,𝔽₇)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#124	C₂×C₃
prime radical	C₂×D₄×AGL(1,𝔽₇)	C₂
hypercenter	#124	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×#124
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×AGL(1,𝔽₇)	C₂	semidirect
3. ΓL(1,𝔽₈)×#124	C₂	semidirect
4. #6510⋊C₃	C₂	semidirect
5. #6499⋊C₃	C₂	semidirect
6. #6448⋊C₃	C₂	semidirect
7. C₂²⋊C₄×AGL(1,𝔽₇)	C₂	semidirect
8. #6438⋊C₃	C₂	semidirect
9. C₄²×AGL(1,𝔽₇)	C₂	semidirect
10. D₇×#124	C₃	semidirect
12. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
21. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
22. C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	semidirect
23. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
24. C₄²×ΓL(1,𝔽₈)	C₂²	semidirect
25. (C₂²⋊Dic₇)⋊C₃	C₂²	semidirect
26. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
27. (C₂×AGL(1,𝔽₇))⋊C₄	C₂²	semidirect
28. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
29. C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	semidirect
30. C₂×Q₈×D₇	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6510	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. #6448	C₂×C₃	semidirect
36. D₇×C₂²⋊C₄	C₂×C₃	semidirect
37. #6438	C₂×C₃	semidirect
38. C₄²×D₇	C₂×C₃	semidirect
61. C₂×C₇×Q₈	C₂²×C₃	semidirect
70. C₂×Dic₁₄	C₂²×C₃	semidirect
71. C₂×Dic₁₄	C₂²×C₃	semidirect
72. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
73. C₄²×C₇	C₂²×C₃	semidirect
74. C₂²⋊Dic₇	C₂²×C₃	semidirect
75. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
76. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
77. C₄×Dic₇	C₂²×C₃	semidirect
78. C₄×Dic₇	C₂²×C₃	semidirect
146. AGL(1,𝔽₇)	#124	direct (N × H)
153. #124	AGL(1,𝔽₇)	direct (N × H)
190. ΓL(1,𝔽₈)	C₂×#124	semidirect
195. C₂×Q₈	C₂×AGL(1,𝔽₇)	semidirect
206. C₂²⋊C₄	C₂×AGL(1,𝔽₇)	semidirect
209. C₄²	C₂×AGL(1,𝔽₇)	semidirect
214. D₇	C₃×#124	semidirect
249. C₇	C₂×C₃×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,𝔽₇)	2
C₂×D₄×AGL(1,𝔽₇)	2
ΓL(1,𝔽₈)×#124	2
#6510⋊C₃	2
#6499⋊C₃	2
#6448⋊C₃	2
C₂²⋊C₄×AGL(1,𝔽₇)	2
#6438⋊C₃	2
C₄²×AGL(1,𝔽₇)	2
D₇×#124	3
C₂×C₃×#124	7

Subgroups — #124×AGL(1,𝔽7) #13057408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #124×AGL(1,𝔽₇) #13057408