Subgroups — C₂×M₁₆×AGL(1,𝔽₇) #13057477

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 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	1	yes	maximal, characteristic
2	C₂×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1136⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	M₁₆×AGL(1,𝔽₇)	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1131⋊C₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, complemented
7	C₂×D₇×M₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂³×AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
9	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₂²×ΓL(1,𝔽₈)⋊C₄	C₄	1	1	yes	characteristic
11	C₂×C₄×AGL(1,𝔽₇)	C₄	1	4	yes	—
12	C₂×C₄×AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
13	C₂×C₄×AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
14	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
15	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
16	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
17	M₁₆×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
18	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
19	#1136⋊C₃	C₂²	1	4	yes	complemented
20	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	2	yes	complemented
21	#1131⋊C₃	C₂²	1	8	yes	complemented
22	C₈×AGL(1,𝔽₇)	C₂²	1	8	yes	complemented
23	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
24	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#1136	C₂×C₃	1	1	yes	characteristic, complemented
26	D₇×M₁₆	C₂×C₃	1	8	yes	complemented
27	C₂×#1131	C₂×C₃	1	2	yes	complemented
28	C₂×C₈×D₇	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃×M₁₆	—	7	1	no	maximal, Hall
30	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
31	C₂²×AGL(1,𝔽₇)	—	2	4	no	—
32	C₂²×AGL(1,𝔽₇)	C₂×C₄	1	1	yes	characteristic
33	C₂²×AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
34	C₂²×AGL(1,𝔽₇)	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
36	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
38	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
39	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×ΓL(1,𝔽₈)⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₄×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
43	C₄×AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
44	C₄×AGL(1,𝔽₇)	C₂×C₄	1	2	yes	—
45	C₄×AGL(1,𝔽₇)	C₂×C₄	1	4	yes	—
46	C₄×AGL(1,𝔽₇)	C₂³	1	2	yes	—
47	C₄×AGL(1,𝔽₇)	C₂³	1	2	yes	—
48	C₈×ΓL(1,𝔽₈)	C₂³	1	4	yes	complemented
49	ΓL(1,𝔽₈)⋊C₈	C₂³	1	4	yes	complemented
50	C₂³×D₇	C₃×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
52	C₂²×Dic₇	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×D₇	C₃×C₄	1	4	yes	—
54	C₂×C₄×D₇	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×D₇	C₃×C₄	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
59	C₇×M₁₆	C₂²×C₃	1	4	yes	complemented
60	C₂×C₇×C₈	C₂²×C₃	1	2	yes	complemented
61	#1136	C₂²×C₃	1	4	yes	complemented
62	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	complemented
63	#1131	C₂²×C₃	1	8	yes	complemented
64	C₈×D₇	C₂²×C₃	1	8	yes	complemented
65	C₂³×C₃×C₄	—	7	1	no	—
66	C₂×C₃×M₁₆	—	7	2	no	—
67	C₂×C₃×M₁₆	—	7	8	no	—
68	C₂×C₃×M₁₆	—	7	1	no	—
69	C₂×C₃×M₁₆	—	7	1	no	—
70	C₂²×C₃×C₈	—	7	2	no	—
71	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
72	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
73	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
74	C₂×AGL(1,𝔽₇)	—	2	2	no	—
75	C₂×AGL(1,𝔽₇)	—	2	8	no	—
76	C₂×AGL(1,𝔽₇)	M₁₆	1	4	yes	direct product factor
77	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	2	yes	—
78	C₂×AGL(1,𝔽₇)	C₂²×C₄	1	2	yes	—
79	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
80	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	2	yes	—
81	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
82	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	2	yes	—
83	ΓL(1,𝔽₈)⋊C₄	C₂²×C₄	1	2	yes	—
84	C₂²×M₁₆	—	7	1	no	Sylow
85	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²×D₇	—	2	4	no	—
87	C₂²×D₇	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×D₇	C₂×C₃×C₄	1	4	yes	—
89	C₂²×D₇	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
93	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
94	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
95	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
97	C₄×D₇	C₂×C₃×C₄	1	2	yes	—
98	C₄×D₇	C₂×C₃×C₄	1	4	yes	—
99	C₄×D₇	C₂×C₃×C₄	1	2	yes	—
100	C₄×D₇	C₂×C₃×C₄	1	4	yes	—
101	C₄×D₇	C₂³×C₃	1	2	yes	—
102	C₄×D₇	C₂³×C₃	1	2	yes	—
103	C₇×C₈	C₂³×C₃	1	4	yes	complemented
104	C₇⋊C₈	C₂³×C₃	1	4	yes	complemented
105	C₂⁴×C₃	—	7	1	no	—
106	C₂²×C₃×C₄	—	7	1	no	—
107	C₂²×C₃×C₄	—	7	4	no	—
108	C₂²×C₃×C₄	—	7	1	no	—
109	C₂²×C₃×C₄	—	7	1	no	—
110	C₂²×C₃×C₄	—	7	4	no	—
111	C₂²×C₃×C₄	—	7	1	no	—
112	C₂²×C₃×C₄	—	7	1	no	—
113	C₂²×C₃×C₄	—	7	1	no	—
114	C₃×M₁₆	—	7	8	no	—
115	C₃×M₁₆	—	7	4	no	—
116	C₃×M₁₆	—	7	4	no	—
117	C₂×C₃×C₈	—	7	8	no	—
118	C₂×C₃×C₈	—	7	2	no	—
119	C₂×C₃×C₈	—	7	2	no	—
120	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
121	C₂×ΓL(1,𝔽₈)	C₂×M₁₆	1	2	yes	complemented
122	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
123	AGL(1,𝔽₇)	—	2	2	no	—
124	AGL(1,𝔽₇)	C₂×M₁₆	1	4	yes	direct product factor
125	C₂³×C₄	—	7	1	no	—
126	C₂×M₁₆	—	7	1	no	—
127	C₂×M₁₆	—	7	8	no	—
128	C₂×M₁₆	—	7	2	no	—
129	C₂×M₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
130	C₂²×C₈	—	7	2	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
133	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×D₇	—	2	2	no	—
135	C₂×D₇	—	2	8	no	—
136	C₂×D₇	C₃×M₁₆	1	4	yes	complemented
137	C₂×D₇	C₂²×C₃×C₄	1	2	yes	—
138	C₂×D₇	C₂²×C₃×C₄	1	2	yes	—
139	C₄×C₇	C₂²×C₃×C₄	1	2	yes	—
140	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
142	Dic₇	C₂²×C₃×C₄	1	2	yes	—
143	Dic₇	C₂²×C₃×C₄	1	2	yes	—
144	C₂³×C₃	—	7	1	no	—
145	C₂³×C₃	—	7	4	no	—
146	C₂³×C₃	—	7	1	no	—
147	C₂³×C₃	—	7	1	no	—
148	C₂³×C₃	—	14	4	no	—
149	C₂×C₃×C₄	—	7	2	no	—
150	C₂×C₃×C₄	—	7	4	no	—
151	C₂×C₃×C₄	—	7	4	no	—
152	C₂×C₃×C₄	—	7	2	no	—
153	C₂×C₃×C₄	—	7	2	no	—
154	C₂×C₃×C₄	—	7	2	no	—
155	C₂×C₃×C₄	—	7	1	no	—
156	C₂×C₃×C₄	—	7	1	no	—
157	C₂×C₃×C₄	—	7	4	no	—
158	C₂×C₃×C₄	—	7	1	no	—
159	C₂×C₃×C₄	—	7	2	no	—
160	C₂×C₃×C₄	—	7	2	no	—
161	C₂×C₃×C₄	—	7	1	no	—
162	C₃×C₈	—	7	4	no	—
163	C₃×C₈	—	7	4	no	—
164	ΓL(1,𝔽₈)	C₂²×M₁₆	1	1	yes	characteristic, complemented, Hall
165	C₂⁴	—	7	1	no	—
166	C₂²×C₄	—	7	1	no	—
167	C₂²×C₄	—	7	1	no	—
168	C₂²×C₄	—	7	4	no	—
169	C₂²×C₄	—	7	1	no	—
170	C₂²×C₄	—	7	4	no	—
171	C₂²×C₄	—	7	1	no	—
172	C₂²×C₄	—	7	1	no	—
173	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
174	M₁₆	—	7	4	no	—
175	M₁₆	—	7	8	no	—
176	M₁₆	C₂×AGL(1,𝔽₇)	1	4	yes	direct product factor
177	C₂×C₈	—	7	2	no	—
178	C₂×C₈	—	7	8	no	—
179	C₂×C₈	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	C₂×C₃×M₁₆	1	2	yes	complemented
182	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
183	D₇	—	2	2	no	—
184	D₇	C₂×C₃×M₁₆	1	4	yes	complemented
185	C₂²×C₃	—	7	2	no	—
186	C₂²×C₃	—	7	4	no	—
187	C₂²×C₃	—	7	2	no	—
188	C₂²×C₃	—	7	1	no	—
189	C₂²×C₃	—	7	2	no	—
190	C₂²×C₃	—	14	2	no	—
191	C₂²×C₃	—	14	8	no	—
192	C₂²×C₃	—	14	2	no	—
193	C₃×C₄	—	7	2	no	—
194	C₃×C₄	—	7	2	no	—
195	C₃×C₄	—	7	2	no	—
196	C₃×C₄	—	7	1	no	—
197	C₃×C₄	—	7	1	no	—
198	C₂³	—	7	1	no	—
199	C₂³	—	7	4	no	—
200	C₂³	—	7	1	no	—
201	C₂³	—	14	4	no	—
202	C₂³	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
203	C₂×C₄	—	7	4	no	—
204	C₂×C₄	—	7	1	no	—
205	C₂×C₄	—	7	1	no	—
206	C₂×C₄	—	7	4	no	—
207	C₂×C₄	—	7	4	no	—
208	C₂×C₄	—	7	2	no	—
209	C₂×C₄	—	7	2	no	—
210	C₂×C₄	—	7	2	no	—
211	C₂×C₄	—	7	2	no	—
212	C₂×C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
213	C₂×C₄	C₄×AGL(1,𝔽₇)	1	2	yes	—
214	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
215	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
216	C₈	—	7	4	no	—
217	C₈	C₂²×AGL(1,𝔽₇)	1	4	yes	complemented
218	C₇	C₂²×C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
219	C₂×C₃	—	7	4	no	—
220	C₂×C₃	—	7	1	no	—
221	C₂×C₃	—	7	2	no	—
222	C₂×C₃	—	14	2	no	—
223	C₂×C₃	—	14	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	7	2	no	—
226	C₂²	—	7	4	no	—
227	C₂²	—	7	2	no	—
228	C₂²	—	14	8	no	—
229	C₂²	—	14	2	no	—
230	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
231	C₂²	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
232	C₄	—	7	2	no	—
233	C₄	—	7	2	no	—
234	C₄	C₂×C₄×AGL(1,𝔽₇)	1	2	yes	—
235	C₄	C₂×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
236	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
237	C₃	—	7	1	no	Sylow
238	C₂	—	2	2	no	—
239	C₂	—	7	4	no	—
240	C₂	—	14	2	no	—
241	C₂	M₁₆×AGL(1,𝔽₇)	1	2	yes	direct product factor
242	C₂	C₂²×C₄×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₂×C₇×M₁₆	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×C₄
prime radical	C₂³×AGL(1,𝔽₇)	C₄
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	C₂×M₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₂²×C₃×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×#1136⋊C₃	C₂	semidirect
4. M₁₆×AGL(1,𝔽₇)	C₂	direct (N × H)
5. C₂×#1131⋊C₃	C₂	semidirect
6. C₂×C₈×AGL(1,𝔽₇)	C₂	semidirect
7. C₂×D₇×M₁₆	C₃	semidirect
17. M₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
18. C₂×C₈×ΓL(1,𝔽₈)	C₂²	semidirect
19. #1136⋊C₃	C₂²	semidirect
20. C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	semidirect
21. #1131⋊C₃	C₂²	semidirect
22. C₈×AGL(1,𝔽₇)	C₂²	semidirect
24. C₂×C₇×M₁₆	C₂×C₃	semidirect
25. C₂×#1136	C₂×C₃	semidirect
26. D₇×M₁₆	C₂×C₃	semidirect
27. C₂×#1131	C₂×C₃	semidirect
28. C₂×C₈×D₇	C₂×C₃	semidirect
48. C₈×ΓL(1,𝔽₈)	C₂³	semidirect
49. ΓL(1,𝔽₈)⋊C₈	C₂³	semidirect
59. C₇×M₁₆	C₂²×C₃	semidirect
60. C₂×C₇×C₈	C₂²×C₃	semidirect
61. #1136	C₂²×C₃	semidirect
62. C₂×C₇⋊C₈	C₂²×C₃	semidirect
63. #1131	C₂²×C₃	semidirect
64. C₈×D₇	C₂²×C₃	semidirect
76. C₂×AGL(1,𝔽₇)	M₁₆	direct (N × H)
103. C₇×C₈	C₂³×C₃	semidirect
104. C₇⋊C₈	C₂³×C₃	semidirect
121. C₂×ΓL(1,𝔽₈)	C₂×M₁₆	semidirect
124. AGL(1,𝔽₇)	C₂×M₁₆	direct (N × H)
129. C₂×M₁₆	AGL(1,𝔽₇)	direct (N × H)
136. C₂×D₇	C₃×M₁₆	semidirect
164. ΓL(1,𝔽₈)	C₂²×M₁₆	semidirect
176. M₁₆	C₂×AGL(1,𝔽₇)	direct (N × H)
179. C₂×C₈	C₂×AGL(1,𝔽₇)	semidirect
181. C₂×C₇	C₂×C₃×M₁₆	semidirect
184. D₇	C₂×C₃×M₁₆	semidirect
217. C₈	C₂²×AGL(1,𝔽₇)	semidirect
218. C₇	C₂²×C₃×M₁₆	semidirect
241. C₂	M₁₆×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×M₁₆×ΓL(1,𝔽₈)	2
C₂×#1136⋊C₃	2
M₁₆×AGL(1,𝔽₇)	2
C₂×#1131⋊C₃	2
C₂×C₈×AGL(1,𝔽₇)	2
C₂×D₇×M₁₆	3
C₂²×C₃×M₁₆	7

Subgroups — C2×M16×AGL(1,𝔽7) #13057477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×AGL(1,𝔽₇) #13057477