Subgroups — ΓL(1,𝔽₈)×#435 #13054966

Order 1344.

248 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
4	ΓL(1,𝔽₈)×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	ΓL(1,𝔽₈)×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	ΓL(1,𝔽₈)×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#435	C₃	1	1	yes	maximal, characteristic, complemented, Hall
9	C₂×D₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
10	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	M₁₆×ΓL(1,𝔽₈)	—	2	2	no	—
13	C₂×C₈×ΓL(1,𝔽₈)	—	2	2	no	—
14	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
19	C₂²⋊C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
21	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
24	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
25	C₇×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₇×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₇×#97	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#435	—	7	1	no	maximal, Hall
30	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
31	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
32	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
33	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
34	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
36	C₂×C₄×ΓL(1,𝔽₈)	—	2	1	no	—
37	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
42	C₈×ΓL(1,𝔽₈)	—	2	1	no	—
43	C₈×ΓL(1,𝔽₈)	—	2	1	no	—
44	C₈×ΓL(1,𝔽₈)	—	2	1	no	—
45	C₈×ΓL(1,𝔽₈)	D₄	1	2	yes	complemented
46	C₂×C₇×D₄	C₃×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
49	C₇×M₁₆	—	2	2	no	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₇×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
56	C₇×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
57	C₇×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
58	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×M₁₆	—	7	1	no	—
60	C₂²×C₃×C₈	—	7	1	no	—
61	C₃×C₄×D₄	—	7	1	no	—
62	C₃×C₄⋊C₈	—	7	1	no	—
63	C₃×C₂²⋊C₈	—	7	1	no	—
64	C₃×C₂²⋊C₈	—	7	1	no	—
65	C₃×#97	—	7	1	no	—
66	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
67	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
68	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
69	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
70	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
71	C₂²×ΓL(1,𝔽₈)	M₁₆	1	2	yes	complemented
72	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
73	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
74	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
75	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
76	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
77	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
78	C₄×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic
79	#435	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
80	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
82	C₇×D₄	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₇×C₈	—	2	1	no	—
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	2	1	no	—
95	C₇×C₈	C₃×D₄	1	2	yes	complemented
96	C₂×C₃×D₄	—	7	1	no	—
97	C₂²×C₃×C₄	—	7	1	no	—
98	C₂²×C₃×C₄	—	7	1	no	—
99	C₃×M₁₆	—	14	2	no	—
100	C₂×C₃×C₈	—	7	1	no	—
101	C₂×C₃×C₈	—	7	1	no	—
102	C₂×C₃×C₈	—	7	1	no	—
103	C₂×C₃×C₈	—	7	1	no	—
104	C₂×C₃×C₈	—	14	2	no	—
105	C₃×C₄⋊C₄	—	7	1	no	—
106	C₃×C₂²⋊C₄	—	7	1	no	—
107	C₃×C₂²⋊C₄	—	7	1	no	—
108	C₃×C₄²	—	7	1	no	—
109	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
110	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
111	C₂×ΓL(1,𝔽₈)	C₄×D₄	1	1	yes	characteristic
112	C₂×ΓL(1,𝔽₈)	C₂×M₁₆	1	1	yes	characteristic
113	C₂×ΓL(1,𝔽₈)	#131	1	1	yes	characteristic
114	C₂×M₁₆	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
115	C₂²×C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
116	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
117	C₄⋊C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
118	C₂²⋊C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
119	C₂²⋊C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
120	#97	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	1	no	—
124	C₂²×C₇	—	2	1	no	—
125	C₂²×C₇	—	2	1	no	—
126	C₂²×C₇	C₃×M₁₆	1	2	yes	complemented
127	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	—	2	1	no	—
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
133	C₄×C₇	C₃×#41	1	1	yes	characteristic
134	C₂³×C₃	—	7	1	no	—
135	C₂³×C₃	—	7	1	no	—
136	C₃×D₄	—	14	2	no	—
137	C₂×C₃×C₄	—	7	1	no	—
138	C₂×C₃×C₄	—	7	1	no	—
139	C₂×C₃×C₄	—	7	1	no	—
140	C₂×C₃×C₄	—	7	1	no	—
141	C₂×C₃×C₄	—	7	1	no	—
142	C₂×C₃×C₄	—	14	1	no	—
143	C₂×C₃×C₄	—	14	1	no	—
144	C₂×C₃×C₄	—	14	1	no	—
145	C₂×C₃×C₄	—	14	1	no	—
146	C₃×C₈	—	7	2	no	—
147	C₃×C₈	—	14	1	no	—
148	C₃×C₈	—	14	1	no	—
149	C₃×C₈	—	14	1	no	—
150	ΓL(1,𝔽₈)	#435	1	1	yes	characteristic, direct product factor, Hall
151	C₂×D₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
152	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
153	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
157	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
158	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
159	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
160	C₄⋊C₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
161	C₂²⋊C₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
162	C₂²⋊C₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
163	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	—	4	1	no	—
166	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
167	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
168	C₂×C₇	C₃×#131	1	1	yes	characteristic
169	C₂²×C₃	—	7	2	no	—
170	C₂²×C₃	—	7	1	no	—
171	C₂²×C₃	—	14	1	no	—
172	C₂²×C₃	—	14	1	no	—
173	C₂²×C₃	—	14	1	no	—
174	C₂²×C₃	—	14	1	no	—
175	C₂²×C₃	—	14	1	no	—
176	C₃×C₄	—	7	1	no	—
177	C₃×C₄	—	7	1	no	—
178	C₃×C₄	—	14	1	no	—
179	C₃×C₄	—	14	1	no	—
180	C₃×C₄	—	14	1	no	—
181	C₃×C₄	—	14	1	no	—
182	C₂³	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
183	C₂³	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
184	D₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
193	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	D₄×ΓL(1,𝔽₈)	1	2	yes	complemented
198	C₇	C₃×#435	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₃	—	7	1	no	—
200	C₂×C₃	—	7	1	no	—
201	C₂×C₃	—	7	1	no	—
202	C₂×C₃	—	14	2	no	—
203	C₂×C₃	—	28	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	M₁₆×ΓL(1,𝔽₈)	1	2	yes	complemented
210	C₂²	C₂²×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
216	C₄	#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
217	C₃	—	7	1	no	Sylow
218	C₂	—	2	2	no	—
219	C₂	—	4	1	no	—
220	C₂	C₄×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
221	C₂	C₂×M₁₆×ΓL(1,𝔽₈)	1	1	yes	characteristic
222	C₂	ΓL(1,𝔽₈)×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
commutator, socle	C₂²×C₇	C₂²×C₃×C₄
Fitting	C₇×#435	C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#435	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#435
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂²×C₈×ΓL(1,𝔽₈)	C₂	semidirect
4. ΓL(1,𝔽₈)×C₄⋊C₈	C₂	semidirect
5. ΓL(1,𝔽₈)×C₂²⋊C₈	C₂	semidirect
6. ΓL(1,𝔽₈)×C₂²⋊C₈	C₂	semidirect
7. ΓL(1,𝔽₈)×#97	C₂	semidirect
8. C₇×#435	C₃	semidirect
22. C₂×C₇×M₁₆	C₂×C₃	semidirect
23. C₂²×C₇×C₈	C₂×C₃	semidirect
25. C₇×C₄⋊C₈	C₂×C₃	semidirect
26. C₇×C₂²⋊C₈	C₂×C₃	semidirect
27. C₇×C₂²⋊C₈	C₂×C₃	semidirect
28. C₇×#97	C₂×C₃	semidirect
45. C₈×ΓL(1,𝔽₈)	D₄	semidirect
71. C₂²×ΓL(1,𝔽₈)	M₁₆	semidirect
79. #435	ΓL(1,𝔽₈)	direct (N × H)
95. C₇×C₈	C₃×D₄	semidirect
114. C₂×M₁₆	C₂×ΓL(1,𝔽₈)	semidirect
115. C₂²×C₈	C₂×ΓL(1,𝔽₈)	semidirect
117. C₄⋊C₈	C₂×ΓL(1,𝔽₈)	semidirect
118. C₂²⋊C₈	C₂×ΓL(1,𝔽₈)	semidirect
119. C₂²⋊C₈	C₂×ΓL(1,𝔽₈)	semidirect
120. #97	C₂×ΓL(1,𝔽₈)	semidirect
126. C₂²×C₇	C₃×M₁₆	semidirect
150. ΓL(1,𝔽₈)	#435	direct (N × H)
197. C₈	D₄×ΓL(1,𝔽₈)	semidirect
198. C₇	C₃×#435	semidirect
209. C₂²	M₁₆×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×ΓL(1,𝔽₈)	2
C₂²×C₈×ΓL(1,𝔽₈)	2
C₄×D₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×C₄⋊C₈	2
ΓL(1,𝔽₈)×C₂²⋊C₈	2
ΓL(1,𝔽₈)×#97	2
C₇×#435	3
C₃×#435	7

Subgroups — ΓL(1,𝔽8)×#435 #13054966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#435 #13054966