Subgroups — ΓL(1,𝔽₈)×#539 #13060700

Order 1344.

436 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#126	C₂	1	2	yes	maximal, complemented
2	ΓL(1,𝔽₈)×#125	C₂	1	2	yes	maximal, complemented
3	ΓL(1,𝔽₈)×#123	C₂	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#123	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	ΓL(1,𝔽₈)×#121	C₂	1	1	yes	maximal, characteristic
8	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#539	C₃	1	1	yes	maximal, characteristic, complemented, Hall
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
23	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
26	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
27	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
28	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
29	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
30	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
31	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
32	C₇×#126	C₂×C₃	1	2	yes	complemented
33	C₇×#125	C₂×C₃	1	2	yes	complemented
34	C₇×#123	C₂×C₃	1	2	yes	complemented
35	C₇×#123	C₂×C₃	1	2	yes	complemented
36	C₇×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
38	C₇×#121	C₂×C₃	1	1	yes	characteristic
39	C₄×C₇×D₄	C₂×C₃	1	2	yes	complemented
40	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#539	—	7	1	no	maximal, Hall
42	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
43	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
44	Q₈×ΓL(1,𝔽₈)	—	4	1	no	—
45	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
46	D₄×ΓL(1,𝔽₈)	—	4	1	no	—
47	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
48	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
49	C₂×C₄×ΓL(1,𝔽₈)	—	4	2	no	—
50	C₂×C₄×ΓL(1,𝔽₈)	—	4	1	no	—
51	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
52	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
53	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
54	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
55	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
56	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
57	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
60	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
65	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
74	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
75	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₄²×C₇	C₂²×C₃	1	2	yes	—
79	C₃×#126	—	7	2	no	—
80	C₃×#125	—	7	2	no	—
81	C₃×#123	—	7	2	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×C₂²⋊Q₈	—	7	1	no	—
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₃×#121	—	7	1	no	—
86	C₃×C₄×D₄	—	7	2	no	—
87	C₂×C₃×C₄⋊C₄	—	7	1	no	—
88	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
89	C₂²×ΓL(1,𝔽₈)	—	2	1	no	—
90	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
91	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
92	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
93	C₂²×ΓL(1,𝔽₈)	#41	1	2	yes	—
94	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
95	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
96	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
97	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
98	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
99	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
100	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
101	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
102	#539	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₇	C₂³×C₃	1	2	yes	—
104	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₇×Q₈	—	4	1	no	—
106	C₇×D₄	—	2	4	no	—
107	C₇×D₄	—	4	1	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	4	1	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	—	4	1	no	—
112	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
113	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
114	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
116	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
117	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×D₄	—	7	2	no	—
121	C₂×C₃×D₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	—	7	1	no	—
124	C₂²×C₃×C₄	—	7	1	no	—
125	C₂²×C₃×C₄	—	7	2	no	—
126	C₃×C₄⋊C₄	—	7	2	no	—
127	C₃×C₄⋊C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	1	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	1	no	—
133	C₃×C₄⋊C₄	—	7	2	no	—
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	2	no	—
136	C₃×C₂²⋊C₄	—	7	2	no	—
137	C₃×C₂²⋊C₄	—	7	2	no	—
138	C₃×C₂²⋊C₄	—	7	2	no	—
139	C₃×C₄²	—	7	2	no	—
140	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
141	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
142	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
143	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
145	#126	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
146	#125	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
147	#123	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
148	#123	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
149	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
150	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
151	#121	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
152	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
153	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	C₃×#41	1	2	yes	—
160	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	2	2	no	—
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	2	2	no	—
168	C₂³×C₃	—	7	1	no	—
169	C₂³×C₃	—	7	2	no	—
170	C₃×Q₈	—	28	1	no	—
171	C₃×D₄	—	14	4	no	—
172	C₃×D₄	—	28	1	no	—
173	C₂×C₃×C₄	—	7	2	no	—
174	C₂×C₃×C₄	—	7	2	no	—
175	C₂×C₃×C₄	—	7	2	no	—
176	C₂×C₃×C₄	—	7	2	no	—
177	C₂×C₃×C₄	—	7	2	no	—
178	C₂×C₃×C₄	—	7	1	no	—
179	C₂×C₃×C₄	—	7	1	no	—
180	C₂×C₃×C₄	—	14	2	no	—
181	C₂×C₃×C₄	—	28	1	no	—
182	C₂×C₃×C₄	—	28	2	no	—
183	C₂×C₃×C₄	—	28	1	no	—
184	ΓL(1,𝔽₈)	#539	1	1	yes	characteristic, direct product factor, Hall
185	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
186	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
187	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
188	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
189	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
190	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
191	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
192	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
193	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
194	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
195	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
196	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
197	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
198	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
199	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
200	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
201	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
202	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
203	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
204	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
205	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
206	C₂×C₇	—	2	2	no	—
207	C₂×C₇	—	4	2	no	—
208	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
209	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂²×C₃	—	7	2	no	—
212	C₂²×C₃	—	7	1	no	—
213	C₂²×C₃	—	14	1	no	—
214	C₂²×C₃	—	14	1	no	—
215	C₂²×C₃	—	14	2	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₃×C₄	—	14	2	no	—
219	C₃×C₄	—	14	2	no	—
220	C₃×C₄	—	14	1	no	—
221	C₃×C₄	—	14	2	no	—
222	C₃×C₄	—	14	1	no	—
223	C₃×C₄	—	14	2	no	—
224	C₃×C₄	—	14	2	no	—
225	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
226	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
227	Q₈	—	4	1	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
235	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
236	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
237	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
238	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
239	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
240	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
241	C₇	C₃×#539	1	1	yes	characteristic, complemented, Sylow
242	C₂×C₃	—	7	1	no	—
243	C₂×C₃	—	7	1	no	—
244	C₂×C₃	—	7	1	no	—
245	C₂×C₃	—	14	2	no	—
246	C₂×C₃	—	28	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#41×ΓL(1,𝔽₈)	1	2	yes	—
253	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₃	—	7	1	no	Sylow
262	C₂	—	2	2	no	—
263	C₂	—	4	2	no	—
264	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
265	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂	ΓL(1,𝔽₈)×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#539	C₃
prime radical	ΓL(1,𝔽₈)×#121	C₂
hypercenter	#539	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#539
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #539

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#126	C₂	semidirect
2. ΓL(1,𝔽₈)×#125	C₂	semidirect
3. ΓL(1,𝔽₈)×#123	C₂	semidirect
4. ΓL(1,𝔽₈)×#123	C₂	semidirect
5. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
6. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
8. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
9. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
10. C₇×#539	C₃	semidirect
32. C₇×#126	C₂×C₃	semidirect
33. C₇×#125	C₂×C₃	semidirect
34. C₇×#123	C₂×C₃	semidirect
35. C₇×#123	C₂×C₃	semidirect
36. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
37. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
39. C₄×C₇×D₄	C₂×C₃	semidirect
40. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
102. #539	ΓL(1,𝔽₈)	direct (N × H)
145. #126	C₂×ΓL(1,𝔽₈)	semidirect
146. #125	C₂×ΓL(1,𝔽₈)	semidirect
147. #123	C₂×ΓL(1,𝔽₈)	semidirect
148. #123	C₂×ΓL(1,𝔽₈)	semidirect
149. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
150. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
152. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
153. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
184. ΓL(1,𝔽₈)	#539	direct (N × H)
241. C₇	C₃×#539	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×#126	2
ΓL(1,𝔽₈)×#125	2
ΓL(1,𝔽₈)×#123	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
ΓL(1,𝔽₈)×#121	2
C₄×D₄×ΓL(1,𝔽₈)	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₇×#539	3
C₃×#539	7

Subgroups — ΓL(1,𝔽8)×#539 #13060700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#539 #13060700