Subgroups — ΓL(1,𝔽₈)×#547 #13060708

Order 1344.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
2	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#123	C₂	1	4	yes	maximal, complemented
4	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)×#121	C₂	1	2	yes	maximal, complemented
6	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
9	C₇×#547	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	#41×ΓL(1,𝔽₈)	—	2	8	no	—
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
13	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
17	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
23	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
25	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
26	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
27	C₂×C₇×#41	C₂×C₃	1	2	yes	complemented
28	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₇×#123	C₂×C₃	1	4	yes	complemented
30	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
31	C₇×#121	C₂×C₃	1	2	yes	complemented
32	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
33	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₃×#547	—	7	1	no	maximal, Hall
36	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
37	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
38	Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
39	D₄×ΓL(1,𝔽₈)	—	2	8	no	—
40	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
41	D₄×ΓL(1,𝔽₈)	D₄	1	4	yes	complemented
42	C₂×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
43	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
44	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
48	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
49	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
50	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
51	C₇×#41	—	2	8	no	—
52	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
55	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
58	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
59	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
61	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
62	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
65	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
66	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
67	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×#41	—	7	2	no	—
69	C₃×C₄⋊Q₈	—	7	1	no	—
70	C₃×#123	—	7	4	no	—
71	C₃×C₂²⋊Q₈	—	7	2	no	—
72	C₃×#121	—	7	2	no	—
73	C₃×C₄×D₄	—	7	1	no	—
74	C₃×C₄×D₄	—	7	1	no	—
75	C₂×C₃×C₄⋊C₄	—	7	2	no	—
76	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
77	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
78	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
79	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
80	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
81	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	4	yes	—
82	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
83	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
84	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
85	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
86	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
87	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
88	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	complemented
89	#547	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×C₇	C₂³×C₃	1	2	yes	—
91	C₂³×C₇	C₂³×C₃	1	2	yes	—
92	C₇×Q₈	—	2	4	no	—
93	C₇×D₄	—	2	8	no	—
94	C₇×D₄	—	2	2	no	—
95	C₇×D₄	C₃×D₄	1	4	yes	complemented
96	C₂×C₄×C₇	—	2	8	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
101	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
102	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
103	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₃×#41	—	14	8	no	—
106	C₂×C₃×Q₈	—	7	2	no	—
107	C₂×C₃×D₄	—	7	1	no	—
108	C₂×C₃×D₄	—	7	1	no	—
109	C₂×C₃×D₄	—	7	4	no	—
110	C₂²×C₃×C₄	—	7	2	no	—
111	C₂²×C₃×C₄	—	7	4	no	—
112	C₂²×C₃×C₄	—	7	2	no	—
113	C₃×C₄⋊C₄	—	7	1	no	—
114	C₃×C₄⋊C₄	—	7	2	no	—
115	C₃×C₄⋊C₄	—	7	4	no	—
116	C₃×C₄⋊C₄	—	7	1	no	—
117	C₃×C₄⋊C₄	—	7	2	no	—
118	C₃×C₂²⋊C₄	—	7	2	no	—
119	C₃×C₂²⋊C₄	—	7	2	no	—
120	C₃×C₂²⋊C₄	—	7	4	no	—
121	C₃×C₄²	—	7	1	no	—
122	C₂×ΓL(1,𝔽₈)	—	2	4	no	—
123	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
124	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
125	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
126	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×#41	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
128	C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
129	#123	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
130	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
131	#121	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
132	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
133	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
134	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	—	2	2	no	—
137	C₂²×C₇	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	C₂×C₃×D₄	1	4	yes	—
141	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	4	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	2	1	no	—
146	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
147	C₄×C₇	C₃×#41	1	2	yes	complemented
148	C₂³×C₃	—	7	2	no	—
149	C₂³×C₃	—	7	2	no	—
150	C₃×Q₈	—	14	4	no	—
151	C₃×D₄	—	7	4	no	—
152	C₃×D₄	—	14	2	no	—
153	C₃×D₄	—	14	8	no	—
154	C₂×C₃×C₄	—	7	1	no	—
155	C₂×C₃×C₄	—	7	1	no	—
156	C₂×C₃×C₄	—	7	1	no	—
157	C₂×C₃×C₄	—	7	2	no	—
158	C₂×C₃×C₄	—	7	2	no	—
159	C₂×C₃×C₄	—	7	4	no	—
160	C₂×C₃×C₄	—	14	4	no	—
161	C₂×C₃×C₄	—	14	4	no	—
162	C₂×C₃×C₄	—	14	8	no	—
163	ΓL(1,𝔽₈)	#547	1	1	yes	characteristic, direct product factor, Hall
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
166	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
167	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
168	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
169	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
170	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
171	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
172	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
173	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
174	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
175	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
176	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
177	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
178	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
179	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
180	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
181	C₂×C₇	—	2	4	no	—
182	C₂×C₇	—	4	2	no	—
183	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
185	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂²×C₃	—	7	4	no	—
187	C₂²×C₃	—	7	1	no	—
188	C₂²×C₃	—	14	2	no	—
189	C₂²×C₃	—	14	2	no	—
190	C₂²×C₃	—	14	2	no	—
191	C₂²×C₃	—	14	2	no	—
192	C₂²×C₃	—	14	2	no	—
193	C₃×C₄	—	7	2	no	—
194	C₃×C₄	—	7	2	no	—
195	C₃×C₄	—	14	2	no	—
196	C₃×C₄	—	14	4	no	—
197	C₃×C₄	—	14	2	no	—
198	C₃×C₄	—	14	1	no	—
199	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
200	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	2	no	—
204	D₄	D₄×ΓL(1,𝔽₈)	1	4	yes	complemented
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
209	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
210	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
211	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
212	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
213	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
214	C₇	C₃×#547	1	1	yes	characteristic, complemented, Sylow
215	C₂×C₃	—	7	1	no	—
216	C₂×C₃	—	7	1	no	—
217	C₂×C₃	—	7	1	no	—
218	C₂×C₃	—	14	4	no	—
219	C₂×C₃	—	28	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	4	yes	—
226	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
232	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	complemented
233	C₃	—	7	1	no	Sylow
234	C₂	—	2	4	no	—
235	C₂	—	4	2	no	—
236	C₂	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
237	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
238	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×#547	C₃
hypercenter	#547	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#547
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #547

Complements

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

N	H ≅ G/N	type
1. C₂×#41×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	semidirect
3. ΓL(1,𝔽₈)×#123	C₂	semidirect
4. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
5. ΓL(1,𝔽₈)×#121	C₂	semidirect
6. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
7. C₄×D₄×ΓL(1,𝔽₈)	C₂	semidirect
8. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
9. C₇×#547	C₃	semidirect
27. C₂×C₇×#41	C₂×C₃	semidirect
28. C₇×C₄⋊Q₈	C₂×C₃	semidirect
29. C₇×#123	C₂×C₃	semidirect
30. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
31. C₇×#121	C₂×C₃	semidirect
32. C₄×C₇×D₄	C₂×C₃	semidirect
33. C₄×C₇×D₄	C₂×C₃	semidirect
34. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
41. D₄×ΓL(1,𝔽₈)	D₄	semidirect
88. C₄×ΓL(1,𝔽₈)	#41	semidirect
89. #547	ΓL(1,𝔽₈)	direct (N × H)
95. C₇×D₄	C₃×D₄	semidirect
127. C₂×#41	C₂×ΓL(1,𝔽₈)	semidirect
128. C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
129. #123	C₂×ΓL(1,𝔽₈)	semidirect
130. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
131. #121	C₂×ΓL(1,𝔽₈)	semidirect
132. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
133. C₄×D₄	C₂×ΓL(1,𝔽₈)	semidirect
134. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
147. C₄×C₇	C₃×#41	semidirect
163. ΓL(1,𝔽₈)	#547	direct (N × H)
204. D₄	D₄×ΓL(1,𝔽₈)	semidirect
214. C₇	C₃×#547	semidirect
232. C₄	#41×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×#41×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×C₄⋊Q₈	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×#547	3
C₃×#547	7

Subgroups — ΓL(1,𝔽8)×#547 #13060708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#547 #13060708