Subgroups — ΓL(1,𝔽₈)×#536 #13060697

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 212 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	1	yes	maximal, characteristic
2	C₂²×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#124	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₈)×#123	C₂	1	4	yes	maximal, complemented
6	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	ΓL(1,𝔽₈)×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#536	C₃	1	1	yes	maximal, characteristic, complemented, Hall
9	#41×ΓL(1,𝔽₈)	—	2	4	no	—
10	C₂×Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
15	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
17	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
18	C₂²×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	8	yes	—
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	8	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₇×C₄⋊Q₈	C₂×C₃	1	2	yes	complemented
27	C₇×#124	C₂×C₃	1	2	yes	complemented
28	C₇×#123	C₂×C₃	1	4	yes	complemented
29	C₇×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
30	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#536	—	7	1	no	maximal, Hall
32	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
33	C₂³×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
34	Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
35	Q₈×ΓL(1,𝔽₈)	—	2	4	no	—
36	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
37	D₄×ΓL(1,𝔽₈)	—	2	4	no	—
38	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
39	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
40	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
41	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	4	yes	—
42	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
43	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
44	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
47	C₇×#41	—	2	4	no	—
48	C₂×C₇×Q₈	—	2	4	no	—
49	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
50	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
51	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
53	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
55	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
56	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
57	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₇×C₄⋊C₄	C₂²×C₃	1	8	yes	—
59	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
60	C₇×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
61	C₄²×C₇	C₂²×C₃	1	2	yes	—
62	C₂×C₃×#41	—	7	1	no	—
63	C₂²×C₃×Q₈	—	7	1	no	—
64	C₃×C₄⋊Q₈	—	7	2	no	—
65	C₃×#124	—	7	2	no	—
66	C₃×#123	—	7	4	no	—
67	C₃×C₂²⋊Q₈	—	7	4	no	—
68	C₃×#117	—	7	1	no	—
69	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
70	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
71	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
72	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
73	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
74	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
75	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
76	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
77	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
78	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
79	#536	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
80	C₂³×C₇	C₂³×C₃	1	2	yes	—
81	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
82	C₇×Q₈	—	2	4	no	—
83	C₇×Q₈	—	2	4	no	—
84	C₇×Q₈	—	2	2	no	—
85	C₇×D₄	—	2	4	no	—
86	C₇×D₄	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	C₃×D₄	1	4	yes	—
90	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
91	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
92	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
93	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
95	C₃×#41	—	14	4	no	—
96	C₂×C₃×Q₈	—	7	2	no	—
97	C₂×C₃×Q₈	—	7	2	no	—
98	C₂×C₃×Q₈	—	7	1	no	—
99	C₂×C₃×Q₈	—	14	4	no	—
100	C₂×C₃×D₄	—	7	2	no	—
101	C₂×C₃×D₄	—	7	1	no	—
102	C₂²×C₃×C₄	—	7	1	no	—
103	C₂²×C₃×C₄	—	7	2	no	—
104	C₂²×C₃×C₄	—	7	2	no	—
105	C₃×C₄⋊C₄	—	7	2	no	—
106	C₃×C₄⋊C₄	—	7	8	no	—
107	C₃×C₂²⋊C₄	—	7	2	no	—
108	C₃×C₂²⋊C₄	—	7	8	no	—
109	C₃×C₄²	—	7	2	no	—
110	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
111	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
112	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
113	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	2	yes	—
114	C₂×#41	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
115	C₂²×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
116	C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
117	#124	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
118	#123	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
119	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
120	#117	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	2	4	no	—
124	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
125	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
126	C₄×C₇	—	2	2	no	—
127	C₄×C₇	—	2	4	no	—
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
130	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
131	C₂³×C₃	—	7	1	no	—
132	C₂³×C₃	—	7	2	no	—
133	C₃×Q₈	—	14	4	no	—
134	C₃×Q₈	—	14	4	no	—
135	C₃×Q₈	—	14	2	no	—
136	C₃×D₄	—	14	2	no	—
137	C₃×D₄	—	14	4	no	—
138	C₂×C₃×C₄	—	7	4	no	—
139	C₂×C₃×C₄	—	7	1	no	—
140	C₂×C₃×C₄	—	7	1	no	—
141	C₂×C₃×C₄	—	7	4	no	—
142	C₂×C₃×C₄	—	7	4	no	—
143	C₂×C₃×C₄	—	7	2	no	—
144	C₂×C₃×C₄	—	14	4	no	—
145	C₂×C₃×C₄	—	14	4	no	—
146	ΓL(1,𝔽₈)	#536	1	1	yes	characteristic, direct product factor, Hall
147	#41	—	2	4	no	—
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
150	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
151	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
152	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
153	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
154	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
155	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
156	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
157	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
158	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
159	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
160	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
161	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	—	4	2	no	—
164	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
165	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
166	C₂²×C₃	—	7	2	no	—
167	C₂²×C₃	—	7	1	no	—
168	C₂²×C₃	—	14	2	no	—
169	C₂²×C₃	—	14	2	no	—
170	C₂²×C₃	—	14	4	no	—
171	C₃×C₄	—	7	2	no	—
172	C₃×C₄	—	7	2	no	—
173	C₃×C₄	—	14	2	no	—
174	C₃×C₄	—	14	4	no	—
175	C₃×C₄	—	14	4	no	—
176	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
177	C₂³	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
178	Q₈	—	2	4	no	—
179	Q₈	—	2	4	no	—
180	Q₈	—	2	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	4	yes	—
186	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
187	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
188	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
189	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
190	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
191	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
192	C₂×C₃	—	7	1	no	—
193	C₂×C₃	—	7	2	no	—
194	C₂×C₃	—	14	2	no	—
195	C₂×C₃	—	28	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
200	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	4	no	—
203	C₄	—	2	4	no	—
204	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
205	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
206	C₃	—	7	1	no	Sylow
207	C₂	—	2	2	no	—
208	C₂	—	4	2	no	—
209	C₂	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
210	C₂	ΓL(1,𝔽₈)×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#536	C₃
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
hypercenter	#536	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#536
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #536

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
3. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	semidirect
4. ΓL(1,𝔽₈)×#124	C₂	semidirect
5. ΓL(1,𝔽₈)×#123	C₂	semidirect
6. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
7. ΓL(1,𝔽₈)×#117	C₂	semidirect
8. C₇×#536	C₃	semidirect
25. C₂²×C₇×Q₈	C₂×C₃	semidirect
26. C₇×C₄⋊Q₈	C₂×C₃	semidirect
27. C₇×#124	C₂×C₃	semidirect
28. C₇×#123	C₂×C₃	semidirect
29. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
30. C₇×#117	C₂×C₃	semidirect
79. #536	ΓL(1,𝔽₈)	direct (N × H)
115. C₂²×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
116. C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
117. #124	C₂×ΓL(1,𝔽₈)	semidirect
118. #123	C₂×ΓL(1,𝔽₈)	semidirect
119. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
120. #117	C₂×ΓL(1,𝔽₈)	semidirect
146. ΓL(1,𝔽₈)	#536	direct (N × H)
191. C₇	C₃×#536	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
C₂²×Q₈×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×C₄⋊Q₈	2
ΓL(1,𝔽₈)×#124	2
ΓL(1,𝔽₈)×#123	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
ΓL(1,𝔽₈)×#117	2
C₇×#536	3
C₃×#536	7

Subgroups — ΓL(1,𝔽8)×#536 #13060697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#536 #13060697