Subgroups — ΓL(1,𝔽₈)×#556 #13060717

Order 1344.

424 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	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×#126	C₂	1	4	yes	maximal, complemented
3	ΓL(1,𝔽₈)×#124	C₂	1	2	yes	maximal
4	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
6	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
8	ΓL(1,𝔽₈)×#117	C₂	1	2	yes	maximal, complemented
9	C₇×#556	C₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
13	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
14	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
16	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
18	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
19	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
20	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
21	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
23	C₄²×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
25	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
26	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₇×#126	C₂×C₃	1	4	yes	complemented
28	C₇×#124	C₂×C₃	1	2	yes	—
29	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
30	C₄×C₇×Q₈	C₂×C₃	1	2	yes	complemented
31	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
33	C₇×#117	C₂×C₃	1	2	yes	complemented
34	C₃×#556	—	7	1	no	maximal, Hall
35	C₂³×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
36	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
37	Q₈×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)	C₂³	1	2	yes	—
41	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
42	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
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	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
49	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
51	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
54	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
58	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
59	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
60	C₄²×C₇	C₂²×C₃	1	4	yes	—
61	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
62	C₄²×C₇	C₂²×C₃	1	2	yes	—
63	C₃×C₄⋊Q₈	—	7	1	no	—
64	C₃×#126	—	7	4	no	—
65	C₃×#124	—	7	2	no	—
66	C₃×C₂²⋊Q₈	—	7	2	no	—
67	C₃×C₄×Q₈	—	7	2	no	—
68	C₃×C₄×Q₈	—	7	1	no	—
69	C₃×C₄×D₄	—	7	1	no	—
70	C₃×#117	—	7	2	no	—
71	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
72	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
73	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
74	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
75	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
76	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
77	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
78	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
79	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
80	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
81	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
82	#556	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
83	C₂³×C₇	C₂³×C₃	1	2	yes	—
84	C₇×Q₈	—	2	2	no	—
85	C₇×Q₈	—	2	4	no	—
86	C₇×D₄	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
89	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
90	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
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	C₂×C₃×Q₈	—	7	1	no	—
96	C₂×C₃×Q₈	—	7	2	no	—
97	C₂×C₃×D₄	—	7	1	no	—
98	C₂²×C₃×C₄	—	7	2	no	—
99	C₃×C₄⋊C₄	—	7	1	no	—
100	C₃×C₄⋊C₄	—	7	2	no	—
101	C₃×C₄⋊C₄	—	7	1	no	—
102	C₃×C₄⋊C₄	—	7	4	no	—
103	C₃×C₄⋊C₄	—	7	2	no	—
104	C₃×C₄⋊C₄	—	7	2	no	—
105	C₃×C₂²⋊C₄	—	7	4	no	—
106	C₃×C₂²⋊C₄	—	7	2	no	—
107	C₃×C₂²⋊C₄	—	7	4	no	—
108	C₃×C₄²	—	7	4	no	—
109	C₃×C₄²	—	7	1	no	—
110	C₃×C₄²	—	7	2	no	—
111	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
112	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
113	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
114	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
116	#126	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
117	#124	C₂×ΓL(1,𝔽₈)	1	2	yes	—
118	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
119	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
120	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
121	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
122	#117	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
127	C₄×C₇	—	2	2	no	—
128	C₄×C₇	—	2	2	no	—
129	C₄×C₇	—	2	4	no	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	C₃×#41	1	2	yes	—
133	C₄×C₇	C₃×#41	1	2	yes	—
134	C₂³×C₃	—	7	2	no	—
135	C₃×Q₈	—	14	2	no	—
136	C₃×Q₈	—	14	4	no	—
137	C₃×D₄	—	14	2	no	—
138	C₂×C₃×C₄	—	7	1	no	—
139	C₂×C₃×C₄	—	7	1	no	—
140	C₂×C₃×C₄	—	7	2	no	—
141	C₂×C₃×C₄	—	7	2	no	—
142	C₂×C₃×C₄	—	7	1	no	—
143	C₂×C₃×C₄	—	7	4	no	—
144	C₂×C₃×C₄	—	7	2	no	—
145	C₂×C₃×C₄	—	14	4	no	—
146	ΓL(1,𝔽₈)	#556	1	1	yes	characteristic, direct product factor, Hall
147	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
148	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
149	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
150	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
151	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
152	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
153	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
154	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
155	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
156	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
157	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
158	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
159	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
160	C₄²	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
161	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
162	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
163	C₂×C₇	—	4	2	no	—
164	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
165	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
166	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
167	C₂²×C₃	—	7	1	no	—
168	C₂²×C₃	—	14	2	no	—
169	C₂²×C₃	—	14	2	no	—
170	C₂²×C₃	—	14	2	no	—
171	C₃×C₄	—	7	2	no	—
172	C₃×C₄	—	7	2	no	—
173	C₃×C₄	—	14	2	no	—
174	C₃×C₄	—	14	2	no	—
175	C₃×C₄	—	14	4	no	—
176	C₃×C₄	—	14	2	no	—
177	C₃×C₄	—	14	1	no	—
178	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
179	Q₈	—	2	2	no	—
180	Q₈	—	2	4	no	—
181	D₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
184	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
185	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
186	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
187	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
188	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
189	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
190	C₇	C₃×#556	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₃	—	7	1	no	—
192	C₂×C₃	—	7	1	no	—
193	C₂×C₃	—	7	1	no	—
194	C₂×C₃	—	28	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	—
205	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	—
206	C₃	—	7	1	no	Sylow
207	C₂	—	4	2	no	—
208	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
209	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
210	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×#556	C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#556	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #556

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	semidirect
2. ΓL(1,𝔽₈)×#126	C₂	semidirect
4. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
5. C₄×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
6. C₄×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
8. ΓL(1,𝔽₈)×#117	C₂	semidirect
9. C₇×#556	C₃	semidirect
26. C₇×C₄⋊Q₈	C₂×C₃	semidirect
27. C₇×#126	C₂×C₃	semidirect
29. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
30. C₄×C₇×Q₈	C₂×C₃	semidirect
31. C₄×C₇×Q₈	C₂×C₃	semidirect
33. C₇×#117	C₂×C₃	semidirect
82. #556	ΓL(1,𝔽₈)	direct (N × H)
115. C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
116. #126	C₂×ΓL(1,𝔽₈)	semidirect
118. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
119. C₄×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
120. C₄×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
122. #117	C₂×ΓL(1,𝔽₈)	semidirect
146. ΓL(1,𝔽₈)	#556	direct (N × H)
190. C₇	C₃×#556	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)×C₄⋊Q₈	2
ΓL(1,𝔽₈)×#126	2
ΓL(1,𝔽₈)×#124	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
C₄×Q₈×ΓL(1,𝔽₈)	2
C₄×D₄×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#117	2
C₇×#556	3
C₃×#556	7

Subgroups — ΓL(1,𝔽8)×#556 #13060717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#556 #13060717