Subgroups — ΓL(1,𝔽₈)×#554 #13060715

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 216 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)×#125	C₂	1	2	yes	maximal, complemented
3	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
4	ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal
7	C₄×D₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
9	C₇×#554	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	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
15	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
17	C₄⋊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	2	yes	—
21	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
22	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
23	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
24	C₄²×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
25	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
26	C₇×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₇×#125	C₂×C₃	1	2	yes	complemented
28	C₇×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
29	C₇×C₂²⋊Q₈	C₂×C₃	1	2	yes	complemented
30	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₄×C₇×D₄	C₂×C₃	1	2	yes	—
32	C₄×C₇×D₄	C₂×C₃	1	1	yes	characteristic
33	C₂×C₇×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
34	C₃×#554	—	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)	—	4	2	no	—
38	D₄×ΓL(1,𝔽₈)	Q₈	1	4	yes	complemented
39	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
40	C₂×C₄×ΓL(1,𝔽₈)	—	4	4	no	—
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	2	yes	—
44	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
45	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
46	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
47	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
48	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
50	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
52	C₂²×C₄×C₇	C₂²×C₃	1	4	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	1	yes	characteristic
57	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
61	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₄²×C₇	C₂²×C₃	1	2	yes	—
63	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊Q₈	—	7	1	no	—
65	C₃×#125	—	7	2	no	—
66	C₃×C₂²⋊Q₈	—	7	4	no	—
67	C₃×C₂²⋊Q₈	—	7	2	no	—
68	C₃×C₄×Q₈	—	7	1	no	—
69	C₃×C₄×D₄	—	7	2	no	—
70	C₃×C₄×D₄	—	7	1	no	—
71	C₂×C₃×C₄⋊C₄	—	7	2	no	—
72	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
73	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
74	C₂²×ΓL(1,𝔽₈)	C₂×Q₈	1	4	yes	—
75	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
76	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
77	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
78	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
79	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
80	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
81	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	2	yes	—
82	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
83	#554	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
84	C₂³×C₇	C₂³×C₃	1	2	yes	—
85	C₇×Q₈	—	2	2	no	—
86	C₇×Q₈	—	4	2	no	—
87	C₇×D₄	C₃×Q₈	1	4	yes	complemented
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	—	4	4	no	—
90	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
91	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
93	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
94	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
95	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₃×Q₈	—	7	1	no	—
98	C₂×C₃×Q₈	—	7	2	no	—
99	C₂×C₃×D₄	—	7	1	no	—
100	C₂²×C₃×C₄	—	7	2	no	—
101	C₂²×C₃×C₄	—	7	4	no	—
102	C₃×C₄⋊C₄	—	7	1	no	—
103	C₃×C₄⋊C₄	—	7	4	no	—
104	C₃×C₄⋊C₄	—	7	2	no	—
105	C₃×C₄⋊C₄	—	7	1	no	—
106	C₃×C₄⋊C₄	—	7	2	no	—
107	C₃×C₄⋊C₄	—	7	2	no	—
108	C₃×C₄⋊C₄	—	7	4	no	—
109	C₃×C₂²⋊C₄	—	7	4	no	—
110	C₃×C₂²⋊C₄	—	7	2	no	—
111	C₃×C₄²	—	7	2	no	—
112	C₃×C₄²	—	7	1	no	—
113	C₂×ΓL(1,𝔽₈)	—	2	4	no	—
114	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	1	yes	characteristic
115	C₂×ΓL(1,𝔽₈)	C₂×#41	1	1	yes	characteristic
116	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
117	C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
118	#125	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
119	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
120	C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
121	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
122	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	2	yes	—
123	C₄×D₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	C₂×C₃×Q₈	1	4	yes	—
128	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	2	4	no	—
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	C₂×C₃×Q₈	1	2	yes	—
135	C₄×C₇	C₃×#41	1	2	yes	—
136	C₂³×C₃	—	7	2	no	—
137	C₃×Q₈	—	14	2	no	—
138	C₃×Q₈	—	28	2	no	—
139	C₃×D₄	—	7	4	no	—
140	C₂×C₃×C₄	—	7	1	no	—
141	C₂×C₃×C₄	—	7	1	no	—
142	C₂×C₃×C₄	—	7	2	no	—
143	C₂×C₃×C₄	—	7	1	no	—
144	C₂×C₃×C₄	—	7	2	no	—
145	C₂×C₃×C₄	—	7	2	no	—
146	C₂×C₃×C₄	—	7	4	no	—
147	C₂×C₃×C₄	—	14	4	no	—
148	C₂×C₃×C₄	—	28	4	no	—
149	ΓL(1,𝔽₈)	#554	1	1	yes	characteristic, direct product factor, Hall
150	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
151	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
152	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
153	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
154	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
155	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
156	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
157	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
158	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
159	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
160	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
161	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
162	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
163	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
164	C₄²	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
165	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
166	C₂×C₇	—	2	4	no	—
167	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
168	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
169	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²×C₃	—	7	4	no	—
171	C₂²×C₃	—	7	1	no	—
172	C₂²×C₃	—	14	2	no	—
173	C₂²×C₃	—	14	2	no	—
174	C₃×C₄	—	7	2	no	—
175	C₃×C₄	—	7	2	no	—
176	C₃×C₄	—	14	2	no	—
177	C₃×C₄	—	14	2	no	—
178	C₃×C₄	—	14	4	no	—
179	C₃×C₄	—	14	2	no	—
180	C₃×C₄	—	14	1	no	—
181	C₂³	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
182	Q₈	—	2	2	no	—
183	Q₈	—	4	2	no	—
184	D₄	Q₈×ΓL(1,𝔽₈)	1	4	yes	complemented
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
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	2	yes	—
190	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
191	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
192	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
193	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
194	C₇	C₃×#554	1	1	yes	characteristic, complemented, Sylow
195	C₂×C₃	—	7	1	no	—
196	C₂×C₃	—	7	1	no	—
197	C₂×C₃	—	7	1	no	—
198	C₂×C₃	—	14	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	C₂×Q₈×ΓL(1,𝔽₈)	1	4	yes	—
202	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	4	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×Q₈×ΓL(1,𝔽₈)	1	2	yes	—
209	C₄	#41×ΓL(1,𝔽₈)	1	2	yes	—
210	C₃	—	7	1	no	Sylow
211	C₂	—	2	4	no	—
212	C₂	C₂²×Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
213	C₂	C₂×#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
214	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#554	C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#554	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#554
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #554

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)×#125	C₂	semidirect
3. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
4. ΓL(1,𝔽₈)×C₂²⋊Q₈	C₂	semidirect
5. C₄×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
8. C₂×C₄⋊C₄×ΓL(1,𝔽₈)	C₂	semidirect
9. C₇×#554	C₃	semidirect
26. C₇×C₄⋊Q₈	C₂×C₃	semidirect
27. C₇×#125	C₂×C₃	semidirect
28. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
29. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
30. C₄×C₇×Q₈	C₂×C₃	semidirect
33. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
38. D₄×ΓL(1,𝔽₈)	Q₈	semidirect
83. #554	ΓL(1,𝔽₈)	direct (N × H)
87. C₇×D₄	C₃×Q₈	semidirect
117. C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
118. #125	C₂×ΓL(1,𝔽₈)	semidirect
119. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
120. C₂²⋊Q₈	C₂×ΓL(1,𝔽₈)	semidirect
121. C₄×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
124. C₂×C₄⋊C₄	C₂×ΓL(1,𝔽₈)	semidirect
149. ΓL(1,𝔽₈)	#554	direct (N × H)
184. D₄	Q₈×ΓL(1,𝔽₈)	semidirect
194. C₇	C₃×#554	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)×#125	2
ΓL(1,𝔽₈)×C₂²⋊Q₈	2
C₄×Q₈×ΓL(1,𝔽₈)	2
C₄×D₄×ΓL(1,𝔽₈)	2
C₂×C₄⋊C₄×ΓL(1,𝔽₈)	2
C₇×#554	3
C₃×#554	7

Subgroups — ΓL(1,𝔽8)×#554 #13060715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#554 #13060715