Subgroups — C₂×ΓL(1,𝔽₈)×#135 #13060733

Order 1344.

532 conjugacy classes of subgroups.

The subgroup lattice has 196 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)×#135	C₂	1	8	yes	maximal, direct product factor
3	C₂×Q₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#135	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	#41×ΓL(1,𝔽₈)	—	2	4	no	—
9	#41×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
10	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
11	C₂×D₄×ΓL(1,𝔽₈)	—	2	2	no	—
12	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₂²×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
14	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	Q₁₆×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
16	SD₁₆×ΓL(1,𝔽₈)	C₂²	1	8	yes	complemented
17	D₈×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
18	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
19	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
20	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	4	yes	complemented
21	C₂×C₇×#41	C₂×C₃	1	2	yes	complemented
22	C₇×#135	C₂×C₃	1	8	yes	complemented
23	C₂×C₇×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₇×SD₁₆	C₂×C₃	1	2	yes	complemented
25	C₂×C₇×D₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×C₃×#135	—	7	1	no	maximal, Hall
28	C₂³×ΓL(1,𝔽₈)	—	2	2	no	—
29	C₂³×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
30	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
31	Q₈×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
32	D₄×ΓL(1,𝔽₈)	—	2	8	no	—
33	D₄×ΓL(1,𝔽₈)	—	2	2	no	—
34	D₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
35	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
36	C₂×C₄×ΓL(1,𝔽₈)	—	2	8	no	—
37	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
38	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
39	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
40	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
41	C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
42	C₈×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
43	C₇×#41	—	2	4	no	—
44	C₇×#41	C₂²×C₃	1	8	yes	complemented
45	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
46	C₂×C₇×D₄	—	2	2	no	—
47	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
50	C₇×Q₁₆	C₂²×C₃	1	4	yes	complemented
51	C₇×SD₁₆	C₂²×C₃	1	8	yes	complemented
52	C₇×D₈	C₂²×C₃	1	4	yes	complemented
53	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂²×C₃	1	4	yes	complemented
56	C₂×C₃×#41	—	7	2	no	—
57	C₃×#135	—	7	8	no	—
58	C₂×C₃×Q₁₆	—	7	1	no	—
59	C₂×C₃×SD₁₆	—	7	2	no	—
60	C₂×C₃×D₈	—	7	1	no	—
61	C₂²×C₃×C₈	—	7	1	no	—
62	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
63	C₂²×ΓL(1,𝔽₈)	—	2	4	no	—
64	C₂²×ΓL(1,𝔽₈)	—	4	4	no	—
65	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
66	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
67	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
68	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	2	yes	—
69	C₄×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
70	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
71	C₂×#135	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
74	C₇×Q₈	—	2	2	no	—
75	C₇×Q₈	C₂³×C₃	1	4	yes	—
76	C₇×D₄	—	2	8	no	—
77	C₇×D₄	—	2	2	no	—
78	C₇×D₄	C₂³×C₃	1	4	yes	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	8	no	—
81	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
82	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
85	C₇×C₈	C₂³×C₃	1	2	yes	—
86	C₇×C₈	C₂³×C₃	1	2	yes	—
87	C₃×#41	—	7	8	no	—
88	C₃×#41	—	14	4	no	—
89	C₂×C₃×Q₈	—	7	2	no	—
90	C₂×C₃×D₄	—	7	2	no	—
91	C₂×C₃×D₄	—	14	2	no	—
92	C₂²×C₃×C₄	—	7	1	no	—
93	C₂²×C₃×C₄	—	14	2	no	—
94	C₃×Q₁₆	—	7	4	no	—
95	C₃×SD₁₆	—	7	8	no	—
96	C₃×D₈	—	7	4	no	—
97	C₂×C₃×C₈	—	7	1	no	—
98	C₂×C₃×C₈	—	7	1	no	—
99	C₂×C₃×C₈	—	7	4	no	—
100	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
101	C₂×ΓL(1,𝔽₈)	—	4	4	no	—
102	C₂×ΓL(1,𝔽₈)	#135	1	2	yes	direct product factor
103	C₂×ΓL(1,𝔽₈)	C₂²×D₄	1	1	yes	characteristic
104	C₂×#41	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
105	#135	C₂×ΓL(1,𝔽₈)	1	8	yes	direct product factor
106	C₂×Q₁₆	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
107	C₂×SD₁₆	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
108	C₂×D₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
109	C₂²×C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
110	C₂²×C₇	—	2	2	no	—
111	C₂²×C₇	—	2	4	no	—
112	C₂²×C₇	—	4	4	no	—
113	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
114	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
115	C₄×C₇	—	2	4	no	—
116	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
117	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
118	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
119	C₂³×C₃	—	7	1	no	—
120	C₂³×C₃	—	14	2	no	—
121	C₃×Q₈	—	7	4	no	—
122	C₃×Q₈	—	14	2	no	—
123	C₃×D₄	—	7	4	no	—
124	C₃×D₄	—	14	2	no	—
125	C₃×D₄	—	14	8	no	—
126	C₂×C₃×C₄	—	7	2	no	—
127	C₂×C₃×C₄	—	7	1	no	—
128	C₂×C₃×C₄	—	7	1	no	—
129	C₂×C₃×C₄	—	7	2	no	—
130	C₂×C₃×C₄	—	14	8	no	—
131	C₂×C₃×C₄	—	14	2	no	—
132	C₃×C₈	—	7	2	no	—
133	C₃×C₈	—	7	2	no	—
134	ΓL(1,𝔽₈)	C₂×#135	1	1	yes	characteristic, direct product factor, Hall
135	#41	—	2	4	no	—
136	#41	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
137	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
142	Q₁₆	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
143	SD₁₆	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
144	D₈	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
145	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
146	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
147	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
148	C₂×C₇	—	2	2	no	—
149	C₂×C₇	—	4	4	no	—
150	C₂×C₇	C₃×#135	1	2	yes	complemented
151	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
152	C₂²×C₃	—	7	1	no	—
153	C₂²×C₃	—	7	2	no	—
154	C₂²×C₃	—	14	2	no	—
155	C₂²×C₃	—	14	4	no	—
156	C₂²×C₃	—	28	4	no	—
157	C₃×C₄	—	7	2	no	—
158	C₃×C₄	—	7	1	no	—
159	C₃×C₄	—	7	1	no	—
160	C₃×C₄	—	14	4	no	—
161	C₂³	—	2	2	no	—
162	C₂³	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
163	Q₈	—	2	2	no	—
164	Q₈	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
165	D₄	—	2	8	no	—
166	D₄	—	2	2	no	—
167	D₄	C₂³×ΓL(1,𝔽₈)	1	4	yes	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	2	yes	—
171	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
172	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
173	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
174	C₈	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
175	C₈	C₂³×ΓL(1,𝔽₈)	1	2	yes	—
176	C₇	C₂×C₃×#135	1	1	yes	characteristic, complemented, Sylow
177	C₂×C₃	—	7	1	no	—
178	C₂×C₃	—	7	2	no	—
179	C₂×C₃	—	14	2	no	—
180	C₂×C₃	—	28	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	4	4	no	—
184	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
185	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	2	yes	—
188	C₄	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
189	C₄	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
190	C₃	—	7	1	no	Sylow
191	C₂	—	2	2	no	—
192	C₂	—	4	4	no	—
193	C₂	ΓL(1,𝔽₈)×#135	1	2	yes	direct product factor
194	C₂	C₂²×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₂×C₇×#135	C₃
socle	C₂²×C₇	C₂×C₃×D₄
hypercenter	C₂×#135	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×C₃×#135
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

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)×#135	C₂	direct (N × H)
3. C₂×Q₁₆×ΓL(1,𝔽₈)	C₂	semidirect
4. C₂×SD₁₆×ΓL(1,𝔽₈)	C₂	semidirect
5. C₂×D₈×ΓL(1,𝔽₈)	C₂	semidirect
6. C₂²×C₈×ΓL(1,𝔽₈)	C₂	semidirect
7. C₂×C₇×#135	C₃	semidirect
9. #41×ΓL(1,𝔽₈)	C₂²	semidirect
15. Q₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
16. SD₁₆×ΓL(1,𝔽₈)	C₂²	semidirect
17. D₈×ΓL(1,𝔽₈)	C₂²	semidirect
20. C₂×C₈×ΓL(1,𝔽₈)	C₂²	semidirect
21. C₂×C₇×#41	C₂×C₃	semidirect
22. C₇×#135	C₂×C₃	semidirect
23. C₂×C₇×Q₁₆	C₂×C₃	semidirect
24. C₂×C₇×SD₁₆	C₂×C₃	semidirect
25. C₂×C₇×D₈	C₂×C₃	semidirect
26. C₂²×C₇×C₈	C₂×C₃	semidirect
44. C₇×#41	C₂²×C₃	semidirect
50. C₇×Q₁₆	C₂²×C₃	semidirect
51. C₇×SD₁₆	C₂²×C₃	semidirect
52. C₇×D₈	C₂²×C₃	semidirect
55. C₂×C₇×C₈	C₂²×C₃	semidirect
71. C₂×#135	ΓL(1,𝔽₈)	direct (N × H)
102. C₂×ΓL(1,𝔽₈)	#135	direct (N × H)
104. C₂×#41	C₂×ΓL(1,𝔽₈)	semidirect
105. #135	C₂×ΓL(1,𝔽₈)	direct (N × H)
106. C₂×Q₁₆	C₂×ΓL(1,𝔽₈)	semidirect
107. C₂×SD₁₆	C₂×ΓL(1,𝔽₈)	semidirect
108. C₂×D₈	C₂×ΓL(1,𝔽₈)	semidirect
109. C₂²×C₈	C₂×ΓL(1,𝔽₈)	semidirect
134. ΓL(1,𝔽₈)	C₂×#135	direct (N × H)
136. #41	C₂²×ΓL(1,𝔽₈)	semidirect
142. Q₁₆	C₂²×ΓL(1,𝔽₈)	semidirect
143. SD₁₆	C₂²×ΓL(1,𝔽₈)	semidirect
144. D₈	C₂²×ΓL(1,𝔽₈)	semidirect
147. C₂×C₈	C₂²×ΓL(1,𝔽₈)	semidirect
150. C₂×C₇	C₃×#135	semidirect
176. C₇	C₂×C₃×#135	semidirect
193. C₂	ΓL(1,𝔽₈)×#135	direct (N × H)

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)×#135	2
C₂×Q₁₆×ΓL(1,𝔽₈)	2
C₂×SD₁₆×ΓL(1,𝔽₈)	2
C₂×D₈×ΓL(1,𝔽₈)	2
C₂²×C₈×ΓL(1,𝔽₈)	2
C₂×C₇×#135	3
C₂×C₃×#135	7

Subgroups — C2×ΓL(1,𝔽8)×#135 #13060733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×#135 #13060733