Subgroups — C₂×C₄⋊C₈×D₂₇₁ #46844913

Order 34688.

640 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₂₇₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₂₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₂₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₂₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×D₂₇₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄⋊C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₂₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₂₇₁	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₂₇₁⋊C₈	C₂²	1	2	yes	—
11	C₂×C₈×D₂₇₁	—	2	8	no	—
12	C₂×C₈×D₂₇₁	C₂²	1	8	yes	—
13	C₂×C₄×Dic₂₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₂₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
16	C₄²×D₂₇₁	C₄	1	4	yes	—
17	C₄²×D₂₇₁	C₂²	1	4	yes	—
18	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	complemented
19	Dic₂₇₁⋊C₈	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₇₁	C₂²	1	4	yes	complemented
23	C₂³×D₂₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₂₇₁	—	2	4	no	—
28	C₂×C₄×D₂₇₁	C₈	1	8	yes	complemented
29	C₂×C₄×D₂₇₁	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₂₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₂₇₁	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₂₇₁	D₄	1	4	yes	—
33	C₂×C₄×D₂₇₁	Q₈	1	4	yes	—
34	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
35	C₂×C₂₇₁⋊C₈	—	2	4	no	—
36	C₂×C₂₇₁⋊C₈	C₂³	1	4	yes	—
37	C₈×D₂₇₁	—	2	16	no	—
38	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
39	C₄×Dic₂₇₁	C₂×C₄	1	4	yes	—
40	C₄×Dic₂₇₁	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₇₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₂₇₁	—	2	4	no	—
46	C₂×C₈×C₂₇₁	C₂³	1	4	yes	—
47	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
48	C₄²×C₂₇₁	C₂³	1	2	yes	—
49	C₂²×D₂₇₁	C₄⋊C₄	1	4	yes	—
50	C₂²×D₂₇₁	C₂×C₈	1	4	yes	—
51	C₂²×D₂₇₁	M₁₆	1	4	yes	—
52	C₂²×D₂₇₁	C₂²×C₄	1	2	yes	—
53	C₂×Dic₂₇₁	—	2	2	no	—
54	C₂×Dic₂₇₁	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₂₇₁	C₂×C₈	1	4	yes	complemented
56	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
57	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
58	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
59	C₄×D₂₇₁	—	2	8	no	—
60	C₄×D₂₇₁	C₄⋊C₄	1	8	yes	—
61	C₄×D₂₇₁	C₂×C₈	1	16	yes	complemented
62	C₄×D₂₇₁	C₂×D₄	1	4	yes	—
63	C₄×D₂₇₁	C₂×Q₈	1	4	yes	—
64	C₂₇₁⋊C₈	—	2	4	no	—
65	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₇₁	—	2	2	no	—
67	C₂×C₄×C₂₇₁	C₂×C₈	1	4	yes	complemented
68	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
74	C₈×C₂₇₁	—	2	4	no	—
75	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₂₇₁	C₂²×C₈	1	2	yes	—
77	C₂²×C₂₇₁	C₂×M₁₆	1	2	yes	—
78	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×D₂₇₁	C₄⋊C₈	1	16	yes	direct product factor
80	C₂×D₂₇₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×D₂₇₁	C₂²×C₈	1	4	yes	—
82	C₂×D₂₇₁	C₂×M₁₆	1	4	yes	—
83	C₄×C₂₇₁	—	2	2	no	—
84	C₄×C₂₇₁	C₂×C₄⋊C₄	1	2	yes	—
85	C₄×C₂₇₁	C₂²×C₈	1	4	yes	complemented
86	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
88	Dic₂₇₁	—	2	2	no	—
89	Dic₂₇₁	C₂×C₄⋊C₄	1	4	yes	—
90	Dic₂₇₁	C₂²×C₈	1	4	yes	complemented
91	C₂×C₂₇₁	C₂×C₄⋊C₈	1	4	yes	complemented
92	C₂×C₂₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₂₇₁	C₂³×C₈	1	1	yes	characteristic
94	C₂×C₂₇₁	C₂²×M₁₆	1	1	yes	characteristic
95	D₂₇₁	C₂×C₄⋊C₈	1	8	yes	direct product factor
96	C₂₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄⋊C₈	—	271	1	no	maximal, Sylow
98	C₂³×C₈	—	271	2	no	—
99	C₂²×C₄²	—	271	1	no	—
100	C₂×C₄⋊C₈	—	271	2	no	—
101	C₂×C₄⋊C₈	—	271	1	no	—
102	C₂×C₄⋊C₈	—	271	8	no	—
103	C₂×C₄⋊C₈	D₂₇₁	1	1	yes	characteristic, direct product factor
104	C₂³×C₄	—	271	1	no	—
105	C₂³×C₄	—	271	1	no	—
106	C₂³×C₄	—	271	1	no	—
107	C₂²×C₈	—	271	2	no	—
108	C₂²×C₈	—	271	8	no	—
109	C₂²×C₈	—	542	8	no	—
110	C₂²×C₈	C₂×D₂₇₁	1	2	yes	—
111	C₂×C₄²	—	271	1	no	—
112	C₂×C₄²	—	271	1	no	—
113	C₂×C₄²	—	271	1	no	—
114	C₂×C₄²	—	271	4	no	—
115	C₂×C₄²	—	271	4	no	—
116	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
117	C₄⋊C₈	—	271	8	no	—
118	C₄⋊C₈	—	271	4	no	—
119	C₄⋊C₈	C₂×D₂₇₁	1	4	yes	direct product factor
120	C₂⁴	—	271	1	no	—
121	C₂²×C₄	—	271	1	no	—
122	C₂²×C₄	—	271	1	no	—
123	C₂²×C₄	—	271	8	no	—
124	C₂²×C₄	—	271	4	no	—
125	C₂²×C₄	—	271	2	no	—
126	C₂²×C₄	—	271	4	no	—
127	C₂²×C₄	—	271	4	no	—
128	C₂²×C₄	—	271	2	no	—
129	C₂²×C₄	—	271	1	no	—
130	C₂²×C₄	—	271	4	no	—
131	C₂²×C₄	—	542	4	no	—
132	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	271	4	no	—
137	C₂×C₈	—	542	16	no	—
138	C₂×C₈	—	542	4	no	—
139	C₂×C₈	C₂²×D₂₇₁	1	4	yes	—
140	C₄²	—	271	4	no	—
141	C₄²	—	271	4	no	—
142	C₄²	—	271	2	no	—
143	C₄²	—	271	2	no	—
144	C₄²	C₄×D₂₇₁	1	2	yes	—
145	C₄²	C₂²×D₂₇₁	1	2	yes	—
146	C₂³	—	271	4	no	—
147	C₂³	—	271	4	no	—
148	C₂³	—	271	2	no	—
149	C₂³	—	271	4	no	—
150	C₂³	C₂×C₄×D₂₇₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	271	4	no	—
153	C₂×C₄	—	271	2	no	—
154	C₂×C₄	—	271	2	no	—
155	C₂×C₄	—	271	4	no	—
156	C₂×C₄	—	271	16	no	—
157	C₂×C₄	—	271	4	no	—
158	C₂×C₄	—	271	4	no	—
159	C₂×C₄	—	271	8	no	—
160	C₂×C₄	—	271	2	no	—
161	C₂×C₄	—	542	8	no	—
162	C₂×C₄	—	542	2	no	—
163	C₂×C₄	C₈×D₂₇₁	1	4	yes	complemented
164	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
167	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
168	C₂×C₄	Q₈×D₂₇₁	1	2	yes	—
169	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	542	4	no	—
172	C₂²	—	271	4	no	—
173	C₂²	—	271	4	no	—
174	C₂²	—	271	4	no	—
175	C₂²	—	271	16	no	—
176	C₂²	C₄⋊C₄×D₂₇₁	1	2	yes	—
177	C₂²	C₂×C₈×D₂₇₁	1	2	yes	—
178	C₂²	M₁₆×D₂₇₁	1	2	yes	—
179	C₂²	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	271	4	no	—
182	C₄	—	271	4	no	—
183	C₄	—	542	2	no	—
184	C₄	C₄⋊C₄×D₂₇₁	1	2	yes	—
185	C₄	C₂×C₈×D₂₇₁	1	4	yes	complemented
186	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
187	C₄	C₂×Q₈×D₂₇₁	1	1	yes	characteristic
188	C₂	—	271	8	no	—
189	C₂	C₄⋊C₈×D₂₇₁	1	4	yes	direct product factor
190	C₂	C₂×C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
191	C₂	C₂²×C₈×D₂₇₁	1	1	yes	characteristic
192	C₂	C₂×M₁₆×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₂×C₄⋊C₈×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂²×C₄
prime radical	C₂³×D₂₇₁	C₂×C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₇₁
hypocenter	C₂₇₁	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊(C₂₇₁⋊C₈)	C₂	semidirect
4. C₂×Dic₂₇₁⋊C₈	C₂	semidirect
5. C₄⋊C₈×D₂₇₁	C₂	direct (N × H)
6. C₂×C₄⋊C₈×C₂₇₁	C₂	semidirect
18. C₄⋊(C₂₇₁⋊C₈)	C₂²	semidirect
19. Dic₂₇₁⋊C₈	C₂²	semidirect
22. C₄⋊C₈×C₂₇₁	C₂²	semidirect
28. C₂×C₄×D₂₇₁	C₈	semidirect
55. C₂×Dic₂₇₁	C₂×C₈	semidirect
61. C₄×D₂₇₁	C₂×C₈	semidirect
67. C₂×C₄×C₂₇₁	C₂×C₈	semidirect
79. C₂×D₂₇₁	C₄⋊C₈	direct (N × H)
85. C₄×C₂₇₁	C₂²×C₈	semidirect
90. Dic₂₇₁	C₂²×C₈	semidirect
91. C₂×C₂₇₁	C₂×C₄⋊C₈	semidirect
95. D₂₇₁	C₂×C₄⋊C₈	direct (N × H)
96. C₂₇₁	C₂²×C₄⋊C₈	semidirect
103. C₂×C₄⋊C₈	D₂₇₁	direct (N × H)
119. C₄⋊C₈	C₂×D₂₇₁	direct (N × H)
163. C₂×C₄	C₈×D₂₇₁	semidirect
185. C₄	C₂×C₈×D₂₇₁	semidirect
189. C₂	C₄⋊C₈×D₂₇₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₈×D₂₇₁	2
C₂×C₄²×D₂₇₁	2
C₂×C₄⋊(C₂₇₁⋊C₈)	2
C₂×Dic₂₇₁⋊C₈	2
C₄⋊C₈×D₂₇₁	2
C₂×C₄⋊C₈×C₂₇₁	2
C₂²×C₄⋊C₈	271

Subgroups — C2×C4⋊C8×D271 #46844913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₈×D₂₇₁ #46844913