Subgroups — C₂×C₄×Q₈×D₂₇₁ #46849537

Order 34688.

1664 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×Dic₅₄₂	C₂	1	3	yes	maximal, complemented
4	C₂×#22576125	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×D₂₇₁	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×D₂₇₁	C₂	1	3	yes	maximal
7	C₂×C₄²×D₂₇₁	C₂	1	3	yes	maximal
8	C₂×C₄×Q₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₅₄₂	C₂²	1	3	yes	—
10	C₂×Q₈×D₂₇₁	C₄	1	16	yes	direct product factor
11	C₂×Q₈×D₂₇₁	C₂²	1	8	yes	—
12	C₂²×C₄×D₂₇₁	C₂²	1	3	yes	—
13	C₂²×C₄×D₂₇₁	C₂²	1	3	yes	—
14	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
15	Q₈×Dic₂₇₁	C₂²	1	8	yes	complemented
16	C₄×Dic₅₄₂	C₂²	1	24	yes	complemented
17	#22576125	C₂²	1	24	yes	complemented
18	C₂×C₄⋊Dic₂₇₁	C₂²	1	3	yes	—
19	C₂×Dic₂₇₁⋊C₄	C₂²	1	6	yes	—
20	C₄⋊C₄×D₂₇₁	C₂²	1	24	yes	—
21	C₂×C₄×Dic₂₇₁	C₂²	1	3	yes	—
22	C₂×C₄×Dic₂₇₁	C₂²	1	3	yes	—
23	C₂×C₄×Dic₂₇₁	C₂²	1	3	yes	—
24	C₄²×D₂₇₁	C₂²	1	24	yes	—
25	C₂²×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₂₇₁	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	3	yes	—
28	C₂×C₄²×C₂₇₁	C₂²	1	3	yes	—
29	C₂³×D₂₇₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₅₄₂	C₂×C₄	1	24	yes	complemented
31	C₂×Dic₅₄₂	C₂³	1	12	yes	—
32	Q₈×D₂₇₁	C₂×C₄	1	64	yes	direct product factor
33	C₂²×Dic₂₇₁	C₂³	1	3	yes	—
34	C₂²×Dic₂₇₁	C₂³	1	3	yes	—
35	C₂²×Dic₂₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₂₇₁	—	2	12	no	—
37	C₂×C₄×D₂₇₁	C₂×C₄	1	24	yes	—
38	C₂×C₄×D₂₇₁	Q₈	1	8	yes	direct product factor
39	C₂×C₄×D₂₇₁	C₂³	1	12	yes	—
40	C₂×C₄×D₂₇₁	C₂³	1	12	yes	—
41	C₂×C₄×D₂₇₁	C₂³	1	4	yes	—
42	C₄⋊Dic₂₇₁	C₂³	1	12	yes	—
43	Dic₂₇₁⋊C₄	C₂³	1	24	yes	—
44	C₄×Dic₂₇₁	C₂³	1	12	yes	—
45	C₄×Dic₂₇₁	C₂³	1	12	yes	—
46	C₄×Dic₂₇₁	C₂³	1	12	yes	—
47	C₂×Q₈×C₂₇₁	C₂×C₄	1	8	yes	complemented
48	C₂×Q₈×C₂₇₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
50	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
51	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₂₇₁	C₂³	1	12	yes	—
53	C₄²×C₂₇₁	C₂³	1	12	yes	—
54	C₂²×D₂₇₁	C₂²×C₄	1	4	yes	—
55	C₂²×D₂₇₁	C₂×Q₈	1	4	yes	—
56	C₂²×D₂₇₁	#41	1	4	yes	—
57	C₂²×D₂₇₁	C₂⁴	1	2	yes	—
58	Dic₅₄₂	C₂²×C₄	1	48	yes	complemented
59	C₂×Dic₂₇₁	—	2	6	no	—
60	C₂×Dic₂₇₁	C₂²×C₄	1	12	yes	—
61	C₂×Dic₂₇₁	C₂×Q₈	1	4	yes	complemented
62	C₂×Dic₂₇₁	C₂⁴	1	6	yes	—
63	C₂×Dic₂₇₁	C₂⁴	1	6	yes	—
64	C₂×Dic₂₇₁	C₂⁴	1	2	yes	—
65	C₄×D₂₇₁	—	2	24	no	—
66	C₄×D₂₇₁	C₂²×C₄	1	48	yes	—
67	C₄×D₂₇₁	C₂×Q₈	1	16	yes	direct product factor
68	C₂³×C₂₇₁	C₂⁴	1	1	yes	characteristic
69	Q₈×C₂₇₁	C₂²×C₄	1	16	yes	complemented
70	C₂×C₄×C₂₇₁	—	2	6	no	—
71	C₂×C₄×C₂₇₁	C₂²×C₄	1	12	yes	—
72	C₂×C₄×C₂₇₁	C₂×Q₈	1	4	yes	complemented
73	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
74	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
75	C₂×C₄×C₂₇₁	C₂⁴	1	2	yes	—
76	C₂²×C₂₇₁	C₂³×C₄	1	2	yes	—
77	C₂²×C₂₇₁	C₂²×Q₈	1	2	yes	—
78	C₂²×C₂₇₁	C₂×#41	1	2	yes	—
79	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
80	C₂×D₂₇₁	C₄×Q₈	1	16	yes	direct product factor
81	C₂×D₂₇₁	C₂³×C₄	1	4	yes	—
82	C₂×D₂₇₁	C₂²×Q₈	1	4	yes	—
83	C₂×D₂₇₁	C₂×#41	1	4	yes	—
84	C₄×C₂₇₁	—	2	6	no	—
85	C₄×C₂₇₁	C₂³×C₄	1	12	yes	—
86	C₄×C₂₇₁	C₂²×Q₈	1	4	yes	complemented
87	Dic₂₇₁	—	2	6	no	—
88	Dic₂₇₁	C₂³×C₄	1	12	yes	—
89	Dic₂₇₁	C₂²×Q₈	1	4	yes	complemented
90	C₂×C₂₇₁	C₂×C₄×Q₈	1	4	yes	complemented
91	C₂×C₂₇₁	C₂⁴×C₄	1	1	yes	characteristic
92	C₂×C₂₇₁	C₂³×Q₈	1	1	yes	characteristic
93	C₂×C₂₇₁	C₂²×#41	1	1	yes	characteristic
94	D₂₇₁	C₂×C₄×Q₈	1	8	yes	direct product factor
95	C₂₇₁	C₂²×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄×Q₈	—	271	1	no	maximal, Sylow
97	C₂³×Q₈	—	271	1	no	—
98	C₂×C₄×Q₈	—	271	1	no	—
99	C₂×C₄×Q₈	—	271	16	no	—
100	C₂×C₄×Q₈	—	271	3	no	—
101	C₂×C₄×Q₈	—	271	3	no	—
102	C₂×C₄×Q₈	D₂₇₁	1	1	yes	characteristic, direct product factor
103	C₂²×C₄⋊C₄	—	271	3	no	—
104	C₂²×C₄²	—	271	3	no	—
105	C₂²×Q₈	—	271	16	no	—
106	C₂²×Q₈	—	271	8	no	—
107	C₂²×Q₈	—	271	3	no	—
108	C₂²×Q₈	C₂×D₂₇₁	1	1	yes	characteristic
109	C₂³×C₄	—	271	3	no	—
110	C₂³×C₄	—	271	1	no	—
111	C₂³×C₄	—	271	3	no	—
112	C₄×Q₈	—	271	8	no	—
113	C₄×Q₈	—	271	24	no	—
114	C₄×Q₈	—	271	24	no	—
115	C₄×Q₈	C₂×D₂₇₁	1	8	yes	direct product factor
116	C₂×C₄⋊C₄	—	271	6	no	—
117	C₂×C₄⋊C₄	—	271	3	no	—
118	C₂×C₄⋊C₄	—	271	24	no	—
119	C₂×C₄⋊C₄	C₂×D₂₇₁	1	3	yes	—
120	C₂×C₄²	—	271	3	no	—
121	C₂×C₄²	—	271	3	no	—
122	C₂×C₄²	—	271	3	no	—
123	C₂×C₄²	—	271	24	no	—
124	C₂×C₄²	C₂×D₂₇₁	1	3	yes	—
125	C₂⁴	—	271	1	no	—
126	C₂×Q₈	—	271	64	no	—
127	C₂×Q₈	—	271	24	no	—
128	C₂×Q₈	—	271	12	no	—
129	C₂×Q₈	C₄×D₂₇₁	1	8	yes	direct product factor
130	C₂×Q₈	C₂²×D₂₇₁	1	4	yes	—
131	C₂²×C₄	—	271	1	no	—
132	C₂²×C₄	—	271	3	no	—
133	C₂²×C₄	—	271	24	no	—
134	C₂²×C₄	—	271	8	no	—
135	C₂²×C₄	—	271	4	no	—
136	C₂²×C₄	—	271	12	no	—
137	C₂²×C₄	—	271	3	no	—
138	C₂²×C₄	—	271	12	no	—
139	C₂²×C₄	—	542	12	no	—
140	C₂²×C₄	C₂²×D₂₇₁	1	3	yes	—
141	C₂²×C₄	C₂²×D₂₇₁	1	3	yes	—
142	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
143	C₄⋊C₄	—	271	24	no	—
144	C₄⋊C₄	—	271	12	no	—
145	C₄⋊C₄	C₂²×D₂₇₁	1	12	yes	—
146	C₄²	—	271	12	no	—
147	C₄²	—	271	12	no	—
148	C₄²	—	271	12	no	—
149	C₄²	C₂²×D₂₇₁	1	12	yes	—
150	C₂³	—	271	4	no	—
151	C₂³	—	271	4	no	—
152	C₂³	—	271	4	no	—
153	C₂³	—	271	2	no	—
154	C₂³	C₂³×D₂₇₁	1	1	yes	characteristic
155	Q₈	—	271	48	no	—
156	Q₈	C₂×C₄×D₂₇₁	1	16	yes	direct product factor
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	271	12	no	—
159	C₂×C₄	—	271	4	no	—
160	C₂×C₄	—	271	2	no	—
161	C₂×C₄	—	271	6	no	—
162	C₂×C₄	—	271	48	no	—
163	C₂×C₄	—	271	16	no	—
164	C₂×C₄	—	271	6	no	—
165	C₂×C₄	—	542	24	no	—
166	C₂×C₄	—	542	6	no	—
167	C₂×C₄	C₂×C₄×D₂₇₁	1	12	yes	—
168	C₂×C₄	Q₈×D₂₇₁	1	4	yes	direct product factor
169	C₂×C₄	C₂³×D₂₇₁	1	6	yes	—
170	C₂×C₄	C₂³×D₂₇₁	1	6	yes	—
171	C₂×C₄	C₂³×D₂₇₁	1	2	yes	—
172	C₂²	—	271	4	no	—
173	C₂²	—	271	4	no	—
174	C₂²	—	271	16	no	—
175	C₂²	—	271	4	no	—
176	C₂²	C₂²×C₄×D₂₇₁	1	2	yes	—
177	C₂²	C₂×Q₈×D₂₇₁	1	2	yes	—
178	C₂²	#41×D₂₇₁	1	2	yes	—
179	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
180	C₄	—	2	6	no	—
181	C₄	—	271	4	no	—
182	C₄	—	271	12	no	—
183	C₄	—	542	6	no	—
184	C₄	C₂²×C₄×D₂₇₁	1	12	yes	—
185	C₄	C₂×Q₈×D₂₇₁	1	4	yes	direct product factor
186	C₂	—	271	8	no	—
187	C₂	C₄×Q₈×D₂₇₁	1	4	yes	direct product factor
188	C₂	C₂³×C₄×D₂₇₁	1	1	yes	characteristic
189	C₂	C₂²×Q₈×D₂₇₁	1	1	yes	characteristic
190	C₂	C₂×#41×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₇₁
Fitting	C₂×C₄×Q₈×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂⁴
prime radical	C₂³×D₂₇₁	C₂³
cosocle	C₂²×C₂₇₁	C₂⁵
hypercenter	C₂×C₄×Q₈	D₂₇₁
hypocenter	C₂₇₁	C₂²×C₄×Q₈
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₂₇₁	C₂	semidirect
3. C₂×C₄×Dic₅₄₂	C₂	semidirect
4. C₂×#22576125	C₂	semidirect
5. C₄×Q₈×D₂₇₁	C₂	direct (N × H)
8. C₂×C₄×Q₈×C₂₇₁	C₂	semidirect
10. C₂×Q₈×D₂₇₁	C₄	direct (N × H)
15. Q₈×Dic₂₇₁	C₂²	semidirect
16. C₄×Dic₅₄₂	C₂²	semidirect
17. #22576125	C₂²	semidirect
26. C₄×Q₈×C₂₇₁	C₂²	semidirect
30. C₂×Dic₅₄₂	C₂×C₄	semidirect
32. Q₈×D₂₇₁	C₂×C₄	direct (N × H)
38. C₂×C₄×D₂₇₁	Q₈	direct (N × H)
47. C₂×Q₈×C₂₇₁	C₂×C₄	semidirect
58. Dic₅₄₂	C₂²×C₄	semidirect
61. C₂×Dic₂₇₁	C₂×Q₈	semidirect
67. C₄×D₂₇₁	C₂×Q₈	direct (N × H)
69. Q₈×C₂₇₁	C₂²×C₄	semidirect
72. C₂×C₄×C₂₇₁	C₂×Q₈	semidirect
80. C₂×D₂₇₁	C₄×Q₈	direct (N × H)
86. C₄×C₂₇₁	C₂²×Q₈	semidirect
89. Dic₂₇₁	C₂²×Q₈	semidirect
90. C₂×C₂₇₁	C₂×C₄×Q₈	semidirect
94. D₂₇₁	C₂×C₄×Q₈	direct (N × H)
95. C₂₇₁	C₂²×C₄×Q₈	semidirect
102. C₂×C₄×Q₈	D₂₇₁	direct (N × H)
115. C₄×Q₈	C₂×D₂₇₁	direct (N × H)
129. C₂×Q₈	C₄×D₂₇₁	direct (N × H)
156. Q₈	C₂×C₄×D₂₇₁	direct (N × H)
168. C₂×C₄	Q₈×D₂₇₁	direct (N × H)
185. C₄	C₂×Q₈×D₂₇₁	direct (N × H)
187. C₂	C₄×Q₈×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×Q₈×D₂₇₁	2
C₂×Q₈×Dic₂₇₁	2
C₂×C₄×Dic₅₄₂	2
C₂×#22576125	2
C₄×Q₈×D₂₇₁	2
C₂×C₄⋊C₄×D₂₇₁	2
C₂×C₄²×D₂₇₁	2
C₂×C₄×Q₈×C₂₇₁	2
C₂²×C₄×Q₈	271

Subgroups — C2×C4×Q8×D271 #46849537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₈×D₂₇₁ #46849537