Subgroups — #1770×C₂₇₁ #46832356

Order 34688.

272 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#97×C₂₇₁	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	#336×C₂₇₁	C₂	1	2	yes	maximal
6	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₇₁	C₄	1	2	yes	—
10	C₂×C₄²×C₂₇₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₇₁	C₄	1	4	yes	complemented
13	#97×C₂₇₁	—	2	4	no	—
14	C₄×C₈×C₂₇₁	C₄	1	4	yes	complemented
15	#95×C₂₇₁	C₄	1	2	yes	—
16	C₂²×C₄×C₂₇₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₂₇₁	—	2	2	no	—
21	C₂×C₈×C₂₇₁	—	2	1	no	—
22	C₂×C₈×C₂₇₁	—	2	1	no	—
23	C₂×C₈×C₂₇₁	—	2	2	no	—
24	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
28	C₂×C₈×C₂₇₁	Q₈	1	2	yes	—
29	C₄⋊C₄×C₂₇₁	—	2	4	no	—
30	C₄²×C₂₇₁	—	2	2	no	—
31	C₄²×C₂₇₁	—	2	2	no	—
32	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
33	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
34	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₇₁	—	2	2	no	—
36	C₂×C₄×C₂₇₁	—	2	4	no	—
37	C₂×C₄×C₂₇₁	—	2	4	no	—
38	C₂×C₄×C₂₇₁	—	2	4	no	—
39	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
40	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
41	C₂×C₄×C₂₇₁	M₁₆	1	4	yes	—
42	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₇₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₂₇₁	#41	1	1	yes	characteristic
48	C₈×C₂₇₁	—	2	2	no	—
49	C₈×C₂₇₁	—	2	4	no	—
50	C₈×C₂₇₁	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₂₇₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₇₁	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₂₇₁	C₂×M₁₆	1	2	yes	—
55	C₂²×C₂₇₁	#131	1	2	yes	—
56	C₄×C₂₇₁	—	2	2	no	—
57	C₄×C₂₇₁	—	4	4	no	—
58	C₄×C₂₇₁	#97	1	4	yes	complemented
59	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₂₇₁	#117	1	1	yes	characteristic
61	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
62	C₄×C₂₇₁	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₂₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₂₇₁	C₂×#97	1	1	yes	characteristic
65	C₂×C₂₇₁	#405	1	1	yes	characteristic
66	C₂×C₂₇₁	#436	1	2	yes	—
67	C₂×C₂₇₁	#446	1	2	yes	—
68	C₂₇₁	#1770	1	1	yes	characteristic, direct product factor, Sylow
69	#1770	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₄⋊C₈	C₂×C₂₇₁	1	1	yes	characteristic
71	C₂×#97	C₂×C₂₇₁	1	2	yes	—
72	C₂×C₄×C₈	C₂×C₂₇₁	1	1	yes	characteristic
73	C₄×C₄⋊C₄	C₂×C₂₇₁	1	1	yes	characteristic
74	#336	C₂×C₂₇₁	1	2	yes	—
75	C₂²×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₂₇₁	1	2	yes	—
77	C₂²×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
78	C₂×C₄⋊C₄	C₄×C₂₇₁	1	2	yes	—
79	C₂×C₄²	C₂²×C₂₇₁	1	2	yes	—
80	C₂×C₄²	C₂²×C₂₇₁	1	1	yes	characteristic
81	C₄⋊C₈	C₄×C₂₇₁	1	4	yes	complemented
82	#97	—	2	4	no	—
83	C₄×C₈	C₄×C₂₇₁	1	4	yes	complemented
84	#95	C₄×C₂₇₁	1	2	yes	—
85	C₂²×C₄	C₂×C₄×C₂₇₁	1	4	yes	—
86	C₂²×C₄	C₂×C₄×C₂₇₁	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₄×C₂₇₁	1	1	yes	characteristic
88	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	C₂×C₄×C₂₇₁	1	2	yes	—
94	C₂×C₈	C₂×C₄×C₂₇₁	1	4	yes	—
95	C₂×C₈	C₂×C₄×C₂₇₁	1	2	yes	—
96	C₂×C₈	D₄×C₂₇₁	1	2	yes	—
97	C₂×C₈	Q₈×C₂₇₁	1	2	yes	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×C₄×C₂₇₁	1	2	yes	—
102	C₄²	C₂×C₄×C₂₇₁	1	2	yes	—
103	C₂³	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	C₄²×C₂₇₁	1	2	yes	—
109	C₂×C₄	C₄²×C₂₇₁	1	2	yes	—
110	C₂×C₄	M₁₆×C₂₇₁	1	4	yes	—
111	C₂×C₄	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₂₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₈×C₂₇₁	1	1	yes	characteristic
115	C₂×C₄	#41×C₂₇₁	1	1	yes	characteristic
116	C₂×C₄	#41×C₂₇₁	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	4	no	—
119	C₈	C₄⋊C₄×C₂₇₁	1	4	yes	complemented
120	C₂²	C₂×C₄²×C₂₇₁	1	1	yes	characteristic
121	C₂²	C₄×D₄×C₂₇₁	1	1	yes	characteristic
122	C₂²	C₄×Q₈×C₂₇₁	1	1	yes	characteristic
123	C₂²	C₂×M₁₆×C₂₇₁	1	2	yes	—
124	C₂²	#131×C₂₇₁	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	#97×C₂₇₁	1	4	yes	complemented
128	C₄	C₂×C₄⋊C₄×C₂₇₁	1	1	yes	characteristic
129	C₄	#117×C₂₇₁	1	1	yes	characteristic
130	C₄	C₄×D₄×C₂₇₁	1	1	yes	characteristic
131	C₄	C₄×Q₈×C₂₇₁	1	1	yes	characteristic
132	C₂	C₄×C₄⋊C₄×C₂₇₁	1	1	yes	characteristic
133	C₂	C₂×#97×C₂₇₁	1	1	yes	characteristic
134	C₂	#405×C₂₇₁	1	1	yes	characteristic
135	C₂	#436×C₂₇₁	1	2	yes	—
136	C₂	#446×C₂₇₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₇₁	C₂³
commutator	C₂²	C₂×C₄²×C₂₇₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₇₁
socle, prime radical	C₂³×C₂₇₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₄⋊C₈×C₂₇₁	C₄	semidirect
14. C₄×C₈×C₂₇₁	C₄	semidirect
50. C₈×C₂₇₁	C₄⋊C₄	semidirect
58. C₄×C₂₇₁	#97	semidirect
68. C₂₇₁	#1770	direct (N × H)
69. #1770	C₂₇₁	direct (N × H)
81. C₄⋊C₈	C₄×C₂₇₁	semidirect
83. C₄×C₈	C₄×C₂₇₁	semidirect
119. C₈	C₄⋊C₄×C₂₇₁	semidirect
127. C₄	#97×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×C₂₇₁	2
C₂×#97×C₂₇₁	2
C₂×C₄×C₈×C₂₇₁	2
C₄×C₄⋊C₄×C₂₇₁	2
#336×C₂₇₁	2
#1770	271

Subgroups — #1770×C271 #46832356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1770×C₂₇₁ #46832356