Subgroups — #46836062

Order 34688.

416 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#22576119	C₂	1	1	yes	maximal, characteristic
3	#31773651	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₇₁⋊C₈	—	2	2	no	—
6	C₂²×C₂₇₁⋊C₈	—	2	2	no	—
7	C₂²×C₂₇₁⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₂₇₁⋊C₈	C₂²	1	2	yes	—
9	#22576119	—	2	2	no	—
10	#22576119	—	2	2	no	—
11	C₂×C₄⋊Dic₂₇₁	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₂₇₁	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₇₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₇₁	—	2	2	no	—
16	C₂×M₁₆×C₂₇₁	—	2	2	no	—
17	C₂×M₁₆×C₂₇₁	C₄	1	4	yes	complemented
18	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
19	C₂²×Dic₂₇₁	C₂×C₄	1	4	yes	—
20	C₂×C₂₇₁⋊C₈	—	2	8	no	—
21	C₂×C₂₇₁⋊C₈	—	2	2	no	—
22	C₂×C₂₇₁⋊C₈	—	2	2	no	—
23	C₂×C₂₇₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₂₇₁	—	2	4	no	—
25	C₂²⋊Dic₂₇₁	—	2	4	no	—
26	C₄×Dic₂₇₁	—	2	2	no	—
27	C₄×Dic₂₇₁	—	2	2	no	—
28	C₂⁴×C₂₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
34	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
35	C₂²×C₄×C₂₇₁	Q₈	1	2	yes	—
36	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₂₇₁	—	2	8	no	—
38	C₂×C₈×C₂₇₁	—	2	2	no	—
39	C₂×C₈×C₂₇₁	—	2	2	no	—
40	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	—
41	C₂×Dic₂₇₁	—	2	4	no	—
42	C₂×Dic₂₇₁	—	2	8	no	—
43	C₂₇₁⋊C₈	—	2	4	no	—
44	C₂³×C₂₇₁	—	2	4	no	—
45	C₂³×C₂₇₁	C₄²	1	2	yes	—
46	C₂³×C₂₇₁	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
48	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
50	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₇₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₂₇₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₈×C₂₇₁	—	2	4	no	—
63	C₂²×C₂₇₁	—	2	4	no	—
64	C₂²×C₂₇₁	—	2	4	no	—
65	C₂²×C₂₇₁	—	2	4	no	—
66	C₂²×C₂₇₁	#95	1	4	yes	—
67	C₂²×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
68	C₂²×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₇₁	#131	1	4	yes	—
71	C₄×C₂₇₁	#95	1	4	yes	—
72	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	Dic₂₇₁	—	4	4	no	—
77	C₂×C₂₇₁	—	2	4	no	—
78	C₂×C₂₇₁	C₂×#95	1	1	yes	characteristic
79	C₂×C₂₇₁	#405	1	2	yes	—
80	C₂×C₂₇₁	#408	1	2	yes	—
81	C₂×C₂₇₁	#424	1	2	yes	—
82	C₂₇₁	#1729	1	1	yes	characteristic, complemented, Sylow
83	#1729	—	271	1	no	maximal, Sylow
84	C₂²×M₁₆	D₂₇₁	1	1	yes	characteristic
85	C₂³×C₈	—	271	1	no	—
86	C₂×#117	—	271	1	no	—
87	#336	—	271	4	no	—
88	C₂³×C₄	C₂×D₂₇₁	1	1	yes	characteristic
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	Dic₂₇₁	1	4	yes	complemented
92	C₂²×C₈	—	271	4	no	—
93	C₂²×C₈	—	271	2	no	—
94	C₂²×C₈	—	542	2	no	—
95	C₂²×C₈	—	542	2	no	—
96	C₂²×C₈	C₂×D₂₇₁	1	2	yes	—
97	#117	—	542	2	no	—
98	#117	—	542	2	no	—
99	C₂×C₄⋊C₄	—	271	2	no	—
100	C₂×C₂²⋊C₄	—	271	2	no	—
101	C₂×C₄²	—	271	2	no	—
102	C₂⁴	C₄×D₂₇₁	1	1	yes	characteristic
103	C₂²×C₄	—	271	4	no	—
104	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
106	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
107	C₂²×C₄	C₂²⋊D₂₇₁	1	2	yes	—
108	C₂²×C₄	C₂²⋊D₂₇₁	1	2	yes	—
109	C₂²×C₄	D₁₀₈₄	1	2	yes	—
110	C₂²×C₄	Dic₅₄₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
112	M₁₆	—	2	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	271	4	no	—
116	C₂×C₈	—	542	8	no	—
117	C₂×C₈	—	542	2	no	—
118	C₂×C₈	—	542	2	no	—
119	C₂×C₈	C₂×Dic₂₇₁	1	4	yes	—
120	C₄⋊C₄	—	542	4	no	—
121	C₂²⋊C₄	—	542	4	no	—
122	C₄²	—	542	2	no	—
123	C₄²	—	542	2	no	—
124	C₂³	—	2	4	no	—
125	C₂³	C₄×Dic₂₇₁	1	2	yes	—
126	C₂³	(C₂×D₂₇₁)⋊C₄	1	2	yes	—
127	C₂³	Dic₂₇₁⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄×D₂₇₁	1	1	yes	characteristic
129	C₂×C₄	—	542	8	no	—
130	C₂×C₄	—	542	4	no	—
131	C₂×C₄	C₄×Dic₂₇₁	1	2	yes	—
132	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	2	yes	—
133	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	4	yes	—
134	C₂×C₄	C₂²⋊Dic₂₇₁	1	4	yes	—
135	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
136	C₂×C₄	Dic₂₇₁⋊C₄	1	4	yes	—
137	C₂×C₄	C₄⋊Dic₂₇₁	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₅₄₂	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	542	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	4	no	—
149	C₂²	C₂₇₁⋊#95	1	4	yes	—
150	C₂²	C₂×C₄×Dic₂₇₁	1	1	yes	characteristic
151	C₂²	C₂×(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
152	C₂²	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
153	C₂²	#22576168	1	4	yes	—
154	C₄	—	1084	4	no	—
155	C₄	C₂₇₁⋊#95	1	4	yes	—
156	C₄	C₂×(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
157	C₄	C₂×C₂²⋊Dic₂₇₁	1	1	yes	characteristic
158	C₄	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
159	C₄	C₂×C₄⋊Dic₂₇₁	1	1	yes	characteristic
160	C₂	—	2	4	no	—
161	C₂	C₂×C₂₇₁⋊#95	1	1	yes	characteristic
162	C₂	#31773855	1	2	yes	—
163	C₂	#31773865	1	2	yes	—
164	C₂	#31773925	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₁
commutator	C₂²×C₂₇₁	C₂×C₄²
Fitting	C₂²×M₁₆×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂²×C₄
prime radical	C₂⁴×C₂₇₁	C₂×C₄
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	C₂²×M₁₆	D₂₇₁
hypocenter	C₂₇₁	#1729
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #31773651	C₂	semidirect
7. C₂²×C₂₇₁⋊C₈	C₄	semidirect
17. C₂×M₁₆×C₂₇₁	C₄	semidirect
82. C₂₇₁	#1729	semidirect
91. C₂×M₁₆	Dic₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₂₇₁⋊C₈	2
C₂×#22576119	2
#31773651	2
C₂²×M₁₆×C₂₇₁	2
#1729	271