Subgroups — #46845296

Order 34688.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774450	C₂	1	1	yes	maximal, characteristic
2	#31774025	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₅₄₂	C₂	1	4	yes	maximal, complemented
4	#31773925	C₂	1	1	yes	maximal, characteristic
5	#31773924	C₂	1	1	yes	maximal, characteristic
6	#31773920	C₂	1	1	yes	maximal, characteristic
7	#31773852	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic
9	#405×C₂₇₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#16864638	C₂²	1	1	yes	characteristic
11	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#22576159	C₄	1	2	yes	—
13	#22576151	C₄	1	2	yes	—
14	#22576137	C₄	1	4	yes	—
15	C₄×Dic₅₄₂	C₂²	1	4	yes	—
16	#22576119	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	—
19	Dic₂₇₁⋊C₈	C₂²	1	4	yes	—
20	Dic₂₇₁⋊C₈	C₂²	1	4	yes	—
21	#22576070	C₂²	1	2	yes	—
22	C₈×Dic₂₇₁	C₂²	1	2	yes	—
23	C₈×Dic₂₇₁	C₂²	1	2	yes	—
24	C₈×Dic₂₇₁	C₂²	1	2	yes	—
25	C₂×M₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
27	#117×C₂₇₁	C₂²	1	1	yes	characteristic
28	#97×C₂₇₁	C₂²	1	2	yes	—
29	C₄×C₈×C₂₇₁	C₂²	1	2	yes	—
30	C₂×Dic₅₄₂	C₂×C₄	1	4	yes	—
31	C₂²×Dic₂₇₁	C₂×C₄	1	2	yes	—
32	#16864638	—	2	2	no	—
33	C₂×C₂₇₁⋊C₈	—	2	2	no	—
34	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
36	C₄⋊Dic₂₇₁	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₂₇₁	C₂×C₄	1	2	yes	—
38	Dic₂₇₁⋊C₄	C₂×C₄	1	4	yes	—
39	Dic₂₇₁⋊C₄	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₂₇₁	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₂₇₁	C₂³	1	2	yes	—
44	C₄×Dic₂₇₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
46	M₁₆×C₂₇₁	—	2	2	no	—
47	C₂×C₈×C₂₇₁	Q₈	1	4	yes	complemented
48	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
49	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
53	C₄²×C₂₇₁	C₂³	1	2	yes	—
54	Dic₅₄₂	—	2	8	no	—
55	C₂×Dic₂₇₁	—	2	4	no	—
56	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
57	C₂×Dic₂₇₁	C₂²×C₄	1	4	yes	—
58	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
59	C₂₇₁⋊C₈	—	2	2	no	—
60	C₂₇₁⋊C₈	—	2	2	no	—
61	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₇₁	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₂₇₁	#41	1	2	yes	—
66	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₇₁	—	2	2	no	—
68	C₈×C₂₇₁	C₂×Q₈	1	2	yes	—
69	C₈×C₂₇₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₂₇₁	—	2	2	no	—
71	C₂²×C₂₇₁	C₄×Q₈	1	2	yes	—
72	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₂₇₁	—	2	4	no	—
74	C₄×C₂₇₁	C₄×Q₈	1	2	yes	—
75	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
77	Dic₂₇₁	—	2	2	no	—
78	Dic₂₇₁	—	2	4	no	—
79	Dic₂₇₁	#131	1	4	yes	—
80	C₂×C₂₇₁	—	2	2	no	—
81	C₂×C₂₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₂₇₁	C₂×#131	1	2	yes	—
83	C₂₇₁	#2960	1	1	yes	characteristic, complemented, Sylow
84	#2960	—	271	1	no	maximal, Sylow
85	#533	—	271	1	no	—
86	#446	—	271	4	no	—
87	C₈×Q₈	—	271	4	no	—
88	#424	—	271	1	no	—
89	#424	—	271	1	no	—
90	#423	—	271	1	no	—
91	#405	—	271	1	no	—
92	#405	D₂₇₁	1	1	yes	characteristic
93	C₂×C₄×C₈	—	271	1	no	—
94	C₂×M₁₆	—	271	1	no	—
95	C₂×M₁₆	C₂×D₂₇₁	1	1	yes	characteristic
96	C₂²×C₈	—	271	1	no	—
97	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
98	C₄⋊Q₈	—	271	2	no	—
99	#125	—	271	2	no	—
100	C₂²⋊Q₈	—	271	4	no	—
101	C₄×Q₈	—	271	4	no	—
102	#117	—	271	1	no	—
103	#117	C₂×D₂₇₁	1	1	yes	characteristic
104	C₂×C₄²	—	271	1	no	—
105	C₄⋊C₈	—	271	4	no	—
106	C₄⋊C₈	—	271	4	no	—
107	C₄⋊C₈	—	271	4	no	—
108	#97	—	271	2	no	—
109	#97	C₂×D₂₇₁	1	2	yes	—
110	C₄×C₈	—	271	2	no	—
111	C₄×C₈	—	271	2	no	—
112	C₄×C₈	—	271	2	no	—
113	C₄×C₈	C₂×D₂₇₁	1	2	yes	—
114	C₂×Q₈	—	271	4	no	—
115	C₂²×C₄	—	271	2	no	—
116	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	542	2	no	—
119	C₂×C₈	—	271	2	no	—
120	C₂×C₈	—	271	2	no	—
121	C₂×C₈	—	542	2	no	—
122	C₂×C₈	Dic₅₄₂	1	4	yes	complemented
123	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
124	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
126	C₄⋊C₄	—	271	2	no	—
127	C₄⋊C₄	—	271	4	no	—
128	C₄⋊C₄	—	271	4	no	—
129	C₄⋊C₄	—	271	4	no	—
130	C₄⋊C₄	C₄×D₂₇₁	1	2	yes	—
131	C₂²⋊C₄	—	271	2	no	—
132	C₂²⋊C₄	C₄×D₂₇₁	1	2	yes	—
133	C₄²	—	271	1	no	—
134	C₄²	—	271	1	no	—
135	C₄²	—	271	2	no	—
136	C₄²	—	271	2	no	—
137	C₄²	C₂²×D₂₇₁	1	2	yes	—
138	C₂³	C₂×C₄×D₂₇₁	1	1	yes	characteristic
139	Q₈	—	542	8	no	—
140	C₂×C₄	—	271	2	no	—
141	C₂×C₄	—	271	2	no	—
142	C₂×C₄	—	271	4	no	—
143	C₂×C₄	—	542	4	no	—
144	C₂×C₄	C₂×C₄×D₂₇₁	1	4	yes	—
145	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
146	C₂×C₄	C₂×Dic₅₄₂	1	2	yes	—
147	C₂×C₄	#16864654	1	2	yes	—
148	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	542	2	no	—
151	C₈	—	542	2	no	—
152	C₈	C₂×Dic₅₄₂	1	2	yes	—
153	C₈	C₂×Dic₅₄₂	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₅₄₂	1	2	yes	—
156	C₂²	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	271	4	no	—
159	C₄	—	542	2	no	—
160	C₄	—	542	4	no	—
161	C₄	C₄×Dic₅₄₂	1	2	yes	—
162	C₄	C₂²×Dic₅₄₂	1	1	yes	characteristic
163	C₄	C₂×#16864654	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₅₄₂	1	1	yes	characteristic
166	C₂	#131×D₂₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#405×C₂₇₁	C₂
prime radical	C₂³×C₂₇₁	C₂²×C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	#405	D₂₇₁
hypocenter	C₂₇₁	#2960
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #31774025	C₂	semidirect
3. C₈×Dic₅₄₂	C₂	semidirect
47. C₂×C₈×C₂₇₁	Q₈	semidirect
83. C₂₇₁	#2960	semidirect
122. C₂×C₈	Dic₅₄₂	semidirect

Maximal subgroups

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

subgroup	index
#31774450	2
#31774025	2
C₈×Dic₅₄₂	2
#31773925	2
#31773924	2
#31773920	2
#31773852	2
C₂×C₈×Dic₂₇₁	2
#405×C₂₇₁	2
#2960	271