Subgroups — #16845431

Order 3968.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520812	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂³×Dic₃₁	—	2	1	no	—
7	C₂×M₁₆×C₃₁	—	2	2	no	—
8	C₂×M₁₆×C₃₁	—	2	2	no	—
9	C₂×M₁₆×C₃₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
12	#11758503	—	2	2	no	—
13	#11758503	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₃₁	—	2	1	no	—
15	C₂×C₄⋊Dic₃₁	—	2	2	no	—
16	C₂×C₄⋊Dic₃₁	C₄	1	4	yes	complemented
17	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂×C₄×Dic₃₁	—	2	1	no	—
20	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	D₄	1	2	yes	—
26	C₂²×C₄×C₃₁	D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₁	—	2	4	no	—
30	C₂²×Dic₃₁	—	2	2	no	—
31	C₂²×Dic₃₁	—	2	2	no	—
32	C₂²×Dic₃₁	—	4	4	no	—
33	M₁₆×C₃₁	—	2	8	no	—
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	—	2	2	no	—
36	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
37	C₂²⋊Dic₃₁	—	2	4	no	—
38	C₄⋊Dic₃₁	—	2	4	no	—
39	C₄⋊Dic₃₁	—	2	2	no	—
40	C₄⋊Dic₃₁	—	2	2	no	—
41	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	complemented
43	C₄×Dic₃₁	—	2	4	no	—
44	C₂³×C₃₁	—	2	2	no	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₄²	1	4	yes	—
51	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	—	2	4	no	—
65	C₂×Dic₃₁	—	4	8	no	—
66	C₂×Dic₃₁	—	4	2	no	—
67	C₂×Dic₃₁	—	4	4	no	—
68	C₂×Dic₃₁	—	4	2	no	—
69	C₈×C₃₁	—	2	4	no	—
70	#1736	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	2	4	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	2	4	no	—
75	C₂²×C₃₁	#95	1	4	yes	—
76	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
80	C₂²×C₃₁	#137	1	2	yes	—
81	C₄×C₃₁	#95	1	4	yes	—
82	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
83	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Dic₃₁	—	4	4	no	—
87	Dic₃₁	—	4	4	no	—
88	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
89	C₂×#117	—	31	1	no	—
90	C₂²×C₄⋊C₄	—	31	1	no	—
91	#340	—	31	4	no	—
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
94	C₂×C₃₁	#417	1	2	yes	—
95	C₂×C₃₁	#418	1	1	yes	characteristic
96	C₂×C₃₁	#419	1	1	yes	characteristic
97	C₂×C₃₁	#428	1	2	yes	—
98	C₂³×C₄	—	62	1	no	—
99	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	Dic₃₁	1	4	yes	complemented
103	C₂²×C₈	C₂×D₃₁	1	2	yes	—
104	#117	—	31	4	no	—
105	#117	—	62	2	no	—
106	C₂×C₄⋊C₄	—	31	4	no	—
107	C₂×C₄⋊C₄	—	31	2	no	—
108	C₂×C₄⋊C₄	—	31	2	no	—
109	C₂×C₄⋊C₄	—	62	1	no	—
110	C₂×C₄⋊C₄	—	62	2	no	—
111	C₂×C₂²⋊C₄	—	62	1	no	—
112	C₂×C₄²	—	62	1	no	—
113	C₃₁	#1736	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	D₁₂₄	1	1	yes	characteristic
115	C₂²×C₄	—	62	4	no	—
116	C₂²×C₄	—	62	2	no	—
117	C₂²×C₄	—	62	2	no	—
118	C₂²×C₄	—	124	4	no	—
119	C₂²×C₄	Dic₆₂	1	2	yes	—
120	C₂²×C₄	C₄×D₃₁	1	2	yes	—
121	C₂²×C₄	C₄×D₃₁	1	2	yes	—
122	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
125	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂×Dic₃₁	1	4	yes	complemented
131	C₄⋊C₄	—	31	4	no	—
132	C₄⋊C₄	—	31	4	no	—
133	C₄⋊C₄	—	62	4	no	—
134	C₄⋊C₄	—	62	2	no	—
135	C₄⋊C₄	—	62	2	no	—
136	C₂²⋊C₄	—	62	4	no	—
137	C₄²	—	62	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₄⋊Dic₃₁	1	2	yes	—
141	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
142	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
143	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
144	C₂×C₄	—	62	4	no	—
145	C₂×C₄	—	62	4	no	—
146	C₂×C₄	—	124	8	no	—
147	C₂×C₄	—	124	2	no	—
148	C₂×C₄	—	124	4	no	—
149	C₂×C₄	—	124	2	no	—
150	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
151	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
152	C₂×C₄	Dic₃₁⋊C₄	1	4	yes	—
153	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
154	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
155	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
157	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	C₃₁⋊#95	1	4	yes	—
169	C₂²	#11758485	1	2	yes	—
170	C₂²	#11758486	1	2	yes	—
171	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
172	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
173	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
174	C₄	—	124	4	no	—
175	C₄	—	124	4	no	—
176	C₄	C₃₁⋊#95	1	4	yes	—
177	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
178	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
179	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
180	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₃₁⋊#95	1	1	yes	characteristic
183	C₂	#14521051	1	2	yes	—
184	C₂	#14521055	1	1	yes	characteristic
185	C₂	#14521059	1	1	yes	characteristic
186	C₂	#14521091	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	#1736
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14520812	C₂	semidirect
9. C₂×M₁₆×C₃₁	C₄	semidirect
13. #11758503	C₄	semidirect
16. C₂×C₄⋊Dic₃₁	C₄	semidirect
36. C₂×C₈×C₃₁	C₂×C₄	semidirect
42. C₄⋊Dic₃₁	C₂×C₄	semidirect
102. C₂×M₁₆	Dic₃₁	semidirect
113. C₃₁	#1736	semidirect
130. C₂×C₈	C₂×Dic₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758503	2
C₂²×C₄⋊Dic₃₁	2
#14520812	2
C₂²×M₁₆×C₃₁	2
#1736	31