Subgroups — C₂×#14521091 #16854296

Order 3968.

536 conjugacy classes of subgroups.

The subgroup lattice has 182 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	#14521091	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758406	C₂	1	2	yes	maximal, complemented
5	C₂×#11758405	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₃₁	—	2	1	no	—
9	C₂×M₁₆×C₃₁	C₄	1	8	yes	complemented
10	C₂×M₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
13	#11758503	—	2	2	no	—
14	#11758503	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₃₁	—	2	1	no	—
16	C₂×C₄⋊Dic₃₁	—	2	2	no	—
17	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₁	—	2	1	no	—
22	#11758406	C₂²	1	8	yes	complemented
23	#11758405	C₂²	1	8	yes	complemented
24	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	D₄	1	2	yes	—
28	C₂²×C₄×C₃₁	Q₈	1	4	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×Dic₃₁	—	2	1	no	—
32	C₂²×Dic₃₁	—	2	1	no	—
33	C₂²×Dic₃₁	—	2	4	no	—
34	C₂²×Dic₃₁	—	2	2	no	—
35	C₂²×Dic₃₁	—	4	4	no	—
36	M₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
37	C₂×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₃₁	C₂³	1	4	yes	—
39	C₂²⋊Dic₃₁	—	2	4	no	—
40	C₄⋊Dic₃₁	—	2	4	no	—
41	C₄⋊Dic₃₁	—	2	2	no	—
42	C₄⋊Dic₃₁	—	2	1	no	—
43	C₄⋊Dic₃₁	—	2	1	no	—
44	C₄⋊Dic₃₁	C₂³	1	2	yes	—
45	C₄⋊Dic₃₁	C₂³	1	4	yes	—
46	C₄⋊Dic₃₁	C₂³	1	2	yes	—
47	C₄×Dic₃₁	—	2	4	no	—
48	C₂³×C₃₁	—	2	2	no	—
49	C₂³×C₃₁	—	2	2	no	—
50	C₂³×C₃₁	C₄⋊C₄	1	4	yes	—
51	C₂³×C₃₁	C₂×D₄	1	2	yes	—
52	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
54	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₃₁	—	2	2	no	—
63	C₂×Dic₃₁	—	2	2	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	C₂²×C₄	1	8	yes	complemented
70	C₂×#428	—	31	1	no	maximal, Sylow
71	C₂²×C₃₁	—	2	2	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	8	no	—
74	C₂²×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂²×C₃₁	#137	1	2	yes	—
78	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
83	Dic₃₁	—	4	2	no	—
84	Dic₃₁	—	4	4	no	—
85	Dic₃₁	—	4	2	no	—
86	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
87	C₂×#117	—	31	1	no	—
88	C₂²×C₄⋊C₄	—	31	1	no	—
89	#428	—	31	8	no	—
90	C₂×#107	—	31	2	no	—
91	C₂×#106	—	31	2	no	—
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	#428	1	4	yes	complemented
94	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
97	C₂³×C₄	—	62	1	no	—
98	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
99	C₂×M₁₆	Dic₃₁	1	8	yes	complemented
100	C₂×M₁₆	C₂×D₃₁	1	4	yes	—
101	C₂²×C₈	C₂×D₃₁	1	2	yes	—
102	#117	—	31	4	no	—
103	#117	—	62	2	no	—
104	C₂×C₄⋊C₄	—	31	2	no	—
105	C₂×C₄⋊C₄	—	31	4	no	—
106	C₂×C₄⋊C₄	—	31	1	no	—
107	C₂×C₄⋊C₄	—	31	1	no	—
108	C₂×C₄⋊C₄	—	62	2	no	—
109	C₂×C₄⋊C₄	—	62	1	no	—
110	C₂×C₂²⋊C₄	—	62	1	no	—
111	C₂×C₄²	—	62	1	no	—
112	#107	—	31	8	no	—
113	#106	—	31	8	no	—
114	C₃₁	C₂×#428	1	1	yes	characteristic, complemented, Sylow
115	C₂⁴	D₁₂₄	1	1	yes	characteristic
116	C₂²×C₄	—	62	4	no	—
117	C₂²×C₄	—	62	1	no	—
118	C₂²×C₄	—	62	1	no	—
119	C₂²×C₄	—	62	2	no	—
120	C₂²×C₄	—	124	4	no	—
121	C₂²×C₄	Dic₆₂	1	4	yes	—
122	C₂²×C₄	D₁₂₄	1	2	yes	—
123	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
125	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
127	M₁₆	C₂×Dic₃₁	1	16	yes	complemented
128	C₂×C₈	C₂×Dic₃₁	1	8	yes	complemented
129	C₂×C₈	C₂²×D₃₁	1	4	yes	—
130	C₄⋊C₄	—	31	2	no	—
131	C₄⋊C₄	—	31	4	no	—
132	C₄⋊C₄	—	31	2	no	—
133	C₄⋊C₄	—	62	4	no	—
134	C₄⋊C₄	—	62	2	no	—
135	C₄⋊C₄	—	62	1	no	—
136	C₄⋊C₄	—	62	1	no	—
137	C₂²⋊C₄	—	62	4	no	—
138	C₄²	—	62	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	C₄⋊Dic₃₁	1	4	yes	—
142	C₂³	C₂×D₁₂₄	1	2	yes	—
143	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
144	C₂×C₄	—	62	2	no	—
145	C₂×C₄	—	62	2	no	—
146	C₂×C₄	—	62	4	no	—
147	C₂×C₄	—	124	8	no	—
148	C₂×C₄	—	124	2	no	—
149	C₂×C₄	—	124	4	no	—
150	C₂×C₄	—	124	2	no	—
151	C₂×C₄	C₄⋊Dic₃₁	1	8	yes	—
152	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
153	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
154	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
155	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
156	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
158	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
159	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
160	C₈	C₂²×Dic₃₁	1	8	yes	complemented
161	C₂²	—	2	8	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	#11758485	1	2	yes	—
165	C₂²	#11758486	1	2	yes	—
166	C₂²	C₂×C₄⋊Dic₃₁	1	4	yes	—
167	C₂²	C₂×C₄⋊Dic₃₁	1	2	yes	—
168	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
169	C₄	—	124	4	no	—
170	C₄	—	124	2	no	—
171	C₄	—	124	2	no	—
172	C₄	C₂×C₄⋊Dic₃₁	1	4	yes	—
173	C₄	C₂×C₄⋊Dic₃₁	1	2	yes	—
174	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
175	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#14521091	1	4	yes	direct product factor
178	C₂	C₂²×C₄⋊Dic₃₁	1	1	yes	characteristic
179	C₂	C₂×#11758485	1	1	yes	characteristic
180	C₂	C₂×#11758486	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	C₂×#428
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. #14521091	C₂	direct (N × H)
4. C₂×#11758406	C₂	semidirect
5. C₂×#11758405	C₂	semidirect
9. C₂×M₁₆×C₃₁	C₄	semidirect
22. #11758406	C₂²	semidirect
23. #11758405	C₂²	semidirect
36. M₁₆×C₃₁	C₂×C₄	semidirect
37. C₂×C₈×C₃₁	C₂×C₄	semidirect
69. C₈×C₃₁	C₂²×C₄	semidirect
93. C₂×C₃₁	#428	semidirect
99. C₂×M₁₆	Dic₃₁	semidirect
114. C₃₁	C₂×#428	semidirect
127. M₁₆	C₂×Dic₃₁	semidirect
128. C₂×C₈	C₂×Dic₃₁	semidirect
160. C₈	C₂²×Dic₃₁	semidirect
177. C₂	#14521091	direct (N × H)

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
#14521091	2
C₂×#11758406	2
C₂×#11758405	2
C₂²×M₁₆×C₃₁	2
C₂×#428	31

Subgroups — C2×#14521091 #16854296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521091 #16854296