Subgroups — C₂×#12019 #483392

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#1566	C₂	1	1	yes	maximal, characteristic
4	#12019	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1368	C₂	1	4	yes	maximal, complemented
6	C₂³×C₁₆	C₄	1	2	yes	—
7	C₂×#432	—	2	1	no	—
8	C₂×#423	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#346	C₄	1	1	yes	characteristic
11	C₂×#346	C₄	1	4	yes	—
12	C₂×#346	C₄	1	1	yes	characteristic
13	C₂×#346	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	4	yes	—
15	#1568	—	2	2	no	—
16	#1568	C₂²	1	4	yes	—
17	#1566	—	2	2	no	—
18	#1566	C₂²	1	4	yes	—
19	C₂×#335	—	2	1	no	—
20	C₂×#334	—	2	1	no	—
21	C₂×#334	C₂²	1	2	yes	—
22	C₂×#334	C₂²	1	2	yes	—
23	#1368	C₂²	1	16	yes	complemented
24	C₂²×M₁₆	—	4	1	no	—
25	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₄²	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	—	2	8	no	—
29	C₂²×C₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₁₆	C₂×C₄	1	8	yes	—
31	C₂²×C₁₆	C₂×C₄	1	2	yes	—
32	#432	—	2	4	no	—
33	#432	—	4	2	no	—
34	#423	—	2	4	no	—
35	#423	—	4	2	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	2	2	no	—
38	C₂×C₄⋊C₈	—	4	1	no	—
39	C₂×C₂²⋊C₈	—	4	1	no	—
40	C₂×#97	—	4	1	no	—
41	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	2	yes	—
45	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈	Q₈	1	2	yes	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	2	yes	—
49	#346	C₂×C₄	1	4	yes	—
50	#346	C₂×C₄	1	4	yes	—
51	#346	C₂×C₄	1	4	yes	—
52	#346	C₂³	1	4	yes	—
53	#335	—	2	4	no	—
54	#334	—	2	4	no	—
55	#334	—	2	2	no	—
56	#334	—	2	2	no	—
57	#334	C₂³	1	4	yes	—
58	#334	C₂³	1	4	yes	—
59	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₄	SD₁₆	1	1	yes	characteristic
61	C₂³×C₄	SD₁₆	1	1	yes	characteristic
62	C₂×M₁₆	—	4	4	no	—
63	C₂×M₁₆	—	8	4	no	—
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	—	4	2	no	—
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₄⋊C₄	1	2	yes	—
71	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	2	yes	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₈	C₂×Q₈	1	2	yes	—
76	C₂×C₄²	C₄⋊C₄	1	2	yes	—
77	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄²	SD₁₆	1	1	yes	characteristic
79	C₂×C₄²	SD₁₆	1	1	yes	characteristic
80	C₂×C₄²	SD₁₆	1	2	yes	—
81	C₂×C₄²	SD₁₆	1	2	yes	—
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₄	1	2	yes	—
84	C₂×C₁₆	—	2	4	no	—
85	C₂×C₁₆	—	2	16	no	—
86	C₂×C₁₆	M₁₆	1	8	yes	complemented
87	C₂×C₁₆	C₂²×C₄	1	4	yes	—
88	C₂×C₁₆	C₂²×C₄	1	4	yes	—
89	C₄⋊C₈	—	2	4	no	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	4	no	—
94	C₂²⋊C₈	—	4	4	no	—
95	#97	—	4	4	no	—
96	C₄×C₈	C₄⋊C₄	1	2	yes	—
97	C₄×C₈	C₄⋊C₄	1	2	yes	—
98	C₄×C₈	C₄⋊C₄	1	2	yes	—
99	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₄×C₈	C₂²×C₄	1	2	yes	—
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	C₂×D₄	1	2	yes	—
103	C₄×C₈	C₂×Q₈	1	2	yes	—
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₄×C₈	C₂⁴	1	1	yes	characteristic
106	C₂⁴	#106	1	1	yes	characteristic
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	#106	1	2	yes	—
112	C₂²×C₄	#106	1	2	yes	—
113	C₂²×C₄	#106	1	1	yes	characteristic
114	C₂²×C₄	#106	1	1	yes	characteristic
115	C₂²×C₄	#106	1	1	yes	characteristic
116	C₂²×C₄	#106	1	2	yes	—
117	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
118	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
120	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
123	M₁₆	—	8	8	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	8	4	no	—
133	C₂×C₈	—	8	4	no	—
134	C₂×C₈	#108	1	4	yes	—
135	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₈	C₂×M₁₆	1	4	yes	—
140	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
142	C₄²	#106	1	2	yes	—
143	C₄²	#106	1	2	yes	—
144	C₄²	#106	1	2	yes	—
145	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₄²	C₂×C₄⋊C₄	1	2	yes	—
147	C₄²	C₂×SD₁₆	1	2	yes	—
148	C₄²	C₂×SD₁₆	1	2	yes	—
149	C₄²	C₂×SD₁₆	1	1	yes	characteristic
150	C₄²	C₂×SD₁₆	1	1	yes	characteristic
151	C₄²	C₂²×D₄	1	1	yes	characteristic
152	C₁₆	—	2	4	no	—
153	C₁₆	C₂×M₁₆	1	8	yes	complemented
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	#365	1	4	yes	—
157	C₂³	C₂×#106	1	2	yes	—
158	C₂³	C₂×#106	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	#423	1	2	yes	—
172	C₂×C₄	C₂×#106	1	2	yes	—
173	C₂×C₄	C₂×#106	1	1	yes	characteristic
174	C₂×C₄	C₂×#106	1	2	yes	—
175	C₂×C₄	C₂×#106	1	1	yes	characteristic
176	C₂×C₄	C₂×#106	1	2	yes	—
177	C₂×C₄	C₂×#106	1	1	yes	characteristic
178	C₂×C₄	C₂×#108	1	2	yes	—
179	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
181	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	8	4	no	—
186	C₈	—	8	4	no	—
187	C₈	#423	1	2	yes	—
188	C₈	C₂×#108	1	4	yes	—
189	C₈	C₂²×M₁₆	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	8	no	—
193	C₂²	#1566	1	2	yes	—
194	C₂²	C₂×#365	1	4	yes	—
195	C₂²	C₂×#365	1	2	yes	—
196	C₂²	#2155	1	2	yes	—
197	C₂²	C₂²×#106	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	#1566	1	2	yes	—
206	C₄	C₂×#423	1	1	yes	characteristic
207	C₄	C₂²×#108	1	1	yes	characteristic
208	C₂	—	2	4	no	—
209	C₂	#12019	1	4	yes	direct product factor
210	C₂	C₂×#1566	1	1	yes	characteristic
211	C₂	C₂²×#365	1	1	yes	characteristic
212	C₂	C₂×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #12019	C₂	direct (N × H)
5. C₂×#1368	C₂	semidirect
23. #1368	C₂²	semidirect
86. C₂×C₁₆	M₁₆	semidirect
153. C₁₆	C₂×M₁₆	semidirect
209. C₂	#12019	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#346	2
C₂×#1568	2
C₂×#1566	2
#12019	2
C₂×#1368	2

Subgroups — C2×#12019 #483392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12019 #483392