Subgroups — C₂×C₁₉×#3483 #18068675

Order 4864.

8418 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3483	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#555	C₂	1	9	yes	maximal, complemented
4	C₂×C₁₉×#550	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#532	C₂	1	9	yes	maximal, complemented
6	C₂×C₄×#41×C₁₉	C₂	1	6	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₂²	1	6	yes	—
8	C₂²×#41×C₁₉	C₂²	1	6	yes	—
9	Q₈²×C₁₉	C₂²	1	16	yes	complemented
10	C₁₉×#555	C₂²	1	144	yes	complemented
11	C₁₉×#550	C₂²	1	96	yes	complemented
12	C₁₉×#532	C₂²	1	144	yes	complemented
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	9	yes	—
14	C₂×C₁₉×#127	C₂²	1	9	yes	—
15	C₂×C₁₉×#124	C₂²	1	18	yes	—
16	C₂×C₁₉×#121	C₂²	1	36	yes	—
17	C₄×#41×C₁₉	C₂²	1	96	yes	complemented
18	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	6	yes	—
20	C₂×C₄×D₄×C₁₉	C₂²	1	18	yes	—
21	C₂×C₄×D₄×C₁₉	C₂²	1	18	yes	—
22	C₂×C₁₉×#117	C₂²	1	18	yes	—
23	C₂²×C₄²×C₁₉	C₂²	1	9	yes	—
24	C₂×#41×C₁₉	—	2	48	no	—
25	C₂×#41×C₁₉	—	4	24	no	—
26	C₂×#41×C₁₉	C₂³	1	48	yes	—
27	C₂×#41×C₁₉	C₂³	1	48	yes	—
28	C₂²×Q₈×C₁₉	C₂³	1	6	yes	—
29	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
30	C₂²×D₄×C₁₉	C₂³	1	18	yes	—
31	C₂²×D₄×C₁₉	C₂³	1	18	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	18	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	9	yes	—
34	C₁₉×C₄⋊Q₈	C₂³	1	72	yes	—
35	C₁₉×#127	C₂³	1	72	yes	—
36	C₁₉×#124	C₂³	1	144	yes	—
37	C₁₉×#121	C₂³	1	288	yes	—
38	C₄×Q₈×C₁₉	C₂³	1	16	yes	—
39	C₄×Q₈×C₁₉	C₂³	1	48	yes	—
40	C₄×D₄×C₁₉	C₂³	1	144	yes	—
41	C₄×D₄×C₁₉	C₂³	1	144	yes	—
42	C₁₉×#117	C₂³	1	144	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	18	yes	—
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	36	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	9	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	9	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	72	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
50	C₂⁴×C₁₉	C₂⁴	1	9	yes	—
51	#41×C₁₉	—	2	192	no	—
52	#41×C₁₉	—	4	96	no	—
53	C₂×Q₈×C₁₉	—	4	12	no	—
54	C₂×Q₈×C₁₉	#41	1	16	yes	complemented
55	C₂×Q₈×C₁₉	C₂⁴	1	24	yes	—
56	C₂×Q₈×C₁₉	C₂⁴	1	8	yes	—
57	C₂×D₄×C₁₉	—	2	72	no	—
58	C₂×D₄×C₁₉	—	4	36	no	—
59	C₂×D₄×C₁₉	C₂⁴	1	72	yes	—
60	C₂×D₄×C₁₉	C₂⁴	1	72	yes	—
61	C₂²×C₄×C₁₉	—	2	72	no	—
62	C₂²×C₄×C₁₉	—	2	36	no	—
63	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	9	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	72	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	36	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	C₂⁴	1	24	yes	—
70	C₄⋊C₄×C₁₉	C₂⁴	1	72	yes	—
71	C₂²⋊C₄×C₁₉	C₂⁴	1	144	yes	—
72	C₄²×C₁₉	C₂⁴	1	24	yes	—
73	C₄²×C₁₉	C₂⁴	1	36	yes	—
74	C₄²×C₁₉	C₂⁴	1	36	yes	—
75	C₂×#3483	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₉	—	2	36	no	—
77	C₂³×C₁₉	—	2	18	no	—
78	C₂³×C₁₉	C₂⁵	1	18	yes	—
79	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
80	Q₈×C₁₉	—	4	24	no	—
81	Q₈×C₁₉	C₂×#41	1	32	yes	complemented
82	D₄×C₁₉	—	2	144	no	—
83	D₄×C₁₉	—	4	72	no	—
84	C₂×C₄×C₁₉	—	2	12	no	—
85	C₂×C₄×C₁₉	—	2	18	no	—
86	C₂×C₄×C₁₉	—	2	144	no	—
87	C₂×C₄×C₁₉	—	2	72	no	—
88	C₂×C₄×C₁₉	C₂×#41	1	24	yes	—
89	C₂×C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
90	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
91	C₂×C₄×C₁₉	C₂⁵	1	18	yes	—
92	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
93	C₂×C₄×C₁₉	C₂⁵	1	2	yes	—
94	#3483	C₂×C₁₉	1	32	yes	direct product factor
95	C₂×Q₈²	C₂×C₁₉	1	1	yes	characteristic, complemented
96	C₂×#555	C₂×C₁₉	1	9	yes	complemented
97	C₂×#550	C₂×C₁₉	1	6	yes	complemented
98	C₂×#532	C₂×C₁₉	1	9	yes	complemented
99	C₂×C₄×#41	C₂×C₁₉	1	6	yes	complemented
100	C₂²×C₁₉	—	2	36	no	—
101	C₂²×C₁₉	—	2	18	no	—
102	C₂²×C₁₉	—	4	36	no	—
103	C₂²×C₁₉	C₂²×#41	1	4	yes	—
104	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
105	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
106	C₄×C₁₉	—	2	12	no	—
107	C₄×C₁₉	—	2	18	no	—
108	C₄×C₁₉	C₂²×#41	1	24	yes	—
109	C₄×C₁₉	C₂×H(5,𝔽₂)	1	4	yes	—
110	C₂²×#41	C₂²×C₁₉	1	6	yes	—
111	C₂²×#41	C₂²×C₁₉	1	6	yes	—
112	Q₈²	C₂²×C₁₉	1	16	yes	complemented
113	#555	C₂²×C₁₉	1	144	yes	complemented
114	#550	C₂²×C₁₉	1	96	yes	complemented
115	#532	C₂²×C₁₉	1	144	yes	complemented
116	C₂×C₄⋊Q₈	C₂²×C₁₉	1	9	yes	—
117	C₂×#127	C₂²×C₁₉	1	9	yes	—
118	C₂×#124	C₂²×C₁₉	1	18	yes	—
119	C₂×#121	C₂²×C₁₉	1	36	yes	—
120	C₄×#41	C₂²×C₁₉	1	96	yes	complemented
121	C₂×C₄×Q₈	C₂²×C₁₉	1	2	yes	—
122	C₂×C₄×Q₈	C₂²×C₁₉	1	6	yes	—
123	C₂×C₄×D₄	C₂²×C₁₉	1	18	yes	—
124	C₂×C₄×D₄	C₂²×C₁₉	1	18	yes	—
125	C₂×#117	C₂²×C₁₉	1	18	yes	—
126	C₂²×C₄²	C₂²×C₁₉	1	9	yes	—
127	C₂×C₁₉	—	4	18	no	—
128	C₂×C₁₉	#3483	1	4	yes	direct product factor
129	C₂×C₁₉	C₂³×#41	1	2	yes	—
130	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×#41	—	2	48	no	—
132	C₂×#41	—	4	24	no	—
133	C₂×#41	C₂³×C₁₉	1	48	yes	—
134	C₂×#41	C₂³×C₁₉	1	48	yes	—
135	C₂²×Q₈	C₂³×C₁₉	1	6	yes	—
136	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
137	C₂²×D₄	C₂³×C₁₉	1	18	yes	—
138	C₂²×D₄	C₂³×C₁₉	1	18	yes	—
139	C₂³×C₄	C₂³×C₁₉	1	18	yes	—
140	C₂³×C₄	C₂³×C₁₉	1	9	yes	—
141	C₄⋊Q₈	C₂³×C₁₉	1	72	yes	—
142	#127	C₂³×C₁₉	1	72	yes	—
143	#124	C₂³×C₁₉	1	144	yes	—
144	#121	C₂³×C₁₉	1	288	yes	—
145	C₄×Q₈	C₂³×C₁₉	1	16	yes	—
146	C₄×Q₈	C₂³×C₁₉	1	48	yes	—
147	C₄×D₄	C₂³×C₁₉	1	144	yes	—
148	C₄×D₄	C₂³×C₁₉	1	144	yes	—
149	#117	C₂³×C₁₉	1	144	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₉	1	18	yes	—
152	C₂×C₂²⋊C₄	C₂³×C₁₉	1	36	yes	—
153	C₂×C₄²	C₂³×C₁₉	1	9	yes	—
154	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
155	C₂×C₄²	C₂³×C₁₉	1	9	yes	—
156	C₂×C₄²	C₂³×C₁₉	1	72	yes	—
157	C₁₉	C₂×#3483	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	C₂⁴×C₁₉	1	9	yes	—
159	#41	—	2	192	no	—
160	#41	—	4	96	no	—
161	C₂×Q₈	—	4	12	no	—
162	C₂×Q₈	#41×C₁₉	1	16	yes	complemented
163	C₂×Q₈	C₂⁴×C₁₉	1	24	yes	—
164	C₂×Q₈	C₂⁴×C₁₉	1	8	yes	—
165	C₂×D₄	—	2	72	no	—
166	C₂×D₄	—	4	36	no	—
167	C₂×D₄	C₂⁴×C₁₉	1	72	yes	—
168	C₂×D₄	C₂⁴×C₁₉	1	72	yes	—
169	C₂²×C₄	—	2	72	no	—
170	C₂²×C₄	—	2	36	no	—
171	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₁₉	1	9	yes	—
173	C₂²×C₄	C₂⁴×C₁₉	1	72	yes	—
174	C₂²×C₄	C₂⁴×C₁₉	1	36	yes	—
175	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
176	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
177	C₄⋊C₄	C₂⁴×C₁₉	1	24	yes	—
178	C₄⋊C₄	C₂⁴×C₁₉	1	72	yes	—
179	C₂²⋊C₄	C₂⁴×C₁₉	1	144	yes	—
180	C₄²	C₂⁴×C₁₉	1	24	yes	—
181	C₄²	C₂⁴×C₁₉	1	36	yes	—
182	C₄²	C₂⁴×C₁₉	1	36	yes	—
183	C₂³	—	2	36	no	—
184	C₂³	—	2	18	no	—
185	C₂³	C₂⁵×C₁₉	1	18	yes	—
186	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
187	Q₈	—	4	24	no	—
188	Q₈	C₂×#41×C₁₉	1	32	yes	complemented
189	D₄	—	2	144	no	—
190	D₄	—	4	72	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	18	no	—
193	C₂×C₄	—	2	144	no	—
194	C₂×C₄	—	2	72	no	—
195	C₂×C₄	C₂×#41×C₁₉	1	24	yes	—
196	C₂×C₄	C₁₉×H(5,𝔽₂)	1	4	yes	—
197	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
198	C₂×C₄	C₂⁵×C₁₉	1	18	yes	—
199	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
200	C₂×C₄	C₂⁵×C₁₉	1	2	yes	—
201	C₂²	—	2	36	no	—
202	C₂²	—	2	18	no	—
203	C₂²	—	4	36	no	—
204	C₂²	C₂²×#41×C₁₉	1	4	yes	—
205	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
206	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
207	C₄	—	2	12	no	—
208	C₄	—	2	18	no	—
209	C₄	C₂²×#41×C₁₉	1	24	yes	—
210	C₄	C₂×C₁₉×H(5,𝔽₂)	1	4	yes	—
211	C₂	—	4	18	no	—
212	C₂	C₁₉×#3483	1	4	yes	direct product factor
213	C₂	C₂³×#41×C₁₉	1	2	yes	—
214	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
socle	C₂³×C₁₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3483	C₂	direct (N × H)
2. C₂×Q₈²×C₁₉	C₂	semidirect
3. C₂×C₁₉×#555	C₂	semidirect
4. C₂×C₁₉×#550	C₂	semidirect
5. C₂×C₁₉×#532	C₂	semidirect
6. C₂×C₄×#41×C₁₉	C₂	semidirect
9. Q₈²×C₁₉	C₂²	semidirect
10. C₁₉×#555	C₂²	semidirect
11. C₁₉×#550	C₂²	semidirect
12. C₁₉×#532	C₂²	semidirect
17. C₄×#41×C₁₉	C₂²	semidirect
54. C₂×Q₈×C₁₉	#41	semidirect
75. C₂×#3483	C₁₉	direct (N × H)
81. Q₈×C₁₉	C₂×#41	semidirect
94. #3483	C₂×C₁₉	direct (N × H)
95. C₂×Q₈²	C₂×C₁₉	semidirect
96. C₂×#555	C₂×C₁₉	semidirect
97. C₂×#550	C₂×C₁₉	semidirect
98. C₂×#532	C₂×C₁₉	semidirect
99. C₂×C₄×#41	C₂×C₁₉	semidirect
112. Q₈²	C₂²×C₁₉	semidirect
113. #555	C₂²×C₁₉	semidirect
114. #550	C₂²×C₁₉	semidirect
115. #532	C₂²×C₁₉	semidirect
120. C₄×#41	C₂²×C₁₉	semidirect
128. C₂×C₁₉	#3483	direct (N × H)
157. C₁₉	C₂×#3483	direct (N × H)
162. C₂×Q₈	#41×C₁₉	semidirect
188. Q₈	C₂×#41×C₁₉	semidirect
212. C₂	C₁₉×#3483	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3483	2
C₂×Q₈²×C₁₉	2
C₂×C₁₉×#555	2
C₂×C₁₉×#550	2
C₂×C₁₉×#532	2
C₂×C₄×#41×C₁₉	2
C₂×#3483	19

Subgroups — C2×C19×#3483 #18068675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#3483 #18068675