Subgroups — #14566547

Order 2432.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10597333	C₂	1	2	yes	maximal
3	#11926440	C₂	1	2	yes	maximal
4	#11926439	C₂	1	4	yes	maximal
5	#11926413	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₁₉	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#384	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉	C₂²	1	4	yes	—
22	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
23	#10597333	—	2	8	no	—
24	C₄×Dic₃₈	C₄	1	8	yes	complemented
25	C₁₉⋊#95	C₂²	1	2	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₉	C₂³	1	2	yes	—
32	C₂²×Dic₁₉	C₂³	1	2	yes	—
33	C₂²×Dic₁₉	C₂³	1	2	yes	—
34	C₂²×Dic₁₉	C₂³	1	2	yes	—
35	C₂×Dic₃₈	C₂×C₄	1	4	yes	—
36	C₂×Dic₃₈	D₄	1	4	yes	—
37	C₂×Dic₃₈	Q₈	1	4	yes	—
38	C₄⋊C₄×C₁₉	—	2	4	no	—
39	C₄⋊C₄×C₁₉	—	2	4	no	—
40	C₄⋊C₄×C₁₉	—	4	2	no	—
41	C₄²×C₁₉	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₉	—	2	2	no	—
43	C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₉	Q₈	1	4	yes	complemented
45	Dic₁₉⋊C₄	—	2	8	no	—
46	Dic₁₉⋊C₄	—	4	4	no	—
47	Dic₁₉⋊C₄	C₂×C₄	1	8	yes	—
48	C₄×Dic₁₉	—	2	4	no	—
49	C₄×Dic₁₉	—	2	8	no	—
50	C₄×Dic₁₉	C₂×C₄	1	8	yes	—
51	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	4	no	—
55	C₂×C₄×C₁₉	—	2	4	no	—
56	C₂×C₄×C₁₉	—	2	4	no	—
57	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₉	#41	1	2	yes	—
64	C₂×Dic₁₉	—	2	2	no	—
65	C₂×Dic₁₉	—	2	2	no	—
66	C₂×Dic₁₉	—	2	2	no	—
67	C₂×Dic₁₉	—	2	2	no	—
68	C₂×Dic₁₉	—	2	2	no	—
69	C₂×Dic₁₉	—	2	2	no	—
70	C₂×Dic₁₉	—	2	2	no	—
71	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
73	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
74	C₂×Dic₁₉	C₂×Q₈	1	4	yes	—
75	C₂×Dic₁₉	C₂×Q₈	1	4	yes	—
76	Dic₃₈	C₄⋊C₄	1	16	yes	complemented
77	Q₈×C₄⋊C₄	—	19	1	no	maximal, Sylow
78	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
83	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₁₉	—	2	2	no	—
86	C₄×C₁₉	—	4	4	no	—
87	C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
88	C₄×C₁₉	C₄×Q₈	1	4	yes	complemented
89	Dic₁₉	—	2	4	no	—
90	Dic₁₉	—	2	4	no	—
91	Dic₁₉	—	4	2	no	—
92	Dic₁₉	C₂×C₄⋊C₄	1	8	yes	—
93	C₂×C₄×Q₈	—	19	2	no	—
94	C₂×C₄×Q₈	—	19	1	no	—
95	#389	—	19	4	no	—
96	#389	—	19	2	no	—
97	#384	—	19	2	no	—
98	#384	D₁₉	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	19	2	no	—
100	C₄×C₄⋊C₄	—	19	1	no	—
101	C₂×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₂×C₁₉	#520	1	1	yes	characteristic
104	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
105	C₂×C₁₉	#550	1	1	yes	characteristic
106	C₂×C₁₉	#557	1	1	yes	characteristic
107	C₂×C₁₉	Q₈²	1	1	yes	characteristic
108	C₂²×Q₈	—	19	1	no	—
109	C₄×Q₈	—	19	8	no	—
110	C₄×Q₈	—	38	8	no	—
111	C₂×C₄⋊C₄	—	19	2	no	—
112	C₂×C₄⋊C₄	—	19	1	no	—
113	C₂×C₄⋊C₄	—	19	1	no	—
114	C₂×C₄⋊C₄	—	19	4	no	—
115	C₂×C₄⋊C₄	—	19	1	no	—
116	C₂×C₄⋊C₄	—	19	4	no	—
117	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
120	C₂×C₄²	—	19	2	no	—
121	C₂×C₄²	—	19	4	no	—
122	C₂×C₄²	—	19	2	no	—
123	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
124	#95	—	19	4	no	—
125	#95	—	19	2	no	—
126	C₁₉	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₈	—	19	4	no	—
128	C₂×Q₈	—	19	4	no	—
129	C₂×Q₈	—	19	4	no	—
130	C₂²×C₄	—	19	2	no	—
131	C₂²×C₄	—	19	2	no	—
132	C₂²×C₄	—	19	2	no	—
133	C₂²×C₄	—	19	2	no	—
134	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	19	8	no	—
142	C₄⋊C₄	—	19	4	no	—
143	C₄⋊C₄	—	19	4	no	—
144	C₄⋊C₄	—	38	8	no	—
145	C₄⋊C₄	—	38	2	no	—
146	C₄⋊C₄	—	76	4	no	—
147	C₄²	—	19	8	no	—
148	C₄²	—	38	8	no	—
149	C₄²	—	38	4	no	—
150	C₄²	C₄×D₁₉	1	4	yes	—
151	C₂³	C₂³×D₁₉	1	1	yes	characteristic
152	Q₈	—	19	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	19	4	no	—
159	C₂×C₄	—	19	4	no	—
160	C₂×C₄	—	19	4	no	—
161	C₂×C₄	—	19	4	no	—
162	C₂×C₄	—	19	4	no	—
163	C₂×C₄	—	38	2	no	—
164	C₂×C₄	—	38	2	no	—
165	C₂×C₄	—	38	2	no	—
166	C₂×C₄	—	38	2	no	—
167	C₂×C₄	—	38	2	no	—
168	C₂×C₄	—	38	2	no	—
169	C₂×C₄	—	38	2	no	—
170	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
173	C₂×C₄	D₄×D₁₉	1	2	yes	—
174	C₂×C₄	#64543	1	2	yes	—
175	C₂×C₄	Q₈×D₁₉	1	2	yes	—
176	C₂×C₄	Q₈×D₁₉	1	2	yes	—
177	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
179	C₂²	C₂×#64543	1	1	yes	characteristic
180	C₂²	#10597430	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
182	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
183	C₂²	#10597433	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	19	8	no	—
187	C₄	—	38	4	no	—
188	C₄	—	38	4	no	—
189	C₄	—	76	2	no	—
190	C₄	#10597333	1	4	yes	complemented
191	C₄	C₄⋊C₄×D₁₉	1	4	yes	—
192	C₂	C₂×C₄⋊C₄×D₁₉	1	1	yes	characteristic
193	C₂	C₂×#10597333	1	1	yes	characteristic
194	C₂	#11927025	1	1	yes	characteristic
195	C₂	#11927209	1	1	yes	characteristic
196	C₂	#11927214	1	1	yes	characteristic
197	C₂	#11927269	1	1	yes	characteristic
198	C₂	#11927274	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#384	C₂
socle, prime radical, cosocle	C₂³×C₁₉	C₂⁴
hypercenter	#384	D₁₉
hypocenter	C₁₉	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
24. C₄×Dic₃₈	C₄	semidirect
44. C₄⋊Dic₁₉	Q₈	semidirect
76. Dic₃₈	C₄⋊C₄	semidirect
88. C₄×C₁₉	C₄×Q₈	semidirect
126. C₁₉	Q₈×C₄⋊C₄	semidirect
190. C₄	#10597333	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₈	2
C₂×#10597333	2
#11926440	2
#11926439	2
#11926413	2
C₄⋊C₄×Dic₁₉	2
C₄×C₄⋊Dic₁₉	2
C₁₉×#384	2
Q₈×C₄⋊C₄	19