Subgroups — #14568661

Order 2432.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597315	C₂	1	2	yes	maximal
2	C₂×#10597334	C₂	1	2	yes	maximal
3	#11926465	C₂	1	4	yes	maximal
4	#11926443	C₂	1	2	yes	maximal
5	#11926442	C₂	1	2	yes	maximal
6	#11926414	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₈	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₉	C₂²	1	4	yes	—
17	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
22	#10597334	—	2	8	no	—
23	#10597315	—	2	8	no	—
24	C₁₉×#95	C₂²	1	2	yes	—
25	C₁₉⋊#95	C₂²	1	2	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₉	C₂³	1	4	yes	—
33	C₂²×Dic₁₉	C₂³	1	2	yes	—
34	C₂²×Dic₁₉	C₂³	1	2	yes	—
35	C₂×Dic₃₈	—	2	8	no	—
36	C₂×Dic₃₈	—	2	8	no	—
37	C₂×Dic₃₈	—	4	4	no	—
38	C₄⋊C₄×C₁₉	D₄	1	4	yes	—
39	C₄⋊C₄×C₁₉	Q₈	1	4	yes	complemented
40	C₄²×C₁₉	—	2	4	no	—
41	C₄²×C₁₉	Q₈	1	4	yes	complemented
42	C₄⋊Dic₁₉	—	2	4	no	—
43	C₄⋊Dic₁₉	—	2	4	no	—
44	C₄⋊Dic₁₉	—	2	8	no	—
45	C₄⋊Dic₁₉	—	4	2	no	—
46	C₄⋊Dic₁₉	D₄	1	4	yes	—
47	C₄⋊Dic₁₉	Q₈	1	4	yes	complemented
48	Dic₁₉⋊C₄	—	2	8	no	—
49	C₄×Dic₁₉	—	2	4	no	—
50	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
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	4	no	—
71	C₂×Dic₁₉	—	2	4	no	—
72	C₂×Dic₁₉	—	2	4	no	—
73	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
74	C₂×Dic₁₉	C₂×Q₈	1	4	yes	—
75	Dic₃₈	—	4	16	no	—
76	#2551	—	19	1	no	maximal, Sylow
77	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₁₉	C₂²×Q₈	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	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₁₉	—	2	2	no	—
85	C₄×C₁₉	—	2	4	no	—
86	C₄×C₁₉	—	2	4	no	—
87	C₄×C₁₉	C₄⋊Q₈	1	4	yes	complemented
88	C₄×C₁₉	C₄⋊Q₈	1	4	yes	complemented
89	Dic₁₉	—	2	4	no	—
90	Dic₁₉	—	2	4	no	—
91	Dic₁₉	—	4	4	no	—
92	C₂×C₄⋊Q₈	—	19	2	no	—
93	C₂×C₄⋊Q₈	—	19	2	no	—
94	#395	—	19	4	no	—
95	#389	—	19	2	no	—
96	#389	—	19	2	no	—
97	#384	—	19	1	no	—
98	C₄×C₄⋊C₄	—	19	1	no	—
99	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
100	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₁₉	#544	1	1	yes	characteristic
103	C₂×C₁₉	D₄²	1	1	yes	characteristic
104	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
105	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
106	C₂×C₁₉	Q₈²	1	1	yes	characteristic
107	C₂²×Q₈	—	19	4	no	—
108	C₄⋊Q₈	—	38	8	no	—
109	C₄⋊Q₈	—	38	8	no	—
110	C₂×C₄⋊C₄	—	19	1	no	—
111	C₂×C₄⋊C₄	—	19	1	no	—
112	C₂×C₄⋊C₄	—	19	2	no	—
113	C₂×C₄⋊C₄	—	19	2	no	—
114	C₂×C₄⋊C₄	—	19	4	no	—
115	C₂×C₄⋊C₄	—	19	4	no	—
116	C₂×C₄⋊C₄	—	19	2	no	—
117	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
119	C₂×C₄²	—	19	2	no	—
120	C₂×C₄²	C₂×D₁₉	1	2	yes	—
121	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
122	#95	—	19	2	no	—
123	#95	—	19	4	no	—
124	#95	C₂×D₁₉	1	2	yes	—
125	C₁₉	#2551	1	1	yes	characteristic, complemented, Sylow
126	C₂×Q₈	—	38	8	no	—
127	C₂×Q₈	—	38	8	no	—
128	C₂×Q₈	—	76	4	no	—
129	C₂²×C₄	—	19	2	no	—
130	C₂²×C₄	—	19	2	no	—
131	C₂²×C₄	—	19	4	no	—
132	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
134	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
135	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
137	C₄⋊C₄	—	19	4	no	—
138	C₄⋊C₄	—	19	4	no	—
139	C₄⋊C₄	—	38	8	no	—
140	C₄⋊C₄	—	38	8	no	—
141	C₄⋊C₄	—	38	4	no	—
142	C₄⋊C₄	—	38	4	no	—
143	C₄⋊C₄	—	76	2	no	—
144	C₄⋊C₄	Dic₃₈	1	4	yes	complemented
145	C₄⋊C₄	D₇₆	1	4	yes	—
146	C₄²	—	2	4	no	—
147	C₄²	—	38	4	no	—
148	C₄²	Dic₃₈	1	4	yes	complemented
149	C₂³	C₂³×D₁₉	1	1	yes	characteristic
150	Q₈	—	76	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	19	4	no	—
158	C₂×C₄	—	19	4	no	—
159	C₂×C₄	—	38	2	no	—
160	C₂×C₄	—	38	2	no	—
161	C₂×C₄	—	38	2	no	—
162	C₂×C₄	—	38	2	no	—
163	C₂×C₄	—	38	4	no	—
164	C₂×C₄	—	38	4	no	—
165	C₂×C₄	—	38	4	no	—
166	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
167	C₂×C₄	C₂×Dic₃₈	1	4	yes	—
168	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
169	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
170	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
171	C₂×C₄	C₂×D₇₆	1	4	yes	—
172	C₂×C₄	C₂×D₇₆	1	2	yes	—
173	C₂×C₄	D₄×D₁₉	1	2	yes	—
174	C₂×C₄	Q₈×D₁₉	1	2	yes	—
175	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
176	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
177	C₂²	C₂²×D₇₆	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
180	C₂²	#10597435	1	1	yes	characteristic
181	C₂²	#10597436	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	38	4	no	—
186	C₄	—	38	4	no	—
187	C₄	—	76	4	no	—
188	C₄	#10597315	1	4	yes	complemented
189	C₄	#10597334	1	4	yes	complemented
190	C₂	C₂×#10597334	1	1	yes	characteristic
191	C₂	C₂×#10597315	1	1	yes	characteristic
192	C₂	#11927172	1	1	yes	characteristic
193	C₂	D₄×D₇₆	1	1	yes	characteristic
194	C₂	D₄×Dic₃₈	1	1	yes	characteristic
195	C₂	Q₈×D₇₆	1	1	yes	characteristic
196	C₂	Q₈×Dic₃₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, prime radical, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2551
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
39. C₄⋊C₄×C₁₉	Q₈	semidirect
41. C₄²×C₁₉	Q₈	semidirect
47. C₄⋊Dic₁₉	Q₈	semidirect
87. C₄×C₁₉	C₄⋊Q₈	semidirect
88. C₄×C₁₉	C₄⋊Q₈	semidirect
125. C₁₉	#2551	semidirect
144. C₄⋊C₄	Dic₃₈	semidirect
148. C₄²	Dic₃₈	semidirect
188. C₄	#10597315	semidirect
189. C₄	#10597334	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597315	2
C₂×#10597334	2
#11926465	2
#11926443	2
#11926442	2
#11926414	2
C₄×C₄⋊Dic₁₉	2
C₄×C₄⋊C₄×C₁₉	2
#2551	19