Subgroups — #14578130

Order 2432.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊#564	C₂	1	4	yes	maximal, complemented
2	#11927273	C₂	1	4	yes	maximal, complemented
3	#11927272	C₂	1	4	yes	maximal, complemented
4	#11927269	C₂	1	4	yes	maximal, complemented
5	#11927156	C₂	1	4	yes	maximal
6	#11927154	C₂	1	8	yes	maximal
7	#11927101	C₂	1	2	yes	maximal
8	C₁₉×#533	C₂	1	1	yes	maximal, characteristic
9	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₉×#125	C₂²	1	2	yes	—
11	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₄×Q₈×C₁₉	C₂²	1	4	yes	—
13	C₁₉×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	Q₈×Dic₁₉	C₂²	1	8	yes	—
16	#10597390	C₂²	1	8	yes	—
17	#10597372	C₂²	1	4	yes	—
18	#10597370	C₂²	1	4	yes	—
19	#10597370	C₂²	1	4	yes	—
20	#10597336	C₂²	1	16	yes	—
21	#10597336	C₂²	1	8	yes	—
22	#10597335	C₂²	1	16	yes	—
23	#10597335	C₂²	1	16	yes	—
24	#10597333	C₂²	1	8	yes	—
25	#10597325	C₂²	1	16	yes	—
26	#10597316	C₂²	1	4	yes	—
27	#10597316	C₂²	1	4	yes	—
28	#10597315	C₂²	1	2	yes	—
29	#10597315	C₂²	1	2	yes	—
30	#10597315	C₂²	1	4	yes	—
31	C₄×Dic₃₈	C₂²	1	4	yes	—
32	C₄×Dic₃₈	C₂²	1	4	yes	—
33	C₄×Dic₃₈	C₂²	1	8	yes	—
34	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂×Dic₃₈	C₂³	1	4	yes	—
38	C₂×Dic₃₈	C₂³	1	4	yes	—
39	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
40	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
41	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
43	C₄²×C₁₉	C₂³	1	2	yes	—
44	C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₄²×C₁₉	C₂³	1	4	yes	—
47	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
48	C₄⋊Dic₁₉	C₂³	1	4	yes	—
49	C₄⋊Dic₁₉	C₂³	1	16	yes	—
50	C₄⋊Dic₁₉	C₂³	1	4	yes	—
51	C₄⋊Dic₁₉	C₂³	1	4	yes	—
52	C₄⋊Dic₁₉	C₂³	1	4	yes	—
53	Dic₁₉⋊C₄	C₂³	1	16	yes	—
54	Dic₁₉⋊C₄	C₂³	1	16	yes	—
55	Dic₁₉⋊C₄	C₂³	1	8	yes	—
56	Dic₁₉⋊C₄	C₂³	1	8	yes	—
57	C₄×Dic₁₉	C₂³	1	8	yes	—
58	C₄×Dic₁₉	C₂³	1	16	yes	—
59	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
60	Q₈×C₁₉	—	2	8	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	4	no	—
63	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
64	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
66	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
67	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₉	C₂⁴	1	8	yes	—
69	C₂×Dic₁₉	C₂⁴	1	8	yes	—
70	Dic₃₈	—	2	8	no	—
71	Dic₃₈	—	2	8	no	—
72	#3556	—	19	1	no	maximal, Sylow
73	C₂²×C₁₉	—	2	2	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
76	C₄×C₁₉	—	2	2	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	—	2	8	no	—
79	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₁₉	2₋¹⁺⁴	1	4	yes	—
81	Dic₁₉	—	2	8	no	—
82	Dic₁₉	—	2	8	no	—
83	#564	—	19	4	no	—
84	#557	—	19	4	no	—
85	#557	—	19	4	no	—
86	#557	—	19	4	no	—
87	#541	—	19	8	no	—
88	#541	—	19	4	no	—
89	#533	—	19	2	no	—
90	#533	D₁₉	1	1	yes	characteristic
91	C₂×C₁₉	—	4	1	no	—
92	C₂×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
93	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
94	C₄⋊Q₈	—	19	8	no	—
95	C₄⋊Q₈	—	19	2	no	—
96	C₄⋊Q₈	—	19	2	no	—
97	C₄⋊Q₈	—	19	4	no	—
98	C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
99	#126	—	19	16	no	—
100	#125	—	19	8	no	—
101	#125	—	19	16	no	—
102	#125	—	19	16	no	—
103	#125	—	19	16	no	—
104	#125	—	19	4	no	—
105	#125	—	19	4	no	—
106	#125	C₂×D₁₉	1	2	yes	—
107	C₂²⋊Q₈	—	19	4	no	—
108	C₂²⋊Q₈	—	19	4	no	—
109	C₂²⋊Q₈	C₂×D₁₉	1	4	yes	—
110	C₄×Q₈	—	19	8	no	—
111	C₄×Q₈	—	19	8	no	—
112	C₄×Q₈	—	19	4	no	—
113	C₄×Q₈	—	19	4	no	—
114	C₄×Q₈	—	19	8	no	—
115	C₄×Q₈	C₂×D₁₉	1	4	yes	—
116	#117	—	19	4	no	—
117	#117	C₂×D₁₉	1	2	yes	—
118	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
119	C₁₉	#3556	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₈	—	19	4	no	—
121	C₂×Q₈	—	19	4	no	—
122	C₂×Q₈	C₂²×D₁₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
125	C₄⋊C₄	—	19	8	no	—
126	C₄⋊C₄	—	19	8	no	—
127	C₄⋊C₄	—	19	16	no	—
128	C₄⋊C₄	—	19	16	no	—
129	C₄⋊C₄	—	19	4	no	—
130	C₄⋊C₄	—	19	4	no	—
131	C₄⋊C₄	—	19	4	no	—
132	C₄⋊C₄	—	19	4	no	—
133	C₄⋊C₄	—	19	16	no	—
134	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
135	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
136	C₄⋊C₄	C₂²×D₁₉	1	8	yes	—
137	C₂²⋊C₄	—	19	8	no	—
138	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
139	C₄²	—	19	8	no	—
140	C₄²	—	19	16	no	—
141	C₄²	C₂²×D₁₉	1	1	yes	characteristic
142	C₄²	C₂²×D₁₉	1	2	yes	—
143	C₄²	C₂²×D₁₉	1	1	yes	characteristic
144	C₄²	C₂²×D₁₉	1	4	yes	—
145	C₂³	C₂³×D₁₉	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	38	8	no	—
148	Q₈	—	38	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	19	8	no	—
152	C₂×C₄	—	19	8	no	—
153	C₂×C₄	C₂³×D₁₉	1	8	yes	—
154	C₂×C₄	C₂³×D₁₉	1	2	yes	—
155	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₁₉	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	1	no	—
164	C₄	—	38	8	no	—
165	C₄	—	38	8	no	—
166	C₄	#10597433	1	2	yes	—
167	C₄	#10597436	1	4	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#10597436	1	2	yes	—
170	C₂	C₂×#10597433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#533	C₂
prime radical	C₂³×C₁₉	C₂⁴
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3556
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₉⋊#564	C₂	semidirect
2. #11927273	C₂	semidirect
3. #11927272	C₂	semidirect
4. #11927269	C₂	semidirect
119. C₁₉	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₁₉⋊#564	2
#11927273	2
#11927272	2
#11927269	2
#11927156	2
#11927154	2
#11927101	2
C₁₉×#533	2
#3556	19