Finite Group Databasebeta

Subgroups — #14565613

Order 2432.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11926984	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597286	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926819	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926360	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926351	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926328	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#578	C₂	1	1	yes	maximal, characteristic
8	C₁₉×2₋¹⁺⁴	—	2	1	no	—
9	C₁₉×#137	—	2	2	no	—
10	C₁₉×#137	—	2	1	no	—
11	C₁₉×#135	—	2	1	no	—
12	C₁₉×#135	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₉	—	2	1	no	—
14	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#131	C₂²	1	1	yes	characteristic
16	#10597397	—	2	1	no	—
17	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
18	#10597286	—	2	2	no	—
19	#10597286	—	2	1	no	—
20	#10597284	—	2	1	no	—
21	#10597281	—	2	1	no	—
22	#10597276	C₂²	1	1	yes	characteristic
23	#10597269	C₂²	1	1	yes	characteristic
24	C₂×C₁₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	#41×C₁₉	—	2	2	no	—
26	#41×C₁₉	—	2	1	no	—
27	#41×C₁₉	—	4	2	no	—
28	#41×C₁₉	—	4	1	no	—
29	#41×C₁₉	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₁₉	—	2	2	no	—
31	C₂×Q₈×C₁₉	—	2	1	no	—
32	C₂×Q₈×C₁₉	—	4	1	no	—
33	C₂×Dic₃₈	—	2	1	no	—
34	Q₁₆×C₁₉	—	2	2	no	—
35	Q₁₆×C₁₉	—	2	1	no	—
36	Q₁₆×C₁₉	D₄	1	1	yes	characteristic
37	Q₁₆×C₁₉	D₄	1	2	yes	—
38	SD₁₆×C₁₉	—	2	2	no	—
39	SD₁₆×C₁₉	—	2	1	no	—
40	D₈×C₁₉	D₄	1	1	yes	characteristic
41	M₁₆×C₁₉	—	2	1	no	—
42	M₁₆×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	—	2	1	no	—
44	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	#64528	—	4	1	no	—
46	#64526	—	4	1	no	—
47	#64520	—	2	1	no	—
48	Dic₇₆	—	2	1	no	—
49	Dic₇₆	—	2	1	no	—
50	C₁₉⋊C₁₆	—	2	1	no	—
51	C₁₉⋊C₁₆	—	2	1	no	—
52	Q₈×C₁₉	—	2	2	no	—
53	Q₈×C₁₉	—	2	1	no	—
54	Q₈×C₁₉	—	4	2	no	—
55	Q₈×C₁₉	—	4	1	no	—
56	Q₈×C₁₉	D₈	1	1	yes	characteristic
57	D₄×C₁₉	—	2	1	no	—
58	D₄×C₁₉	—	2	1	no	—
59	D₄×C₁₉	—	4	2	no	—
60	D₄×C₁₉	—	4	1	no	—
61	D₄×C₁₉	D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	4	2	no	—
64	C₂×C₄×C₁₉	—	4	2	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₁₉	—	4	1	no	—
69	Dic₃₈	—	4	1	no	—
70	Dic₃₈	—	4	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₁₉⋊C₈	—	4	1	no	—
75	#2195	—	19	1	no	maximal, Sylow
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	4	1	no	—
78	C₂²×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	—	4	2	no	—
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	#120	1	1	yes	characteristic
83	C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
84	Dic₁₉	—	8	1	no	—
85	#578	D₁₉	1	1	yes	characteristic
86	#510	—	19	1	no	—
87	C₂×Q₃₂	—	19	1	no	—
88	#473	—	19	1	no	—
89	#362	—	19	1	no	—
90	#359	—	19	1	no	—
91	#350	—	19	1	no	—
92	C₂×C₁₉	—	2	1	no	—
93	C₂×C₁₉	—	4	1	no	—
94	C₂×C₁₉	—	8	1	no	—
95	C₂×C₁₉	#447	1	1	yes	characteristic
96	2₋¹⁺⁴	—	2	1	no	—
97	#137	—	2	2	no	—
98	#137	—	2	1	no	—
99	#137	—	38	1	no	—
100	#135	—	2	1	no	—
101	#135	C₂×D₁₉	1	1	yes	characteristic
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	19	1	no	—
104	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
105	#131	C₂×D₁₉	1	1	yes	characteristic
106	Q₃₂	—	38	2	no	—
107	Q₃₂	—	38	1	no	—
108	SD₃₂	—	38	1	no	—
109	M₃₂	—	19	1	no	—
110	C₂×C₁₆	—	19	1	no	—
111	#108	—	19	1	no	—
112	#101	—	38	1	no	—
113	C₁₉	#2195	1	1	yes	characteristic, complemented, Sylow
114	#41	—	2	2	no	—
115	#41	—	2	1	no	—
116	#41	—	4	2	no	—
117	#41	—	4	1	no	—
118	#41	C₂²⋊D₁₉	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	38	1	no	—
123	Q₁₆	—	2	2	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	38	1	no	—
126	Q₁₆	—	38	1	no	—
127	Q₁₆	—	76	1	no	—
128	Q₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
129	Q₁₆	C₂²⋊D₁₉	1	2	yes	—
130	SD₁₆	—	2	2	no	—
131	SD₁₆	—	2	1	no	—
132	SD₁₆	—	76	1	no	—
133	D₈	C₂²⋊D₁₉	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	38	1	no	—
136	M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
139	C₁₆	—	38	1	no	—
140	C₁₆	—	38	1	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	76	1	no	—
146	Q₈	—	76	1	no	—
147	Q₈	#64525	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	1	no	—
152	D₄	#64525	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	76	1	no	—
159	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	76	1	no	—
162	C₈	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#10597378	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	152	1	no	—
171	C₄	C₂×#64525	1	1	yes	characteristic
172	C₄	#10597399	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	8	1	no	—
176	C₂	#11926678	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11926678
commutator, cosocle	C₂×C₈×C₁₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₉
Fitting, prime radical	C₁₉×#578	C₂
socle	C₂×C₁₉	#447
hypercenter	#578	D₁₉
hypocenter	C₁₉	#2195
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #11926984	C₂	semidirect
2. C₂×#10597286	C₂	semidirect
3. #11926819	C₂	semidirect
4. #11926360	C₂	semidirect
5. #11926351	C₂	semidirect
6. #11926328	C₂	semidirect
113. C₁₉	#2195	semidirect

Maximal subgroups

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

subgroup	index
#11926984	2
C₂×#10597286	2
#11926819	2
#11926360	2
#11926351	2
#11926328	2
C₁₉×#578	2
#2195	19