Subgroups — #14567000

Order 2432.

840 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597370	C₂	1	2	yes	maximal, complemented
3	#11926464	C₂	1	2	yes	maximal, complemented
4	#11926456	C₂	1	2	yes	maximal, complemented
5	#11926450	C₂	1	2	yes	maximal, complemented
6	#11926424	C₂	1	2	yes	maximal
7	#11926416	C₂	1	2	yes	maximal, complemented
8	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₉	C₂²	1	2	yes	—
14	C₂²×Dic₃₈	—	2	4	no	—
15	C₂²×Dic₃₈	—	2	8	no	—
16	C₂²×Dic₃₈	C₂²	1	2	yes	—
17	C₂²×Dic₃₈	C₂²	1	2	yes	—
18	C₄×D₄×C₁₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	—
24	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
26	#10597370	—	2	8	no	—
27	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
28	C₁₉⋊#95	C₂²	1	2	yes	—
29	C₁₉⋊#95	C₂²	1	4	yes	—
30	C₁₉⋊#95	C₂²	1	4	yes	—
31	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
32	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×D₄×C₁₉	—	2	2	no	—
34	C₂×D₄×C₁₉	—	2	2	no	—
35	C₂×D₄×C₁₉	Q₈	1	4	yes	complemented
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	4	no	—
39	C₂²×C₄×C₁₉	D₄	1	4	yes	—
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₉	—	2	4	no	—
46	C₂²×Dic₁₉	—	2	8	no	—
47	C₂²×Dic₁₉	C₂³	1	4	yes	—
48	C₂²×Dic₁₉	C₂³	1	4	yes	—
49	C₂×Dic₃₈	—	2	32	no	—
50	C₂×Dic₃₈	—	2	4	no	—
51	C₂×Dic₃₈	—	2	4	no	—
52	C₂×Dic₃₈	—	2	4	no	—
53	C₂×Dic₃₈	—	2	8	no	—
54	C₂×Dic₃₈	D₄	1	8	yes	complemented
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₉	—	2	2	no	—
58	C₄²×C₁₉	—	2	2	no	—
59	C₂²⋊Dic₁₉	—	2	4	no	—
60	C₂²⋊Dic₁₉	—	2	8	no	—
61	C₄⋊Dic₁₉	—	2	4	no	—
62	Dic₁₉⋊C₄	—	2	8	no	—
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	—	2	2	no	—
65	C₂³×C₁₉	—	2	2	no	—
66	C₂³×C₁₉	—	2	1	no	—
67	C₂³×C₁₉	—	2	1	no	—
68	C₂³×C₁₉	—	2	2	no	—
69	C₂³×C₁₉	C₂×D₄	1	2	yes	—
70	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
71	C₂³×C₁₉	C₂×Q₈	1	2	yes	complemented
72	C₂³×C₁₉	#41	1	2	yes	—
73	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
74	D₄×C₁₉	—	2	8	no	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	8	no	—
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	4	4	no	—
84	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	complemented
88	C₂×Dic₁₉	—	2	16	no	—
89	C₂×Dic₁₉	—	2	4	no	—
90	C₂×Dic₁₉	—	2	4	no	—
91	C₂×Dic₁₉	—	2	4	no	—
92	C₂×Dic₁₉	—	2	4	no	—
93	C₂×Dic₁₉	—	2	4	no	—
94	C₂×Dic₁₉	C₂×D₄	1	8	yes	—
95	Dic₃₈	—	2	16	no	—
96	Dic₃₈	—	2	16	no	—
97	#2409	—	19	1	no	maximal, Sylow
98	C₂²×C₁₉	—	2	2	no	—
99	C₂²×C₁₉	—	2	2	no	—
100	C₂²×C₁₉	—	2	4	no	—
101	C₂²×C₁₉	—	2	4	no	—
102	C₂²×C₁₉	—	2	2	no	—
103	C₂²×C₁₉	—	2	4	no	—
104	C₂²×C₁₉	—	4	1	no	—
105	C₂²×C₁₉	—	4	1	no	—
106	C₂²×C₁₉	—	4	2	no	—
107	C₂²×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
108	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
109	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
113	C₄×C₁₉	—	2	2	no	—
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	4	1	no	—
116	C₄×C₁₉	—	4	1	no	—
117	C₄×C₁₉	#120	1	4	yes	—
118	Dic₁₉	—	2	8	no	—
119	Dic₁₉	—	4	4	no	—
120	C₂³×Q₈	—	19	1	no	—
121	C₂×C₂²⋊Q₈	—	19	2	no	—
122	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic
123	#395	—	19	2	no	—
124	#393	—	19	2	no	—
125	#391	—	19	2	no	—
126	#386	—	19	2	no	—
127	#385	—	19	2	no	—
128	#380	—	19	1	no	—
129	C₂×C₁₉	—	2	4	no	—
130	C₂×C₁₉	—	4	2	no	—
131	C₂×C₁₉	C₂×#120	1	1	yes	characteristic
132	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₁₉	#536	1	1	yes	characteristic
134	C₂×C₁₉	#548	1	2	yes	—
135	C₂×C₁₉	D₄×Q₈	1	2	yes	—
136	C₂²×Q₈	—	19	2	no	—
137	C₂²×Q₈	—	19	2	no	—
138	C₂²×Q₈	—	38	8	no	—
139	C₂²×Q₈	—	38	4	no	—
140	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
141	C₂³×C₄	—	19	2	no	—
142	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
143	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
144	C₂²⋊Q₈	—	38	8	no	—
145	C₄×D₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	19	2	no	—
147	C₂×C₄⋊C₄	—	19	4	no	—
148	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	—	19	4	no	—
150	C₂×C₂²⋊C₄	—	19	2	no	—
151	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
154	#95	—	19	2	no	—
155	#95	—	19	4	no	—
156	#95	—	19	4	no	—
157	C₁₉	#2409	1	1	yes	characteristic, complemented, Sylow
158	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
159	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
160	C₂×Q₈	—	19	8	no	—
161	C₂×Q₈	—	38	8	no	—
162	C₂×Q₈	—	38	32	no	—
163	C₂×Q₈	—	38	4	no	—
164	C₂×Q₈	—	38	4	no	—
165	C₂×Q₈	—	38	4	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	Dic₃₈	1	4	yes	complemented
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	19	4	no	—
173	C₂²×C₄	—	19	4	no	—
174	C₂²×C₄	—	38	8	no	—
175	C₂²×C₄	—	38	4	no	—
176	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	—
177	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	38	8	no	—
184	C₄⋊C₄	—	38	4	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	38	8	no	—
188	C₂²⋊C₄	—	38	4	no	—
189	C₄²	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	C₂×Dic₃₈	1	2	yes	—
197	C₂³	C₂×Dic₃₈	1	2	yes	complemented
198	C₂³	#64541	1	2	yes	—
199	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
200	C₂³	C₂³×D₁₉	1	1	yes	characteristic
201	Q₈	—	38	16	no	—
202	Q₈	—	38	16	no	—
203	D₄	—	2	8	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	19	8	no	—
214	C₂×C₄	—	38	16	no	—
215	C₂×C₄	—	38	4	no	—
216	C₂×C₄	—	38	4	no	—
217	C₂×C₄	—	38	4	no	—
218	C₂×C₄	—	38	4	no	—
219	C₂×C₄	—	38	4	no	—
220	C₂×C₄	C₂×Dic₃₈	1	2	yes	complemented
221	C₂×C₄	D₄×D₁₉	1	4	yes	—
222	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	#10597370	1	4	yes	complemented
234	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
235	C₂²	C₂×#64541	1	1	yes	characteristic
236	C₂²	C₂×D₄×D₁₉	1	2	yes	—
237	C₂²	#10597436	1	2	yes	—
238	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	38	8	no	—
244	C₄	—	76	4	no	—
245	C₄	#10597383	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	4	2	no	—
248	C₂	C₂×#10597383	1	1	yes	characteristic
249	C₂	C₂×#10597370	1	1	yes	characteristic
250	C₂	#11927115	1	1	yes	characteristic
251	C₂	#11927203	1	2	yes	—
252	C₂	D₄×Dic₃₈	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₄×D₄×C₁₉	C₂
prime radical	C₂²×D₄×C₁₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	#2409
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₃₈	C₂	semidirect
2. C₂×#10597370	C₂	semidirect
3. #11926464	C₂	semidirect
4. #11926456	C₂	semidirect
5. #11926450	C₂	semidirect
7. #11926416	C₂	semidirect
8. #11926404	C₂	semidirect
35. C₂×D₄×C₁₉	Q₈	semidirect
54. C₂×Dic₃₈	D₄	semidirect
71. C₂³×C₁₉	C₂×Q₈	semidirect
87. C₂×C₄×C₁₉	C₂×Q₈	semidirect
107. C₂²×C₁₉	C₂²⋊Q₈	semidirect
157. C₁₉	#2409	semidirect
168. C₂×D₄	Dic₃₈	semidirect
197. C₂³	C₂×Dic₃₈	semidirect
220. C₂×C₄	C₂×Dic₃₈	semidirect
233. C₂²	#10597370	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₈	2
C₂×#10597370	2
#11926464	2
#11926456	2
#11926450	2
#11926424	2
#11926416	2
#11926404	2
C₂×C₄×D₄×C₁₉	2
#2409	19