Subgroups — #18071978

Order 4864.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	2	yes	maximal
2	#14561468	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
4	#11926597	—	2	2	no	—
5	#11926597	—	2	2	no	—
6	#11926597	C₄	1	4	yes	complemented
7	#11926597	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
11	C₂×C₁₉×#117	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₁₉	—	2	2	no	—
15	C₁₉×#376	C₄	1	2	yes	—
16	C₄³×C₁₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	D₄	1	2	yes	—
19	C₁₉×#117	—	2	4	no	—
20	C₁₉×#117	C₈	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	D₄	1	2	yes	—
30	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉⋊C₈	—	2	4	no	—
33	C₁₉×#95	—	2	4	no	—
34	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
35	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
37	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
38	C₄×C₁₉⋊C₈	—	2	4	no	—
39	C₄×C₁₉⋊C₈	—	2	4	no	—
40	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	—	2	4	no	—
42	C₂²×C₄×C₁₉	—	4	4	no	—
43	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
55	C₄⋊C₄×C₁₉	—	2	8	no	—
56	C₂²⋊C₄×C₁₉	—	2	8	no	—
57	C₄²×C₁₉	—	2	4	no	—
58	C₄²×C₁₉	—	4	8	no	—
59	C₄²×C₁₉	C₄²	1	1	yes	characteristic
60	C₄²×C₁₉	C₄²	1	1	yes	characteristic
61	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₉	C₂×C₈	1	8	yes	complemented
67	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₁₉⋊C₈	—	2	8	no	—
71	C₂×C₁₉⋊C₈	—	4	8	no	—
72	#7866	—	19	1	no	maximal, Sylow
73	C₂³×C₁₉	—	4	2	no	—
74	C₂³×C₁₉	#95	1	2	yes	—
75	C₂³×C₁₉	C₂²⋊C₈	1	4	yes	—
76	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	—	2	4	no	—
78	C₂×C₄×C₁₉	—	2	4	no	—
79	C₂×C₄×C₁₉	—	2	4	no	—
80	C₂×C₄×C₁₉	—	2	8	no	—
81	C₂×C₄×C₁₉	—	4	2	no	—
82	C₂×C₄×C₁₉	—	4	2	no	—
83	C₂×C₄×C₁₉	—	4	2	no	—
84	C₂×C₄×C₁₉	—	4	8	no	—
85	C₂×C₄×C₁₉	#95	1	2	yes	—
86	C₂×C₄×C₁₉	#95	1	2	yes	—
87	C₂×C₄×C₁₉	#95	1	2	yes	—
88	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
89	C₂×C₄×C₁₉	#97	1	4	yes	—
90	C₂×C₄×C₁₉	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
92	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
98	C₁₉⋊C₈	—	4	8	no	—
99	C₂×#431	—	19	2	no	—
100	C₄×#117	D₁₉	1	1	yes	characteristic
101	#1276	—	19	4	no	—
102	C₂²×C₁₉	—	2	2	no	—
103	C₂²×C₁₉	—	2	2	no	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	—	4	4	no	—
106	C₂²×C₁₉	#336	1	4	yes	—
107	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
108	C₂²×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	2	2	no	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	—	4	8	no	—
113	C₄×C₁₉	#336	1	4	yes	—
114	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
115	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
116	C₄×C₁₉	#421	1	4	yes	—
117	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
118	C₂×#117	Dic₁₉	1	2	yes	—
119	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
120	#431	—	19	4	no	—
121	#431	—	19	4	no	—
122	#431	—	38	2	no	—
123	#431	—	38	2	no	—
124	C₂×C₄⋊C₈	—	19	4	no	—
125	C₂×C₂²⋊C₈	—	38	2	no	—
126	C₂×C₄×C₈	—	38	2	no	—
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄×C₂²⋊C₄	—	2	2	no	—
129	#376	Dic₁₉	1	2	yes	—
130	C₄³	C₂×D₁₉	1	2	yes	—
131	C₂×C₁₉	—	2	2	no	—
132	C₂×C₁₉	—	2	4	no	—
133	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
134	C₂×C₁₉	#1733	1	2	yes	—
135	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
136	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
137	C₂²×C₈	—	38	4	no	—
138	#117	—	2	4	no	—
139	#117	C₁₉⋊C₈	1	8	yes	complemented
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₂²⋊C₄	—	2	2	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	Dic₃₈	1	2	yes	—
144	C₂×C₄²	C₄×D₁₉	1	2	yes	—
145	C₂×C₄²	C₄×D₁₉	1	2	yes	—
146	C₂×C₄²	D₇₆	1	2	yes	—
147	C₂×C₄²	C₂×Dic₁₉	1	4	yes	—
148	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
150	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
151	C₄⋊C₈	—	19	8	no	—
152	C₄⋊C₈	—	38	4	no	—
153	C₂²⋊C₈	—	38	4	no	—
154	C₂²⋊C₈	—	38	4	no	—
155	C₄×C₈	—	38	4	no	—
156	C₄×C₈	—	38	4	no	—
157	#95	—	2	4	no	—
158	C₁₉	#7866	1	1	yes	characteristic, complemented, Sylow
159	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
163	C₂²×C₄	#64520	1	4	yes	—
164	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
165	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
166	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
168	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
169	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
171	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
174	C₂×C₈	—	38	8	no	—
175	C₂×C₈	—	76	8	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₂×C₁₉⋊C₈	1	8	yes	complemented
181	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
182	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
183	C₄²	Dic₁₉⋊C₄	1	2	yes	—
184	C₄²	Dic₁₉⋊C₄	1	2	yes	—
185	C₄²	C₄⋊Dic₁₉	1	2	yes	—
186	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
187	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
188	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₁₉	1	2	yes	—
190	C₄²	C₂×D₇₆	1	1	yes	characteristic
191	C₂³	—	4	2	no	—
192	C₂³	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
193	C₂³	C₁₉⋊#95	1	2	yes	—
194	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
204	C₂×C₄	#10597260	1	4	yes	—
205	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
206	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
207	C₂×C₄	C₁₉⋊#95	1	2	yes	—
208	C₂×C₄	C₁₉⋊#95	1	2	yes	—
209	C₂×C₄	C₁₉⋊#95	1	2	yes	—
210	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
211	C₂×C₄	C₂×#64520	1	2	yes	—
212	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
215	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
216	C₈	—	76	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#11926299	1	4	yes	—
222	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
223	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	#11926299	1	4	yes	—
229	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
230	C₄	C₂×#10597260	1	1	yes	characteristic
231	C₄	#11926562	1	4	yes	—
232	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×#11926299	1	1	yes	characteristic
236	C₂	#14563345	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	#7866
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #14561468	C₂	semidirect
6. #11926597	C₄	semidirect
8. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
20. C₁₉×#117	C₈	semidirect
66. C₄²×C₁₉	C₂×C₈	semidirect
139. #117	C₁₉⋊C₈	semidirect
158. C₁₉	#7866	semidirect
180. C₄²	C₂×C₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
#14561468	2
C₄×C₁₉×#117	2
#7866	19