Subgroups — #18088275

Order 4864.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926309	C₂	1	2	yes	maximal
2	C₂×#11926309	C₂	1	2	yes	maximal
3	#14563329	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	4	yes	—
6	#11926309	—	2	4	no	—
7	#11926309	C₄	1	8	yes	complemented
8	#11926299	—	2	4	no	—
9	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	—	2	4	no	—
13	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	—
14	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
15	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	2	2	no	—
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	D₄	1	4	yes	—
24	C₂×C₄²×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×#64520	—	2	4	no	—
29	C₂×#64520	—	2	4	no	—
30	C₂×#64520	—	2	4	no	—
31	C₂×#64520	—	2	4	no	—
32	C₂×#64520	—	2	4	no	—
33	C₂×#64520	C₂×C₄	1	8	yes	—
34	C₂²×C₁₉⋊C₈	—	2	4	no	—
35	C₄×C₈×C₁₉	—	2	8	no	—
36	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	—	2	4	no	—
38	C₂²×C₄×C₁₉	—	2	8	no	—
39	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₉	#41	1	2	yes	—
45	C₂²×C₄×C₁₉	#41	1	2	yes	—
46	C₂²×C₄×C₁₉	#41	1	2	yes	—
47	C₂²×C₄×C₁₉	#41	1	2	yes	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	2	2	no	—
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	2	8	no	—
53	C₂×C₈×C₁₉	—	2	8	no	—
54	C₂×C₈×C₁₉	C₄²	1	4	yes	—
55	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₁₉	—	2	2	no	—
57	C₄²×C₁₉	—	2	2	no	—
58	C₄²×C₁₉	—	2	4	no	—
59	#64520	—	2	8	no	—
60	#64520	—	2	8	no	—
61	#64520	—	4	4	no	—
62	#64520	—	4	4	no	—
63	C₂×C₁₉⋊C₈	—	2	8	no	—
64	C₂×C₁₉⋊C₈	—	4	4	no	—
65	C₂×C₁₉⋊C₈	—	4	4	no	—
66	#10822	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	—	2	4	no	—
68	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
70	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
71	C₂³×C₁₉	#125	1	2	yes	—
72	C₂×C₄×C₁₉	—	2	16	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	2	4	no	—
75	C₂×C₄×C₁₉	#108	1	8	yes	—
76	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#117	1	4	yes	—
79	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
85	C₂×C₄×C₁₉	#121	1	2	yes	—
86	C₂×C₄×C₁₉	#123	1	4	yes	—
87	C₂×C₄×C₁₉	#124	1	2	yes	—
88	C₂×C₄×C₁₉	#126	1	4	yes	—
89	C₈×C₁₉	—	2	4	no	—
90	C₈×C₁₉	—	2	4	no	—
91	C₁₉⋊C₈	—	4	8	no	—
92	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
93	C₂×#341	—	19	2	no	—
94	C₂×#341	—	19	2	no	—
95	#1728	—	19	2	no	—
96	C₂²×C₁₉	—	2	4	no	—
97	C₂²×C₁₉	—	2	4	no	—
98	C₂²×C₁₉	—	2	4	no	—
99	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₁₉	#380	1	1	yes	characteristic
101	C₂²×C₁₉	#382	1	1	yes	characteristic
102	C₂²×C₁₉	#387	1	4	yes	—
103	C₂²×C₁₉	C₂×#108	1	4	yes	—
104	C₄×C₁₉	—	2	8	no	—
105	C₄×C₁₉	#376	1	1	yes	characteristic
106	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₁₉	#381	1	1	yes	characteristic
108	C₄×C₁₉	#383	1	1	yes	characteristic
109	C₄×C₁₉	#388	1	4	yes	—
110	C₂²×M₁₆	—	19	4	no	—
111	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
112	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
113	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	Dic₁₉	1	4	yes	—
116	#341	—	19	8	no	—
117	#341	—	38	4	no	—
118	#336	—	38	4	no	—
119	C₂×C₁₉	—	2	4	no	—
120	C₂×C₁₉	#1434	1	1	yes	characteristic
121	C₂×C₁₉	C₄×#108	1	2	yes	—
122	C₂×C₁₉	#1810	1	2	yes	—
123	C₂×C₁₉	#1837	1	2	yes	—
124	C₂³×C₄	Dic₃₈	1	2	yes	—
125	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	C₂×M₁₆	—	19	8	no	—
127	C₂×M₁₆	—	38	4	no	—
128	C₂×M₁₆	—	38	4	no	—
129	C₂×M₁₆	—	38	4	no	—
130	C₂×M₁₆	—	38	4	no	—
131	C₂×M₁₆	—	38	4	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	38	4	no	—
137	C₂²×C₈	C₄×D₁₉	1	4	yes	—
138	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
139	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
140	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	D₇₆	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
145	C₄×C₈	—	2	8	no	—
146	C₁₉	#10822	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
152	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
153	C₂²×C₄	#64541	1	2	yes	—
154	C₂²×C₄	#64541	1	2	yes	—
155	C₂²×C₄	#64541	1	2	yes	—
156	C₂²×C₄	#64541	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
159	M₁₆	—	38	8	no	—
160	M₁₆	—	38	8	no	—
161	M₁₆	—	76	4	no	—
162	M₁₆	—	76	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	38	8	no	—
170	C₂×C₈	—	76	4	no	—
171	C₂×C₈	—	76	4	no	—
172	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
173	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	C₄×Dic₃₈	1	2	yes	—
179	C₂³	#10597316	1	2	yes	—
180	C₂³	#10597370	1	2	yes	—
181	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	#10597276	1	8	yes	—
186	C₂×C₄	#10597318	1	4	yes	—
187	C₂×C₄	C₄×D₇₆	1	2	yes	—
188	C₂×C₄	#10597321	1	2	yes	—
189	C₂×C₄	#10597322	1	4	yes	—
190	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
191	C₂×C₄	#10597372	1	1	yes	characteristic
192	C₂×C₄	#10597372	1	1	yes	characteristic
193	C₂×C₄	#10597372	1	1	yes	characteristic
194	C₂×C₄	#10597372	1	1	yes	characteristic
195	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
196	C₂×C₄	#10597375	1	4	yes	—
197	C₂×C₄	#10597376	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	76	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
206	C₂²	#11926404	1	1	yes	characteristic
207	C₂²	#11926410	1	1	yes	characteristic
208	C₂²	#11926428	1	4	yes	—
209	C₂²	C₂×#10597276	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#11926390	1	1	yes	characteristic
212	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
213	C₄	#11926407	1	1	yes	characteristic
214	C₄	#11926412	1	1	yes	characteristic
215	C₄	#11926435	1	4	yes	—
216	C₂	—	2	4	no	—
217	C₂	#14561862	1	1	yes	characteristic
218	C₂	C₄×#10597276	1	2	yes	—
219	C₂	#14563673	1	2	yes	—
220	C₂	#14563773	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10822
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #11926309	C₄	semidirect
146. C₁₉	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926309	2
#14563329	2
C₂²×C₄×C₈×C₁₉	2
#10822	19