Subgroups — #14566699

Order 2432.

832 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597320	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₆	C₂	1	2	yes	maximal, complemented
3	C₂×#10597339	C₂	1	2	yes	maximal, complemented
4	#11926446	C₂	1	2	yes	maximal, complemented
5	#11926445	C₂	1	2	yes	maximal, complemented
6	#11926422	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₆	C₂²	1	2	yes	—
10	C₂²×D₇₆	C₂²	1	2	yes	—
11	C₂²×D₇₆	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
18	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
19	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
23	#10597339	—	2	8	no	—
24	#10597320	C₄	1	8	yes	complemented
25	C₄×D₇₆	—	2	8	no	—
26	C₁₉×#95	C₂²	1	2	yes	—
27	C₁₉⋊#95	C₂²	1	2	yes	—
28	C₂³×D₁₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₉	C₂³	1	2	yes	—
35	C₂²×Dic₁₉	C₂³	1	2	yes	—
36	C₂×D₇₆	—	2	4	no	—
37	C₂×D₇₆	—	2	4	no	—
38	C₂×D₇₆	—	2	4	no	—
39	C₂×D₇₆	—	2	4	no	—
40	C₂×D₇₆	—	4	2	no	—
41	C₂×D₇₆	—	4	2	no	—
42	C₂×D₇₆	C₂×C₄	1	8	yes	complemented
43	C₂×D₇₆	C₂×C₄	1	8	yes	complemented
44	C₂×D₇₆	C₂×C₄	1	8	yes	complemented
45	C₂×C₄×D₁₉	—	2	8	no	—
46	C₂×C₄×D₁₉	—	2	8	no	—
47	C₂×C₄×D₁₉	—	4	4	no	—
48	C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₄⋊C₄×C₁₉	D₄	1	4	yes	complemented
50	C₄²×C₁₉	—	2	4	no	—
51	C₄²×C₁₉	C₂×C₄	1	4	yes	complemented
52	(C₂×D₁₉)⋊C₄	—	2	8	no	—
53	(C₂×D₁₉)⋊C₄	—	2	8	no	—
54	(C₂×D₁₉)⋊C₄	—	4	4	no	—
55	C₄⋊Dic₁₉	—	2	2	no	—
56	C₄⋊Dic₁₉	D₄	1	4	yes	complemented
57	C₄×Dic₁₉	—	2	4	no	—
58	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
59	C₂²×D₁₉	—	2	4	no	—
60	C₂²×D₁₉	—	2	4	no	—
61	C₂²×D₁₉	—	2	4	no	—
62	C₂²×D₁₉	—	2	4	no	—
63	C₂²×D₁₉	—	2	4	no	—
64	C₂²×D₁₉	—	2	4	no	—
65	C₂²×D₁₉	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₉	#41	1	2	yes	—
80	C₂×C₄×C₁₉	#41	1	2	yes	—
81	C₂×Dic₁₉	—	2	2	no	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	—	2	2	no	—
84	C₂×Dic₁₉	—	2	2	no	—
85	C₂×Dic₁₉	—	2	2	no	—
86	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
87	D₇₆	—	2	16	no	—
88	D₇₆	—	2	16	no	—
89	D₇₆	—	4	8	no	—
90	C₄×D₁₉	—	4	16	no	—
91	#2366	—	19	1	no	maximal, Sylow
92	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
94	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
97	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×D₁₉	—	2	8	no	—
100	C₂×D₁₉	—	2	8	no	—
101	C₂×D₁₉	—	2	8	no	—
102	C₂×D₁₉	—	4	4	no	—
103	C₂×D₁₉	—	4	4	no	—
104	C₂×D₁₉	—	4	4	no	—
105	C₂×D₁₉	—	4	4	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	—	2	4	no	—
108	C₄×C₁₉	—	4	2	no	—
109	C₄×C₁₉	C₄×D₄	1	4	yes	complemented
110	C₄×C₁₉	C₄×D₄	1	4	yes	complemented
111	Dic₁₉	—	2	4	no	—
112	Dic₁₉	—	4	2	no	—
113	C₂×#127	—	19	1	no	—
114	C₂×C₄×D₄	—	19	2	no	—
115	C₂×C₄×D₄	—	19	2	no	—
116	#390	—	19	2	no	—
117	#390	—	19	2	no	—
118	#386	—	19	4	no	—
119	C₄×C₄⋊C₄	—	19	1	no	—
120	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic, complemented
121	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×C₁₉	#518	1	1	yes	characteristic
124	C₂×C₁₉	D₄²	1	1	yes	characteristic
125	C₂×C₁₉	#550	1	1	yes	characteristic
126	C₂×C₁₉	#550	1	1	yes	characteristic
127	C₂×C₁₉	#559	1	1	yes	characteristic
128	D₁₉	—	4	8	no	—
129	C₂²×D₄	—	19	2	no	—
130	C₂²×D₄	—	19	2	no	—
131	C₂²×D₄	—	19	2	no	—
132	C₂³×C₄	—	19	4	no	—
133	#127	—	19	8	no	—
134	C₄×D₄	—	38	8	no	—
135	C₄×D₄	—	38	8	no	—
136	C₂×C₄⋊C₄	—	19	1	no	—
137	C₂×C₄⋊C₄	—	19	1	no	—
138	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	19	4	no	—
141	C₂×C₂²⋊C₄	—	19	4	no	—
142	C₂×C₂²⋊C₄	—	19	4	no	—
143	C₂×C₄²	—	19	2	no	—
144	C₂×C₄²	C₂×D₁₉	1	2	yes	—
145	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
146	#95	—	19	2	no	—
147	#95	C₂×D₁₉	1	2	yes	—
148	C₁₉	#2366	1	1	yes	characteristic, complemented, Sylow
149	C₂⁴	—	19	4	no	—
150	C₂×D₄	—	19	8	no	—
151	C₂×D₄	—	19	8	no	—
152	C₂×D₄	—	19	8	no	—
153	C₂×D₄	—	38	4	no	—
154	C₂×D₄	—	38	4	no	—
155	C₂×D₄	—	38	4	no	—
156	C₂×D₄	—	38	4	no	—
157	C₂×D₄	—	76	2	no	—
158	C₂×D₄	—	76	2	no	—
159	C₂²×C₄	—	19	2	no	—
160	C₂²×C₄	—	19	2	no	—
161	C₂²×C₄	—	38	8	no	—
162	C₂²×C₄	—	38	8	no	—
163	C₂²×C₄	—	76	4	no	—
164	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	19	4	no	—
171	C₄⋊C₄	—	38	2	no	—
172	C₄⋊C₄	D₇₆	1	4	yes	complemented
173	C₂²⋊C₄	—	38	8	no	—
174	C₂²⋊C₄	—	38	8	no	—
175	C₂²⋊C₄	—	76	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	38	4	no	—
178	C₄²	C₄×D₁₉	1	4	yes	complemented
179	C₂³	—	19	8	no	—
180	C₂³	—	38	4	no	—
181	C₂³	—	38	4	no	—
182	C₂³	—	38	4	no	—
183	C₂³	—	38	4	no	—
184	C₂³	—	38	4	no	—
185	C₂³	—	38	4	no	—
186	C₂³	C₂³×D₁₉	1	1	yes	characteristic
187	D₄	—	38	16	no	—
188	D₄	—	38	16	no	—
189	D₄	—	76	8	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	19	4	no	—
198	C₂×C₄	—	38	2	no	—
199	C₂×C₄	—	38	2	no	—
200	C₂×C₄	—	38	2	no	—
201	C₂×C₄	—	38	2	no	—
202	C₂×C₄	—	38	2	no	—
203	C₂×C₄	—	76	16	no	—
204	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
207	C₂×C₄	C₂×D₇₆	1	4	yes	—
208	C₂×C₄	C₂×D₇₆	1	2	yes	—
209	C₂×C₄	#64541	1	2	yes	—
210	C₂×C₄	D₄×D₁₉	1	2	yes	—
211	C₂×C₄	#64545	1	2	yes	—
212	C₂²	—	38	8	no	—
213	C₂²	—	38	8	no	—
214	C₂²	—	38	8	no	—
215	C₂²	—	76	4	no	—
216	C₂²	—	76	4	no	—
217	C₂²	—	76	4	no	—
218	C₂²	—	76	4	no	—
219	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
220	C₂²	C₂²×D₇₆	1	1	yes	characteristic
221	C₂²	C₂×#64541	1	1	yes	characteristic
222	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
223	C₂²	C₂×#64545	1	1	yes	characteristic
224	C₂²	#10597435	1	1	yes	characteristic
225	C₂²	#10597435	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	38	4	no	—
230	C₄	—	76	2	no	—
231	C₄	C₄×D₇₆	1	4	yes	complemented
232	C₄	#10597339	1	4	yes	complemented
233	C₂	—	76	8	no	—
234	C₂	C₂×#10597339	1	1	yes	characteristic
235	C₂	C₂×C₄×D₇₆	1	1	yes	characteristic
236	C₂	#11927018	1	1	yes	characteristic
237	C₂	D₄×D₇₆	1	1	yes	characteristic
238	C₂	#11927215	1	1	yes	characteristic
239	C₂	#11927216	1	1	yes	characteristic
240	C₂	#11927279	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
socle, cosocle	C₂³×C₁₉	C₂⁴
prime radical	C₂×#10597320	C₂
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2366
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₇₆	C₂	semidirect
3. C₂×#10597339	C₂	semidirect
4. #11926446	C₂	semidirect
5. #11926445	C₂	semidirect
6. #11926422	C₂	semidirect
7. C₄×C₄⋊Dic₁₉	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
24. #10597320	C₄	semidirect
42. C₂×D₇₆	C₂×C₄	semidirect
43. C₂×D₇₆	C₂×C₄	semidirect
44. C₂×D₇₆	C₂×C₄	semidirect
49. C₄⋊C₄×C₁₉	D₄	semidirect
51. C₄²×C₁₉	C₂×C₄	semidirect
56. C₄⋊Dic₁₉	D₄	semidirect
109. C₄×C₁₉	C₄×D₄	semidirect
110. C₄×C₁₉	C₄×D₄	semidirect
120. C₄×C₄⋊C₄	D₁₉	semidirect
148. C₁₉	#2366	semidirect
172. C₄⋊C₄	D₇₆	semidirect
178. C₄²	C₄×D₁₉	semidirect
231. C₄	C₄×D₇₆	semidirect
232. C₄	#10597339	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597320	2
C₂×C₄×D₇₆	2
C₂×#10597339	2
#11926446	2
#11926445	2
#11926422	2
C₄×C₄⋊Dic₁₉	2
C₄×C₄⋊C₄×C₁₉	2
#2366	19