Subgroups — #14568548

Order 2432.

788 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597376	C₂	1	2	yes	maximal, complemented
2	C₂×#10597384	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926460	C₂	1	4	yes	maximal, complemented
5	#11926414	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926404	C₂	1	2	yes	maximal, complemented
7	#11926401	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	—
12	C₂²×D₇₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
14	C₄×D₄×C₁₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	—
19	#10597384	—	2	8	no	—
20	#10597376	—	2	8	no	—
21	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
22	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
27	C₄×D₇₆	—	2	4	no	—
28	C₁₉⋊#95	C₂²	1	4	yes	—
29	C₂⁴×C₁₉	C₂³	1	2	yes	—
30	C₂³×D₁₉	C₂³	1	2	yes	—
31	C₂×D₄×C₁₉	—	2	2	no	—
32	C₂×D₄×C₁₉	—	2	2	no	—
33	C₂×D₄×C₁₉	D₄	1	4	yes	complemented
34	C₂²×C₄×C₁₉	—	2	4	no	—
35	C₂²×C₄×C₁₉	—	2	4	no	—
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₉	—	2	8	no	—
42	C₂×C₂²⋊D₁₉	—	2	8	no	—
43	C₂×C₂²⋊D₁₉	—	4	4	no	—
44	C₂²×Dic₁₉	C₂³	1	2	yes	—
45	C₂²×Dic₁₉	C₂³	1	4	yes	—
46	C₂×D₇₆	—	2	2	no	—
47	C₂×D₇₆	—	2	2	no	—
48	C₂×D₇₆	D₄	1	4	yes	complemented
49	C₂×C₄×D₁₉	—	2	4	no	—
50	C₂×C₄×D₁₉	—	2	4	no	—
51	C₂×C₄×D₁₉	—	2	4	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₂²⋊C₄×C₁₉	—	2	4	no	—
54	C₄²×C₁₉	D₄	1	4	yes	complemented
55	C₂²⋊Dic₁₉	—	2	8	no	—
56	(C₂×D₁₉)⋊C₄	—	2	8	no	—
57	(C₂×D₁₉)⋊C₄	—	2	4	no	—
58	C₄⋊Dic₁₉	—	2	4	no	—
59	C₄⋊Dic₁₉	—	2	4	no	—
60	C₄⋊Dic₁₉	—	2	2	no	—
61	C₄⋊Dic₁₉	—	4	2	no	—
62	C₂³×C₁₉	—	2	2	no	—
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	—	2	2	no	—
65	C₂³×C₁₉	—	2	2	no	—
66	C₂³×C₁₉	—	2	2	no	—
67	C₂³×C₁₉	—	2	2	no	—
68	C₂³×C₁₉	C₂×D₄	1	4	yes	—
69	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
70	C₂²×D₁₉	—	2	2	no	—
71	C₂²×D₁₉	—	2	2	no	—
72	C₂²×D₁₉	—	2	2	no	—
73	C₂²×D₁₉	—	2	2	no	—
74	C₂²×D₁₉	—	2	2	no	—
75	C₂²×D₁₉	—	2	2	no	—
76	C₂²×D₁₉	C₂×D₄	1	4	yes	—
77	D₄×C₁₉	—	2	8	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	4	8	no	—
84	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₉	#41	1	2	yes	—
90	C₂×C₄×C₁₉	#41	1	2	yes	—
91	C₂²⋊D₁₉	—	4	16	no	—
92	C₂×Dic₁₉	—	2	2	no	—
93	C₂×Dic₁₉	—	2	4	no	—
94	C₂×Dic₁₉	—	2	4	no	—
95	C₂×Dic₁₉	—	2	2	no	—
96	C₂×Dic₁₉	—	2	2	no	—
97	C₂×Dic₁₉	—	2	4	no	—
98	D₇₆	—	2	8	no	—
99	C₄×D₁₉	—	4	8	no	—
100	#2537	—	19	1	no	maximal, Sylow
101	C₂²×C₁₉	—	2	4	no	—
102	C₂²×C₁₉	—	2	4	no	—
103	C₂²×C₁₉	—	2	4	no	—
104	C₂²×C₁₉	—	4	2	no	—
105	C₂²×C₁₉	—	4	2	no	—
106	C₂²×C₁₉	—	4	2	no	—
107	C₂²×C₁₉	—	4	2	no	—
108	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
109	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
111	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×D₁₉	—	2	4	no	—
116	C₂×D₁₉	—	2	4	no	—
117	C₂×D₁₉	—	2	4	no	—
118	C₂×D₁₉	—	4	2	no	—
119	C₂×D₁₉	—	4	2	no	—
120	C₂×D₁₉	—	4	2	no	—
121	C₂×D₁₉	—	4	2	no	—
122	C₄×C₁₉	—	2	2	no	—
123	C₄×C₁₉	—	4	2	no	—
124	C₄×C₁₉	#121	1	4	yes	complemented
125	C₄×C₁₉	#121	1	4	yes	complemented
126	Dic₁₉	—	4	2	no	—
127	Dic₁₉	—	4	4	no	—
128	C₂×#121	—	19	2	no	—
129	C₂×#121	—	19	2	no	—
130	C₂×C₄×D₄	—	19	1	no	—
131	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
132	#394	—	19	4	no	—
133	#384	—	19	1	no	—
134	#380	—	19	2	no	—
135	#380	—	19	2	no	—
136	C₂×C₁₉	—	4	4	no	—
137	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
138	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
139	C₂×C₁₉	#537	1	1	yes	characteristic
140	C₂×C₁₉	D₄²	1	1	yes	characteristic
141	C₂×C₁₉	#547	1	1	yes	characteristic
142	C₂×C₁₉	#547	1	1	yes	characteristic
143	C₂×C₁₉	#555	1	1	yes	characteristic
144	D₁₉	—	4	4	no	—
145	C₂²×D₄	—	19	4	no	—
146	C₂²×D₄	—	19	1	no	—
147	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
148	C₂³×C₄	—	19	2	no	—
149	C₂³×C₄	C₂×D₁₉	1	2	yes	—
150	#121	—	38	8	no	—
151	#121	—	38	8	no	—
152	C₄×D₄	—	2	4	no	—
153	C₄×D₄	—	38	4	no	—
154	C₂×C₄⋊C₄	—	19	1	no	—
155	C₂×C₄⋊C₄	—	19	2	no	—
156	C₂×C₄⋊C₄	—	19	2	no	—
157	C₂×C₄⋊C₄	—	19	2	no	—
158	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	—	19	4	no	—
160	C₂×C₂²⋊C₄	—	19	2	no	—
161	C₂×C₂²⋊C₄	—	19	4	no	—
162	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
163	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
164	#95	—	19	4	no	—
165	C₁₉	#2537	1	1	yes	characteristic, complemented, Sylow
166	C₂⁴	—	19	2	no	—
167	C₂⁴	C₂²×D₁₉	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	19	4	no	—
171	C₂×D₄	—	38	8	no	—
172	C₂×D₄	—	38	8	no	—
173	C₂×D₄	—	38	2	no	—
174	C₂×D₄	—	38	2	no	—
175	C₂×D₄	—	76	4	no	—
176	C₂×D₄	D₇₆	1	4	yes	complemented
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	19	2	no	—
181	C₂²×C₄	—	19	4	no	—
182	C₂²×C₄	—	38	4	no	—
183	C₂²×C₄	—	38	4	no	—
184	C₂²×C₄	—	38	4	no	—
185	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
186	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	38	4	no	—
191	C₄⋊C₄	—	38	4	no	—
192	C₄⋊C₄	—	38	2	no	—
193	C₄⋊C₄	—	76	2	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	38	8	no	—
196	C₂²⋊C₄	—	38	8	no	—
197	C₂²⋊C₄	—	38	4	no	—
198	C₄²	C₂²⋊D₁₉	1	4	yes	complemented
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	19	4	no	—
206	C₂³	—	38	2	no	—
207	C₂³	—	38	2	no	—
208	C₂³	—	38	2	no	—
209	C₂³	—	38	2	no	—
210	C₂³	—	38	2	no	—
211	C₂³	—	38	2	no	—
212	C₂³	C₂×D₇₆	1	4	yes	—
213	C₂³	C₂³×D₁₉	1	1	yes	characteristic
214	D₄	—	2	8	no	—
215	D₄	—	38	8	no	—
216	D₄	—	76	16	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	38	2	no	—
224	C₂×C₄	—	38	4	no	—
225	C₂×C₄	—	38	2	no	—
226	C₂×C₄	—	38	4	no	—
227	C₂×C₄	—	38	4	no	—
228	C₂×C₄	—	38	2	no	—
229	C₂×C₄	—	76	8	no	—
230	C₂×C₄	C₂×D₇₆	1	2	yes	—
231	C₂×C₄	#64541	1	2	yes	—
232	C₂×C₄	D₄×D₁₉	1	2	yes	—
233	C₂×C₄	#64543	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
235	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
236	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	38	4	no	—
245	C₂²	—	38	4	no	—
246	C₂²	—	38	4	no	—
247	C₂²	—	76	2	no	—
248	C₂²	—	76	2	no	—
249	C₂²	—	76	2	no	—
250	C₂²	—	76	2	no	—
251	C₂²	C₂²×D₇₆	1	1	yes	characteristic
252	C₂²	C₂×#64541	1	1	yes	characteristic
253	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
254	C₂²	C₂×#64543	1	1	yes	characteristic
255	C₂²	#10597435	1	1	yes	characteristic
256	C₂²	#10597436	1	1	yes	characteristic
257	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	76	4	no	—
261	C₄	—	76	2	no	—
262	C₄	#10597376	1	4	yes	complemented
263	C₄	#10597384	1	4	yes	complemented
264	C₂	—	4	4	no	—
265	C₂	—	76	4	no	—
266	C₂	C₂×#10597384	1	1	yes	characteristic
267	C₂	C₂×#10597376	1	1	yes	characteristic
268	C₂	#11927122	1	1	yes	characteristic
269	C₂	D₄×D₇₆	1	1	yes	characteristic
270	C₂	#11927198	1	1	yes	characteristic
271	C₂	#11927199	1	1	yes	characteristic
272	C₂	#11927259	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₄×D₄×C₁₉	C₂
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	#2537
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. C₂×#10597376	C₂	semidirect
2. C₂×#10597384	C₂	semidirect
3. C₂×C₄×D₇₆	C₂	semidirect
4. #11926460	C₂	semidirect
5. #11926414	C₂	semidirect
6. #11926404	C₂	semidirect
7. #11926401	C₂	semidirect
8. C₂×C₄×D₄×C₁₉	C₂	semidirect
33. C₂×D₄×C₁₉	D₄	semidirect
48. C₂×D₇₆	D₄	semidirect
54. C₄²×C₁₉	D₄	semidirect
124. C₄×C₁₉	#121	semidirect
125. C₄×C₁₉	#121	semidirect
131. C₂×C₄×D₄	D₁₉	semidirect
165. C₁₉	#2537	semidirect
176. C₂×D₄	D₇₆	semidirect
198. C₄²	C₂²⋊D₁₉	semidirect
262. C₄	#10597376	semidirect
263. C₄	#10597384	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597376	2
C₂×#10597384	2
C₂×C₄×D₇₆	2
#11926460	2
#11926414	2
#11926404	2
#11926401	2
C₂×C₄×D₄×C₁₉	2
#2537	19