Subgroups — #18081586

Order 4864.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572030	C₂	1	1	yes	maximal, characteristic
2	#14563787	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926287	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597320	—	2	2	no	—
6	C₂×#10597320	—	2	1	no	—
7	C₂×#10597320	—	2	1	no	—
8	C₂×#10597320	C₄	1	4	yes	—
9	C₂×#10597320	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
12	#11926287	—	2	8	no	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	—	2	1	no	—
15	C₄³×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#336	—	2	1	no	—
17	C₂²×D₇₆	—	2	4	no	—
18	C₂²×D₇₆	—	2	4	no	—
19	C₂²×D₇₆	—	2	2	no	—
20	C₂²×D₇₆	—	2	2	no	—
21	C₂²×D₇₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	D₄	1	2	yes	—
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₁₉⋊C₈	—	2	2	no	—
29	#10597320	—	2	4	no	—
30	#10597320	—	2	4	no	—
31	#10597320	—	4	4	no	—
32	#10597320	—	4	8	no	—
33	#10597320	—	4	4	no	—
34	#10597320	—	4	2	no	—
35	#10597320	—	4	4	no	—
36	#10597320	—	4	2	no	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₁₉×C₄⋊C₈	D₄	1	4	yes	complemented
39	C₄×C₈×C₁₉	—	4	2	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
41	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
42	C₂³×D₁₉	—	2	4	no	—
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×D₇₆	—	2	8	no	—
49	C₂×D₇₆	—	4	8	no	—
50	C₂×D₇₆	—	4	8	no	—
51	C₂×D₇₆	—	4	8	no	—
52	C₂×D₇₆	—	4	8	no	—
53	C₂×D₇₆	—	4	8	no	—
54	C₂×D₇₆	—	4	8	no	—
55	C₂×D₇₆	—	4	8	no	—
56	C₂×D₇₆	—	4	4	no	—
57	C₂×D₇₆	—	4	2	no	—
58	C₂×D₇₆	—	4	2	no	—
59	C₂×D₇₆	—	4	2	no	—
60	C₂×D₇₆	—	4	4	no	—
61	C₂×D₇₆	—	4	2	no	—
62	C₂×D₇₆	—	4	4	no	—
63	C₂×D₇₆	—	4	4	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	4	2	no	—
67	C₄²×C₁₉	—	2	4	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	D₈	1	4	yes	—
70	C₄²×C₁₉	D₈	1	2	yes	—
71	C₄²×C₁₉	SD₁₆	1	4	yes	—
72	C₄²×C₁₉	SD₁₆	1	2	yes	—
73	C₄²×C₁₉	C₂×D₄	1	2	yes	—
74	C₄²×C₁₉	#41	1	2	yes	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	4	no	—
77	#9695	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×D₁₉	—	4	4	no	—
80	C₂²×D₁₉	—	4	8	no	—
81	C₂²×D₁₉	—	4	4	no	—
82	C₂²×D₁₉	—	4	8	no	—
83	C₂²×D₁₉	—	4	4	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	#100	1	4	yes	—
88	C₂×C₄×C₁₉	#102	1	4	yes	—
89	C₂×C₄×C₁₉	#102	1	4	yes	—
90	C₂×C₄×C₁₉	#102	1	4	yes	—
91	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₉	#121	1	2	yes	—
93	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
100	D₇₆	—	8	8	no	—
101	D₇₆	—	8	16	no	—
102	D₇₆	—	8	16	no	—
103	D₇₆	—	8	4	no	—
104	D₇₆	—	8	4	no	—
105	D₇₆	—	8	4	no	—
106	D₇₆	—	8	2	no	—
107	D₇₆	—	8	2	no	—
108	C₈×C₁₉	—	4	4	no	—
109	C₁₉⋊C₈	—	8	2	no	—
110	#2867	—	19	1	no	—
111	#1842	—	19	1	no	—
112	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic, complemented
113	C₂×#331	—	19	4	no	—
114	C₂²×C₁₉	#390	1	1	yes	characteristic
115	C₂²×C₁₉	C₂×#100	1	2	yes	—
116	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×#102	1	2	yes	—
118	C₂²×C₁₉	#418	1	1	yes	characteristic
119	C₂×D₁₉	—	8	8	no	—
120	C₂×D₁₉	—	8	8	no	—
121	C₂×D₁₉	—	8	4	no	—
122	C₂×D₁₉	—	8	4	no	—
123	C₂×D₁₉	—	8	4	no	—
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	#331	1	8	yes	complemented
127	C₄×C₁₉	#459	1	2	yes	—
128	C₄×C₁₉	#460	1	2	yes	—
129	C₄×C₁₉	#486	1	2	yes	—
130	C₄×C₁₉	#492	1	1	yes	characteristic
131	C₄×C₁₉	#493	1	1	yes	characteristic
132	C₂×#127	—	19	4	no	—
133	C₂×#127	—	19	2	no	—
134	C₂×#127	—	38	2	no	—
135	C₂×#127	—	38	1	no	—
136	C₂×#127	—	38	1	no	—
137	C₂×C₄⋊C₈	—	19	2	no	—
138	C₂×C₄⋊C₈	—	38	2	no	—
139	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
140	C₂×C₄×C₈	—	2	1	no	—
141	C₄³	C₂×D₁₉	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#331	—	38	8	no	—
144	C₂×C₁₉	C₂×#331	1	2	yes	—
145	C₂×C₁₉	#1545	1	2	yes	—
146	C₂×C₁₉	#1957	1	1	yes	characteristic
147	C₂×C₁₉	#1965	1	1	yes	characteristic
148	C₂×C₁₉	#1982	1	1	yes	characteristic
149	D₁₉	—	16	4	no	—
150	C₂²×D₄	—	19	4	no	—
151	C₂²×D₄	—	38	4	no	—
152	C₂²×D₄	—	38	4	no	—
153	C₂²×D₄	—	38	2	no	—
154	C₂²×D₄	—	38	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	38	2	no	—
157	#127	—	38	4	no	—
158	#127	—	38	4	no	—
159	#127	—	76	8	no	—
160	#127	—	76	4	no	—
161	#127	—	76	4	no	—
162	#127	—	76	4	no	—
163	#127	—	76	2	no	—
164	#127	—	76	2	no	—
165	C₂×C₄²	C₄×D₁₉	1	2	yes	—
166	C₂×C₄²	D₇₆	1	2	yes	—
167	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	38	4	no	—
172	C₄⋊C₈	—	76	4	no	—
173	C₄⋊C₈	D₇₆	1	4	yes	complemented
174	C₄×C₈	—	4	2	no	—
175	C₁₉	#9695	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	38	4	no	—
177	C₂×D₄	—	38	8	no	—
178	C₂×D₄	—	76	8	no	—
179	C₂×D₄	—	76	8	no	—
180	C₂×D₄	—	76	8	no	—
181	C₂×D₄	—	76	4	no	—
182	C₂×D₄	—	76	2	no	—
183	C₂×D₄	—	76	2	no	—
184	C₂×D₄	—	76	4	no	—
185	C₂×D₄	—	76	8	no	—
186	C₂×D₄	—	76	8	no	—
187	C₂×D₄	—	76	8	no	—
188	C₂×D₄	—	76	8	no	—
189	C₂×D₄	—	76	2	no	—
190	C₂×D₄	—	76	2	no	—
191	C₂×D₄	—	76	4	no	—
192	C₂×D₄	—	76	4	no	—
193	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
194	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	76	2	no	—
202	C₂×C₈	—	76	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	4	no	—
205	C₄²	#64516	1	4	yes	—
206	C₄²	D₁₅₂	1	4	yes	—
207	C₄²	#64525	1	2	yes	—
208	C₄²	#64527	1	2	yes	—
209	C₄²	C₂×D₇₆	1	2	yes	—
210	C₄²	#64541	1	2	yes	—
211	C₂³	—	76	4	no	—
212	C₂³	—	76	8	no	—
213	C₂³	—	76	4	no	—
214	C₂³	—	76	4	no	—
215	C₂³	—	76	8	no	—
216	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
217	D₄	—	152	8	no	—
218	D₄	—	152	16	no	—
219	D₄	—	152	16	no	—
220	D₄	—	152	4	no	—
221	D₄	—	152	2	no	—
222	D₄	—	152	2	no	—
223	D₄	—	152	4	no	—
224	D₄	—	152	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#10597266	1	4	yes	—
229	C₂×C₄	#10597278	1	4	yes	—
230	C₂×C₄	#10597278	1	4	yes	—
231	C₂×C₄	#10597280	1	4	yes	—
232	C₂×C₄	C₄×D₇₆	1	2	yes	—
233	C₂×C₄	#10597320	1	1	yes	characteristic
234	C₂×C₄	#10597321	1	1	yes	characteristic
235	C₂×C₄	C₂×#64516	1	2	yes	—
236	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
237	C₂×C₄	#10597376	1	2	yes	—
238	C₂×C₄	C₂×#64525	1	1	yes	characteristic
239	C₂×C₄	C₂×#64527	1	1	yes	characteristic
240	C₂×C₄	#10597395	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	152	2	no	—
243	C₂²	—	152	4	no	—
244	C₂²	—	152	4	no	—
245	C₂²	—	152	4	no	—
246	C₂²	—	152	8	no	—
247	C₂²	—	152	8	no	—
248	C₂²	#11926446	1	1	yes	characteristic
249	C₂²	C₂×#10597280	1	2	yes	—
250	C₂²	C₂×#10597266	1	1	yes	characteristic
251	C₂²	C₂×#10597278	1	2	yes	—
252	C₂²	#11926548	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	#11926287	1	8	yes	complemented
256	C₄	#11926739	1	2	yes	—
257	C₄	#11926744	1	2	yes	—
258	C₄	#11926891	1	2	yes	—
259	C₄	#11926917	1	1	yes	characteristic
260	C₄	#11926921	1	1	yes	characteristic
261	C₂	—	304	4	no	—
262	C₂	C₂×#11926287	1	2	yes	—
263	C₂	#14562302	1	2	yes	—
264	C₂	#14564441	1	1	yes	characteristic
265	C₂	#14564475	1	1	yes	characteristic
266	C₂	#14564562	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	#14572030	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#9695
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14563787	C₂	semidirect
3. C₂×#11926287	C₂	semidirect
4. C₄×C₁₉×C₄⋊C₈	C₂	semidirect
38. C₁₉×C₄⋊C₈	D₄	semidirect
112. C₄×C₄⋊C₈	D₁₉	semidirect
126. C₄×C₁₉	#331	semidirect
173. C₄⋊C₈	D₇₆	semidirect
175. C₁₉	#9695	semidirect
255. C₄	#11926287	semidirect

Maximal subgroups

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

subgroup	index
#14572030	2
#14563787	2
C₂×#11926287	2
C₄×C₁₉×C₄⋊C₈	2
#9695	19