Subgroups — #18081833

Order 4864.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926924	C₂	1	1	yes	maximal, characteristic
2	#14564586	C₂	1	2	yes	maximal
3	#14563502	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926289	C₂	1	2	yes	maximal
5	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₆	—	2	1	no	—
7	C₂²×Dic₇₆	—	2	1	no	—
8	C₂×#10597315	C₂²	1	2	yes	—
9	#11926924	—	2	2	no	—
10	#11926924	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597272	—	2	1	no	—
13	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
14	#11926307	—	2	4	no	—
15	#11926298	—	2	1	no	—
16	#11926289	—	2	4	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#334	—	2	2	no	—
20	C₂²×Dic₃₈	—	2	2	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉	—	2	2	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
28	C₂×C₄×Dic₁₉	—	2	1	no	—
29	C₂²×C₁₉⋊C₈	—	2	1	no	—
30	C₂×Dic₇₆	—	2	4	no	—
31	C₂×Dic₇₆	—	2	4	no	—
32	C₂×Dic₇₆	—	4	2	no	—
33	C₂×Dic₇₆	—	4	2	no	—
34	C₂×Dic₇₆	—	4	2	no	—
35	C₂×Dic₇₆	—	4	2	no	—
36	#10597315	—	2	2	no	—
37	#10597315	—	2	2	no	—
38	#10597315	—	2	2	no	—
39	#10597315	C₂×C₄	1	4	yes	—
40	C₁₉×C₄⋊C₈	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	2	no	—
42	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₉	D₄	1	2	yes	—
44	C₁₉⋊#95	—	2	1	no	—
45	#10597272	—	4	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
47	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
51	C₂²×Dic₁₉	—	2	2	no	—
52	C₂²×Dic₁₉	—	4	2	no	—
53	C₂×Dic₃₈	—	2	4	no	—
54	C₂×Dic₃₈	—	4	2	no	—
55	C₂×Dic₃₈	—	4	2	no	—
56	C₂×Dic₃₈	—	4	2	no	—
57	C₂×Dic₃₈	—	4	2	no	—
58	C₂×Dic₃₈	—	4	2	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	—	4	2	no	—
62	C₂×C₈×C₁₉	—	4	4	no	—
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₉	D₈	1	2	yes	—
66	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
67	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	#41	1	1	yes	characteristic
71	C₄⋊Dic₁₉	—	2	4	no	—
72	C₄⋊Dic₁₉	—	2	4	no	—
73	C₄⋊Dic₁₉	—	2	4	no	—
74	C₄⋊Dic₁₉	—	4	2	no	—
75	C₄⋊Dic₁₉	—	4	2	no	—
76	C₄×Dic₁₉	—	4	1	no	—
77	C₄×Dic₁₉	—	4	1	no	—
78	C₂×C₁₉⋊C₈	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	4	1	no	—
80	C₂×C₁₉⋊C₈	—	4	1	no	—
81	Dic₇₆	—	4	8	no	—
82	Dic₇₆	—	4	8	no	—
83	#9742	—	19	1	no	maximal, Sylow
84	C₂³×C₁₉	#121	1	1	yes	characteristic
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#137	1	4	yes	—
97	C₂×Dic₁₉	—	4	4	no	—
98	C₂×Dic₁₉	—	4	2	no	—
99	C₂×Dic₁₉	—	4	2	no	—
100	C₂×Dic₁₉	—	4	2	no	—
101	C₂×Dic₁₉	—	8	2	no	—
102	C₂×Dic₁₉	—	8	2	no	—
103	Dic₃₈	—	4	4	no	—
104	Dic₃₈	—	4	4	no	—
105	Dic₃₈	—	8	2	no	—
106	Dic₃₈	—	8	2	no	—
107	C₈×C₁₉	—	2	2	no	—
108	C₈×C₁₉	—	8	2	no	—
109	C₈×C₁₉	#102	1	4	yes	—
110	C₁₉⋊C₈	—	4	2	no	—
111	C₂×#494	—	19	1	no	—
112	#1988	—	19	2	no	—
113	#1770	—	19	1	no	—
114	C₂×#334	D₁₉	1	1	yes	characteristic
115	C₂×#332	—	19	2	no	—
116	C₂²×C₁₉	#390	1	1	yes	characteristic
117	C₂²×C₁₉	#459	1	1	yes	characteristic
118	C₂²×C₁₉	#460	1	1	yes	characteristic
119	C₂²×C₁₉	#470	1	2	yes	—
120	C₂²×C₁₉	#473	1	2	yes	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
124	C₄×C₁₉	#418	1	1	yes	characteristic
125	C₄×C₁₉	#441	1	2	yes	—
126	C₄×C₁₉	#459	1	1	yes	characteristic
127	C₄×C₁₉	#460	1	1	yes	characteristic
128	C₄×C₁₉	#497	1	2	yes	—
129	Dic₁₉	—	8	2	no	—
130	Dic₁₉	—	8	4	no	—
131	C₂²×Q₁₆	—	38	1	no	—
132	C₂²×Q₁₆	—	38	1	no	—
133	C₂×C₄⋊Q₈	—	19	2	no	—
134	#494	—	19	4	no	—
135	#494	—	38	2	no	—
136	C₂×C₄⋊C₈	—	19	1	no	—
137	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
138	C₂×#97	—	38	1	no	—
139	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
140	C₄×C₄⋊C₄	—	19	1	no	—
141	#340	—	38	4	no	—
142	#336	—	38	1	no	—
143	#334	—	2	2	no	—
144	#332	—	38	4	no	—
145	C₂×C₁₉	#1689	1	2	yes	—
146	C₂×C₁₉	#1690	1	2	yes	—
147	C₂×C₁₉	#1971	1	1	yes	characteristic
148	C₂×C₁₉	#1979	1	1	yes	characteristic
149	C₂×C₁₉	#1982	1	1	yes	characteristic
150	C₂²×Q₈	—	38	2	no	—
151	C₂×Q₁₆	—	38	4	no	—
152	C₂×Q₁₆	—	38	4	no	—
153	C₂×Q₁₆	—	76	2	no	—
154	C₂×Q₁₆	—	76	2	no	—
155	C₂×Q₁₆	—	76	2	no	—
156	C₂×Q₁₆	—	76	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	38	1	no	—
159	C₂²×C₈	D₇₆	1	1	yes	characteristic
160	C₂²×C₈	D₇₆	1	1	yes	characteristic
161	C₄⋊Q₈	—	19	4	no	—
162	C₄⋊Q₈	—	38	2	no	—
163	C₄⋊Q₈	—	38	2	no	—
164	C₄⋊Q₈	—	38	2	no	—
165	C₂×C₄⋊C₄	—	19	2	no	—
166	C₂×C₄⋊C₄	—	38	2	no	—
167	C₂×C₄⋊C₄	—	38	2	no	—
168	C₂×C₄²	—	38	1	no	—
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	19	2	no	—
173	C₄⋊C₈	—	38	1	no	—
174	#97	—	76	2	no	—
175	C₄×C₈	C₄×D₁₉	1	2	yes	—
176	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
177	#95	—	38	1	no	—
178	C₁₉	#9742	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	38	4	no	—
180	C₂×Q₈	—	76	2	no	—
181	C₂×Q₈	—	76	2	no	—
182	C₂×Q₈	—	76	2	no	—
183	C₂×Q₈	—	76	2	no	—
184	C₂×Q₈	—	76	2	no	—
185	C₂²×C₄	—	38	2	no	—
186	C₂²×C₄	—	76	2	no	—
187	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
189	C₂²×C₄	#64545	1	1	yes	characteristic
190	Q₁₆	—	76	8	no	—
191	Q₁₆	—	76	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	38	2	no	—
197	C₂×C₈	—	76	1	no	—
198	C₂×C₈	—	76	1	no	—
199	C₂×C₈	#64516	1	2	yes	—
200	C₂×C₈	D₁₅₂	1	2	yes	—
201	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
202	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
203	C₄⋊C₄	—	38	4	no	—
204	C₄⋊C₄	—	38	4	no	—
205	C₄⋊C₄	—	38	4	no	—
206	C₄⋊C₄	—	76	2	no	—
207	C₄⋊C₄	—	76	2	no	—
208	C₄²	—	76	1	no	—
209	C₄²	—	76	1	no	—
210	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₄²	D₄×D₁₉	1	1	yes	characteristic
212	C₄²	#64545	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂³	#10597341	1	1	yes	characteristic
215	Q₈	—	76	4	no	—
216	Q₈	—	76	4	no	—
217	Q₈	—	152	2	no	—
218	Q₈	—	152	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	76	2	no	—
221	C₂×C₄	—	76	2	no	—
222	C₂×C₄	—	76	4	no	—
223	C₂×C₄	—	76	2	no	—
224	C₂×C₄	—	152	2	no	—
225	C₂×C₄	—	152	2	no	—
226	C₂×C₄	#10597339	1	1	yes	characteristic
227	C₂×C₄	#10597339	1	1	yes	characteristic
228	C₂×C₄	#10597341	1	1	yes	characteristic
229	C₂×C₄	C₂×#64516	1	1	yes	characteristic
230	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
231	C₂×C₄	#10597354	1	1	yes	characteristic
232	C₂×C₄	#10597354	1	1	yes	characteristic
233	C₂×C₄	#10597361	1	4	yes	—
234	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#10597386	1	1	yes	characteristic
236	C₂×C₄	#10597393	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	76	2	no	—
240	C₈	#10597278	1	4	yes	—
241	C₂²	#11926445	1	1	yes	characteristic
242	C₂²	#11926740	1	1	yes	characteristic
243	C₂²	#11926745	1	1	yes	characteristic
244	C₂²	#11926803	1	2	yes	—
245	C₂²	#11926820	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	152	2	no	—
249	C₄	—	152	4	no	—
250	C₄	C₂×#10597278	1	1	yes	characteristic
251	C₄	#11926550	1	1	yes	characteristic
252	C₄	#11926650	1	2	yes	—
253	C₄	#11926740	1	1	yes	characteristic
254	C₄	#11926745	1	1	yes	characteristic
255	C₄	#11926935	1	2	yes	—
256	C₂	#14563125	1	2	yes	—
257	C₂	#14563130	1	2	yes	—
258	C₂	#14564504	1	1	yes	characteristic
259	C₂	#14564548	1	1	yes	characteristic
260	C₂	#14564561	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#334	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#334	D₁₉
hypocenter	C₁₉	#9742
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
10. #11926924	C₄	semidirect
178. C₁₉	#9742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926924	2
#14564586	2
#14563502	2
C₂×#11926289	2
C₂×C₁₉×#334	2
#9742	19