Subgroups — #18081780

Order 4864.

518 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	C₂×#11926924	C₂	1	1	yes	maximal, characteristic
2	#14564587	C₂	1	2	yes	maximal
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926291	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₁₉	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	2	yes	—
12	#11926306	—	2	4	no	—
13	#11926297	—	2	2	no	—
14	#11926291	—	2	4	no	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	—	2	1	no	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#336	—	2	1	no	—
20	C₂²×Dic₃₈	—	2	2	no	—
21	C₂²×C₈×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
25	C₂×C₄²×C₁₉	—	2	1	no	—
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	—	2	2	no	—
28	C₂×C₄⋊Dic₁₉	—	2	2	no	—
29	C₂²×C₁₉⋊C₈	—	2	2	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	1	no	—
41	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
42	C₄×C₈×C₁₉	—	4	2	no	—
43	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₉	D₄	1	2	yes	—
45	C₁₉×#95	—	2	1	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
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	—	2	2	no	—
62	C₂×C₈×C₁₉	—	4	1	no	—
63	C₂×C₈×C₁₉	—	4	1	no	—
64	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₉	D₈	1	2	yes	—
67	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
68	C₄⋊C₄×C₁₉	—	2	2	no	—
69	C₄⋊C₄×C₁₉	Q₁₆	1	4	yes	complemented
70	C₄²×C₁₉	—	4	1	no	—
71	C₄²×C₁₉	—	4	1	no	—
72	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₉	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₉	—	2	4	no	—
77	C₄⋊Dic₁₉	—	2	4	no	—
78	C₄⋊Dic₁₉	—	4	2	no	—
79	C₄⋊Dic₁₉	—	4	2	no	—
80	C₂×C₁₉⋊C₈	—	4	2	no	—
81	C₂×C₁₉⋊C₈	—	4	4	no	—
82	Dic₇₆	—	4	8	no	—
83	Dic₇₆	—	4	8	no	—
84	#9733	—	19	1	no	maximal, Sylow
85	C₂³×C₁₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	4	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₁₉	#135	1	2	yes	—
100	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×Dic₁₉	—	4	4	no	—
103	C₂×Dic₁₉	—	8	2	no	—
104	C₂×Dic₁₉	—	8	2	no	—
105	Dic₃₈	—	4	4	no	—
106	Dic₃₈	—	4	4	no	—
107	Dic₃₈	—	8	2	no	—
108	Dic₃₈	—	8	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	C₈×C₁₉	—	4	2	no	—
111	C₈×C₁₉	#102	1	4	yes	—
112	C₁₉⋊C₈	—	8	2	no	—
113	C₂×#494	—	19	1	no	—
114	#1988	—	19	2	no	—
115	C₈×C₄⋊C₄	D₁₉	1	1	yes	characteristic
116	C₂×#335	—	19	1	no	—
117	C₂×#332	—	19	2	no	—
118	C₂²×C₁₉	#390	1	1	yes	characteristic
119	C₂²×C₁₉	#459	1	1	yes	characteristic
120	C₂²×C₁₉	#460	1	1	yes	characteristic
121	C₂²×C₁₉	#467	1	2	yes	—
122	C₂²×C₁₉	#473	1	2	yes	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	4	4	no	—
126	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
127	C₄×C₁₉	#418	1	1	yes	characteristic
128	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
129	C₄×C₁₉	#459	1	1	yes	characteristic
130	C₄×C₁₉	#460	1	1	yes	characteristic
131	C₄×C₁₉	#494	1	1	yes	characteristic
132	C₄×C₁₉	#495	1	1	yes	characteristic
133	Dic₁₉	—	8	4	no	—
134	C₂²×Q₁₆	—	38	1	no	—
135	C₂²×Q₁₆	—	38	1	no	—
136	C₂×C₄⋊Q₈	—	19	2	no	—
137	#494	—	19	4	no	—
138	#494	—	38	2	no	—
139	C₂×C₄⋊C₈	—	19	2	no	—
140	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
144	#340	—	38	4	no	—
145	#336	—	2	1	no	—
146	#335	—	38	2	no	—
147	#332	—	38	4	no	—
148	C₂×C₁₉	#1669	1	2	yes	—
149	C₂×C₁₉	#1670	1	2	yes	—
150	C₂×C₁₉	#1969	1	1	yes	characteristic
151	C₂×C₁₉	#1979	1	1	yes	characteristic
152	C₂×C₁₉	#1982	1	1	yes	characteristic
153	C₂²×Q₈	—	38	2	no	—
154	C₂×Q₁₆	—	38	4	no	—
155	C₂×Q₁₆	—	38	4	no	—
156	C₂×Q₁₆	—	76	2	no	—
157	C₂×Q₁₆	—	76	2	no	—
158	C₂×Q₁₆	—	76	2	no	—
159	C₂×Q₁₆	—	76	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	38	2	no	—
162	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
164	C₄⋊Q₈	—	19	4	no	—
165	C₄⋊Q₈	—	38	2	no	—
166	C₄⋊Q₈	—	38	2	no	—
167	C₄⋊Q₈	—	38	2	no	—
168	C₂×C₄⋊C₄	—	38	2	no	—
169	C₂×C₄⋊C₄	—	38	2	no	—
170	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	38	2	no	—
175	C₄⋊C₈	—	38	2	no	—
176	C₄⋊C₈	D₇₆	1	2	yes	—
177	C₄×C₈	—	4	2	no	—
178	C₄×C₈	C₄×D₁₉	1	2	yes	—
179	C₄×C₈	D₇₆	1	2	yes	—
180	#95	—	2	1	no	—
181	C₁₉	#9733	1	1	yes	characteristic, complemented, Sylow
182	C₂×Q₈	—	38	4	no	—
183	C₂×Q₈	—	76	2	no	—
184	C₂×Q₈	—	76	2	no	—
185	C₂×Q₈	—	76	2	no	—
186	C₂×Q₈	—	76	2	no	—
187	C₂×Q₈	—	76	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	76	2	no	—
190	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
191	C₂²×C₄	#64541	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
193	Q₁₆	—	76	8	no	—
194	Q₁₆	—	76	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	—	76	4	no	—
202	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
203	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
204	C₂×C₈	#64525	1	2	yes	—
205	C₂×C₈	#64527	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	38	4	no	—
208	C₄⋊C₄	—	38	4	no	—
209	C₄⋊C₄	—	76	2	no	—
210	C₄⋊C₄	—	76	2	no	—
211	C₄⋊C₄	Dic₇₆	1	4	yes	complemented
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₄²	C₂×D₇₆	1	1	yes	characteristic
216	C₄²	C₂×D₇₆	1	1	yes	characteristic
217	C₄²	#64541	1	1	yes	characteristic
218	C₂³	#10597376	1	1	yes	characteristic
219	Q₈	—	76	4	no	—
220	Q₈	—	76	4	no	—
221	Q₈	—	152	2	no	—
222	Q₈	—	152	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	76	4	no	—
228	C₂×C₄	—	152	2	no	—
229	C₂×C₄	—	152	2	no	—
230	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
231	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
232	C₂×C₄	#10597320	1	1	yes	characteristic
233	C₂×C₄	#10597321	1	1	yes	characteristic
234	C₂×C₄	#10597350	1	2	yes	—
235	C₂×C₄	C₂×Dic₇₆	1	2	yes	—
236	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
237	C₂×C₄	#10597376	1	1	yes	characteristic
238	C₂×C₄	C₂×#64525	1	1	yes	characteristic
239	C₂×C₄	C₂×#64527	1	1	yes	characteristic
240	C₂×C₄	#10597395	1	1	yes	characteristic
241	C₂×C₄	#10597395	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	152	2	no	—
245	C₈	#10597266	1	4	yes	—
246	C₂²	#11926446	1	1	yes	characteristic
247	C₂²	#11926739	1	1	yes	characteristic
248	C₂²	#11926744	1	1	yes	characteristic
249	C₂²	#11926785	1	2	yes	—
250	C₂²	#11926819	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	152	4	no	—
255	C₄	C₂×#10597266	1	1	yes	characteristic
256	C₄	#11926548	1	1	yes	characteristic
257	C₄	C₄×Dic₇₆	1	2	yes	—
258	C₄	#11926739	1	1	yes	characteristic
259	C₄	#11926744	1	1	yes	characteristic
260	C₄	#11926924	1	1	yes	characteristic
261	C₄	#11926928	1	1	yes	characteristic
262	C₂	#14563000	1	2	yes	—
263	C₂	#14563005	1	2	yes	—
264	C₂	#14564496	1	1	yes	characteristic
265	C₂	#14564550	1	1	yes	characteristic
266	C₂	#14564562	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×C₄⋊C₄×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#9733
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #11926924	C₄	semidirect
69. C₄⋊C₄×C₁₉	Q₁₆	semidirect
181. C₁₉	#9733	semidirect
211. C₄⋊C₄	Dic₇₆	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
#14564587	2
C₂×#11926297	2
C₂×#11926291	2
C₈×C₄⋊C₄×C₁₉	2
#9733	19