Subgroups — C₁₉×#34365 #18040324

Order 4864.

2238 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2381	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2369	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2344	C₂	1	8	yes	maximal
5	C₁₉×#2331	C₂	1	8	yes	maximal, complemented
6	C₁₉×#2310	C₂	1	4	yes	maximal
7	C₁₉×#2283	C₂	1	2	yes	maximal
8	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×C₄⋊Q₈	C₄	1	16	yes	complemented
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₉×#117	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#391	C₂²	1	16	yes	—
17	C₁₉×#391	C₂²	1	16	yes	—
18	C₁₉×#389	C₂²	1	16	yes	—
19	C₁₉×#387	C₂²	1	32	yes	—
20	C₁₉×#383	C₂²	1	4	yes	—
21	C₁₉×#382	C₂²	1	4	yes	—
22	C₁₉×#381	C₂²	1	4	yes	—
23	C₁₉×#381	C₂²	1	8	yes	—
24	C₁₉×#380	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	16	yes	—
27	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
28	C₁₉×#376	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₉	—	4	8	no	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	16	yes	—
31	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
32	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	32	yes	—
37	C₁₉×C₄⋊Q₈	C₂×C₄	1	16	yes	—
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	16	yes	—
39	C₁₉×#117	—	4	16	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	32	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
46	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×#95	C₂³	1	8	yes	—
53	C₁₉×#95	C₂³	1	8	yes	—
54	C₁₉×#95	C₂³	1	8	yes	—
55	C₁₉×#95	C₂³	1	8	yes	—
56	C₁₉×#95	C₂³	1	32	yes	—
57	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₁₉	—	4	32	no	—
59	C₂×Q₈×C₁₉	—	4	4	no	—
60	C₂×Q₈×C₁₉	—	4	8	no	—
61	C₂×Q₈×C₁₉	—	4	4	no	—
62	C₂×Q₈×C₁₉	C₂²×C₄	1	16	yes	—
63	C₂×Q₈×C₁₉	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₁₉	—	2	16	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	—	2	8	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
76	C₄⋊C₄×C₁₉	—	2	16	no	—
77	C₄⋊C₄×C₁₉	—	4	16	no	—
78	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
79	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
80	C₂²⋊C₄×C₁₉	—	2	16	no	—
81	C₄²×C₁₉	—	2	16	no	—
82	C₄²×C₁₉	—	4	16	no	—
83	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₉	C₂²×C₄	1	8	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
86	#34365	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	4	no	—
88	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
89	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
90	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
91	Q₈×C₁₉	—	4	16	no	—
92	Q₈×C₁₉	—	4	16	no	—
93	C₂×C₄×C₁₉	—	2	32	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	2	8	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	16	no	—
101	C₂×C₄×C₁₉	—	2	16	no	—
102	C₂×C₄×C₁₉	—	2	16	no	—
103	C₂×C₄×C₁₉	—	2	16	no	—
104	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
108	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
109	C₂×C₄×C₁₉	2₋¹⁺⁴	1	8	yes	—
110	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
111	#2381	C₂×C₁₉	1	4	yes	complemented
112	#2369	C₂×C₁₉	1	4	yes	complemented
113	#2344	C₂×C₁₉	1	8	yes	—
114	#2331	C₂×C₁₉	1	8	yes	complemented
115	#2310	C₂×C₁₉	1	4	yes	—
116	#2283	C₂×C₁₉	1	2	yes	—
117	C₂²×C₁₉	—	2	4	no	—
118	C₂²×C₁₉	—	2	8	no	—
119	C₂²×C₁₉	#518	1	4	yes	—
120	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	4	yes	—
123	C₄×C₁₉	—	2	16	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	4	no	—
127	C₄×C₁₉	—	4	16	no	—
128	C₄×C₁₉	#519	1	8	yes	—
129	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
130	C₂×C₄⋊Q₈	C₄×C₁₉	1	16	yes	complemented
131	C₂×C₄⋊Q₈	C₂²×C₁₉	1	4	yes	—
132	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
133	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
134	C₂×#117	C₂²×C₁₉	1	8	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
136	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
137	#391	C₂²×C₁₉	1	16	yes	—
138	#391	C₂²×C₁₉	1	16	yes	—
139	#389	C₂²×C₁₉	1	16	yes	—
140	#387	C₂²×C₁₉	1	32	yes	—
141	#383	C₂²×C₁₉	1	4	yes	—
142	#382	C₂²×C₁₉	1	4	yes	—
143	#381	C₂²×C₁₉	1	4	yes	—
144	#381	C₂²×C₁₉	1	8	yes	—
145	#380	C₂²×C₁₉	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₁₉	1	16	yes	—
147	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
148	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
149	#376	C₂²×C₁₉	1	4	yes	—
150	C₂×C₁₉	—	2	4	no	—
151	C₂×C₁₉	C₂×#518	1	1	yes	characteristic
152	C₂×C₁₉	C₂×#519	1	2	yes	—
153	C₂×C₁₉	#3544	1	4	yes	—
154	C₂²×Q₈	—	4	8	no	—
155	C₂²×Q₈	C₂×C₄×C₁₉	1	16	yes	—
156	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
157	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
158	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
159	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
160	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
161	C₄⋊Q₈	C₂×C₄×C₁₉	1	32	yes	—
162	C₄⋊Q₈	C₂×C₄×C₁₉	1	16	yes	—
163	C₄⋊Q₈	C₂×C₄×C₁₉	1	16	yes	—
164	#117	—	4	16	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	32	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
170	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
171	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	—
172	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
173	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
174	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
175	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
177	#95	C₂³×C₁₉	1	8	yes	—
178	#95	C₂³×C₁₉	1	8	yes	—
179	#95	C₂³×C₁₉	1	8	yes	—
180	#95	C₂³×C₁₉	1	8	yes	—
181	#95	C₂³×C₁₉	1	32	yes	—
182	C₁₉	#34365	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
184	C₂×Q₈	—	4	32	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	C₂²×C₄×C₁₉	1	16	yes	—
189	C₂×Q₈	C₂²×C₄×C₁₉	1	16	yes	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
196	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
197	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
201	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	4	16	no	—
204	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
205	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
206	C₂²⋊C₄	—	2	16	no	—
207	C₄²	—	2	16	no	—
208	C₄²	—	4	16	no	—
209	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
210	C₄²	C₂²×C₄×C₁₉	1	8	yes	—
211	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
212	C₂³	—	2	4	no	—
213	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
214	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
215	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
216	Q₈	—	4	16	no	—
217	Q₈	—	4	16	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	32	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
230	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
232	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
233	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
234	C₂×C₄	C₁₉×2₋¹⁺⁴	1	8	yes	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	C₁₉×#518	1	4	yes	—
238	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
239	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
240	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	4	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	16	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	16	no	—
246	C₄	C₁₉×#519	1	8	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₂×C₁₉×#518	1	1	yes	characteristic
249	C₂	C₂×C₁₉×#519	1	2	yes	—
250	C₂	C₁₉×#3544	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂⁴×C₁₉	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2381	C₂	semidirect
3. C₁₉×#2369	C₂	semidirect
5. C₁₉×#2331	C₂	semidirect
9. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
86. #34365	C₁₉	direct (N × H)
111. #2381	C₂×C₁₉	semidirect
112. #2369	C₂×C₁₉	semidirect
114. #2331	C₂×C₁₉	semidirect
130. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
182. C₁₉	#34365	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×C₄⋊Q₈	2
C₁₉×#2381	2
C₁₉×#2369	2
C₁₉×#2344	2
C₁₉×#2331	2
C₁₉×#2310	2
C₁₉×#2283	2
#34365	19

Subgroups — C19×#34365 #18040324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34365 #18040324