Subgroups — #14566991

Order 2432.

1232 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597376	C₂	1	2	yes	maximal, complemented
3	C₂×#10597383	C₂	1	2	yes	maximal, complemented
4	#11926461	C₂	1	2	yes	maximal, complemented
5	#11926453	C₂	1	2	yes	maximal, complemented
6	#11926446	C₂	1	2	yes	maximal, complemented
7	#11926404	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926399	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
15	C₂²×D₇₆	—	2	4	no	—
16	C₂²×D₇₆	—	2	8	no	—
17	C₂²×D₇₆	C₂²	1	2	yes	—
18	C₂²×D₇₆	C₂²	1	2	yes	—
19	C₄×D₄×C₁₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
25	#10597383	—	2	8	no	—
26	#10597376	—	2	8	no	—
27	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
28	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
29	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
30	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	complemented
31	C₁₉⋊#95	C₂²	1	2	yes	complemented
32	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
33	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
34	C₂³×D₁₉	—	2	4	no	—
35	C₂³×D₁₉	—	2	8	no	—
36	C₂³×D₁₉	—	2	4	no	—
37	C₂³×D₁₉	—	2	4	no	—
38	C₂³×D₁₉	—	4	4	no	—
39	C₂³×D₁₉	C₂³	1	4	yes	—
40	C₂×D₄×C₁₉	—	2	2	no	—
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	D₄	1	4	yes	complemented
43	C₂²×C₄×C₁₉	—	2	4	no	—
44	C₂²×C₄×C₁₉	—	2	4	no	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	D₄	1	4	yes	complemented
47	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₉	—	2	8	no	—
53	C₂×C₂²⋊D₁₉	—	2	8	no	—
54	C₂×C₂²⋊D₁₉	—	4	4	no	—
55	C₂²×Dic₁₉	C₂³	1	4	yes	—
56	C₂×D₇₆	—	2	32	no	—
57	C₂×D₇₆	—	2	8	no	—
58	C₂×D₇₆	—	2	4	no	—
59	C₂×D₇₆	—	2	4	no	—
60	C₂×D₇₆	—	2	4	no	—
61	C₂×D₇₆	D₄	1	8	yes	complemented
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₉	—	2	2	no	—
64	C₂²⋊C₄×C₁₉	—	2	2	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₂²⋊Dic₁₉	—	2	4	no	—
67	(C₂×D₁₉)⋊C₄	—	2	8	no	—
68	(C₂×D₁₉)⋊C₄	—	2	8	no	—
69	(C₂×D₁₉)⋊C₄	—	4	4	no	—
70	C₄⋊Dic₁₉	—	2	4	no	—
71	C₂³×C₁₉	—	2	2	no	—
72	C₂³×C₁₉	—	2	2	no	—
73	C₂³×C₁₉	—	2	1	no	—
74	C₂³×C₁₉	—	2	2	no	—
75	C₂³×C₁₉	—	2	1	no	—
76	C₂³×C₁₉	—	2	2	no	—
77	C₂³×C₁₉	C₂×D₄	1	2	yes	—
78	C₂³×C₁₉	C₂×D₄	1	2	yes	—
79	C₂³×C₁₉	C₂×D₄	1	2	yes	complemented
80	C₂³×C₁₉	#41	1	2	yes	—
81	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
82	C₂²×D₁₉	—	2	16	no	—
83	C₂²×D₁₉	—	2	4	no	—
84	C₂²×D₁₉	—	2	4	no	—
85	C₂²×D₁₉	—	2	4	no	—
86	C₂²×D₁₉	—	2	4	no	—
87	C₂²×D₁₉	—	2	4	no	—
88	C₂²×D₁₉	—	2	4	no	—
89	C₂²×D₁₉	—	4	16	no	—
90	C₂²×D₁₉	—	4	16	no	—
91	C₂²×D₁₉	—	4	8	no	—
92	C₂²×D₁₉	—	4	8	no	—
93	C₂²×D₁₉	C₂×D₄	1	8	yes	—
94	D₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
105	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	complemented
108	C₂²⋊D₁₉	—	4	16	no	—
109	C₂×Dic₁₉	—	2	4	no	—
110	C₂×Dic₁₉	—	2	4	no	—
111	C₂×Dic₁₉	—	2	4	no	—
112	D₇₆	—	2	16	no	—
113	D₇₆	—	2	16	no	—
114	#2408	—	19	1	no	maximal, Sylow
115	C₂²×C₁₉	—	2	2	no	—
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	4	no	—
118	C₂²×C₁₉	—	2	4	no	—
119	C₂²×C₁₉	—	2	2	no	—
120	C₂²×C₁₉	—	2	4	no	—
121	C₂²×C₁₉	—	4	1	no	—
122	C₂²×C₁₉	—	4	1	no	—
123	C₂²×C₁₉	—	4	2	no	—
124	C₂²×C₁₉	#121	1	4	yes	complemented
125	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
127	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
130	C₂×D₁₉	—	2	8	no	—
131	C₂×D₁₉	—	2	8	no	—
132	C₂×D₁₉	—	2	8	no	—
133	C₂×D₁₉	—	4	32	no	—
134	C₂×D₁₉	—	4	4	no	—
135	C₂×D₁₉	—	4	4	no	—
136	C₂×D₁₉	—	4	4	no	—
137	C₂×D₁₉	—	4	4	no	—
138	C₄×C₁₉	—	2	2	no	—
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	4	1	no	—
141	C₄×C₁₉	—	4	1	no	—
142	C₄×C₁₉	#120	1	4	yes	complemented
143	Dic₁₉	—	4	4	no	—
144	C₂³×D₄	—	19	1	no	—
145	C₂×#121	—	19	2	no	—
146	C₂×#120	—	19	2	no	—
147	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
148	#394	—	19	2	no	—
149	#392	—	19	2	no	—
150	#390	—	19	2	no	—
151	#380	—	19	1	no	—
152	#379	—	19	2	no	—
153	C₂×C₁₉	—	2	4	no	—
154	C₂×C₁₉	—	4	2	no	—
155	C₂×C₁₉	C₂×#120	1	1	yes	characteristic
156	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
157	C₂×C₁₉	#535	1	1	yes	characteristic
158	C₂×C₁₉	D₄²	1	2	yes	—
159	C₂×C₁₉	#546	1	2	yes	—
160	D₁₉	—	4	8	no	—
161	C₂⁵	—	19	2	no	—
162	C₂²×D₄	—	19	4	no	—
163	C₂²×D₄	—	19	2	no	—
164	C₂²×D₄	—	19	2	no	—
165	C₂²×D₄	—	38	8	no	—
166	C₂²×D₄	—	38	4	no	—
167	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
168	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
169	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
170	#121	—	38	8	no	—
171	#120	—	38	8	no	—
172	C₄×D₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	19	2	no	—
174	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
175	C₂×C₂²⋊C₄	—	19	2	no	—
176	C₂×C₂²⋊C₄	—	19	4	no	—
177	C₂×C₂²⋊C₄	—	19	4	no	—
178	C₂×C₂²⋊C₄	—	19	4	no	—
179	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
181	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic, complemented
182	#95	—	19	2	no	—
183	C₁₉	#2408	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	19	4	no	—
185	C₂⁴	—	38	8	no	—
186	C₂⁴	—	38	4	no	—
187	C₂⁴	—	38	4	no	—
188	C₂⁴	—	38	4	no	—
189	C₂⁴	—	76	4	no	—
190	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
191	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	19	8	no	—
195	C₂×D₄	—	38	8	no	—
196	C₂×D₄	—	38	32	no	—
197	C₂×D₄	—	38	8	no	—
198	C₂×D₄	—	38	8	no	—
199	C₂×D₄	—	38	4	no	—
200	C₂×D₄	—	38	4	no	—
201	C₂×D₄	—	38	4	no	—
202	C₂×D₄	—	76	4	no	—
203	C₂×D₄	D₇₆	1	4	yes	complemented
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	19	4	no	—
208	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	complemented
209	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	38	4	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	38	8	no	—
219	C₂²⋊C₄	—	38	4	no	—
220	C₂²⋊C₄	—	38	8	no	—
221	C₂²⋊C₄	—	76	4	no	—
222	C₄²	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	19	8	no	—
230	C₂³	—	38	16	no	—
231	C₂³	—	38	4	no	—
232	C₂³	—	38	4	no	—
233	C₂³	—	38	4	no	—
234	C₂³	—	38	4	no	—
235	C₂³	—	38	4	no	—
236	C₂³	—	38	4	no	—
237	C₂³	—	76	16	no	—
238	C₂³	—	76	16	no	—
239	C₂³	—	76	8	no	—
240	C₂³	—	76	8	no	—
241	C₂³	C₂×D₇₆	1	2	yes	—
242	C₂³	C₂×D₇₆	1	2	yes	complemented
243	C₂³	#64541	1	2	yes	—
244	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
245	C₂³	C₂³×D₁₉	1	1	yes	characteristic
246	D₄	—	2	8	no	—
247	D₄	—	38	16	no	—
248	D₄	—	38	16	no	—
249	D₄	—	76	16	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	38	4	no	—
260	C₂×C₄	—	38	4	no	—
261	C₂×C₄	—	38	4	no	—
262	C₂×C₄	C₂×D₇₆	1	2	yes	complemented
263	C₂×C₄	D₄×D₁₉	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	38	8	no	—
276	C₂²	—	38	8	no	—
277	C₂²	—	38	8	no	—
278	C₂²	—	76	32	no	—
279	C₂²	—	76	4	no	—
280	C₂²	—	76	4	no	—
281	C₂²	—	76	4	no	—
282	C₂²	—	76	4	no	—
283	C₂²	#10597376	1	4	yes	complemented
284	C₂²	C₂²×D₇₆	1	1	yes	characteristic
285	C₂²	C₂×#64541	1	1	yes	characteristic
286	C₂²	C₂×D₄×D₁₉	1	2	yes	—
287	C₂²	#10597435	1	2	yes	—
288	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	76	4	no	—
294	C₄	#10597383	1	4	yes	complemented
295	C₂	—	2	4	no	—
296	C₂	—	4	2	no	—
297	C₂	—	76	8	no	—
298	C₂	C₂×#10597383	1	1	yes	characteristic
299	C₂	C₂×#10597376	1	1	yes	characteristic
300	C₂	#11927108	1	1	yes	characteristic
301	C₂	D₄×D₇₆	1	2	yes	—
302	C₂	#11927190	1	2	yes	—

	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	#2408
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. C₂³×D₇₆	C₂	semidirect
2. C₂×#10597376	C₂	semidirect
3. C₂×#10597383	C₂	semidirect
4. #11926461	C₂	semidirect
5. #11926453	C₂	semidirect
6. #11926446	C₂	semidirect
7. #11926404	C₂	semidirect
8. #11926399	C₂	semidirect
9. C₂×C₄×D₄×C₁₉	C₂	semidirect
13. C₂³×C₄×C₁₉	C₂²	semidirect
14. C₂²×C₂²⋊D₁₉	C₂²	semidirect
20. C₂×C₄⋊C₄×C₁₉	C₂²	semidirect
22. C₂×C₂²⋊C₄×C₁₉	C₂²	semidirect
23. C₂×C₄²×C₁₉	C₂²	semidirect
27. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
28. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
29. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
30. C₂×C₄⋊Dic₁₉	C₂²	semidirect
31. C₁₉⋊#95	C₂²	semidirect
42. C₂×D₄×C₁₉	D₄	semidirect
46. C₂²×C₄×C₁₉	D₄	semidirect
61. C₂×D₇₆	D₄	semidirect
79. C₂³×C₁₉	C₂×D₄	semidirect
107. C₂×C₄×C₁₉	C₂×D₄	semidirect
124. C₂²×C₁₉	#121	semidirect
142. C₄×C₁₉	#120	semidirect
147. C₂×C₄×D₄	D₁₉	semidirect
169. C₂³×C₄	C₂×D₁₉	semidirect
174. C₂×C₄⋊C₄	C₂×D₁₉	semidirect
180. C₂×C₂²⋊C₄	C₂×D₁₉	semidirect
181. C₂×C₄²	C₂×D₁₉	semidirect
183. C₁₉	#2408	semidirect
203. C₂×D₄	D₇₆	semidirect
208. C₂²×C₄	C₂²⋊D₁₉	semidirect
242. C₂³	C₂×D₇₆	semidirect
262. C₂×C₄	C₂×D₇₆	semidirect
283. C₂²	#10597376	semidirect
294. C₄	#10597383	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₇₆	2
C₂×#10597376	2
C₂×#10597383	2
#11926461	2
#11926453	2
#11926446	2
#11926404	2
#11926399	2
C₂×C₄×D₄×C₁₉	2
#2408	19