Subgroups — #14577447

Order 2432.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₈	C₂	1	2	yes	maximal, complemented
2	#11927272	C₂	1	2	yes	maximal, complemented
3	#11927254	C₂	1	2	yes	maximal, complemented
4	#11927252	C₂	1	4	yes	maximal, complemented
5	Q₈×D₇₆	C₂	1	2	yes	maximal, complemented
6	#11927207	C₂	1	4	yes	maximal, complemented
7	#11927174	C₂	1	4	yes	maximal, complemented
8	#11927172	C₂	1	2	yes	maximal, complemented
9	#11927099	C₂	1	2	yes	maximal, complemented
10	C₂×#10597315	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊Q₈×D₁₉	C₂	1	2	yes	maximal, complemented
12	#11927024	C₂	1	2	yes	maximal
13	C₄×#64541	C₂	1	1	yes	maximal, characteristic
14	C₁₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×Q₈×D₁₉	C₂²	1	4	yes	—
16	C₂×#64541	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₈	C₂²	1	2	yes	—
18	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₉×#125	C₂²	1	2	yes	—
21	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
25	C₁₉×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
27	#10597392	C₂²	1	4	yes	—
28	Q₈×Dic₁₉	C₂²	1	4	yes	—
29	#10597390	C₂²	1	4	yes	—
30	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
31	C₄×C₂²⋊D₁₉	C₂²	1	2	yes	—
32	#10597372	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
34	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
35	#10597343	C₂²	1	4	yes	—
36	#10597343	C₂²	1	8	yes	—
37	#10597342	C₂²	1	8	yes	—
38	#10597338	C₂²	1	4	yes	—
39	C₄⋊C₄×D₁₉	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₉	C₂²	1	4	yes	—
41	#10597336	C₂²	1	4	yes	—
42	#10597335	C₂²	1	8	yes	—
43	#10597334	C₂²	1	4	yes	—
44	#10597334	C₂²	1	4	yes	—
45	#10597334	C₂²	1	8	yes	—
46	#10597333	C₂²	1	4	yes	—
47	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	—
48	#10597324	C₂²	1	8	yes	—
49	C₄×D₇₆	C₂²	1	1	yes	characteristic
50	C₄×D₇₆	C₂²	1	4	yes	—
51	C₄×D₇₆	C₂²	1	1	yes	characteristic
52	#10597318	C₂²	1	2	yes	—
53	C₄²×D₁₉	C₂²	1	2	yes	—
54	#10597316	C₂²	1	4	yes	—
55	#10597315	C₂²	1	2	yes	—
56	#10597315	C₂²	1	2	yes	—
57	#10597315	C₂²	1	2	yes	—
58	#10597315	C₂²	1	2	yes	—
59	#10597315	C₂²	1	4	yes	—
60	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
61	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
62	C₄×Dic₃₈	C₂²	1	4	yes	—
63	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
64	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
69	C₂²×Dic₁₉	C₂³	1	4	yes	—
70	Q₈×D₁₉	—	4	8	no	—
71	#64541	Q₈	1	8	yes	complemented
72	C₂×D₇₆	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₉	C₂³	1	2	yes	—
74	C₂×C₄×D₁₉	C₂³	1	2	yes	—
75	C₂×C₄×D₁₉	C₂³	1	8	yes	—
76	C₂×C₄×D₁₉	C₂³	1	2	yes	—
77	C₂×Dic₃₈	—	4	4	no	—
78	C₂×Dic₃₈	C₂³	1	4	yes	—
79	C₂×Dic₃₈	C₂³	1	4	yes	—
80	C₂×Dic₃₈	C₂³	1	8	yes	—
81	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
85	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
88	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
89	C₄²×C₁₉	C₂³	1	1	yes	characteristic
90	C₄²×C₁₉	C₂³	1	1	yes	characteristic
91	C₄²×C₁₉	C₂³	1	1	yes	characteristic
92	C₄²×C₁₉	C₂³	1	1	yes	characteristic
93	C₄²×C₁₉	C₂³	1	4	yes	—
94	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
95	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
96	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
97	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
98	C₄⋊Dic₁₉	C₂³	1	4	yes	—
99	C₄⋊Dic₁₉	C₂³	1	8	yes	—
100	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₉	C₂³	1	4	yes	—
102	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₁₉	C₂³	1	4	yes	—
104	C₄⋊Dic₁₉	C₂³	1	4	yes	—
105	Dic₁₉⋊C₄	C₂³	1	8	yes	—
106	Dic₁₉⋊C₄	C₂³	1	8	yes	—
107	Dic₁₉⋊C₄	C₂³	1	8	yes	—
108	Dic₁₉⋊C₄	C₂³	1	2	yes	—
109	Dic₁₉⋊C₄	C₂³	1	2	yes	—
110	C₄×Dic₁₉	C₂³	1	2	yes	—
111	C₄×Dic₁₉	C₂³	1	2	yes	—
112	C₄×Dic₁₉	C₂³	1	8	yes	—
113	C₄×Dic₁₉	C₂³	1	2	yes	—
114	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
115	C₂²×D₁₉	C₂⁴	1	2	yes	—
116	Q₈×C₁₉	—	2	4	no	—
117	Q₈×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	complemented
121	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
122	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
123	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₉	C₂×Q₈	1	8	yes	complemented
130	C₂×Dic₁₉	—	4	4	no	—
131	C₂×Dic₁₉	C₂⁴	1	2	yes	—
132	C₂×Dic₁₉	C₂⁴	1	2	yes	—
133	C₂×Dic₁₉	C₂⁴	1	8	yes	—
134	C₂×Dic₁₉	C₂⁴	1	2	yes	—
135	D₇₆	C₂×Q₈	1	4	yes	complemented
136	C₄×D₁₉	—	2	4	no	—
137	C₄×D₁₉	—	2	4	no	—
138	C₄×D₁₉	—	4	8	no	—
139	C₄×D₁₉	C₂×Q₈	1	8	yes	complemented
140	Dic₃₈	—	2	8	no	—
141	Dic₃₈	—	4	4	no	—
142	Dic₃₈	—	4	8	no	—
143	Dic₃₈	C₂×Q₈	1	4	yes	complemented
144	#3503	—	19	1	no	maximal, Sylow
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
148	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
149	C₂×D₁₉	—	2	2	no	—
150	C₂×D₁₉	—	2	2	no	—
151	C₂×D₁₉	C₂²×Q₈	1	4	yes	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	4	no	—
156	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
157	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
158	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
159	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
160	Dic₁₉	—	2	8	no	—
161	Dic₁₉	—	2	2	no	—
162	Dic₁₉	—	2	2	no	—
163	Dic₁₉	C₂²×Q₈	1	4	yes	—
164	Q₈²	—	19	2	no	—
165	#557	—	19	2	no	—
166	#554	—	19	4	no	—
167	#554	—	19	2	no	—
168	D₄×Q₈	—	19	4	no	—
169	D₄×Q₈	—	19	2	no	—
170	#544	—	19	4	no	—
171	#544	—	19	2	no	—
172	#533	—	19	2	no	—
173	#533	D₁₉	1	1	yes	characteristic, complemented
174	C₂×C₄⋊Q₈	—	19	2	no	—
175	C₂×C₄⋊Q₈	—	19	1	no	—
176	#520	—	19	2	no	—
177	C₄×#41	—	19	1	no	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
180	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
182	D₁₉	—	2	4	no	—
183	C₂×#41	—	19	1	no	—
184	C₂²×Q₈	—	19	4	no	—
185	C₂²×Q₈	—	19	2	no	—
186	C₄⋊Q₈	—	19	4	no	—
187	C₄⋊Q₈	—	19	4	no	—
188	C₄⋊Q₈	—	19	4	no	—
189	C₄⋊Q₈	—	19	8	no	—
190	C₄⋊Q₈	—	19	2	no	—
191	C₄⋊Q₈	—	19	2	no	—
192	C₄⋊Q₈	—	19	2	no	—
193	C₄⋊Q₈	—	19	2	no	—
194	C₄⋊Q₈	—	19	4	no	—
195	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
196	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
197	#125	—	19	4	no	—
198	#125	—	19	8	no	—
199	#125	—	19	4	no	—
200	#125	C₂×D₁₉	1	2	yes	—
201	C₂²⋊Q₈	—	19	4	no	—
202	C₂²⋊Q₈	—	19	4	no	—
203	C₂²⋊Q₈	—	19	8	no	—
204	C₂²⋊Q₈	—	19	8	no	—
205	C₂²⋊Q₈	—	19	8	no	—
206	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
207	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
208	C₄×Q₈	—	19	4	no	—
209	C₄×Q₈	—	19	4	no	—
210	C₄×Q₈	—	19	1	no	—
211	C₄×Q₈	—	19	1	no	—
212	C₄×Q₈	—	19	4	no	—
213	C₄×Q₈	C₂×D₁₉	1	2	yes	—
214	C₄×Q₈	C₂×D₁₉	1	2	yes	—
215	C₄×D₄	—	19	2	no	—
216	C₄×D₄	—	19	2	no	—
217	C₄×D₄	—	19	8	no	—
218	C₄×D₄	—	19	1	no	—
219	C₄×D₄	—	19	1	no	—
220	C₄×D₄	—	19	4	no	—
221	#117	—	19	1	no	—
222	#117	—	19	4	no	—
223	#117	—	19	2	no	—
224	#117	C₂×D₁₉	1	2	yes	—
225	C₂×C₄⋊C₄	—	19	2	no	—
226	C₂×C₄⋊C₄	—	19	2	no	—
227	C₂×C₄⋊C₄	—	19	4	no	—
228	C₂×C₄⋊C₄	—	19	4	no	—
229	C₂×C₄²	—	19	2	no	—
230	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
231	C₁₉	#3503	1	1	yes	characteristic, complemented, Sylow
232	#41	—	19	8	no	—
233	C₂×Q₈	—	19	1	no	—
234	C₂×Q₈	—	19	4	no	—
235	C₂×Q₈	—	19	8	no	—
236	C₂×Q₈	—	19	4	no	—
237	C₂×Q₈	—	76	4	no	—
238	C₂×Q₈	—	76	8	no	—
239	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
240	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
241	C₂×D₄	—	19	2	no	—
242	C₂×D₄	—	19	1	no	—
243	C₂²×C₄	—	19	2	no	—
244	C₂²×C₄	—	19	4	no	—
245	C₂²×C₄	—	19	2	no	—
246	C₂²×C₄	—	19	2	no	—
247	C₂²×C₄	—	19	8	no	—
248	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₄⋊C₄	—	19	8	no	—
252	C₄⋊C₄	—	19	8	no	—
253	C₄⋊C₄	—	19	2	no	—
254	C₄⋊C₄	—	19	2	no	—
255	C₄⋊C₄	—	19	8	no	—
256	C₄⋊C₄	—	19	4	no	—
257	C₄⋊C₄	—	19	4	no	—
258	C₄⋊C₄	—	19	4	no	—
259	C₄⋊C₄	—	19	4	no	—
260	C₄⋊C₄	—	19	1	no	—
261	C₄⋊C₄	—	19	1	no	—
262	C₄⋊C₄	—	19	8	no	—
263	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
265	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
266	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
267	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
268	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
269	C₂²⋊C₄	—	19	2	no	—
270	C₂²⋊C₄	—	19	2	no	—
271	C₂²⋊C₄	—	19	2	no	—
272	C₂²⋊C₄	—	19	8	no	—
273	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
274	C₄²	—	19	2	no	—
275	C₄²	—	19	2	no	—
276	C₄²	—	19	2	no	—
277	C₄²	—	19	8	no	—
278	C₄²	C₂²×D₁₉	1	1	yes	characteristic
279	C₄²	C₂²×D₁₉	1	1	yes	characteristic
280	C₄²	C₂²×D₁₉	1	1	yes	characteristic
281	C₄²	C₂²×D₁₉	1	1	yes	characteristic
282	C₄²	C₂²×D₁₉	1	4	yes	—
283	C₂³	—	19	2	no	—
284	C₂³	C₂³×D₁₉	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	19	4	no	—
288	Q₈	—	38	8	no	—
289	Q₈	—	76	4	no	—
290	Q₈	—	76	8	no	—
291	D₄	—	19	8	no	—
292	D₄	—	19	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	19	2	no	—
296	C₂×C₄	—	19	8	no	—
297	C₂×C₄	—	19	2	no	—
298	C₂×C₄	—	19	2	no	—
299	C₂×C₄	—	19	8	no	—
300	C₂×C₄	—	38	4	no	—
301	C₂×C₄	—	38	4	no	—
302	C₂×C₄	—	76	8	no	—
303	C₂×C₄	—	76	4	no	—
304	C₂×C₄	Q₈×D₁₉	1	4	yes	complemented
305	C₂×C₄	C₂³×D₁₉	1	4	yes	—
306	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₁₉	1	4	yes	—
309	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	19	4	no	—
316	C₂²	—	38	2	no	—
317	C₂²	—	38	2	no	—
318	C₂²	C₂×Q₈×D₁₉	1	2	yes	—
319	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	19	4	no	—
325	C₄	—	38	2	no	—
326	C₄	—	38	8	no	—
327	C₄	—	38	2	no	—
328	C₄	C₂×Q₈×D₁₉	1	2	yes	—
329	C₄	C₂×Q₈×D₁₉	1	2	yes	—
330	C₄	#10597435	1	2	yes	—
331	C₄	#10597436	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	38	4	no	—
334	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
335	C₂	C₂×#10597435	1	1	yes	characteristic
336	C₂	C₂×#10597436	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#533	C₂
prime radical	C₂×#64541	C₂²
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3503
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927272	C₂	semidirect
3. #11927254	C₂	semidirect
4. #11927252	C₂	semidirect
5. Q₈×D₇₆	C₂	semidirect
6. #11927207	C₂	semidirect
7. #11927174	C₂	semidirect
8. #11927172	C₂	semidirect
9. #11927099	C₂	semidirect
10. C₂×#10597315	C₂	semidirect
11. C₄⋊Q₈×D₁₉	C₂	semidirect
14. C₁₉×#533	C₂	semidirect
71. #64541	Q₈	semidirect
120. C₂×C₄×C₁₉	C₂×Q₈	semidirect
129. C₂²⋊D₁₉	C₂×Q₈	semidirect
135. D₇₆	C₂×Q₈	semidirect
139. C₄×D₁₉	C₂×Q₈	semidirect
143. Dic₃₈	C₂×Q₈	semidirect
173. #533	D₁₉	semidirect
231. C₁₉	#3503	semidirect
304. C₂×C₄	Q₈×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₈	2
#11927272	2
#11927254	2
#11927252	2
Q₈×D₇₆	2
#11927207	2
#11927174	2
#11927172	2
#11927099	2
C₂×#10597315	2
C₄⋊Q₈×D₁₉	2
#11927024	2
C₄×#64541	2
C₁₉×#533	2
#3503	19