Subgroups — #14577440

Order 2432.

964 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#11927274	C₂	1	2	yes	maximal, complemented
3	#11927272	C₂	1	2	yes	maximal, complemented
4	#11927272	C₂	1	4	yes	maximal, complemented
5	#11927271	C₂	1	4	yes	maximal, complemented
6	#11927269	C₂	1	2	yes	maximal, complemented
7	#11927173	C₂	1	4	yes	maximal
8	#11927172	C₂	1	2	yes	maximal
9	#11927101	C₂	1	1	yes	maximal, characteristic
10	#11927096	C₂	1	4	yes	maximal
11	#11927021	C₂	1	2	yes	maximal
12	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic
13	C₁₉×#533	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
15	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	2	yes	—
18	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
22	C₁₉×#117	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₉	C₂²	1	4	yes	—
25	#10597390	C₂²	1	4	yes	—
26	#10597390	C₂²	1	8	yes	—
27	#10597372	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	#10597370	C₂²	1	2	yes	—
30	#10597370	C₂²	1	2	yes	—
31	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
33	#10597336	C₂²	1	4	yes	—
34	#10597335	C₂²	1	8	yes	—
35	#10597335	C₂²	1	8	yes	—
36	#10597335	C₂²	1	8	yes	—
37	#10597334	C₂²	1	4	yes	—
38	#10597334	C₂²	1	4	yes	—
39	#10597334	C₂²	1	8	yes	—
40	#10597333	C₂²	1	8	yes	—
41	#10597333	C₂²	1	4	yes	—
42	#10597333	C₂²	1	4	yes	—
43	#10597333	C₂²	1	4	yes	—
44	#10597324	C₂²	1	8	yes	—
45	#10597323	C₂²	1	8	yes	—
46	#10597316	C₂²	1	2	yes	—
47	#10597316	C₂²	1	4	yes	—
48	#10597315	C₂²	1	1	yes	characteristic
49	#10597315	C₂²	1	1	yes	characteristic
50	#10597315	C₂²	1	4	yes	—
51	C₄×Dic₃₈	C₂²	1	2	yes	—
52	C₄×Dic₃₈	C₂²	1	2	yes	—
53	C₄×Dic₃₈	C₂²	1	2	yes	—
54	C₄×Dic₃₈	C₂²	1	2	yes	—
55	C₄×Dic₃₈	C₂²	1	2	yes	—
56	C₄×Dic₃₈	C₂²	1	2	yes	—
57	C₄×Dic₃₈	C₂²	1	4	yes	—
58	C₄×Dic₃₈	C₂²	1	4	yes	—
59	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
60	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₂³	1	2	yes	—
65	C₂²×Dic₁₉	C₂³	1	2	yes	—
66	C₂×Dic₃₈	Q₈	1	8	yes	complemented
67	C₂×Dic₃₈	C₂³	1	2	yes	—
68	C₂×Dic₃₈	C₂³	1	8	yes	—
69	C₂×Dic₃₈	C₂³	1	2	yes	—
70	C₂×Dic₃₈	C₂³	1	2	yes	—
71	C₂×Dic₃₈	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
73	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
75	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
76	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
77	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
79	C₄²×C₁₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
81	C₄²×C₁₉	C₂³	1	1	yes	characteristic
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	4	yes	—
84	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
85	C₄⋊Dic₁₉	C₂³	1	2	yes	—
86	C₄⋊Dic₁₉	C₂³	1	8	yes	—
87	C₄⋊Dic₁₉	C₂³	1	2	yes	—
88	C₄⋊Dic₁₉	C₂³	1	2	yes	—
89	C₄⋊Dic₁₉	C₂³	1	2	yes	—
90	C₄⋊Dic₁₉	C₂³	1	2	yes	—
91	C₄⋊Dic₁₉	C₂³	1	2	yes	—
92	Dic₁₉⋊C₄	C₂³	1	4	yes	—
93	Dic₁₉⋊C₄	C₂³	1	4	yes	—
94	Dic₁₉⋊C₄	C₂³	1	8	yes	—
95	Dic₁₉⋊C₄	C₂³	1	8	yes	—
96	Dic₁₉⋊C₄	C₂³	1	8	yes	—
97	Dic₁₉⋊C₄	C₂³	1	8	yes	—
98	Dic₁₉⋊C₄	C₂³	1	4	yes	—
99	Dic₁₉⋊C₄	C₂³	1	4	yes	—
100	C₄×Dic₁₉	C₂³	1	4	yes	—
101	C₄×Dic₁₉	C₂³	1	8	yes	—
102	C₄×Dic₁₉	C₂³	1	8	yes	—
103	C₄×Dic₁₉	C₂³	1	4	yes	—
104	C₄×Dic₁₉	C₂³	1	4	yes	—
105	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
106	Q₈×C₁₉	—	2	4	no	—
107	Q₈×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
114	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₁₉	—	4	2	no	—
120	C₂×Dic₁₉	C₂×Q₈	1	8	yes	—
121	C₂×Dic₁₉	C₂⁴	1	4	yes	—
122	C₂×Dic₁₉	C₂⁴	1	4	yes	—
123	C₂×Dic₁₉	C₂⁴	1	4	yes	—
124	C₂×Dic₁₉	C₂⁴	1	4	yes	—
125	Dic₃₈	—	2	4	no	—
126	Dic₃₈	—	4	2	no	—
127	Dic₃₈	—	4	8	no	—
128	Dic₃₈	C₂×Q₈	1	8	yes	—
129	Dic₃₈	C₂×Q₈	1	8	yes	—
130	#3502	—	19	1	no	maximal, Sylow
131	C₂²×C₁₉	—	2	1	no	—
132	C₂²×C₁₉	—	2	1	no	—
133	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
134	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	2	4	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	2	4	no	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
141	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
142	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
143	Dic₁₉	—	2	4	no	—
144	Dic₁₉	—	2	4	no	—
145	Dic₁₉	—	2	4	no	—
146	Dic₁₉	C₂²×Q₈	1	8	yes	—
147	Q₈²	—	19	2	no	—
148	Q₈²	—	19	2	no	—
149	#557	—	19	2	no	—
150	#557	—	19	4	no	—
151	#557	—	19	4	no	—
152	#557	—	19	2	no	—
153	#544	—	19	4	no	—
154	#544	—	19	2	no	—
155	#533	—	19	4	no	—
156	#533	—	19	1	no	—
157	#533	D₁₉	1	1	yes	characteristic
158	#519	—	19	2	no	—
159	C₂×C₄×Q₈	—	19	1	no	—
160	C₂×C₁₉	—	2	2	no	—
161	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
162	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
163	C₂×C₁₉	#585	1	1	yes	characteristic
164	C₂²×Q₈	—	19	1	no	—
165	C₄⋊Q₈	—	19	4	no	—
166	C₄⋊Q₈	—	19	8	no	—
167	C₄⋊Q₈	—	19	4	no	—
168	C₄⋊Q₈	—	19	8	no	—
169	C₄⋊Q₈	—	19	4	no	—
170	C₄⋊Q₈	—	19	1	no	—
171	C₄⋊Q₈	—	19	1	no	—
172	C₄⋊Q₈	—	19	4	no	—
173	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
174	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
175	#125	—	19	8	no	—
176	#125	—	19	8	no	—
177	#125	—	19	8	no	—
178	#125	—	19	4	no	—
179	#125	—	19	2	no	—
180	#125	—	19	4	no	—
181	#125	C₂×D₁₉	1	2	yes	—
182	C₂²⋊Q₈	—	19	2	no	—
183	C₂²⋊Q₈	—	19	2	no	—
184	C₂²⋊Q₈	—	19	8	no	—
185	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
186	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
187	C₄×Q₈	—	19	4	no	—
188	C₄×Q₈	—	19	4	no	—
189	C₄×Q₈	—	19	4	no	—
190	C₄×Q₈	—	19	8	no	—
191	C₄×Q₈	—	19	4	no	—
192	C₄×Q₈	—	19	2	no	—
193	C₄×Q₈	—	19	2	no	—
194	C₄×Q₈	—	19	2	no	—
195	C₄×Q₈	—	19	2	no	—
196	C₄×Q₈	—	19	4	no	—
197	C₄×Q₈	—	19	4	no	—
198	C₄×Q₈	—	19	2	no	—
199	C₄×Q₈	—	19	2	no	—
200	C₄×Q₈	C₂×D₁₉	1	2	yes	—
201	C₄×Q₈	C₂×D₁₉	1	2	yes	—
202	#117	—	19	2	no	—
203	#117	—	19	8	no	—
204	#117	C₂×D₁₉	1	2	yes	—
205	C₂×C₄⋊C₄	—	19	2	no	—
206	C₂×C₄⋊C₄	—	19	1	no	—
207	C₂×C₄²	—	19	2	no	—
208	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
209	C₁₉	#3502	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₈	—	19	8	no	—
211	C₂×Q₈	—	19	2	no	—
212	C₂×Q₈	—	19	2	no	—
213	C₂×Q₈	—	19	2	no	—
214	C₂×Q₈	—	19	2	no	—
215	C₂×Q₈	—	19	8	no	—
216	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
217	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
218	C₂²×C₄	—	19	2	no	—
219	C₂²×C₄	—	19	2	no	—
220	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₄	—	19	4	no	—
224	C₄⋊C₄	—	19	4	no	—
225	C₄⋊C₄	—	19	8	no	—
226	C₄⋊C₄	—	19	8	no	—
227	C₄⋊C₄	—	19	4	no	—
228	C₄⋊C₄	—	19	8	no	—
229	C₄⋊C₄	—	19	4	no	—
230	C₄⋊C₄	—	19	8	no	—
231	C₄⋊C₄	—	19	2	no	—
232	C₄⋊C₄	—	19	2	no	—
233	C₄⋊C₄	—	19	2	no	—
234	C₄⋊C₄	—	19	2	no	—
235	C₄⋊C₄	—	19	2	no	—
236	C₄⋊C₄	—	19	2	no	—
237	C₄⋊C₄	—	19	8	no	—
238	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
239	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
240	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
241	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
242	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
243	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
244	C₂²⋊C₄	—	19	4	no	—
245	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
246	C₄²	—	19	4	no	—
247	C₄²	—	19	4	no	—
248	C₄²	—	19	8	no	—
249	C₄²	—	19	8	no	—
250	C₄²	—	19	4	no	—
251	C₄²	C₂²×D₁₉	1	1	yes	characteristic
252	C₄²	C₂²×D₁₉	1	1	yes	characteristic
253	C₄²	C₂²×D₁₉	1	1	yes	characteristic
254	C₄²	C₂²×D₁₉	1	1	yes	characteristic
255	C₄²	C₂²×D₁₉	1	4	yes	—
256	C₂³	C₂³×D₁₉	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	4	2	no	—
259	Q₈	—	19	8	no	—
260	Q₈	—	19	8	no	—
261	Q₈	—	38	4	no	—
262	Q₈	—	76	2	no	—
263	Q₈	—	76	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	19	8	no	—
267	C₂×C₄	—	19	4	no	—
268	C₂×C₄	—	19	4	no	—
269	C₂×C₄	—	19	4	no	—
270	C₂×C₄	—	19	4	no	—
271	C₂×C₄	—	76	2	no	—
272	C₂×C₄	Q₈×D₁₉	1	4	yes	—
273	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₁₉	1	4	yes	—
275	C₂×C₄	C₂³×D₁₉	1	4	yes	—
276	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂×Q₈×D₁₉	1	2	yes	—
284	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	4	no	—
290	C₄	—	19	8	no	—
291	C₄	—	38	4	no	—
292	C₄	—	38	4	no	—
293	C₄	—	38	4	no	—
294	C₄	C₂×Q₈×D₁₉	1	2	yes	—
295	C₄	C₂×Q₈×D₁₉	1	2	yes	—
296	C₄	#10597436	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
299	C₂	C₂×#10597436	1	1	yes	characteristic
300	C₂	#11927380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#533	C₂
prime radical	C₂³×C₁₉	C₂⁴
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3502
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. #11927274	C₂	semidirect
3. #11927272	C₂	semidirect
4. #11927272	C₂	semidirect
5. #11927271	C₂	semidirect
6. #11927269	C₂	semidirect
66. C₂×Dic₃₈	Q₈	semidirect
209. C₁₉	#3502	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
#11927274	2
#11927272	2
#11927271	2
#11927269	2
#11927173	2
#11927172	2
#11927101	2
#11927096	2
#11927021	2
C₂×C₄×Dic₃₈	2
C₁₉×#533	2
#3502	19