Subgroups — #14574303

Order 2432.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927165	C₂	1	1	yes	maximal, characteristic
2	#11927156	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926953	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926938	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926899	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926895	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926802	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926790	C₂	1	1	yes	maximal, characteristic
9	#11926785	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926773	C₂	1	1	yes	maximal, characteristic
11	#11926652	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926646	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#64516	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#117	C₂²	1	1	yes	characteristic
19	#10597376	C₂²	1	1	yes	characteristic
20	#10597370	C₂²	1	1	yes	characteristic
21	#10597370	C₂²	1	1	yes	characteristic
22	#10597370	C₂²	1	1	yes	characteristic
23	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
24	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
25	C₂×#64516	C₂²	1	1	yes	characteristic
26	C₂×#64516	C₂²	1	1	yes	characteristic
27	#10597344	—	2	1	no	—
28	#10597342	—	2	1	no	—
29	#10597335	—	2	1	no	—
30	#10597335	—	2	1	no	—
31	#10597331	—	2	1	no	—
32	#10597329	—	2	1	no	—
33	#10597325	—	2	1	no	—
34	#10597325	—	2	1	no	—
35	#10597321	C₂²	1	1	yes	characteristic
36	C₄×D₇₆	C₂²	1	1	yes	characteristic
37	#10597316	C₂²	1	1	yes	characteristic
38	#10597315	C₂²	1	1	yes	characteristic
39	#10597315	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
43	C₁₉×#97	C₂²	1	1	yes	characteristic
44	C₁₉×#97	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597275	C₂²	1	1	yes	characteristic
50	#10597275	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597274	C₂²	1	1	yes	characteristic
53	#10597273	C₂²	1	1	yes	characteristic
54	#10597273	C₂²	1	1	yes	characteristic
55	#10597273	C₂²	1	1	yes	characteristic
56	#10597273	C₂²	1	1	yes	characteristic
57	#10597273	C₂²	1	1	yes	characteristic
58	#10597273	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₉	—	2	1	no	—
61	C₂×D₇₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₉	—	2	1	no	—
63	C₂×Dic₃₈	—	2	1	no	—
64	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂³	1	1	yes	characteristic
78	C₄²×C₁₉	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₉	—	2	1	no	—
80	C₂²⋊Dic₁₉	—	2	1	no	—
81	C₂²⋊Dic₁₉	—	2	1	no	—
82	(C₂×D₁₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	—	2	1	no	—
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	—	2	1	no	—
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	—	2	1	no	—
101	C₄×Dic₁₉	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	Dic₇₆	—	2	2	no	—
105	Dic₇₆	—	2	2	no	—
106	#64516	—	2	2	no	—
107	#64516	—	2	2	no	—
108	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₉	—	8	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	D₇₆	—	2	1	no	—
127	D₇₆	—	2	1	no	—
128	C₄×D₁₉	—	8	1	no	—
129	Dic₃₈	—	2	1	no	—
130	Dic₃₈	—	2	1	no	—
131	Dic₃₈	—	2	1	no	—
132	Dic₃₈	—	2	1	no	—
133	Dic₃₈	—	2	1	no	—
134	Dic₃₈	—	2	1	no	—
135	Dic₃₈	—	8	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	#41	1	2	yes	—
139	C₈×C₁₉	#41	1	2	yes	—
140	#3178	—	19	1	no	maximal, Sylow
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₉	—	4	1	no	—
146	C₂×D₁₉	—	4	1	no	—
147	C₂×D₁₉	—	4	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
155	Dic₁₉	—	4	1	no	—
156	Dic₁₉	—	4	1	no	—
157	Dic₁₉	—	4	1	no	—
158	Dic₁₉	—	4	1	no	—
159	Dic₁₉	—	4	1	no	—
160	Dic₁₉	—	4	1	no	—
161	Dic₁₉	—	4	1	no	—
162	#542	—	19	1	no	—
163	#541	—	19	1	no	—
164	#501	—	19	1	no	—
165	#497	—	19	1	no	—
166	#488	—	19	1	no	—
167	#487	—	19	1	no	—
168	#470	—	19	1	no	—
169	#468	—	19	1	no	—
170	#467	—	19	1	no	—
171	#465	—	19	1	no	—
172	#441	—	19	1	no	—
173	#440	—	19	1	no	—
174	C₄×Q₁₆	—	19	1	no	—
175	C₄×SD₁₆	—	19	1	no	—
176	#405	D₁₉	1	1	yes	characteristic, complemented
177	C₂×C₁₉	—	4	1	no	—
178	C₂×C₁₉	#532	1	1	yes	characteristic
179	C₂×C₁₉	#577	1	1	yes	characteristic
180	C₂×C₁₉	#578	1	1	yes	characteristic
181	D₁₉	—	8	1	no	—
182	C₂×Q₁₆	—	19	1	no	—
183	C₂×Q₁₆	—	19	1	no	—
184	C₂×SD₁₆	—	19	1	no	—
185	C₂×SD₁₆	—	19	1	no	—
186	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
187	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	19	1	no	—
189	C₄⋊Q₈	—	19	1	no	—
190	#126	—	38	1	no	—
191	#126	—	38	1	no	—
192	#126	—	38	1	no	—
193	#125	—	19	1	no	—
194	#125	—	38	1	no	—
195	#125	—	38	1	no	—
196	#124	—	19	1	no	—
197	#124	—	38	1	no	—
198	#123	—	38	1	no	—
199	C₂²⋊Q₈	—	19	1	no	—
200	C₂²⋊Q₈	—	19	1	no	—
201	C₂²⋊Q₈	—	19	1	no	—
202	C₂²⋊Q₈	—	38	1	no	—
203	#121	—	19	1	no	—
204	C₄×Q₈	—	19	1	no	—
205	C₄×Q₈	—	19	1	no	—
206	C₄×Q₈	—	19	1	no	—
207	C₄×D₄	—	19	1	no	—
208	#117	C₂×D₁₉	1	1	yes	characteristic
209	#107	—	19	1	no	—
210	#107	—	19	1	no	—
211	#106	—	19	1	no	—
212	#106	—	19	1	no	—
213	#103	—	19	1	no	—
214	#103	—	19	1	no	—
215	#103	—	19	1	no	—
216	#103	—	19	1	no	—
217	#103	—	19	1	no	—
218	#103	—	19	1	no	—
219	#102	—	19	1	no	—
220	#102	—	19	1	no	—
221	#97	C₂×D₁₉	1	1	yes	characteristic
222	#97	C₂×D₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
224	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
225	C₁₉	#3178	1	1	yes	characteristic, complemented, Sylow
226	C₂×Q₈	—	19	1	no	—
227	C₂×Q₈	—	19	1	no	—
228	C₂×Q₈	—	19	1	no	—
229	C₂×Q₈	—	38	1	no	—
230	C₂×D₄	—	19	1	no	—
231	C₂×D₄	—	38	1	no	—
232	C₂²×C₄	—	38	1	no	—
233	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
234	Q₁₆	—	38	2	no	—
235	Q₁₆	—	38	2	no	—
236	SD₁₆	—	38	2	no	—
237	SD₁₆	—	38	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
244	C₄⋊C₄	—	19	1	no	—
245	C₄⋊C₄	—	19	1	no	—
246	C₄⋊C₄	—	19	1	no	—
247	C₄⋊C₄	—	19	1	no	—
248	C₄⋊C₄	—	38	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	—	38	1	no	—
252	C₄⋊C₄	—	38	1	no	—
253	C₄⋊C₄	—	38	1	no	—
254	C₄⋊C₄	—	38	1	no	—
255	C₄⋊C₄	—	38	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	D₇₆	1	1	yes	characteristic
260	C₄⋊C₄	D₇₆	1	1	yes	characteristic
261	C₂²⋊C₄	—	38	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₂²⋊C₄	—	38	1	no	—
264	C₂²⋊C₄	—	38	1	no	—
265	C₂²⋊C₄	—	38	1	no	—
266	C₂²⋊C₄	—	38	1	no	—
267	C₂²⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
269	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
270	C₄²	—	38	1	no	—
271	C₄²	—	38	1	no	—
272	C₄²	—	38	1	no	—
273	C₄²	C₂²×D₁₉	1	1	yes	characteristic
274	C₄²	C₂²×D₁₉	1	1	yes	characteristic
275	C₂³	—	38	1	no	—
276	C₂³	C₂×D₇₆	1	1	yes	characteristic
277	Q₈	—	38	1	no	—
278	Q₈	—	38	1	no	—
279	Q₈	—	38	1	no	—
280	Q₈	—	38	1	no	—
281	Q₈	—	38	1	no	—
282	Q₈	—	38	1	no	—
283	Q₈	—	152	1	no	—
284	D₄	—	38	1	no	—
285	D₄	—	38	1	no	—
286	D₄	—	152	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	38	1	no	—
291	C₂×C₄	—	38	1	no	—
292	C₂×C₄	—	38	1	no	—
293	C₂×C₄	—	38	1	no	—
294	C₂×C₄	—	38	1	no	—
295	C₂×C₄	—	38	1	no	—
296	C₂×C₄	—	152	1	no	—
297	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
298	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
300	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
301	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#64541	1	2	yes	—
306	C₈	#64541	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	76	1	no	—
311	C₂²	—	76	1	no	—
312	C₂²	—	76	1	no	—
313	C₂²	C₂²×D₇₆	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	76	1	no	—
320	C₄	—	76	1	no	—
321	C₄	—	76	1	no	—
322	C₄	—	76	1	no	—
323	C₄	—	76	1	no	—
324	C₄	—	76	1	no	—
325	C₄	—	76	1	no	—
326	C₄	C₂×#64541	1	1	yes	characteristic
327	C₄	C₂×#64541	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	152	1	no	—
330	C₂	#11927094	1	1	yes	characteristic
331	C₂	#11927356	1	1	yes	characteristic
332	C₂	#11927361	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#405	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#10597376	C₂²
hypercenter	#405	D₁₉
hypocenter	C₁₉	#3178
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #11927156	C₂	semidirect
3. #11926953	C₂	semidirect
4. #11926938	C₂	semidirect
5. #11926899	C₂	semidirect
6. #11926895	C₂	semidirect
7. #11926802	C₂	semidirect
9. #11926785	C₂	semidirect
11. #11926652	C₂	semidirect
12. #11926646	C₂	semidirect
13. C₄×Dic₇₆	C₂	semidirect
14. C₄×#64516	C₂	semidirect
15. C₁₉×#405	C₂	semidirect
176. #405	D₁₉	semidirect
225. C₁₉	#3178	semidirect

Maximal subgroups

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

subgroup	index
#11927165	2
#11927156	2
#11926953	2
#11926938	2
#11926899	2
#11926895	2
#11926802	2
#11926790	2
#11926785	2
#11926773	2
#11926652	2
#11926646	2
C₄×Dic₇₆	2
C₄×#64516	2
C₁₉×#405	2
#3178	19