Subgroups — #14574890

Order 2432.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927172	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927149	C₂	1	1	yes	maximal, characteristic
3	#11926953	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926946	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926909	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926891	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926869	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926869	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926863	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926858	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926845	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926845	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926835	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926825	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#117	C₂²	1	1	yes	characteristic
17	#10597376	C₂²	1	1	yes	characteristic
18	#10597376	C₂²	1	1	yes	characteristic
19	#10597372	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	#10597340	—	2	1	no	—
22	#10597340	—	2	1	no	—
23	#10597336	—	2	1	no	—
24	#10597334	—	2	1	no	—
25	#10597330	—	2	1	no	—
26	#10597330	—	2	1	no	—
27	#10597324	—	2	1	no	—
28	#10597324	—	2	1	no	—
29	#10597320	C₂²	1	1	yes	characteristic
30	C₄×D₇₆	C₂²	1	1	yes	characteristic
31	C₄×D₇₆	C₂²	1	1	yes	characteristic
32	#10597316	C₂²	1	1	yes	characteristic
33	#10597316	C₂²	1	1	yes	characteristic
34	#10597316	C₂²	1	1	yes	characteristic
35	#10597315	C₂²	1	1	yes	characteristic
36	#10597315	C₂²	1	1	yes	characteristic
37	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₉×#97	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
43	#10597278	C₂²	1	1	yes	characteristic
44	#10597278	C₂²	1	1	yes	characteristic
45	#10597278	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597278	C₂²	1	1	yes	characteristic
51	#10597275	C₂²	1	1	yes	characteristic
52	#10597275	C₂²	1	1	yes	characteristic
53	#10597275	C₂²	1	1	yes	characteristic
54	#10597275	C₂²	1	1	yes	characteristic
55	#10597274	C₂²	1	1	yes	characteristic
56	#10597274	C₂²	1	1	yes	characteristic
57	#10597274	C₂²	1	1	yes	characteristic
58	#10597274	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₉	—	2	1	no	—
61	C₂×C₂²⋊D₁₉	—	2	1	no	—
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂×D₇₆	—	2	1	no	—
64	C₂×D₇₆	—	2	1	no	—
65	C₂×D₇₆	C₂³	1	1	yes	characteristic
66	C₂×D₇₆	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	C₂×Dic₃₈	—	2	1	no	—
71	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
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	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×D₁₉	—	2	1	no	—
104	C₂²×D₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₁₉	—	8	1	no	—
114	C₂²⋊D₁₉	—	8	1	no	—
115	C₂×Dic₁₉	—	2	1	no	—
116	C₂×Dic₁₉	—	2	1	no	—
117	C₂×Dic₁₉	—	2	1	no	—
118	C₂×Dic₁₉	—	2	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	8	1	no	—
122	D₇₆	—	2	1	no	—
123	D₇₆	—	2	1	no	—
124	D₇₆	—	2	1	no	—
125	D₇₆	—	2	1	no	—
126	D₇₆	—	8	1	no	—
127	D₇₆	—	8	1	no	—
128	C₄×D₁₉	—	8	1	no	—
129	C₄×D₁₉	—	8	1	no	—
130	Dic₃₈	—	8	1	no	—
131	Dic₃₈	—	8	1	no	—
132	C₈×C₁₉	—	2	1	no	—
133	C₈×C₁₉	—	2	1	no	—
134	C₈×C₁₉	—	2	1	no	—
135	C₈×C₁₉	—	2	1	no	—
136	#3241	—	19	1	no	maximal, Sylow
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₁₉	—	4	1	no	—
142	C₂×D₁₉	—	4	1	no	—
143	C₂×D₁₉	—	4	1	no	—
144	C₂×D₁₉	—	4	1	no	—
145	C₂×D₁₉	—	4	1	no	—
146	C₂×D₁₉	—	4	1	no	—
147	C₄×C₁₉	—	2	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⁺⁴	1	1	yes	characteristic
153	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₉	—	4	1	no	—
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	#544	—	19	1	no	—
161	#540	—	19	1	no	—
162	#501	—	19	1	no	—
163	#499	—	19	1	no	—
164	#490	—	19	1	no	—
165	#486	—	19	1	no	—
166	#482	—	19	1	no	—
167	#482	—	19	1	no	—
168	#481	—	19	1	no	—
169	#480	—	19	1	no	—
170	#478	—	19	1	no	—
171	#478	—	19	1	no	—
172	#476	—	19	1	no	—
173	#474	—	19	1	no	—
174	#433	D₁₉	1	1	yes	characteristic, complemented
175	C₂×C₁₉	—	4	1	no	—
176	C₂×C₁₉	#536	1	1	yes	characteristic
177	C₂×C₁₉	#576	1	1	yes	characteristic
178	C₂×C₁₉	#577	1	1	yes	characteristic
179	D₁₉	—	8	1	no	—
180	D₁₉	—	8	1	no	—
181	C₄⋊Q₈	—	19	1	no	—
182	C₄⋊Q₈	—	19	1	no	—
183	C₄⋊Q₈	—	38	1	no	—
184	#127	—	19	1	no	—
185	#125	—	19	1	no	—
186	#125	—	19	1	no	—
187	#125	—	19	1	no	—
188	#125	—	38	1	no	—
189	#123	—	38	1	no	—
190	#123	—	38	1	no	—
191	C₂²⋊Q₈	—	38	1	no	—
192	C₂²⋊Q₈	—	38	1	no	—
193	#121	—	19	1	no	—
194	#121	—	19	1	no	—
195	#121	—	38	1	no	—
196	#121	—	38	1	no	—
197	C₄×D₄	—	19	1	no	—
198	C₄×D₄	—	19	1	no	—
199	#117	—	19	1	no	—
200	#117	C₂×D₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	19	1	no	—
202	#107	—	19	1	no	—
203	#107	—	19	1	no	—
204	#107	—	19	1	no	—
205	#107	—	19	1	no	—
206	#106	—	19	1	no	—
207	#106	—	19	1	no	—
208	#106	—	19	1	no	—
209	#106	—	19	1	no	—
210	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
211	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
212	#102	—	19	1	no	—
213	#102	—	19	1	no	—
214	#102	—	19	1	no	—
215	#102	—	19	1	no	—
216	#102	—	19	1	no	—
217	#102	—	19	1	no	—
218	#102	—	19	1	no	—
219	#102	—	19	1	no	—
220	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
221	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
222	#97	C₂×D₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
224	C₁₉	#3241	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	38	1	no	—
226	C₂×Q₈	—	38	1	no	—
227	C₂×D₄	—	19	1	no	—
228	C₂×D₄	—	19	1	no	—
229	C₂×D₄	—	38	1	no	—
230	C₂×D₄	—	38	1	no	—
231	C₂×D₄	—	38	1	no	—
232	C₂×D₄	—	38	1	no	—
233	C₂²×C₄	—	38	1	no	—
234	C₂²×C₄	—	38	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
241	C₄⋊C₄	—	19	1	no	—
242	C₄⋊C₄	—	19	1	no	—
243	C₄⋊C₄	—	19	1	no	—
244	C₄⋊C₄	—	19	1	no	—
245	C₄⋊C₄	—	19	1	no	—
246	C₄⋊C₄	—	19	1	no	—
247	C₄⋊C₄	—	38	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	D₇₆	1	1	yes	characteristic
257	C₄⋊C₄	D₇₆	1	1	yes	characteristic
258	C₂²⋊C₄	—	38	1	no	—
259	C₂²⋊C₄	—	38	1	no	—
260	C₂²⋊C₄	—	38	1	no	—
261	C₂²⋊C₄	—	38	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
264	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
265	C₄²	—	38	1	no	—
266	C₄²	C₂²×D₁₉	1	1	yes	characteristic
267	C₄²	C₂²×D₁₉	1	1	yes	characteristic
268	C₂³	—	38	1	no	—
269	C₂³	—	38	1	no	—
270	C₂³	C₂×D₇₆	1	1	yes	characteristic
271	Q₈	—	152	1	no	—
272	Q₈	—	152	1	no	—
273	D₄	—	38	1	no	—
274	D₄	—	38	1	no	—
275	D₄	—	38	1	no	—
276	D₄	—	38	1	no	—
277	D₄	—	152	1	no	—
278	D₄	—	152	1	no	—
279	D₄	—	152	1	no	—
280	D₄	—	152	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	38	1	no	—
284	C₂×C₄	—	38	1	no	—
285	C₂×C₄	—	38	1	no	—
286	C₂×C₄	—	38	1	no	—
287	C₂×C₄	—	38	1	no	—
288	C₂×C₄	—	38	1	no	—
289	C₂×C₄	—	152	1	no	—
290	C₂×C₄	—	152	1	no	—
291	C₂×C₄	—	152	1	no	—
292	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
293	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
294	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
296	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	76	1	no	—
306	C₂²	—	76	1	no	—
307	C₂²	—	76	1	no	—
308	C₂²	—	76	1	no	—
309	C₂²	—	76	1	no	—
310	C₂²	—	76	1	no	—
311	C₂²	C₂²×D₇₆	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	76	1	no	—
318	C₄	—	76	1	no	—
319	C₄	—	76	1	no	—
320	C₄	—	76	1	no	—
321	C₄	—	76	1	no	—
322	C₄	—	76	1	no	—
323	C₄	#10597436	1	1	yes	characteristic
324	C₄	#10597436	1	1	yes	characteristic
325	C₂	—	4	1	no	—
326	C₂	—	152	1	no	—
327	C₂	—	152	1	no	—
328	C₂	#11927120	1	1	yes	characteristic
329	C₂	#11927349	1	1	yes	characteristic
330	C₂	#11927356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#433	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927149	C₂
hypercenter	#433	D₁₉
hypocenter	C₁₉	#3241
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #11927172	C₂	semidirect
3. #11926953	C₂	semidirect
4. #11926946	C₂	semidirect
5. #11926909	C₂	semidirect
6. #11926891	C₂	semidirect
7. #11926869	C₂	semidirect
8. #11926869	C₂	semidirect
9. #11926863	C₂	semidirect
10. #11926858	C₂	semidirect
11. #11926845	C₂	semidirect
12. #11926845	C₂	semidirect
13. #11926835	C₂	semidirect
14. #11926825	C₂	semidirect
15. C₁₉×#433	C₂	semidirect
174. #433	D₁₉	semidirect
224. C₁₉	#3241	semidirect

Maximal subgroups

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

subgroup	index
#11927172	2
#11927149	2
#11926953	2
#11926946	2
#11926909	2
#11926891	2
#11926869	2
#11926863	2
#11926858	2
#11926845	2
#11926835	2
#11926825	2
C₁₉×#433	2
#3241	19