Subgroups — D₃×C₂²⋊#674 #14312813

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×#168	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×#674	C₂	1	2	yes	maximal, complemented
3	#11746782	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₃)⋊#674	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊#674	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₄	1	1	yes	characteristic
9	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
10	C₂×D₃×#168	—	2	4	no	—
11	C₂×D₃×#168	C₄	1	1	yes	characteristic
12	C₂×D₃×#168	C₄	1	1	yes	characteristic
13	C₂×D₃×#168	C₂²	1	1	yes	characteristic
14	C₂×D₃×#168	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊#674	C₂²	1	2	yes	complemented
16	C₂×C₃×#674	C₂²	1	2	yes	complemented
17	D₃×#674	—	2	4	no	—
18	D₃×C₂²⋊C₁₆	—	5	1	no	maximal, Hall
19	C₂³×#168	—	3	1	no	—
20	C₂²×#674	—	3	2	no	—
21	C₂²⋊#674	—	3	1	no	—
22	C₂²⋊#674	—	3	2	no	—
23	C₂²⋊#674	D₃	1	1	yes	characteristic, direct product factor
24	C₂³×C₅×D₃	C₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₈	1	1	yes	characteristic
27	C₂×C₄×C₅×D₃	—	2	2	no	—
28	C₂×C₄×C₅×D₃	—	2	2	no	—
29	C₂×C₄×C₅×D₃	C₈	1	1	yes	characteristic
30	C₂×C₄×C₅×D₃	C₈	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂³	1	1	yes	characteristic
39	D₃×#168	—	2	2	no	—
40	D₃×#168	—	2	2	no	—
41	D₃×#168	—	2	2	no	—
42	D₃×#168	D₄	1	4	yes	—
43	C₃⋊#674	—	2	2	no	—
44	C₃×#674	—	2	2	no	—
45	C₂²×D₃×C₈	—	5	1	no	—
46	C₂×D₃×C₁₆	—	5	2	no	—
47	C₃×C₂²⋊C₁₆	—	5	1	no	—
48	#4549	—	5	1	no	—
49	(C₂×D₃)⋊C₁₆	—	5	2	no	—
50	C₂³×C₄×C₅	—	3	1	no	—
51	C₂²×#168	—	3	1	no	—
52	C₂²×#168	—	3	1	no	—
53	C₂²×#168	—	3	1	no	—
54	C₂²×#168	—	3	1	no	—
55	C₂²×#168	—	3	1	no	—
56	C₂²×#168	—	6	4	no	—
57	C₂²×#168	C₂×D₃	1	1	yes	characteristic
58	C₂×#674	—	3	2	no	—
59	C₂×#674	—	6	4	no	—
60	C₂×#674	C₂×D₃	1	2	yes	complemented
61	C₂×C₂²⋊C₁₆	—	15	1	no	Sylow
62	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₅×D₃	—	2	2	no	—
64	C₂²×C₅×D₃	—	2	2	no	—
65	C₂²×C₅×D₃	C₁₆	1	4	yes	complemented
66	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	1	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₅×Dic₃	—	2	2	no	—
73	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
75	C₄×C₅×D₃	—	2	2	no	—
76	C₄×C₅×D₃	—	2	1	no	—
77	C₄×C₅×D₃	—	2	1	no	—
78	C₄×C₅×D₃	—	2	2	no	—
79	C₄×C₅×D₃	C₂²⋊C₄	1	2	yes	—
80	C₄×C₅×D₃	M₁₆	1	2	yes	—
81	C₃⋊#168	—	2	1	no	—
82	C₃⋊#168	C₂²⋊C₄	1	2	yes	—
83	C₃×#168	—	2	1	no	—
84	C₃×#168	C₂×D₄	1	2	yes	—
85	C₂²×C₄×D₃	AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₂²×C₃×C₈	—	5	1	no	—
87	C₂²×C₃⋊C₈	—	5	1	no	—
88	C₂×D₃×C₈	—	5	1	no	—
89	C₂×D₃×C₈	—	5	1	no	—
90	C₂×D₃×C₈	—	5	1	no	—
91	C₂×D₃×C₈	—	5	1	no	—
92	C₂×D₃×C₈	—	10	4	no	—
93	C₂×C₃×C₁₆	—	5	2	no	—
94	C₂×C₃⋊C₁₆	—	5	2	no	—
95	D₃×C₁₆	—	10	4	no	—
96	C₂⁴×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂²×C₄×C₅	—	3	1	no	—
99	C₂²×C₄×C₅	—	3	1	no	—
100	C₂²×C₄×C₅	—	3	1	no	—
101	C₂²×C₄×C₅	—	3	1	no	—
102	C₂²×C₄×C₅	—	6	2	no	—
103	C₂²×C₄×C₅	—	6	2	no	—
104	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
105	C₂×#168	—	2	2	no	—
106	C₂×#168	—	3	1	no	—
107	C₂×#168	—	3	4	no	—
108	C₂×#168	—	3	1	no	—
109	C₂×#168	—	6	2	no	—
110	C₂×#168	—	6	2	no	—
111	C₂×#168	—	6	2	no	—
112	C₂×#168	—	6	2	no	—
113	C₂×#168	C₄×D₃	1	1	yes	characteristic
114	C₂×#168	C₂²×D₃	1	1	yes	characteristic
115	#674	—	2	2	no	—
116	#674	—	6	2	no	—
117	C₂³×C₈	—	15	1	no	—
118	C₂²×C₁₆	—	15	2	no	—
119	C₂²⋊C₁₆	—	5	1	no	—
120	C₂²⋊C₁₆	—	15	2	no	—
121	C₂²⋊C₁₆	—	15	1	no	—
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	C₂×C₁₆	1	2	yes	complemented
125	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₅×D₃	—	2	1	no	—
127	C₂×C₅×D₃	—	2	1	no	—
128	C₂×C₅×D₃	—	2	8	no	—
129	C₂×C₅×D₃	C₂²⋊C₈	1	2	yes	—
130	C₂×C₅×D₃	C₂×C₁₆	1	2	yes	—
131	C₂×C₅×D₃	C₂×C₁₆	1	2	yes	complemented
132	C₂×C₅×D₃	M₃₂	1	2	yes	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	C₅×Dic₃	—	2	1	no	—
137	C₅×Dic₃	C₂²⋊C₈	1	2	yes	—
138	C₂³×D₃	#170	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂²×Dic₃	#170	1	1	yes	characteristic
141	C₂×C₄×D₃	—	2	2	no	—
142	C₂×C₄×D₃	—	2	2	no	—
143	C₂×C₄×D₃	#170	1	1	yes	characteristic
144	C₂×C₄×D₃	#170	1	1	yes	characteristic
145	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₃×C₈	—	5	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	10	2	no	—
150	C₂×C₃⋊C₈	—	5	1	no	—
151	C₂×C₃⋊C₈	—	5	1	no	—
152	C₂×C₃⋊C₈	—	10	2	no	—
153	D₃×C₈	—	5	4	no	—
154	D₃×C₈	—	10	2	no	—
155	D₃×C₈	—	10	2	no	—
156	D₃×C₈	—	10	2	no	—
157	C₃×C₁₆	—	10	2	no	—
158	C₃⋊C₁₆	—	10	2	no	—
159	C₂³×C₅	—	3	1	no	—
160	C₂³×C₅	—	3	4	no	—
161	C₂³×C₅	—	3	1	no	—
162	C₂³×C₅	—	6	2	no	—
163	C₂³×C₅	—	6	2	no	—
164	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	2	no	—
169	C₂×C₄×C₅	—	3	2	no	—
170	C₂×C₄×C₅	—	3	1	no	—
171	C₂×C₄×C₅	—	6	2	no	—
172	C₂×C₄×C₅	—	6	2	no	—
173	C₂×C₄×C₅	—	6	2	no	—
174	C₂×C₄×C₅	—	6	1	no	—
175	C₂×C₄×C₅	—	6	1	no	—
176	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	#168	—	2	1	no	—
179	#168	—	3	2	no	—
180	#168	—	6	1	no	—
181	#168	D₃×D₄	1	2	yes	—
182	C₂³×C₄	—	3	1	no	—
183	C₂²×C₈	—	5	1	no	—
184	C₂²×C₈	—	15	1	no	—
185	C₂²×C₈	—	15	1	no	—
186	C₂²×C₈	—	15	1	no	—
187	C₂²×C₈	—	15	1	no	—
188	C₂²×C₈	—	15	1	no	—
189	C₂²×C₈	—	30	4	no	—
190	C₂×C₁₆	—	5	2	no	—
191	C₂×C₁₆	—	15	2	no	—
192	C₂×C₁₆	—	30	4	no	—
193	C₂×C₃×C₅	—	2	2	no	—
194	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
197	C₅×D₃	—	2	2	no	—
198	C₅×D₃	C₂²⋊C₁₆	1	4	yes	complemented
199	C₂³×C₃	C₂×#170	1	1	yes	characteristic
200	C₂²×D₃	—	2	2	no	—
201	C₂²×D₃	—	2	2	no	—
202	C₂²×D₃	#674	1	4	yes	complemented
203	C₂²×D₃	C₂×#170	1	1	yes	characteristic
204	C₂²×D₃	C₂×#170	1	1	yes	characteristic
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
211	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
212	C₄×D₃	—	2	2	no	—
213	C₄×D₃	—	2	1	no	—
214	C₄×D₃	—	2	1	no	—
215	C₄×D₃	—	2	2	no	—
216	C₄×D₃	#704	1	2	yes	—
217	C₄×D₃	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
218	C₃×C₈	—	5	2	no	—
219	C₃×C₈	—	10	1	no	—
220	C₃⋊C₈	—	5	2	no	—
221	C₃⋊C₈	—	10	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	3	2	no	—
225	C₂²×C₅	—	3	2	no	—
226	C₂²×C₅	—	3	2	no	—
227	C₂²×C₅	—	3	2	no	—
228	C₂²×C₅	—	6	8	no	—
229	C₂²×C₅	—	6	1	no	—
230	C₂²×C₅	—	6	1	no	—
231	C₂²×C₅	D₃×C₁₆	1	2	yes	complemented
232	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	3	2	no	—
235	C₄×C₅	—	6	1	no	—
236	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
238	C₂⁴	—	3	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	—	3	1	no	—
241	C₂²×C₄	—	3	1	no	—
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	1	no	—
244	C₂²×C₄	—	6	2	no	—
245	C₂²×C₄	—	6	2	no	—
246	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	1	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	15	4	no	—
251	C₂×C₈	—	15	1	no	—
252	C₂×C₈	—	15	1	no	—
253	C₂×C₈	—	30	2	no	—
254	C₂×C₈	—	30	2	no	—
255	C₂×C₈	—	30	2	no	—
256	C₂×C₈	—	30	2	no	—
257	C₁₆	—	10	2	no	—
258	C₁₆	—	30	2	no	—
259	C₃×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Hall
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	C₂×#674	1	2	yes	complemented
263	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	—	2	8	no	—
267	C₂×D₃	C₂×#674	1	2	yes	—
268	C₂×D₃	C₂×#674	1	2	yes	complemented
269	C₂×D₃	#3998	1	2	yes	—
270	C₂×D₃	C₂²⋊#170	1	2	yes	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	C₂×#704	1	1	yes	characteristic
273	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
274	Dic₃	—	2	1	no	—
275	Dic₃	C₂²⋊#170	1	2	yes	—
276	C₂×C₅	—	2	2	no	—
277	C₂×C₅	—	3	4	no	—
278	C₂×C₅	—	6	2	no	—
279	C₂×C₅	D₃×C₂²⋊C₈	1	1	yes	characteristic
280	C₂×C₅	C₂×D₃×C₁₆	1	1	yes	characteristic
281	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
282	C₂³	—	3	1	no	—
283	C₂³	—	3	1	no	—
284	C₂³	—	3	4	no	—
285	C₂³	—	6	2	no	—
286	C₂³	—	6	2	no	—
287	C₂³	D₃×#170	1	1	yes	characteristic
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	3	2	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	6	2	no	—
295	C₂×C₄	—	6	2	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	D₃×#170	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₈	—	5	2	no	—
302	C₈	—	10	1	no	—
303	C₈	—	15	2	no	—
304	C₈	—	30	1	no	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	C₂²×#674	1	1	yes	characteristic
307	C₂×C₃	C₂×#3998	1	1	yes	characteristic
308	C₂×C₃	C₂×C₂²⋊#170	1	1	yes	characteristic
309	D₃	—	2	2	no	—
310	D₃	C₂²⋊#674	1	4	yes	direct product factor
311	C₅	D₃×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	3	2	no	—
315	C₂²	—	3	2	no	—
316	C₂²	—	3	2	no	—
317	C₂²	—	3	2	no	—
318	C₂²	—	6	8	no	—
319	C₂²	—	6	1	no	—
320	C₂²	—	6	1	no	—
321	C₂²	D₃×#674	1	2	yes	complemented
322	C₂²	C₂×D₃×#170	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	3	2	no	—
325	C₄	—	6	1	no	—
326	C₄	D₃×#704	1	1	yes	characteristic
327	C₄	D₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
328	C₃	C₂×C₂²⋊#674	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	3	4	no	—
331	C₂	—	6	2	no	—
332	C₂	C₂×D₃×#674	1	1	yes	characteristic
333	C₂	D₃×#3998	1	1	yes	characteristic
334	C₂	D₃×C₂²⋊#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂²×C₁₆
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₅×D₃	C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×D₃×#674	C₂	semidirect
3. #11746782	C₂	semidirect
4. C₃×C₂²⋊#674	C₂	semidirect
5. (C₂×D₃)⋊#674	C₂	semidirect
15. C₂×C₃⋊#674	C₂²	semidirect
16. C₂×C₃×#674	C₂²	semidirect
23. C₂²⋊#674	D₃	direct (N × H)
60. C₂×#674	C₂×D₃	semidirect
65. C₂²×C₅×D₃	C₁₆	semidirect
124. C₂²×C₃×C₅	C₂×C₁₆	semidirect
131. C₂×C₅×D₃	C₂×C₁₆	semidirect
198. C₅×D₃	C₂²⋊C₁₆	semidirect
202. C₂²×D₃	#674	semidirect
231. C₂²×C₅	D₃×C₁₆	semidirect
259. C₃×C₅	C₂×C₂²⋊C₁₆	semidirect
262. C₂²×C₃	C₂×#674	semidirect
268. C₂×D₃	C₂×#674	semidirect
310. D₃	C₂²⋊#674	direct (N × H)
311. C₅	D₃×C₂²⋊C₁₆	semidirect
321. C₂²	D₃×#674	semidirect
328. C₃	C₂×C₂²⋊#674	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₃×#168	2
C₂×D₃×#674	2
#11746782	2
C₃×C₂²⋊#674	2
(C₂×D₃)⋊#674	2
C₂×C₂²⋊#674	3
D₃×C₂²⋊C₁₆	5

Subgroups — D3×C22⋊#674 #14312813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₂²⋊#674 #14312813