Subgroups — C₂×C₂₃×#2214 #19181219

Order 5888.

686 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	C₂×C₂₃×#510	C₂	1	2	yes	maximal
2	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2214	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#362	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
8	C₂²×Q₁₆×C₂₃	—	2	1	no	—
9	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×#131	—	2	1	no	—
11	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
12	C₂₃×#510	—	2	4	no	—
13	C₂₃×#510	C₂²	1	8	yes	—
14	C₂×C₂₃×Q₃₂	—	2	2	no	—
15	C₂×C₂₃×SD₃₂	—	2	2	no	—
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂₃×#494	—	2	2	no	—
18	C₂₃×#494	C₂²	1	4	yes	complemented
19	C₂₃×#443	—	2	2	no	—
20	C₂₃×#443	C₂²	1	4	yes	—
21	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#104	—	2	1	no	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#364	C₂²	1	4	yes	complemented
25	C₂₃×#362	C₂²	1	8	yes	complemented
26	C₂×#41×C₂₃	—	2	1	no	—
27	C₂²×Q₈×C₂₃	—	2	2	no	—
28	C₂₃×#135	—	2	1	no	—
29	C₂₃×#135	—	2	1	no	—
30	C₂₃×#135	D₄	1	2	yes	—
31	C₂₃×#135	C₂³	1	2	yes	—
32	C₂×Q₁₆×C₂₃	—	2	2	no	—
33	C₂×Q₁₆×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	4	no	—
35	C₂×Q₁₆×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	—	2	2	no	—
37	C₂×Q₁₆×C₂₃	—	2	2	no	—
38	C₂×Q₁₆×C₂₃	—	4	1	no	—
39	C₂×Q₁₆×C₂₃	—	4	1	no	—
40	C₂×Q₁₆×C₂₃	—	4	2	no	—
41	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
43	C₂×SD₁₆×C₂₃	—	2	1	no	—
44	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂₃×#131	—	2	4	no	—
46	C₂₃×#131	—	4	2	no	—
47	C₂×M₁₆×C₂₃	—	2	1	no	—
48	C₂×M₁₆×C₂₃	—	4	1	no	—
49	C₂²×C₈×C₂₃	—	4	1	no	—
50	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×C₄⋊Q₈	—	2	4	no	—
53	C₂₃×C₄⋊Q₈	—	4	2	no	—
54	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
56	C₂₃×Q₃₂	—	2	8	no	—
57	C₂₃×SD₃₂	—	2	8	no	—
58	C₂₃×M₃₂	—	2	2	no	—
59	C₂₃×M₃₂	C₂³	1	4	yes	—
60	C₂×C₁₆×C₂₃	—	2	2	no	—
61	C₂₃×#108	—	2	1	no	—
62	C₂₃×#108	C₂³	1	2	yes	—
63	C₂₃×#104	—	2	4	no	—
64	C₄×C₈×C₂₃	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	#41×C₂₃	—	2	2	no	—
67	#41×C₂₃	—	2	2	no	—
68	#41×C₂₃	—	4	1	no	—
69	#41×C₂₃	—	4	1	no	—
70	C₂×Q₈×C₂₃	—	2	1	no	—
71	C₂×Q₈×C₂₃	—	2	4	no	—
72	C₂×Q₈×C₂₃	—	4	2	no	—
73	C₂×Q₈×C₂₃	—	4	2	no	—
74	C₂×Q₈×C₂₃	—	4	2	no	—
75	C₂×Q₈×C₂₃	—	4	2	no	—
76	C₂×Q₈×C₂₃	—	4	2	no	—
77	C₂×D₄×C₂₃	—	2	1	no	—
78	C₂×D₄×C₂₃	—	4	1	no	—
79	C₂²×C₄×C₂₃	—	2	1	no	—
80	C₂²×C₄×C₂₃	—	4	1	no	—
81	C₂²×C₄×C₂₃	—	4	2	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₂₃	—	2	1	no	—
84	Q₁₆×C₂₃	—	2	4	no	—
85	Q₁₆×C₂₃	—	2	4	no	—
86	Q₁₆×C₂₃	—	4	2	no	—
87	Q₁₆×C₂₃	—	4	2	no	—
88	Q₁₆×C₂₃	—	4	4	no	—
89	Q₁₆×C₂₃	—	4	4	no	—
90	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
91	SD₁₆×C₂₃	—	2	2	no	—
92	SD₁₆×C₂₃	—	2	2	no	—
93	D₈×C₂₃	—	2	1	no	—
94	D₈×C₂₃	C₂×D₄	1	2	yes	—
95	M₁₆×C₂₃	—	2	2	no	—
96	M₁₆×C₂₃	—	4	1	no	—
97	M₁₆×C₂₃	—	4	4	no	—
98	C₂×C₈×C₂₃	—	4	1	no	—
99	C₂×C₈×C₂₃	—	4	4	no	—
100	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
101	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
102	C₂×C₈×C₂₃	D₈	1	2	yes	—
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
108	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₂₃	—	4	8	no	—
111	C₄²×C₂₃	—	2	1	no	—
112	C₄²×C₂₃	C₂×D₄	1	2	yes	—
113	C₁₆×C₂₃	—	2	4	no	—
114	C₂×#2214	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₃	—	4	1	no	—
116	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	Q₈×C₂₃	—	2	2	no	—
118	Q₈×C₂₃	—	4	1	no	—
119	Q₈×C₂₃	—	4	4	no	—
120	Q₈×C₂₃	—	4	4	no	—
121	Q₈×C₂₃	—	8	2	no	—
122	Q₈×C₂₃	—	8	2	no	—
123	D₄×C₂₃	—	2	2	no	—
124	D₄×C₂₃	—	4	2	no	—
125	D₄×C₂₃	—	4	2	no	—
126	D₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	2	4	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	4	1	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	4	no	—
133	C₂×C₄×C₂₃	—	8	2	no	—
134	C₂×C₄×C₂₃	—	8	2	no	—
135	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₂₃	—	4	2	no	—
142	C₈×C₂₃	#121	1	1	yes	characteristic
143	C₈×C₂₃	#121	1	1	yes	characteristic
144	C₈×C₂₃	C₂×D₈	1	2	yes	—
145	C₈×C₂₃	C₂×D₈	1	2	yes	—
146	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
148	C₂×#510	C₂×C₂₃	1	2	yes	—
149	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
151	#2214	C₂×C₂₃	1	8	yes	direct product factor
152	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂×#362	C₂×C₂₃	1	2	yes	complemented
154	C₂²×C₂₃	—	2	2	no	—
155	C₂²×C₂₃	—	4	2	no	—
156	C₂²×C₂₃	—	8	2	no	—
157	C₂²×C₂₃	#459	1	1	yes	characteristic
158	C₂²×C₂₃	#459	1	1	yes	characteristic
159	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄×C₂₃	—	2	4	no	—
161	C₄×C₂₃	—	4	2	no	—
162	C₄×C₂₃	—	8	4	no	—
163	C₄×C₂₃	#459	1	1	yes	characteristic
164	C₄×C₂₃	#459	1	1	yes	characteristic
165	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
166	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
168	C₂²×Q₁₆	—	2	1	no	—
169	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
170	C₂×#131	—	2	1	no	—
171	C₂×C₄⋊Q₈	—	2	1	no	—
172	#510	—	2	4	no	—
173	#510	C₂²×C₂₃	1	8	yes	—
174	C₂×Q₃₂	—	2	2	no	—
175	C₂×SD₃₂	—	2	2	no	—
176	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
177	#494	—	2	2	no	—
178	#494	C₂²×C₂₃	1	4	yes	complemented
179	#443	—	2	2	no	—
180	#443	C₂²×C₂₃	1	4	yes	—
181	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×#104	—	2	1	no	—
183	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
184	#364	C₂²×C₂₃	1	4	yes	complemented
185	#362	C₂²×C₂₃	1	8	yes	complemented
186	C₂×C₂₃	—	2	1	no	—
187	C₂×C₂₃	—	2	1	no	—
188	C₂×C₂₃	—	8	2	no	—
189	C₂×C₂₃	#2214	1	2	yes	direct product factor
190	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	2	2	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	D₄×C₂₃	1	2	yes	—
196	#135	C₂³×C₂₃	1	2	yes	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	4	1	no	—
204	C₂×Q₁₆	—	4	1	no	—
205	C₂×Q₁₆	—	4	2	no	—
206	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
207	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
208	C₂×SD₁₆	—	2	1	no	—
209	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
210	#131	—	2	4	no	—
211	#131	—	4	2	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	4	1	no	—
214	C₂²×C₈	—	4	1	no	—
215	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	4	2	no	—
219	C₂×C₄⋊C₄	—	4	2	no	—
220	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
221	Q₃₂	—	2	8	no	—
222	SD₃₂	—	2	8	no	—
223	M₃₂	—	2	2	no	—
224	M₃₂	C₂³×C₂₃	1	4	yes	—
225	C₂×C₁₆	—	2	2	no	—
226	#108	—	2	1	no	—
227	#108	C₂³×C₂₃	1	2	yes	—
228	#104	—	2	4	no	—
229	C₄×C₈	D₄×C₂₃	1	2	yes	—
230	C₄×C₈	C₂³×C₂₃	1	2	yes	—
231	C₂₃	C₂×#2214	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	4	2	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	—	4	4	no	—
256	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
257	SD₁₆	—	2	2	no	—
258	SD₁₆	—	2	2	no	—
259	D₈	—	2	1	no	—
260	D₈	C₂×D₄×C₂₃	1	2	yes	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
267	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
268	C₂×C₈	D₈×C₂₃	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
273	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
274	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
275	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
276	C₄⋊C₄	—	4	8	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂×D₄×C₂₃	1	2	yes	—
279	C₁₆	—	2	4	no	—
280	C₂³	—	4	1	no	—
281	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
301	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	C₂₃×#121	1	1	yes	characteristic
308	C₈	C₂₃×#121	1	1	yes	characteristic
309	C₈	C₂×D₈×C₂₃	1	2	yes	—
310	C₈	C₂×D₈×C₂₃	1	2	yes	—
311	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
312	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂₃×#459	1	1	yes	characteristic
317	C₂²	C₂₃×#459	1	1	yes	characteristic
318	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	4	no	—
322	C₄	C₂₃×#459	1	1	yes	characteristic
323	C₄	C₂₃×#459	1	1	yes	characteristic
324	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
325	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	2	no	—
329	C₂	C₂₃×#2214	1	2	yes	direct product factor
330	C₂	C₂×C₂₃×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#494	C₂	semidirect
4. C₂₃×#2214	C₂	direct (N × H)
5. C₂×C₂₃×#364	C₂	semidirect
6. C₂×C₂₃×#362	C₂	semidirect
18. C₂₃×#494	C₂²	semidirect
24. C₂₃×#364	C₂²	semidirect
25. C₂₃×#362	C₂²	semidirect
114. C₂×#2214	C₂₃	direct (N × H)
149. C₂×#494	C₂×C₂₃	semidirect
151. #2214	C₂×C₂₃	direct (N × H)
152. C₂×#364	C₂×C₂₃	semidirect
153. C₂×#362	C₂×C₂₃	semidirect
178. #494	C₂²×C₂₃	semidirect
184. #364	C₂²×C₂₃	semidirect
185. #362	C₂²×C₂₃	semidirect
189. C₂×C₂₃	#2214	direct (N × H)
231. C₂₃	C₂×#2214	direct (N × H)
329. C₂	C₂₃×#2214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#510	2
C₂×C₂₃×#494	2
C₂×C₂₃×#443	2
C₂₃×#2214	2
C₂×C₂₃×#364	2
C₂×C₂₃×#362	2
C₂×#2214	23

Subgroups — C2×C23×#2214 #19181219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2214 #19181219