Subgroups — C₄×C₂₃×D₄⋊C₈ #19155794

Order 5888.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×D₄⋊C₈	C₂	1	2	yes	maximal
5	C₂₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₂₃	—	2	1	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	—	2	2	no	—
11	C₂×C₄×C₈×C₂₃	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
16	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
17	C₄³×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×D₄⋊C₈	C₄	1	8	yes	direct product factor
20	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	2	no	—
22	C₂³×C₄×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₄×D₄×C₂₃	—	2	4	no	—
26	C₄×D₄×C₂₃	—	2	2	no	—
27	C₄×D₄×C₂₃	C₈	1	8	yes	complemented
28	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	—	4	2	no	—
36	C₂×C₄²×C₂₃	—	4	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	2	yes	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₄×C₈×C₂₃	—	2	8	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₄×C₈×C₂₃	—	4	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	1	no	—
50	C₂⁴×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	C₄²	1	2	yes	—
54	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	8	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	4	4	no	—
60	C₂²×C₄×C₂₃	—	4	4	no	—
61	C₂²×C₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	C₄²	1	2	yes	—
77	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
78	C₂²⋊C₄×C₂₃	—	2	8	no	—
79	C₂²⋊C₄×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	—	2	2	no	—
82	C₄²×C₂₃	—	4	4	no	—
83	C₄²×C₂₃	—	4	1	no	—
84	C₄²×C₂₃	—	4	1	no	—
85	C₄²×C₂₃	C₂×C₈	1	4	yes	—
86	C₄²×C₂₃	M₁₆	1	4	yes	—
87	C₄²×C₂₃	D₈	1	2	yes	—
88	C₄²×C₂₃	SD₁₆	1	2	yes	—
89	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₂₃	#41	1	2	yes	—
91	C₄×D₄⋊C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	4	no	—
94	C₂³×C₂₃	—	4	1	no	—
95	C₂³×C₂₃	—	4	1	no	—
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	4	no	—
100	D₄×C₂₃	C₄×C₈	1	4	yes	complemented
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	16	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
115	C₂×C₄×C₂₃	#102	1	4	yes	—
116	C₂×C₄×C₂₃	#104	1	4	yes	—
117	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#135	1	2	yes	—
126	C₈×C₂₃	—	2	8	no	—
127	C₈×C₂₃	—	4	4	no	—
128	C₄²×D₄	C₂×C₂₃	1	1	yes	characteristic
129	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₂×D₄⋊C₈	C₂×C₂₃	1	2	yes	—
132	#1279	C₂×C₂₃	1	2	yes	complemented
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	4	4	no	—
135	C₂²×C₂₃	—	4	4	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
140	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
141	C₂²×C₂₃	#416	1	1	yes	characteristic
142	C₂²×C₂₃	C₂×#104	1	2	yes	—
143	C₄×C₂₃	—	2	4	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	4	4	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	D₄⋊C₈	1	8	yes	direct product factor
148	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
149	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
150	C₄×C₂₃	#431	1	2	yes	—
151	C₄×C₂₃	C₄×D₈	1	2	yes	—
152	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
153	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	—
154	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×C₄²	—	2	1	no	—
156	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄×C₈	—	2	2	no	—
158	C₂×C₄×C₈	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
160	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
161	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	C₄×C₂²⋊C₄	—	2	2	no	—
164	C₄³	C₂²×C₂₃	1	1	yes	characteristic
165	#336	—	2	1	no	—
166	D₄⋊C₈	C₄×C₂₃	1	8	yes	direct product factor
167	C₂×C₂₃	—	4	4	no	—
168	C₂×C₂₃	C₂×D₄⋊C₈	1	2	yes	—
169	C₂×C₂₃	#1476	1	2	yes	—
170	C₂×C₂₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₂₃	C₄×#104	1	1	yes	characteristic
172	C₂×C₂₃	C₄×#102	1	1	yes	characteristic
173	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₈×C₂₃	1	8	yes	complemented
181	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	—	4	2	no	—
190	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₂₃	1	2	yes	—
193	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
197	C₄×C₈	—	2	8	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	4	2	no	—
200	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
201	#95	—	2	2	no	—
202	#95	—	2	1	no	—
203	C₂₃	C₄×D₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₄²×C₂₃	1	2	yes	—
208	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₄²×C₂₃	1	2	yes	—
231	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₈×C₂₃	1	4	yes	—
240	C₄²	M₁₆×C₂₃	1	4	yes	—
241	C₄²	D₈×C₂₃	1	2	yes	—
242	C₄²	SD₁₆×C₂₃	1	2	yes	—
243	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
244	C₄²	#41×C₂₃	1	2	yes	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	D₄	—	2	2	no	—
252	D₄	—	2	4	no	—
253	D₄	C₄×C₈×C₂₃	1	4	yes	complemented
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
267	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
268	C₂×C₄	C₂₃×#102	1	4	yes	—
269	C₂×C₄	C₂₃×#104	1	4	yes	—
270	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
274	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
275	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
276	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂₃×#135	1	2	yes	—
279	C₈	—	2	8	no	—
280	C₈	—	4	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
287	C₂²	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
288	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
289	C₂²	C₂₃×#416	1	1	yes	characteristic
290	C₂²	C₂×C₂₃×#104	1	2	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	C₂₃×D₄⋊C₈	1	8	yes	direct product factor
296	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
297	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
298	C₄	C₂₃×#431	1	2	yes	—
299	C₄	C₄×D₈×C₂₃	1	2	yes	—
300	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
301	C₂	—	4	4	no	—
302	C₂	C₂×C₂₃×D₄⋊C₈	1	2	yes	—
303	C₂	C₂₃×#1476	1	2	yes	—
304	C₂	C₄×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
305	C₂	C₄×C₂₃×#104	1	1	yes	characteristic
306	C₂	C₄×C₂₃×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₂₃	D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
3. C₄²×C₈×C₂₃	C₂	semidirect
5. C₂₃×#1279	C₂	semidirect
19. C₂₃×D₄⋊C₈	C₄	direct (N × H)
27. C₄×D₄×C₂₃	C₈	semidirect
43. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
47. C₄×C₈×C₂₃	C₂×C₄	semidirect
91. C₄×D₄⋊C₈	C₂₃	direct (N × H)
100. D₄×C₂₃	C₄×C₈	semidirect
129. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
130. C₄²×C₈	C₂×C₂₃	semidirect
132. #1279	C₂×C₂₃	semidirect
147. C₄×C₂₃	D₄⋊C₈	direct (N × H)
166. D₄⋊C₈	C₄×C₂₃	direct (N × H)
180. C₄×D₄	C₈×C₂₃	semidirect
196. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
200. C₄×C₈	C₂×C₄×C₂₃	semidirect
203. C₂₃	C₄×D₄⋊C₈	direct (N × H)
253. D₄	C₄×C₈×C₂₃	semidirect
295. C₄	C₂₃×D₄⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₂₃	2
C₄×C₂₃×C₄⋊C₈	2
C₄²×C₈×C₂₃	2
C₂×C₂₃×D₄⋊C₈	2
C₂₃×#1279	2
C₄×D₄⋊C₈	23

Subgroups — C4×C23×D4⋊C8 #19155794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×D₄⋊C₈ #19155794