Subgroups — C₈×C₂₃×#117 #19168097

Order 5888.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₂₃	C₂	1	4	yes	maximal
6	C₂₃×#1744	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#431	C₄	1	8	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₂₃	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#376	C₄	1	2	yes	—
25	C₄³×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#336	C₂²	1	8	yes	—
27	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	—	2	8	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂₃×#117	C₈	1	16	yes	direct product factor
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₂₃	—	2	16	no	—
50	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
54	C₂₃×#95	C₂×C₄	1	4	yes	—
55	C₂₃×#95	C₂×C₄	1	4	yes	—
56	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	4	no	—
59	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
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	8	yes	—
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	8	no	—
73	C₂×C₈×C₂₃	—	2	16	no	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₂₃	#41	1	8	yes	—
78	C₄⋊C₄×C₂₃	C₂×C₈	1	16	yes	complemented
79	C₂²⋊C₄×C₂₃	C₂×C₈	1	16	yes	—
80	C₄²×C₂₃	—	2	8	no	—
81	C₄²×C₂₃	—	2	8	no	—
82	C₄²×C₂₃	C₄²	1	4	yes	—
83	C₄²×C₂₃	C₄²	1	4	yes	—
84	C₄²×C₂₃	C₂×C₈	1	16	yes	complemented
85	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
89	C₈×#117	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
93	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
94	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	8	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	C₄×C₈	1	16	yes	complemented
103	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²×C₈	1	16	yes	—
109	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₂₃	#131	1	4	yes	—
112	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	2	8	no	—
117	C₈×C₂₃	#117	1	8	yes	direct product factor
118	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
119	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
120	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
121	C₈×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
122	C₈×C₂²⋊C₄	C₂×C₂₃	1	4	yes	—
123	#1744	C₂×C₂₃	1	2	yes	complemented
124	C₄²×C₈	C₂×C₂₃	1	2	yes	complemented
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	2	8	no	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	C₂×C₄×C₈	1	4	yes	—
129	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
130	C₂²×C₂₃	C₄×#41	1	2	yes	—
131	C₂²×C₂₃	C₂³×C₈	1	2	yes	—
132	C₂²×C₂₃	C₂×#131	1	2	yes	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	16	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	C₂×C₄×C₈	1	4	yes	—
137	C₄×C₂₃	C₂×C₄×C₈	1	4	yes	—
138	C₄×C₂₃	#405	1	4	yes	—
139	C₄×C₂₃	C₂×#117	1	2	yes	—
140	C₄×C₂₃	C₄×#41	1	2	yes	—
141	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
142	C₂×#117	C₄×C₂₃	1	2	yes	—
143	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
144	#431	C₄×C₂₃	1	8	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
147	C₂×C₄×C₈	C₄×C₂₃	1	8	yes	complemented
148	C₂×C₄×C₈	C₂²×C₂₃	1	8	yes	—
149	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
153	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
154	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
156	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
157	#376	C₄×C₂₃	1	2	yes	—
158	C₄³	C₂²×C₂₃	1	2	yes	—
159	#336	C₂²×C₂₃	1	8	yes	—
160	C₂×C₂₃	—	2	4	no	—
161	C₂×C₂₃	C₄×#117	1	1	yes	characteristic
162	C₂×C₂₃	C₂²×C₄×C₈	1	1	yes	characteristic
163	C₂×C₂₃	C₂×#405	1	1	yes	characteristic
164	C₂×C₂₃	C₈×#41	1	4	yes	—
165	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
168	C₂²×C₈	—	2	8	no	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
170	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
171	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
173	#117	C₈×C₂₃	1	16	yes	direct product factor
174	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
181	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
182	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
186	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
187	C₄×C₈	—	2	16	no	—
188	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
189	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
190	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
191	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
192	#95	C₂×C₄×C₂₃	1	4	yes	—
193	#95	C₂×C₄×C₂₃	1	4	yes	—
194	C₂₃	C₈×#117	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
199	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
200	C₂²×C₄	C₂×C₈×C₂₃	1	8	yes	—
201	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	16	no	—
213	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
214	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
215	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
216	C₂×C₈	#41×C₂₃	1	8	yes	—
217	C₄⋊C₄	C₂×C₈×C₂₃	1	16	yes	complemented
218	C₂²⋊C₄	C₂×C₈×C₂₃	1	16	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	C₄²×C₂₃	1	4	yes	—
222	C₄²	C₄²×C₂₃	1	4	yes	—
223	C₄²	C₂×C₈×C₂₃	1	16	yes	complemented
224	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
225	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
226	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
227	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
231	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
232	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₄×C₈×C₂₃	1	16	yes	complemented
241	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
244	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
246	C₂×C₄	C₂²×C₈×C₂₃	1	16	yes	—
247	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
248	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
249	C₂×C₄	C₂₃×#131	1	4	yes	—
250	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₂₃×#117	1	8	yes	direct product factor
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×C₄×C₈×C₂₃	1	4	yes	—
260	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
261	C₂²	C₄×#41×C₂₃	1	2	yes	—
262	C₂²	C₂³×C₈×C₂₃	1	2	yes	—
263	C₂²	C₂×C₂₃×#131	1	2	yes	—
264	C₄	—	2	16	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×C₄×C₈×C₂₃	1	4	yes	—
268	C₄	C₂×C₄×C₈×C₂₃	1	4	yes	—
269	C₄	C₂₃×#405	1	4	yes	—
270	C₄	C₂×C₂₃×#117	1	2	yes	—
271	C₄	C₄×#41×C₂₃	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₄×C₂₃×#117	1	1	yes	characteristic
274	C₂	C₂²×C₄×C₈×C₂₃	1	1	yes	characteristic
275	C₂	C₂×C₂₃×#405	1	1	yes	characteristic
276	C₂	C₈×#41×C₂₃	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1744	C₂	semidirect
7. C₄²×C₈×C₂₃	C₂	semidirect
11. C₂₃×#431	C₄	semidirect
14. C₂×C₄×C₈×C₂₃	C₄	semidirect
35. C₂₃×#117	C₈	direct (N × H)
78. C₄⋊C₄×C₂₃	C₂×C₈	semidirect
84. C₄²×C₂₃	C₂×C₈	semidirect
89. C₈×#117	C₂₃	direct (N × H)
102. C₂×C₄×C₂₃	C₄×C₈	semidirect
117. C₈×C₂₃	#117	direct (N × H)
121. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
123. #1744	C₂×C₂₃	semidirect
124. C₄²×C₈	C₂×C₂₃	semidirect
144. #431	C₄×C₂₃	semidirect
147. C₂×C₄×C₈	C₄×C₂₃	semidirect
173. #117	C₈×C₂₃	direct (N × H)
194. C₂₃	C₈×#117	direct (N × H)
217. C₄⋊C₄	C₂×C₈×C₂₃	semidirect
223. C₄²	C₂×C₈×C₂₃	semidirect
240. C₂×C₄	C₄×C₈×C₂₃	semidirect
255. C₈	C₂₃×#117	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×#431	2
C₂²×C₄×C₈×C₂₃	2
C₄×C₂₃×#117	2
C₈×C₄⋊C₄×C₂₃	2
C₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1744	2
C₄²×C₈×C₂₃	2
C₈×#117	23

Subgroups — C8×C23×#117 #19168097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂₃×#117 #19168097