Subgroups — C₂₃×#8390 #19156016

Order 5888.

706 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1801	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1546	C₂	1	2	yes	maximal
4	C₂₃×#1487	C₂	1	2	yes	maximal
5	C₂₃×#1461	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#557	—	2	4	no	—
7	C₂₃×#557	—	2	4	no	—
8	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#125	—	2	2	no	—
11	C₂×C₂₃×#125	C₄	1	2	yes	—
12	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₂₃	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#431	C₂²	1	2	yes	—
17	C₂₃×#423	C₂²	1	2	yes	—
18	C₂×C₂₃×#101	—	2	1	no	—
19	C₂₃×#343	—	2	1	no	—
20	C₂₃×#332	—	2	4	no	—
21	C₂₃×Q₈⋊C₈	—	2	4	no	—
22	C₂₃×#324	—	2	2	no	—
23	C₂²×Q₈×C₂₃	—	2	1	no	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂₃×C₄⋊Q₈	—	2	8	no	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
30	C₂₃×#125	—	2	16	no	—
31	C₂₃×#125	—	2	4	no	—
32	C₂₃×#125	—	2	4	no	—
33	C₄×Q₈×C₂₃	—	2	8	no	—
34	C₄×Q₈×C₂₃	—	2	4	no	—
35	C₄×Q₈×C₂₃	—	2	4	no	—
36	C₄×Q₈×C₂₃	—	2	2	no	—
37	C₄×Q₈×C₂₃	—	2	2	no	—
38	C₄×Q₈×C₂₃	—	2	2	no	—
39	C₄×Q₈×C₂₃	D₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	—	2	1	no	—
51	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	—	2	2	no	—
54	C₂₃×C₄⋊C₈	—	2	2	no	—
55	C₂₃×C₄⋊C₈	—	2	2	no	—
56	C₂₃×#101	—	4	2	no	—
57	C₂₃×C₂²⋊C₈	—	2	2	no	—
58	C₄×C₈×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	—	2	4	no	—
60	C₂×Q₈×C₂₃	—	2	1	no	—
61	C₂×Q₈×C₂₃	—	2	2	no	—
62	C₂×Q₈×C₂₃	—	2	4	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	2	1	no	—
65	C₂×Q₈×C₂₃	—	2	1	no	—
66	C₂×Q₈×C₂₃	—	4	2	no	—
67	C₂×Q₈×C₂₃	—	4	1	no	—
68	C₂×Q₈×C₂₃	—	4	1	no	—
69	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₂₃	—	2	1	no	—
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	—	2	2	no	—
73	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₂₃	—	4	2	no	—
79	M₁₆×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₄⋊C₄×C₂₃	—	2	8	no	—
83	C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₄⋊C₄×C₂₃	—	2	8	no	—
85	C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
92	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₂₃	—	2	8	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	4	no	—
97	C₄²×C₂₃	C₂×D₄	1	2	yes	—
98	C₄²×C₂₃	C₂×D₄	1	2	yes	—
99	#8390	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈×C₂₃	—	2	4	no	—
102	Q₈×C₂₃	—	2	2	no	—
103	Q₈×C₂₃	—	4	2	no	—
104	Q₈×C₂₃	—	4	4	no	—
105	Q₈×C₂₃	—	4	2	no	—
106	Q₈×C₂₃	—	4	4	no	—
107	Q₈×C₂₃	#101	1	4	yes	complemented
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	4	1	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#120	1	2	yes	—
125	C₂×C₄×C₂₃	#120	1	2	yes	—
126	C₂×C₄×C₂₃	#135	1	2	yes	—
127	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
129	C₈×C₂₃	—	8	2	no	—
130	C₈×C₂₃	—	8	2	no	—
131	C₂×#557	C₂×C₂₃	1	1	yes	characteristic
132	#1801	C₂×C₂₃	1	1	yes	characteristic
133	#1546	C₂×C₂₃	1	2	yes	—
134	#1487	C₂×C₂₃	1	2	yes	—
135	#1461	C₂×C₂₃	1	1	yes	characteristic
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	#379	1	1	yes	characteristic
139	C₂²×C₂₃	#416	1	1	yes	characteristic
140	C₂²×C₂₃	#417	1	1	yes	characteristic
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	4	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	4	2	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	4	4	no	—
148	C₄×C₂₃	C₂×#101	1	2	yes	—
149	C₄×C₂₃	#449	1	2	yes	—
150	C₄×C₂₃	#452	1	2	yes	—
151	#557	—	2	4	no	—
152	#557	—	2	4	no	—
153	C₂×C₄⋊Q₈	—	2	1	no	—
154	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×#125	—	2	2	no	—
156	C₂×#125	C₄×C₂₃	1	2	yes	—
157	C₂×C₄×Q₈	—	2	1	no	—
158	C₂×C₄×Q₈	C₄×C₂₃	1	2	yes	—
159	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	#431	C₂²×C₂₃	1	2	yes	—
162	#423	C₂²×C₂₃	1	2	yes	—
163	C₂×#101	—	2	1	no	—
164	#343	—	2	1	no	—
165	#332	—	2	4	no	—
166	Q₈⋊C₈	—	2	4	no	—
167	#324	—	2	2	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	#1784	1	1	yes	characteristic
171	C₂×C₂₃	#1788	1	1	yes	characteristic
172	C₂×C₂₃	#1796	1	1	yes	characteristic
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
180	#125	—	2	16	no	—
181	#125	—	2	4	no	—
182	#125	—	2	4	no	—
183	C₄×Q₈	—	2	8	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	4	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	D₄×C₂₃	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	#101	—	4	2	no	—
207	C₂²⋊C₈	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₂₃	#8390	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	2	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
243	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
244	C₄²	—	2	8	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	4	no	—
248	C₄²	C₂×D₄×C₂₃	1	2	yes	—
249	C₄²	C₂×D₄×C₂₃	1	2	yes	—
250	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	2	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	4	no	—
257	Q₈	C₂₃×#101	1	4	yes	complemented
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂₃×#120	1	2	yes	—
275	C₂×C₄	C₂₃×#120	1	2	yes	—
276	C₂×C₄	C₂₃×#135	1	2	yes	—
277	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
279	C₈	—	8	2	no	—
280	C₈	—	8	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂₃×#379	1	1	yes	characteristic
284	C₂²	C₂₃×#416	1	1	yes	characteristic
285	C₂²	C₂₃×#417	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×C₂₃×#101	1	2	yes	—
294	C₄	C₂₃×#449	1	2	yes	—
295	C₄	C₂₃×#452	1	2	yes	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	C₂₃×#1784	1	1	yes	characteristic
299	C₂	C₂₃×#1788	1	1	yes	characteristic
300	C₂	C₂₃×#1796	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
99. #8390	C₂₃	direct (N × H)
107. Q₈×C₂₃	#101	semidirect
209. C₂₃	#8390	direct (N × H)
257. Q₈	C₂₃×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#557	2
C₂₃×#1801	2
C₂₃×#1546	2
C₂₃×#1487	2
C₂₃×#1461	2
#8390	23

Subgroups — C23×#8390 #19156016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8390 #19156016