Subgroups — C₂₃×#8392 #19156018

Order 5888.

706 conjugacy classes of subgroups.

The subgroup lattice has 294 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×#1548	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	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#125	—	2	2	no	—
11	C₂×C₂₃×#125	—	2	1	no	—
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×#333	—	2	4	no	—
21	C₂₃×Q₈⋊C₈	—	2	4	no	—
22	C₂₃×#324	—	2	2	no	—
23	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
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	4	no	—
28	C₂₃×C₄⋊Q₈	—	2	4	no	—
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
31	C₂₃×#125	—	2	16	no	—
32	C₂₃×#125	—	2	8	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	—	2	1	no	—
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	1	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	—	2	2	no	—
62	C₂×Q₈×C₂₃	—	2	2	no	—
63	C₂×Q₈×C₂₃	—	4	2	no	—
64	C₂×Q₈×C₂₃	—	4	4	no	—
65	C₂×Q₈×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
67	C₂×Q₈×C₂₃	SD₁₆	1	2	yes	—
68	C₂×Q₈×C₂₃	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	1	no	—
72	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	M₁₆×C₂₃	—	4	2	no	—
78	M₁₆×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₄⋊C₄×C₂₃	—	2	8	no	—
82	C₄⋊C₄×C₂₃	—	2	8	no	—
83	C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₄⋊C₄×C₂₃	—	2	4	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	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₂₃	—	2	8	no	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	4	no	—
95	C₄²×C₂₃	C₂×D₄	1	2	yes	—
96	C₄²×C₂₃	C₂×D₄	1	2	yes	—
97	#8392	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₂₃	—	2	4	no	—
100	Q₈×C₂₃	—	2	4	no	—
101	Q₈×C₂₃	—	2	4	no	—
102	Q₈×C₂₃	—	4	4	no	—
103	Q₈×C₂₃	—	4	2	no	—
104	Q₈×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	#103	1	4	yes	—
118	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#120	1	2	yes	—
121	C₂×C₄×C₂₃	#120	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#137	1	2	yes	—
125	C₈×C₂₃	—	8	2	no	—
126	C₈×C₂₃	—	8	2	no	—
127	C₂×#557	C₂×C₂₃	1	1	yes	characteristic
128	#1801	C₂×C₂₃	1	1	yes	characteristic
129	#1548	C₂×C₂₃	1	2	yes	—
130	#1487	C₂×C₂₃	1	2	yes	—
131	#1461	C₂×C₂₃	1	1	yes	characteristic
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	#379	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
136	C₂²×C₂₃	#419	1	1	yes	characteristic
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	4	no	—
140	C₄×C₂₃	—	4	2	no	—
141	C₄×C₂₃	—	4	2	no	—
142	C₄×C₂₃	—	4	2	no	—
143	C₄×C₂₃	—	4	4	no	—
144	C₄×C₂₃	#413	1	2	yes	—
145	C₄×C₂₃	#448	1	2	yes	—
146	C₄×C₂₃	#451	1	2	yes	—
147	#557	—	2	4	no	—
148	#557	—	2	4	no	—
149	C₂×C₄⋊Q₈	C₄×C₂₃	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#125	—	2	2	no	—
152	C₂×#125	—	2	1	no	—
153	C₂×C₄×Q₈	—	2	1	no	—
154	C₂×C₄×Q₈	C₄×C₂₃	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
157	#431	C₂²×C₂₃	1	2	yes	—
158	#423	C₂²×C₂₃	1	2	yes	—
159	C₂×#101	—	2	1	no	—
160	#343	—	2	1	no	—
161	#333	—	2	4	no	—
162	Q₈⋊C₈	—	2	4	no	—
163	#324	—	2	2	no	—
164	C₂×C₂₃	—	2	1	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	#1785	1	1	yes	characteristic
167	C₂×C₂₃	#1787	1	1	yes	characteristic
168	C₂×C₂₃	#1796	1	1	yes	characteristic
169	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
170	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₄⋊Q₈	—	2	4	no	—
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
177	#125	—	2	16	no	—
178	#125	—	2	8	no	—
179	C₄×Q₈	—	2	8	no	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	—	2	2	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	D₄×C₂₃	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
194	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	#101	—	4	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₂₃	#8392	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	C₂²⋊C₄×C₂₃	1	2	yes	—
213	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
214	C₂×Q₈	SD₁₆×C₂₃	1	2	yes	—
215	C₂×Q₈	Q₁₆×C₂₃	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	C₂×D₄×C₂₃	1	2	yes	—
243	C₄²	C₂×D₄×C₂₃	1	2	yes	—
244	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂₃×#103	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂₃×#120	1	2	yes	—
267	C₂×C₄	C₂₃×#120	1	2	yes	—
268	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂₃×#137	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	C₂₃×#379	1	1	yes	characteristic
276	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
277	C₂²	C₂₃×#419	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	C₂₃×#413	1	2	yes	—
286	C₄	C₂₃×#448	1	2	yes	—
287	C₄	C₂₃×#451	1	2	yes	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	C₂₃×#1785	1	1	yes	characteristic
291	C₂	C₂₃×#1787	1	1	yes	characteristic
292	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
97. #8392	C₂₃	direct (N × H)
205. C₂₃	#8392	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×#557	2
C₂₃×#1801	2
C₂₃×#1548	2
C₂₃×#1487	2
C₂₃×#1461	2
#8392	23

Subgroups — C23×#8392 #19156018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8392 #19156018