Subgroups — C₂₃×#32679 #19180305

Order 5888.

706 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2568	C₂	1	2	yes	maximal
2	C₂₃×#2056	C₂	1	2	yes	maximal
3	C₂₃×#2016	C₂	1	4	yes	maximal
4	C₂₃×#1985	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1971	C₂	1	2	yes	maximal
6	C₂₃×#1927	C₂	1	4	yes	maximal
7	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#106	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#103	C₂²	1	4	yes	—
14	C₂×C₂₃×#103	C₂²	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
16	C₂₃×#400	—	2	2	no	—
17	C₂₃×#400	—	2	2	no	—
18	C₂₃×#391	—	2	2	no	—
19	C₂₃×#391	C₂²	1	4	yes	—
20	C₂₃×#391	C₂²	1	4	yes	—
21	C₂₃×#382	—	2	2	no	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂₃×#340	C₂²	1	2	yes	—
24	C₂₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
27	C₂×Q₁₆×C₂₃	—	2	4	no	—
28	C₂×Q₁₆×C₂₃	—	4	2	no	—
29	C₂×Q₁₆×C₂₃	—	4	2	no	—
30	C₂×Q₁₆×C₂₃	—	4	4	no	—
31	C₂×Q₁₆×C₂₃	—	4	2	no	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊Q₈	—	4	2	no	—
35	C₂₃×C₄⋊Q₈	—	4	2	no	—
36	C₄×Q₈×C₂₃	—	4	1	no	—
37	C₄×Q₈×C₂₃	—	4	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#106	—	4	2	no	—
47	C₂₃×C₄⋊C₈	—	4	1	no	—
48	C₂₃×#103	—	2	8	no	—
49	C₂₃×#103	—	4	4	no	—
50	C₂₃×#97	—	4	1	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	4	no	—
53	C₂₃×#95	—	2	4	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	4	no	—
56	C₂×Q₈×C₂₃	—	2	4	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	—	2	4	no	—
59	C₂×Q₈×C₂₃	—	2	4	no	—
60	C₂×Q₈×C₂₃	—	2	4	no	—
61	C₂×Q₈×C₂₃	—	2	4	no	—
62	C₂×Q₈×C₂₃	—	2	4	no	—
63	C₂×Q₈×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	Q₁₆×C₂₃	—	4	8	no	—
72	Q₁₆×C₂₃	—	4	8	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	#41	1	4	yes	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₄⋊C₄×C₂₃	—	4	1	no	—
82	C₄⋊C₄×C₂₃	—	4	4	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	—	2	2	no	—
86	C₄²×C₂₃	—	2	2	no	—
87	C₄²×C₂₃	—	2	1	no	—
88	C₄²×C₂₃	—	4	4	no	—
89	#32679	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₂₃	—	4	4	no	—
92	Q₈×C₂₃	—	4	4	no	—
93	Q₈×C₂₃	—	4	4	no	—
94	Q₈×C₂₃	—	4	4	no	—
95	Q₈×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	#137	1	4	yes	—
111	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
112	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
114	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	4	2	no	—
117	#2568	C₂×C₂₃	1	2	yes	—
118	#2056	C₂×C₂₃	1	2	yes	—
119	#2016	C₂×C₂₃	1	4	yes	—
120	#1985	C₂×C₂₃	1	1	yes	characteristic
121	#1971	C₂×C₂₃	1	2	yes	—
122	#1927	C₂×C₂₃	1	4	yes	—
123	C₂²×C₂₃	#534	1	1	yes	characteristic
124	C₂²×C₂₃	#550	1	2	yes	—
125	C₂²×C₂₃	C₂×#137	1	2	yes	—
126	C₂²×C₂₃	#578	1	2	yes	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	2	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	—	8	4	no	—
132	C₄×C₂₃	—	8	4	no	—
133	C₄×C₂₃	#538	1	2	yes	—
134	C₄×C₂₃	#555	1	1	yes	characteristic
135	C₄×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
136	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
137	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#106	C₂²×C₂₃	1	2	yes	—
141	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#103	C₂²×C₂₃	1	4	yes	—
143	C₂×#103	C₂²×C₂₃	1	4	yes	—
144	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
145	#400	—	2	2	no	—
146	#400	—	2	2	no	—
147	#391	—	2	2	no	—
148	#391	C₂²×C₂₃	1	4	yes	—
149	#391	C₂²×C₂₃	1	4	yes	—
150	#382	—	2	2	no	—
151	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	#340	C₂²×C₂₃	1	2	yes	—
153	#336	C₂²×C₂₃	1	4	yes	—
154	C₂×C₂₃	#2733	1	1	yes	characteristic
155	C₂×C₂₃	#3195	1	1	yes	characteristic
156	C₂×C₂₃	#3202	1	1	yes	characteristic
157	C₂×C₂₃	#3398	1	4	yes	—
158	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
159	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	—	4	4	no	—
164	C₂×Q₁₆	—	4	2	no	—
165	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₄⋊Q₈	—	4	2	no	—
169	C₄×Q₈	—	4	1	no	—
170	C₄×Q₈	—	4	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
174	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
178	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
179	#106	—	4	2	no	—
180	C₄⋊C₈	—	4	1	no	—
181	#103	—	2	8	no	—
182	#103	—	4	4	no	—
183	#97	—	4	1	no	—
184	#95	—	2	2	no	—
185	#95	—	2	4	no	—
186	#95	—	2	4	no	—
187	#95	—	2	4	no	—
188	#95	—	2	4	no	—
189	C₂₃	#32679	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
205	Q₁₆	—	4	8	no	—
206	Q₁₆	—	4	8	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	#41×C₂₃	1	4	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	4	no	—
223	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	Q₈	—	4	4	no	—
226	Q₈	—	4	4	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₂₃×#137	1	4	yes	—
244	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
245	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	—	4	2	no	—
250	C₂²	C₂₃×#534	1	1	yes	characteristic
251	C₂²	C₂₃×#550	1	2	yes	—
252	C₂²	C₂×C₂₃×#137	1	2	yes	—
253	C₂²	C₂₃×#578	1	2	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	4	no	—
259	C₄	—	8	4	no	—
260	C₄	C₂₃×#538	1	2	yes	—
261	C₄	C₂₃×#555	1	1	yes	characteristic
262	C₄	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic
263	C₂	C₂₃×#2733	1	1	yes	characteristic
264	C₂	C₂₃×#3195	1	1	yes	characteristic
265	C₂	C₂₃×#3202	1	1	yes	characteristic
266	C₂	C₂₃×#3398	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
89. #32679	C₂₃	direct (N × H)
189. C₂₃	#32679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2568	2
C₂₃×#2056	2
C₂₃×#2016	2
C₂₃×#1985	2
C₂₃×#1971	2
C₂₃×#1927	2
#32679	23

Subgroups — C23×#32679 #19180305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32679 #19180305