Subgroups — C₂₃×#12820 #19160446

Order 5888.

414 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	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2139	C₂	1	2	yes	maximal
4	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1387	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#131	—	2	1	no	—
8	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
9	C₂₃×#443	—	2	2	no	—
10	C₂₃×#443	C₄	1	4	yes	complemented
11	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#106	—	2	1	no	—
14	C₂×C₂₃×#104	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#382	—	2	1	no	—
17	C₂₃×#364	—	2	2	no	—
18	C₂₃×#358	—	2	2	no	—
19	C₂₃×#357	—	2	2	no	—
20	C₂×#41×C₂₃	—	2	1	no	—
21	C₂₃×#135	—	2	1	no	—
22	C₂₃×#135	—	2	1	no	—
23	C₂₃×#135	C₂×C₄	1	2	yes	—
24	C₂₃×#135	Q₈	1	2	yes	—
25	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂₃×#131	—	2	4	no	—
29	C₂₃×#131	—	4	2	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	4	1	no	—
32	C₂²×C₈×C₂₃	—	4	1	no	—
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂₃×M₃₂	—	2	2	no	—
38	C₂₃×M₃₂	—	2	2	no	—
39	C₂×C₁₆×C₂₃	—	2	2	no	—
40	C₂₃×#108	—	2	1	no	—
41	C₂₃×#108	C₂×C₄	1	2	yes	—
42	C₂₃×#107	—	2	2	no	—
43	C₂₃×#107	—	2	2	no	—
44	C₂₃×#106	—	4	1	no	—
45	C₂₃×#106	—	4	1	no	—
46	C₂₃×#104	—	2	4	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₂₃	D₄	1	2	yes	—
49	C₂₃×#95	—	4	2	no	—
50	#41×C₂₃	—	2	2	no	—
51	#41×C₂₃	—	2	2	no	—
52	#41×C₂₃	—	4	1	no	—
53	#41×C₂₃	—	4	1	no	—
54	C₂×Q₈×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	—	2	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	2	1	no	—
62	Q₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
63	SD₁₆×C₂₃	—	2	2	no	—
64	SD₁₆×C₂₃	—	2	2	no	—
65	D₈×C₂₃	—	2	1	no	—
66	D₈×C₂₃	C₄⋊C₄	1	2	yes	—
67	M₁₆×C₂₃	—	2	2	no	—
68	M₁₆×C₂₃	—	4	1	no	—
69	M₁₆×C₂₃	—	4	4	no	—
70	C₂×C₈×C₂₃	—	4	1	no	—
71	C₂×C₈×C₂₃	—	4	4	no	—
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
74	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
75	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
86	C₁₆×C₂₃	—	4	2	no	—
87	#12820	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	4	1	no	—
89	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	2	no	—
91	Q₈×C₂₃	—	4	1	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	4	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
109	C₈×C₂₃	—	4	2	no	—
110	C₈×C₂₃	#102	1	2	yes	—
111	C₈×C₂₃	#102	1	2	yes	—
112	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₂₃	#117	1	1	yes	characteristic
114	C₈×C₂₃	#121	1	1	yes	characteristic
115	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
117	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#2139	C₂×C₂₃	1	2	yes	—
119	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#1387	C₂×C₂₃	1	2	yes	complemented
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	8	2	no	—
124	C₂²×C₂₃	#417	1	1	yes	characteristic
125	C₂²×C₂₃	#459	1	1	yes	characteristic
126	C₂²×C₂₃	#476	1	1	yes	characteristic
127	C₄×C₂₃	—	4	2	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	16	2	no	—
130	C₄×C₂₃	#380	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
132	C₄×C₂₃	#461	1	1	yes	characteristic
133	C₄×C₂₃	#474	1	1	yes	characteristic
134	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×#131	—	2	1	no	—
136	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
137	#443	—	2	2	no	—
138	#443	C₄×C₂₃	1	4	yes	complemented
139	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#107	C₂²×C₂₃	1	2	yes	—
141	C₂×#106	—	2	1	no	—
142	C₂×#104	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	#382	—	2	1	no	—
145	#364	—	2	2	no	—
146	#358	—	2	2	no	—
147	#357	—	2	2	no	—
148	C₂×C₂₃	—	2	1	no	—
149	C₂×C₂₃	—	2	1	no	—
150	C₂×C₂₃	—	8	2	no	—
151	C₂×C₂₃	#1802	1	1	yes	characteristic
152	C₂×C₂₃	#2212	1	2	yes	—
153	C₂×#41	—	2	1	no	—
154	#135	—	2	1	no	—
155	#135	—	2	1	no	—
156	#135	C₂×C₄×C₂₃	1	2	yes	—
157	#135	Q₈×C₂₃	1	2	yes	—
158	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
159	C₂×SD₁₆	—	2	1	no	—
160	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
161	#131	—	2	4	no	—
162	#131	—	4	2	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	4	1	no	—
165	C₂²×C₈	—	4	1	no	—
166	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	#108	—	2	1	no	—
174	#108	C₂×C₄×C₂₃	1	2	yes	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#106	—	4	1	no	—
178	#106	—	4	1	no	—
179	#104	—	2	4	no	—
180	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
181	C₄×C₈	D₄×C₂₃	1	2	yes	—
182	#95	—	4	2	no	—
183	C₂₃	#12820	1	1	yes	characteristic, direct product factor, Sylow
184	#41	—	2	2	no	—
185	#41	—	2	2	no	—
186	#41	—	4	1	no	—
187	#41	—	4	1	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
195	Q₁₆	—	2	1	no	—
196	Q₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	—	2	2	no	—
199	D₈	—	2	1	no	—
200	D₈	C₄⋊C₄×C₂₃	1	2	yes	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	4	1	no	—
203	M₁₆	—	4	4	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
207	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
208	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
209	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
214	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
215	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₁₆	—	4	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
238	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
242	C₈	—	4	2	no	—
243	C₈	C₂₃×#102	1	2	yes	—
244	C₈	C₂₃×#102	1	2	yes	—
245	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
246	C₈	C₂₃×#117	1	1	yes	characteristic
247	C₈	C₂₃×#121	1	1	yes	characteristic
248	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₂₃×#417	1	1	yes	characteristic
253	C₂²	C₂₃×#459	1	1	yes	characteristic
254	C₂²	C₂₃×#476	1	1	yes	characteristic
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	16	2	no	—
258	C₄	C₂₃×#380	1	1	yes	characteristic
259	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
260	C₄	C₂₃×#461	1	1	yes	characteristic
261	C₄	C₂₃×#474	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	8	2	no	—
265	C₂	C₂₃×#1802	1	1	yes	characteristic
266	C₂	C₂₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#459
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#364	C₂	semidirect
4. C₂₃×#1846	C₂	semidirect
5. C₂₃×#1387	C₂	semidirect
10. C₂₃×#443	C₄	semidirect
87. #12820	C₂₃	direct (N × H)
117. C₂×#364	C₂×C₂₃	semidirect
119. #1846	C₂×C₂₃	semidirect
120. #1387	C₂×C₂₃	semidirect
138. #443	C₄×C₂₃	semidirect
183. C₂₃	#12820	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×#443	2
C₂×C₂₃×#364	2
C₂₃×#2139	2
C₂₃×#1846	2
C₂₃×#1387	2
#12820	23

Subgroups — C23×#12820 #19160446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12820 #19160446