Subgroups — C₂₃×#12824 #19160450

Order 5888.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 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	C₂×C₂₃×#360	C₂	1	2	yes	maximal
4	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1383	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	—	2	1	no	—
13	C₂×C₂₃×#106	C₂²	1	2	yes	—
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×#360	—	2	4	no	—
19	C₂×#41×C₂₃	—	2	1	no	—
20	C₂₃×#135	—	2	1	no	—
21	C₂₃×#135	—	2	1	no	—
22	C₂₃×#135	C₂×C₄	1	2	yes	—
23	C₂₃×#135	D₄	1	2	yes	—
24	C₂×Q₁₆×C₂₃	Q₈	1	1	yes	characteristic
25	C₂×SD₁₆×C₂₃	—	2	1	no	—
26	C₂×D₈×C₂₃	Q₈	1	1	yes	characteristic
27	C₂₃×#131	—	2	4	no	—
28	C₂₃×#131	—	4	2	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	4	1	no	—
31	C₂²×C₈×C₂₃	—	4	1	no	—
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂₃×M₃₂	—	2	2	no	—
37	C₂₃×M₃₂	—	2	2	no	—
38	C₂×C₁₆×C₂₃	—	2	2	no	—
39	C₂₃×#108	—	2	1	no	—
40	C₂₃×#108	C₂×C₄	1	2	yes	—
41	C₂₃×#107	—	4	2	no	—
42	C₂₃×#106	—	2	2	no	—
43	C₂₃×#106	—	2	2	no	—
44	C₂₃×#104	—	2	4	no	—
45	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₂₃	D₄	1	2	yes	—
47	C₂₃×#95	—	4	2	no	—
48	#41×C₂₃	—	2	2	no	—
49	#41×C₂₃	—	2	2	no	—
50	#41×C₂₃	—	4	1	no	—
51	#41×C₂₃	—	4	1	no	—
52	C₂×Q₈×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	—	4	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₂₃	—	2	1	no	—
60	Q₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
61	SD₁₆×C₂₃	—	2	2	no	—
62	SD₁₆×C₂₃	—	2	2	no	—
63	D₈×C₂₃	—	2	1	no	—
64	D₈×C₂₃	C₄⋊C₄	1	2	yes	—
65	M₁₆×C₂₃	—	2	2	no	—
66	M₁₆×C₂₃	—	4	1	no	—
67	M₁₆×C₂₃	—	4	4	no	—
68	C₂×C₈×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	—	4	4	no	—
70	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₂₃	D₈	1	2	yes	—
72	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
74	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	—	2	1	no	—
83	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₁₆×C₂₃	—	4	2	no	—
85	#12824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	2	no	—
89	Q₈×C₂₃	—	4	1	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	—	4	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	8	4	no	—
100	C₂×C₄×C₂₃	—	8	2	no	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
107	C₈×C₂₃	—	4	2	no	—
108	C₈×C₂₃	#102	1	2	yes	—
109	C₈×C₂₃	#102	1	2	yes	—
110	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₂₃	#117	1	1	yes	characteristic
112	C₈×C₂₃	#121	1	1	yes	characteristic
113	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
115	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×#360	C₂×C₂₃	1	2	yes	—
117	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#1383	C₂×C₂₃	1	2	yes	complemented
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	8	2	no	—
122	C₂²×C₂₃	#417	1	1	yes	characteristic
123	C₂²×C₂₃	#459	1	1	yes	characteristic
124	C₂²×C₂₃	#476	1	1	yes	characteristic
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	16	2	no	—
128	C₄×C₂₃	#380	1	1	yes	characteristic
129	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
130	C₄×C₂₃	#461	1	1	yes	characteristic
131	C₄×C₂₃	#474	1	1	yes	characteristic
132	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×#131	—	2	1	no	—
134	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
135	#443	—	2	2	no	—
136	#443	C₄×C₂₃	1	4	yes	complemented
137	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×#107	—	2	1	no	—
139	C₂×#106	C₂²×C₂₃	1	2	yes	—
140	C₂×#104	—	2	1	no	—
141	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	#382	—	2	1	no	—
143	#364	—	2	2	no	—
144	#360	—	2	4	no	—
145	C₂×C₂₃	—	2	1	no	—
146	C₂×C₂₃	—	2	1	no	—
147	C₂×C₂₃	—	8	2	no	—
148	C₂×C₂₃	#1802	1	1	yes	characteristic
149	C₂×C₂₃	#2231	1	2	yes	—
150	C₂×#41	—	2	1	no	—
151	#135	—	2	1	no	—
152	#135	—	2	1	no	—
153	#135	C₂×C₄×C₂₃	1	2	yes	—
154	#135	D₄×C₂₃	1	2	yes	—
155	C₂×Q₁₆	Q₈×C₂₃	1	1	yes	characteristic
156	C₂×SD₁₆	—	2	1	no	—
157	C₂×D₈	Q₈×C₂₃	1	1	yes	characteristic
158	#131	—	2	4	no	—
159	#131	—	4	2	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	4	1	no	—
162	C₂²×C₈	—	4	1	no	—
163	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
167	M₃₂	—	2	2	no	—
168	M₃₂	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	#108	—	2	1	no	—
171	#108	C₂×C₄×C₂₃	1	2	yes	—
172	#107	—	4	2	no	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	#104	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
177	C₄×C₈	D₄×C₂₃	1	2	yes	—
178	#95	—	4	2	no	—
179	C₂₃	#12824	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	#41	—	4	1	no	—
183	#41	—	4	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
191	Q₁₆	—	2	1	no	—
192	Q₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	D₈	—	2	1	no	—
196	D₈	C₄⋊C₄×C₂₃	1	2	yes	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	4	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
203	C₂×C₈	D₈×C₂₃	1	2	yes	—
204	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
205	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
210	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
211	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄²	—	2	1	no	—
215	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
216	C₁₆	—	4	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
219	Q₈	—	2	2	no	—
220	Q₈	—	4	1	no	—
221	D₄	—	2	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
234	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
238	C₈	—	4	2	no	—
239	C₈	C₂₃×#102	1	2	yes	—
240	C₈	C₂₃×#102	1	2	yes	—
241	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
242	C₈	C₂₃×#117	1	1	yes	characteristic
243	C₈	C₂₃×#121	1	1	yes	characteristic
244	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₂₃×#417	1	1	yes	characteristic
249	C₂²	C₂₃×#459	1	1	yes	characteristic
250	C₂²	C₂₃×#476	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	16	2	no	—
254	C₄	C₂₃×#380	1	1	yes	characteristic
255	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
256	C₄	C₂₃×#461	1	1	yes	characteristic
257	C₄	C₂₃×#474	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	8	2	no	—
261	C₂	C₂₃×#1802	1	1	yes	characteristic
262	C₂	C₂₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#476
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×#1383	C₂	semidirect
10. C₂₃×#443	C₄	semidirect
85. #12824	C₂₃	direct (N × H)
115. C₂×#364	C₂×C₂₃	semidirect
117. #1846	C₂×C₂₃	semidirect
118. #1383	C₂×C₂₃	semidirect
136. #443	C₄×C₂₃	semidirect
179. C₂₃	#12824	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
C₂×C₂₃×#360	2
C₂₃×#1846	2
C₂₃×#1383	2
#12824	23

Subgroups — C23×#12824 #19160450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12824 #19160450