Subgroups — C₂₃×#11597 #19159223

Order 5888.

394 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1334	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1333	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#135	C₄	1	1	yes	characteristic
7	C₄×#41×C₂₃	—	2	1	no	—
8	C₂²×C₁₆×C₂₃	—	2	1	no	—
9	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
10	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆×C₂₃	—	2	1	no	—
12	C₄×SD₁₆×C₂₃	—	2	1	no	—
13	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic
14	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#431	—	2	1	no	—
16	C₂₃×#427	C₄	1	1	yes	characteristic
17	C₂₃×#416	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#346	—	2	1	no	—
20	C₂₃×#346	C₂²	1	2	yes	—
21	C₂₃×#335	—	2	1	no	—
22	C₂₃×#334	C₂²	1	2	yes	—
23	C₂×#41×C₂₃	—	2	1	no	—
24	C₂₃×#135	—	2	2	no	—
25	C₂₃×#135	C₈	1	4	yes	complemented
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₂₃	—	2	1	no	—
32	C₄×Q₈×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	—	2	1	no	—
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	4	1	no	—
36	C₄×D₄×C₂₃	—	4	1	no	—
37	C₂₃×#117	—	2	1	no	—
38	C₂₃×#117	—	4	1	no	—
39	C₂×C₄²×C₂₃	—	4	1	no	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₂₃	—	2	4	no	—
42	C₂×C₁₆×C₂₃	—	2	2	no	—
43	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
44	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
45	C₂₃×#106	—	2	1	no	—
46	C₂₃×C₄⋊C₈	—	2	2	no	—
47	C₂₃×#103	—	2	1	no	—
48	C₂₃×#103	—	2	1	no	—
49	C₂₃×#102	—	2	1	no	—
50	C₂₃×#102	—	2	1	no	—
51	C₂₃×C₂²⋊C₈	—	4	1	no	—
52	C₄×C₈×C₂₃	—	4	1	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	#41×C₂₃	—	2	4	no	—
58	#41×C₂₃	—	4	2	no	—
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×D₄×C₂₃	—	2	1	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
68	Q₁₆×C₂₃	—	2	1	no	—
69	Q₁₆×C₂₃	C₂×C₈	1	2	yes	—
70	SD₁₆×C₂₃	—	2	4	no	—
71	D₈×C₂₃	—	2	1	no	—
72	D₈×C₂₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₄⋊C₄×C₂₃	—	4	1	no	—
84	C₄⋊C₄×C₂₃	—	4	1	no	—
85	C₂²⋊C₄×C₂₃	—	4	1	no	—
86	C₂²⋊C₄×C₂₃	—	4	1	no	—
87	C₂²⋊C₄×C₂₃	—	4	1	no	—
88	C₄²×C₂₃	—	4	1	no	—
89	C₄²×C₂₃	—	4	1	no	—
90	C₄²×C₂₃	—	4	1	no	—
91	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₂₃	D₈	1	1	yes	characteristic
93	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
94	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₁₆×C₂₃	—	2	4	no	—
96	#11597	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	4	1	no	—
98	C₂³×C₂₃	#102	1	1	yes	characteristic
99	Q₈×C₂₃	—	2	2	no	—
100	Q₈×C₂₃	—	4	1	no	—
101	D₄×C₂₃	—	2	2	no	—
102	D₄×C₂₃	—	4	4	no	—
103	D₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	8	2	no	—
111	C₂×C₄×C₂₃	—	8	2	no	—
112	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#104	1	2	yes	—
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₈×C₂₃	—	2	1	no	—
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	8	2	no	—
123	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
125	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
126	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
127	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#1334	C₂×C₂₃	1	2	yes	complemented
130	#1333	C₂×C₂₃	1	2	yes	complemented
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	—	8	1	no	—
135	C₂²×C₂₃	—	8	1	no	—
136	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
137	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	4	2	no	—
143	C₄×C₂₃	—	8	1	no	—
144	C₄×C₂₃	—	8	1	no	—
145	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
146	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
148	C₂×#135	C₄×C₂₃	1	1	yes	characteristic
149	C₄×#41	—	2	1	no	—
150	C₂²×C₁₆	—	2	1	no	—
151	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
152	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
153	C₄×SD₁₆	—	2	1	no	—
154	C₄×SD₁₆	—	2	1	no	—
155	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic
156	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
157	#431	—	2	1	no	—
158	#427	C₄×C₂₃	1	1	yes	characteristic
159	#416	—	2	1	no	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
161	#346	—	2	1	no	—
162	#346	C₂²×C₂₃	1	2	yes	—
163	#335	—	2	1	no	—
164	#334	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂₃	—	2	2	no	—
166	C₂×C₂₃	—	8	2	no	—
167	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
168	C₂×C₂₃	#2145	1	1	yes	characteristic
169	C₂×C₂₃	#2148	1	1	yes	characteristic
170	C₂×#41	—	2	1	no	—
171	#135	—	2	2	no	—
172	#135	C₈×C₂₃	1	4	yes	complemented
173	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂×SD₁₆	—	2	1	no	—
175	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	—	4	1	no	—
183	C₄×D₄	—	4	1	no	—
184	#117	—	2	1	no	—
185	#117	—	4	1	no	—
186	C₂×C₄²	—	4	1	no	—
187	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
188	C₂×C₁₆	—	2	4	no	—
189	C₂×C₁₆	—	2	2	no	—
190	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
191	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
192	#106	—	2	1	no	—
193	C₄⋊C₈	—	2	2	no	—
194	#103	—	2	1	no	—
195	#103	—	2	1	no	—
196	#102	—	2	1	no	—
197	#102	—	2	1	no	—
198	C₂²⋊C₈	—	4	1	no	—
199	C₄×C₈	—	4	1	no	—
200	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
202	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
204	C₂₃	#11597	1	1	yes	characteristic, direct product factor, Sylow
205	#41	—	2	4	no	—
206	#41	—	4	2	no	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
216	Q₁₆	—	2	1	no	—
217	Q₁₆	C₂×C₈×C₂₃	1	2	yes	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	1	no	—
220	D₈	C₂×C₈×C₂₃	1	2	yes	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
226	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	1	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
240	C₄²	D₈×C₂₃	1	1	yes	characteristic
241	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₁₆	—	2	4	no	—
244	C₂³	—	4	1	no	—
245	C₂³	C₂₃×#102	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	Q₈	—	4	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
260	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
261	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#104	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	8	2	no	—
270	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
271	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
272	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	1	no	—
278	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
279	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	1	no	—
286	C₄	—	8	1	no	—
287	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
288	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
289	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
293	C₂	C₂₃×#2145	1	1	yes	characteristic
294	C₂	C₂₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂×C₂₃×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×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×#346	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
4. C₂₃×#1334	C₂	semidirect
5. C₂₃×#1333	C₂	semidirect
25. C₂₃×#135	C₈	semidirect
96. #11597	C₂₃	direct (N × H)
127. C₂×#346	C₂×C₂₃	semidirect
128. #1564	C₂×C₂₃	semidirect
129. #1334	C₂×C₂₃	semidirect
130. #1333	C₂×C₂₃	semidirect
172. #135	C₈×C₂₃	semidirect
204. C₂₃	#11597	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#135	2
C₂×C₂₃×#346	2
C₂₃×#1564	2
C₂₃×#1334	2
C₂₃×#1333	2
#11597	23

Subgroups — C23×#11597 #19159223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11597 #19159223