Subgroups — C₂₃×#33843 #19181469

Order 5888.

668 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#510	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3285	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2188	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2186	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#135	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#548	—	2	2	no	—
13	C₂₃×#510	—	2	8	no	—
14	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
19	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
21	C₂₃×#495	C₂²	1	2	yes	—
22	C₂₃×#467	C₂²	1	2	yes	—
23	C₂₃×#466	C₂²	1	2	yes	—
24	C₂₃×#464	—	2	2	no	—
25	C₂₃×#452	—	2	2	no	—
26	C₂₃×#448	—	2	2	no	—
27	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
28	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#366	C₂²	1	1	yes	characteristic
30	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₂₃×#358	C₂²	1	2	yes	—
32	C₂₃×#358	C₂²	1	2	yes	—
33	C₂₃×#357	C₂²	1	2	yes	—
34	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×#41×C₂₃	—	2	2	no	—
37	C₂²×Q₈×C₂₃	—	2	2	no	—
38	C₂₃×#135	—	2	4	no	—
39	C₂₃×#135	—	4	2	no	—
40	C₂×Q₁₆×C₂₃	—	2	4	no	—
41	C₂×Q₁₆×C₂₃	—	2	2	no	—
42	C₂×Q₁₆×C₂₃	—	4	2	no	—
43	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
45	C₂×SD₁₆×C₂₃	—	2	2	no	—
46	C₂×SD₁₆×C₂₃	—	2	2	no	—
47	C₂×D₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂₃×#124	—	2	2	no	—
50	C₂₃×#124	—	4	2	no	—
51	C₂₃×C₂²⋊Q₈	—	2	2	no	—
52	C₂₃×C₂²⋊Q₈	—	4	2	no	—
53	C₂₃×#121	—	2	2	no	—
54	C₂₃×#121	—	4	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×D₄×C₂₃	—	4	2	no	—
57	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂₃×Q₃₂	—	2	4	no	—
59	C₂₃×Q₃₂	—	4	1	no	—
60	C₂₃×Q₃₂	D₄	1	4	yes	complemented
61	C₂₃×SD₃₂	—	2	4	no	—
62	C₂₃×SD₃₂	—	2	4	no	—
63	C₂₃×D₁₆	—	4	1	no	—
64	C₂₃×M₃₂	—	2	4	no	—
65	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
67	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#107	C₂³	1	2	yes	—
69	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
70	C₂₃×#103	—	2	2	no	—
71	C₂₃×#103	—	2	2	no	—
72	C₂₃×#102	—	2	2	no	—
73	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
74	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	#41×C₂₃	—	4	4	no	—
76	#41×C₂₃	—	8	2	no	—
77	C₂×Q₈×C₂₃	—	2	2	no	—
78	C₂×Q₈×C₂₃	—	2	2	no	—
79	C₂×Q₈×C₂₃	—	4	4	no	—
80	C₂×Q₈×C₂₃	—	4	2	no	—
81	C₂×Q₈×C₂₃	—	8	2	no	—
82	C₂×D₄×C₂₃	—	2	2	no	—
83	C₂×D₄×C₂₃	—	4	2	no	—
84	C₂×D₄×C₂₃	—	4	2	no	—
85	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	—	4	2	no	—
88	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
89	Q₁₆×C₂₃	—	2	2	no	—
90	Q₁₆×C₂₃	—	2	2	no	—
91	Q₁₆×C₂₃	—	2	2	no	—
92	Q₁₆×C₂₃	—	4	4	no	—
93	Q₁₆×C₂₃	C₂×D₄	1	4	yes	—
94	SD₁₆×C₂₃	—	4	4	no	—
95	SD₁₆×C₂₃	—	8	2	no	—
96	D₈×C₂₃	—	2	2	no	—
97	D₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₂₃	—	4	2	no	—
108	C₂²⋊C₄×C₂₃	—	4	2	no	—
109	C₂²⋊C₄×C₂₃	—	4	2	no	—
110	C₂²⋊C₄×C₂₃	—	4	2	no	—
111	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
112	C₄²×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₂₃	—	2	2	no	—
115	C₁₆×C₂₃	—	2	1	no	—
116	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
117	#33843	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₂₃	—	4	2	no	—
119	C₂³×C₂₃	C₂×D₈	1	2	yes	—
120	Q₈×C₂₃	—	2	4	no	—
121	Q₈×C₂₃	—	4	2	no	—
122	Q₈×C₂₃	—	4	2	no	—
123	Q₈×C₂₃	—	4	2	no	—
124	Q₈×C₂₃	—	8	4	no	—
125	D₄×C₂₃	—	4	1	no	—
126	D₄×C₂₃	—	4	2	no	—
127	D₄×C₂₃	—	4	2	no	—
128	D₄×C₂₃	—	8	4	no	—
129	D₄×C₂₃	—	8	4	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	8	4	no	—
136	C₂×C₄×C₂₃	—	8	4	no	—
137	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₂₃	—	2	2	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×#510	C₂×C₂₃	1	2	yes	complemented
146	#3285	C₂×C₂₃	1	2	yes	complemented
147	#2249	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#2220	C₂×C₂₃	1	2	yes	complemented
149	#2211	C₂×C₂₃	1	2	yes	complemented
150	#2188	C₂×C₂₃	1	2	yes	complemented
151	#2186	C₂×C₂₃	1	2	yes	complemented
152	C₄×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#2169	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂²×C₂₃	—	2	2	no	—
155	C₂²×C₂₃	—	2	2	no	—
156	C₂²×C₂₃	—	2	2	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	2	2	no	—
164	C₄×C₂₃	—	4	4	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	D₄²	1	1	yes	characteristic
168	C₄×C₂₃	#576	1	1	yes	characteristic
169	C₂×#135	C₂²×C₂₃	1	2	yes	—
170	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
171	#548	—	2	2	no	—
172	#510	—	2	8	no	—
173	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
174	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
177	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
178	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
180	#495	C₂²×C₂₃	1	2	yes	—
181	#467	C₂²×C₂₃	1	2	yes	—
182	#466	C₂²×C₂₃	1	2	yes	—
183	#464	—	2	2	no	—
184	#452	—	2	2	no	—
185	#448	—	2	2	no	—
186	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
187	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
188	#366	C₂²×C₂₃	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
190	#358	C₂²×C₂₃	1	2	yes	—
191	#358	C₂²×C₂₃	1	2	yes	—
192	#357	C₂²×C₂₃	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
194	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×C₂₃	—	4	2	no	—
196	C₂×C₂₃	—	16	2	no	—
197	C₂×C₂₃	D₄×D₈	1	1	yes	characteristic
198	C₂×C₂₃	#3414	1	1	yes	characteristic
199	C₂×C₂₃	#3417	1	1	yes	characteristic
200	C₂×#41	—	2	2	no	—
201	C₂²×Q₈	—	2	2	no	—
202	#135	—	2	4	no	—
203	#135	—	4	2	no	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	4	2	no	—
207	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
208	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×D₈	C₂³×C₂₃	1	2	yes	—
212	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
213	#124	—	2	2	no	—
214	#124	—	4	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×D₄	—	4	2	no	—
221	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
222	Q₃₂	—	2	4	no	—
223	Q₃₂	—	4	1	no	—
224	Q₃₂	D₄×C₂₃	1	4	yes	complemented
225	SD₃₂	—	2	4	no	—
226	SD₃₂	—	2	4	no	—
227	D₁₆	—	4	1	no	—
228	M₃₂	—	2	4	no	—
229	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
231	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
232	#107	C₂³×C₂₃	1	2	yes	—
233	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#102	—	2	2	no	—
237	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
238	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
239	C₂₃	#33843	1	1	yes	characteristic, direct product factor, Sylow
240	#41	—	4	4	no	—
241	#41	—	8	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	8	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	4	4	no	—
258	Q₁₆	C₂×D₄×C₂₃	1	4	yes	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	8	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
267	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₁₆	—	2	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₂₃	1	2	yes	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
308	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	D₄²×C₂₃	1	1	yes	characteristic
323	C₄	C₂₃×#576	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	16	2	no	—
326	C₂	D₄×D₈×C₂₃	1	1	yes	characteristic
327	C₂	C₂₃×#3414	1	1	yes	characteristic
328	C₂	C₂₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ C₈×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#510	C₂	semidirect
2. C₂₃×#3285	C₂	semidirect
3. C₂₃×#2249	C₂	semidirect
4. C₂₃×#2220	C₂	semidirect
5. C₂₃×#2211	C₂	semidirect
6. C₂₃×#2188	C₂	semidirect
7. C₂₃×#2186	C₂	semidirect
8. C₄×C₂₃×Q₃₂	C₂	semidirect
9. C₂₃×#2169	C₂	semidirect
60. C₂₃×Q₃₂	D₄	semidirect
117. #33843	C₂₃	direct (N × H)
145. C₂×#510	C₂×C₂₃	semidirect
146. #3285	C₂×C₂₃	semidirect
147. #2249	C₂×C₂₃	semidirect
148. #2220	C₂×C₂₃	semidirect
149. #2211	C₂×C₂₃	semidirect
150. #2188	C₂×C₂₃	semidirect
151. #2186	C₂×C₂₃	semidirect
152. C₄×Q₃₂	C₂×C₂₃	semidirect
153. #2169	C₂×C₂₃	semidirect
224. Q₃₂	D₄×C₂₃	semidirect
239. C₂₃	#33843	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×#510	2
C₂₃×#3285	2
C₂₃×#2249	2
C₂₃×#2220	2
C₂₃×#2211	2
C₂₃×#2188	2
C₂₃×#2186	2
C₄×C₂₃×Q₃₂	2
C₂₃×#2169	2
#33843	23

Subgroups — C23×#33843 #19181469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33843 #19181469