Subgroups — C₂₃×#33610 #19181236

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3061	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2220	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2216	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2139	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
11	C₂₃×#537	—	2	2	no	—
12	C₂₃×#508	—	2	4	no	—
13	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₂₃×#504	—	2	4	no	—
17	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
18	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#468	—	2	4	no	—
22	C₂₃×#467	C₂²	1	4	yes	—
23	C₂₃×#466	C₂²	1	4	yes	—
24	C₂×C₂₃×#107	C₂²	1	2	yes	—
25	C₂₃×#417	—	2	2	no	—
26	C₂₃×#366	C₂²	1	2	yes	—
27	C₂₃×#366	C₂²	1	2	yes	—
28	C₂₃×#358	C₂²	1	4	yes	—
29	C₂₃×#357	C₂²	1	4	yes	—
30	C₂×#41×C₂₃	—	2	2	no	—
31	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
32	C₂₃×#135	—	2	4	no	—
33	C₂₃×#135	—	2	4	no	—
34	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	C₂³	1	2	yes	—
37	C₂₃×#131	—	2	2	no	—
38	C₂₃×#131	—	2	2	no	—
39	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×C₂²⋊Q₈	—	2	4	no	—
44	C₂₃×C₂²⋊Q₈	—	4	2	no	—
45	C₂₃×#121	—	2	4	no	—
46	C₂₃×#121	—	4	2	no	—
47	C₂₃×#121	—	4	4	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂₃×Q₃₂	—	2	2	no	—
50	C₂₃×SD₃₂	—	2	4	no	—
51	C₂₃×D₁₆	—	2	2	no	—
52	C₂₃×M₃₂	—	2	2	no	—
53	C₂₃×M₃₂	—	2	4	no	—
54	C₂×C₁₆×C₂₃	—	2	4	no	—
55	C₂×C₁₆×C₂₃	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
57	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#107	—	2	2	no	—
60	C₂₃×#107	C₂³	1	4	yes	—
61	C₂₃×#103	—	2	4	no	—
62	C₂₃×#102	—	2	4	no	—
63	#41×C₂₃	—	2	2	no	—
64	#41×C₂₃	—	2	2	no	—
65	#41×C₂₃	—	4	4	no	—
66	#41×C₂₃	—	4	4	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	4	2	no	—
71	C₂×D₄×C₂₃	—	4	4	no	—
72	C₂×D₄×C₂₃	D₈	1	2	yes	—
73	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	—	4	2	no	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₂₃	—	2	2	no	—
79	Q₁₆×C₂₃	—	2	2	no	—
80	SD₁₆×C₂₃	—	4	4	no	—
81	D₈×C₂₃	—	2	2	no	—
82	D₈×C₂₃	—	2	2	no	—
83	M₁₆×C₂₃	—	2	2	no	—
84	M₁₆×C₂₃	—	2	4	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₃	#41	1	2	yes	—
91	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	4	2	no	—
94	C₄⋊C₄×C₂₃	—	4	4	no	—
95	C₂²⋊C₄×C₂₃	—	4	4	no	—
96	C₂²⋊C₄×C₂₃	—	4	4	no	—
97	C₁₆×C₂₃	—	2	2	no	—
98	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
99	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
100	#33610	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	C₂×D₈	1	2	yes	—
103	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	2	no	—
105	Q₈×C₂₃	—	4	2	no	—
106	Q₈×C₂₃	—	4	2	no	—
107	D₄×C₂₃	—	2	4	no	—
108	D₄×C₂₃	—	2	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	8	4	no	—
112	D₄×C₂₃	—	16	4	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	4	4	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	8	4	no	—
118	C₂×C₄×C₂₃	—	16	2	no	—
119	C₂×C₄×C₂₃	#121	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#135	1	2	yes	—
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₂₃	—	2	2	no	—
125	C₈×C₂₃	#121	1	2	yes	—
126	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₂×#508	C₂×C₂₃	1	1	yes	characteristic, complemented
129	C₂×#504	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#3061	C₂×C₂₃	1	2	yes	complemented
131	#2220	C₂×C₂₃	1	4	yes	complemented
132	#2216	C₂×C₂₃	1	2	yes	complemented
133	#2215	C₂×C₂₃	1	2	yes	complemented
134	C₂×#366	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#2139	C₂×C₂₃	1	2	yes	complemented
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	2	no	—
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	2	2	no	—
141	C₂²×C₂₃	—	8	2	no	—
142	C₂²×C₂₃	—	8	2	no	—
143	C₂²×C₂₃	—	8	2	no	—
144	C₂²×C₂₃	#466	1	2	yes	—
145	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	8	4	no	—
148	C₄×C₂₃	—	8	2	no	—
149	C₄×C₂₃	#466	1	2	yes	—
150	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
151	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
152	C₂×#135	C₂²×C₂₃	1	2	yes	—
153	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
154	#537	—	2	2	no	—
155	#508	—	2	4	no	—
156	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
158	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
159	#504	—	2	4	no	—
160	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
162	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
163	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
164	#468	—	2	4	no	—
165	#467	C₂²×C₂₃	1	4	yes	—
166	#466	C₂²×C₂₃	1	4	yes	—
167	C₂×#107	C₂²×C₂₃	1	2	yes	—
168	#417	—	2	2	no	—
169	#366	C₂²×C₂₃	1	2	yes	—
170	#366	C₂²×C₂₃	1	2	yes	—
171	#358	C₂²×C₂₃	1	4	yes	—
172	#357	C₂²×C₂₃	1	4	yes	—
173	C₂×C₂₃	—	2	2	no	—
174	C₂×C₂₃	—	4	2	no	—
175	C₂×C₂₃	—	16	2	no	—
176	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
177	C₂×C₂₃	#3415	1	1	yes	characteristic
178	C₂×C₂₃	#3417	1	1	yes	characteristic
179	C₂×#41	—	2	2	no	—
180	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
181	#135	—	2	4	no	—
182	#135	—	2	4	no	—
183	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
184	C₂×SD₁₆	—	2	2	no	—
185	C₂×D₈	C₂³×C₂₃	1	2	yes	—
186	#131	—	2	2	no	—
187	#131	—	2	2	no	—
188	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
189	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	C₂²⋊Q₈	—	2	4	no	—
193	C₂²⋊Q₈	—	4	2	no	—
194	#121	—	2	4	no	—
195	#121	—	4	2	no	—
196	#121	—	4	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	Q₃₂	—	2	2	no	—
199	SD₃₂	—	2	4	no	—
200	D₁₆	—	2	2	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	2	4	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	D₄×C₂₃	1	4	yes	complemented
205	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
206	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
207	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
208	#107	—	2	2	no	—
209	#107	C₂³×C₂₃	1	4	yes	—
210	#103	—	2	4	no	—
211	#102	—	2	4	no	—
212	C₂₃	#33610	1	1	yes	characteristic, direct product factor, Sylow
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	—	4	4	no	—
216	#41	—	4	4	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	D₈×C₂₃	1	2	yes	—
223	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	4	4	no	—
231	D₈	—	2	2	no	—
232	D₈	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
240	C₂×C₈	#41×C₂₃	1	2	yes	—
241	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
249	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₂₃	1	2	yes	—
252	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	16	2	no	—
268	C₂×C₄	C₂₃×#121	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
270	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#135	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	C₂₃×#121	1	2	yes	—
275	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
276	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₂₃×#466	1	2	yes	—
286	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₂₃×#466	1	2	yes	—
291	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
292	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
297	C₂	C₂₃×#3415	1	1	yes	characteristic
298	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 ◁ C₂×#41×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#508	C₂	semidirect
2. C₂×C₂₃×#504	C₂	semidirect
3. C₂₃×#3061	C₂	semidirect
4. C₂₃×#2220	C₂	semidirect
5. C₂₃×#2216	C₂	semidirect
6. C₂₃×#2215	C₂	semidirect
7. C₂×C₂₃×#366	C₂	semidirect
8. C₂₃×#2139	C₂	semidirect
55. C₂×C₁₆×C₂₃	D₄	semidirect
100. #33610	C₂₃	direct (N × H)
128. C₂×#508	C₂×C₂₃	semidirect
129. C₂×#504	C₂×C₂₃	semidirect
130. #3061	C₂×C₂₃	semidirect
131. #2220	C₂×C₂₃	semidirect
132. #2216	C₂×C₂₃	semidirect
133. #2215	C₂×C₂₃	semidirect
134. C₂×#366	C₂×C₂₃	semidirect
135. #2139	C₂×C₂₃	semidirect
204. C₂×C₁₆	D₄×C₂₃	semidirect
212. C₂₃	#33610	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×#508	2
C₂×C₂₃×#504	2
C₂₃×#3061	2
C₂₃×#2220	2
C₂₃×#2216	2
C₂₃×#2215	2
C₂×C₂₃×#366	2
C₂₃×#2139	2
#33610	23

Subgroups — C23×#33610 #19181236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33610 #19181236