Subgroups — C₂₃×#29209 #19176835

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2058	C₂	1	2	yes	maximal
4	C₂₃×#2057	C₂	1	2	yes	maximal
5	C₂₃×#1991	C₂	1	2	yes	maximal
6	C₂₃×#1949	C₂	1	4	yes	maximal
7	C₂₃×#1846	C₂	1	2	yes	maximal
8	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#106	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#391	—	2	2	no	—
16	C₂₃×#391	—	2	2	no	—
17	C₂₃×#389	C₂²	1	2	yes	—
18	C₂₃×#387	—	2	2	no	—
19	C₂₃×#382	C₂²	1	2	yes	—
20	C₂₃×#382	C₂²	1	2	yes	—
21	C₂₃×#382	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₄³×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#340	C₂²	1	4	yes	—
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#336	C₂²	1	1	yes	characteristic
28	C₂₃×#336	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
32	C₂₃×C₄⋊Q₈	—	4	4	no	—
33	C₂₃×#125	—	2	2	no	—
34	C₂₃×#125	Q₈	1	4	yes	complemented
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₂₃	—	2	2	no	—
42	C₂×C₄²×C₂₃	D₄	1	2	yes	—
43	C₂×C₄²×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#107	—	4	4	no	—
48	C₂₃×#106	—	4	4	no	—
49	C₂₃×C₄⋊C₈	Q₈	1	8	yes	complemented
50	C₄×C₈×C₂₃	—	2	4	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	2	no	—
56	C₂₃×#95	—	2	2	no	—
57	C₂×Q₈×C₂₃	—	4	4	no	—
58	C₂×Q₈×C₂₃	—	4	4	no	—
59	C₂×Q₈×C₂₃	—	4	4	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	4	no	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
74	C₄⋊C₄×C₂₃	—	2	8	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
82	C₄²×C₂₃	—	2	4	no	—
83	C₄²×C₂₃	—	2	2	no	—
84	C₄²×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
87	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
88	C₄²×C₂₃	#41	1	2	yes	—
89	#29209	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₂₃	—	8	8	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	#135	1	4	yes	—
111	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	2	4	no	—
117	#2862	C₂×C₂₃	1	1	yes	characteristic
118	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
119	#2058	C₂×C₂₃	1	2	yes	—
120	#2057	C₂×C₂₃	1	2	yes	—
121	#1991	C₂×C₂₃	1	2	yes	—
122	#1949	C₂×C₂₃	1	4	yes	—
123	#1846	C₂×C₂₃	1	2	yes	—
124	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
126	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂²×C₂₃	#535	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×#135	1	2	yes	—
129	C₂²×C₂₃	#575	1	2	yes	—
130	C₄×C₂₃	—	2	4	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	—	4	4	no	—
135	C₄×C₂₃	—	8	2	no	—
136	C₄×C₂₃	—	8	4	no	—
137	C₄×C₂₃	#554	1	2	yes	—
138	C₄×C₂₃	Q₈²	1	2	yes	—
139	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
140	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#107	C₂²×C₂₃	1	4	yes	—
142	C₂×#106	C₂²×C₂₃	1	4	yes	—
143	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
145	#391	—	2	2	no	—
146	#391	—	2	2	no	—
147	#389	C₂²×C₂₃	1	2	yes	—
148	#387	—	2	2	no	—
149	#382	C₂²×C₂₃	1	1	yes	characteristic
150	#382	C₂²×C₂₃	1	2	yes	—
151	#382	C₂²×C₂₃	1	2	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
154	C₄³	C₂²×C₂₃	1	1	yes	characteristic
155	#340	C₂²×C₂₃	1	4	yes	—
156	#340	C₂²×C₂₃	1	4	yes	—
157	#336	C₂²×C₂₃	1	1	yes	characteristic
158	#336	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₂₃	#2567	1	1	yes	characteristic
160	C₂×C₂₃	#3076	1	2	yes	—
161	C₂×C₂₃	#3160	1	1	yes	characteristic
162	C₂×C₂₃	#3166	1	1	yes	characteristic
163	C₂×C₂₃	#3211	1	2	yes	—
164	C₂²×Q₈	—	2	2	no	—
165	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
167	C₄⋊Q₈	—	4	4	no	—
168	#125	—	2	2	no	—
169	#125	Q₈×C₂₃	1	4	yes	complemented
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	D₄×C₂₃	1	2	yes	—
178	C₂×C₄²	D₄×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
182	#107	—	4	4	no	—
183	#106	—	4	4	no	—
184	C₄⋊C₈	Q₈×C₂₃	1	8	yes	complemented
185	C₄×C₈	—	2	4	no	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₂₃	#29209	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
209	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
223	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
224	C₄²	#41×C₂₃	1	2	yes	—
225	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
226	Q₈	—	8	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
243	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
244	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
245	C₂×C₄	C₂₃×#135	1	4	yes	—
246	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
247	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
250	C₈	—	2	4	no	—
251	C₈	—	2	4	no	—
252	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
254	C₂²	C₂₃×#535	1	1	yes	characteristic
255	C₂²	C₂×C₂₃×#135	1	2	yes	—
256	C₂²	C₂₃×#575	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	C₂₃×#554	1	2	yes	—
265	C₄	Q₈²×C₂₃	1	2	yes	—
266	C₂	C₂₃×#2567	1	1	yes	characteristic
267	C₂	C₂₃×#3076	1	2	yes	—
268	C₂	C₂₃×#3160	1	1	yes	characteristic
269	C₂	C₂₃×#3166	1	1	yes	characteristic
270	C₂	C₂₃×#3211	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂₃×#125	Q₈	semidirect
49. C₂₃×C₄⋊C₈	Q₈	semidirect
89. #29209	C₂₃	direct (N × H)
169. #125	Q₈×C₂₃	semidirect
184. C₄⋊C₈	Q₈×C₂₃	semidirect
192. C₂₃	#29209	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2862	2
C₄×C₂₃×#125	2
C₂₃×#2058	2
C₂₃×#2057	2
C₂₃×#1991	2
C₂₃×#1949	2
C₂₃×#1846	2
C₄×C₂₃×C₄⋊C₈	2
#29209	23

Subgroups — C23×#29209 #19176835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29209 #19176835