Subgroups — C₂₃×#29221 #19176847

Order 5888.

638 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	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	2	yes	maximal
4	C₂₃×#2099	C₂	1	2	yes	maximal
5	C₂₃×#1991	C₂	1	2	yes	maximal
6	C₂₃×#1989	C₂	1	4	yes	maximal
7	C₂₃×#1835	C₂	1	2	yes	maximal
8	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×#125	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#389	—	2	2	no	—
14	C₂₃×#389	—	2	2	no	—
15	C₂₃×#389	C₂²	1	2	yes	—
16	C₂₃×#387	—	2	2	no	—
17	C₂₃×#384	C₂²	1	1	yes	characteristic
18	C₂₃×#384	C₂²	1	1	yes	characteristic
19	C₂₃×#382	C₂²	1	2	yes	—
20	C₂₃×#382	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₄³×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#340	C₂²	1	4	yes	—
25	C₂₃×#340	C₂²	1	4	yes	—
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
30	C₂₃×#125	—	2	2	no	—
31	C₂₃×#125	—	2	8	no	—
32	C₂₃×#125	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊C₈	—	2	4	no	—
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	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	2	8	no	—
68	C₄⋊C₄×C₂₃	—	2	2	no	—
69	C₄⋊C₄×C₂₃	—	2	4	no	—
70	C₄⋊C₄×C₂₃	—	2	4	no	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	—	4	4	no	—
73	C₄⋊C₄×C₂₃	—	4	4	no	—
74	C₄⋊C₄×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	—	4	4	no	—
76	C₄⋊C₄×C₂₃	—	4	4	no	—
77	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
78	C₄²×C₂₃	—	2	4	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	4	2	no	—
81	C₄²×C₂₃	—	4	4	no	—
82	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₂₃	#41	1	2	yes	—
84	C₄²×C₂₃	#41	1	4	yes	—
85	#29221	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
104	C₂×C₄×C₂₃	#124	1	2	yes	—
105	C₂×C₄×C₂₃	#124	1	2	yes	—
106	C₂×C₄×C₂₃	#135	1	4	yes	—
107	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
110	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₂₃	—	4	4	no	—
112	#2859	C₂×C₂₃	1	1	yes	characteristic
113	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
114	#2100	C₂×C₂₃	1	2	yes	—
115	#2099	C₂×C₂₃	1	2	yes	—
116	#1991	C₂×C₂₃	1	2	yes	—
117	#1989	C₂×C₂₃	1	4	yes	—
118	#1835	C₂×C₂₃	1	2	yes	—
119	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
122	C₂²×C₂₃	#536	1	1	yes	characteristic
123	C₂²×C₂₃	C₂×#135	1	2	yes	—
124	C₂²×C₂₃	#575	1	2	yes	—
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	4	no	—
129	C₄×C₂₃	—	4	4	no	—
130	C₄×C₂₃	—	8	2	no	—
131	C₄×C₂₃	—	8	2	no	—
132	C₄×C₂₃	—	8	2	no	—
133	C₄×C₂₃	#556	1	2	yes	—
134	C₄×C₂₃	#557	1	2	yes	—
135	C₂×#125	C₂²×C₂₃	1	2	yes	—
136	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
139	#389	—	2	2	no	—
140	#389	—	2	2	no	—
141	#389	C₂²×C₂₃	1	2	yes	—
142	#387	—	2	2	no	—
143	#384	C₂²×C₂₃	1	1	yes	characteristic
144	#384	C₂²×C₂₃	1	1	yes	characteristic
145	#382	C₂²×C₂₃	1	1	yes	characteristic
146	#382	C₂²×C₂₃	1	2	yes	—
147	C₄×C₄⋊C₄	—	2	2	no	—
148	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
149	C₄³	C₂²×C₂₃	1	1	yes	characteristic
150	#340	C₂²×C₂₃	1	4	yes	—
151	#340	C₂²×C₂₃	1	4	yes	—
152	#340	C₂²×C₂₃	1	4	yes	—
153	#340	C₂²×C₂₃	1	4	yes	—
154	#336	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂₃	#2568	1	1	yes	characteristic
156	C₂×C₂₃	#3076	1	2	yes	—
157	C₂×C₂₃	#3131	1	1	yes	characteristic
158	C₂×C₂₃	#3142	1	1	yes	characteristic
159	C₂×C₂₃	#3232	1	2	yes	—
160	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
161	#125	—	2	2	no	—
162	#125	—	2	8	no	—
163	#125	Q₈×C₂₃	1	4	yes	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
171	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	D₄×C₂₃	1	2	yes	—
176	C₂×C₄²	D₄×C₂₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄×C₈	—	2	4	no	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	#95	—	2	4	no	—
186	C₂₃	#29221	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
215	C₄²	#41×C₂₃	1	2	yes	—
216	C₄²	#41×C₂₃	1	4	yes	—
217	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
235	C₂×C₄	C₂₃×#124	1	2	yes	—
236	C₂×C₄	C₂₃×#124	1	2	yes	—
237	C₂×C₄	C₂₃×#135	1	4	yes	—
238	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
241	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
242	C₈	—	4	4	no	—
243	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
245	C₂²	C₂₃×#536	1	1	yes	characteristic
246	C₂²	C₂×C₂₃×#135	1	2	yes	—
247	C₂²	C₂₃×#575	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₂₃×#556	1	2	yes	—
257	C₄	C₂₃×#557	1	2	yes	—
258	C₂	C₂₃×#2568	1	1	yes	characteristic
259	C₂	C₂₃×#3076	1	2	yes	—
260	C₂	C₂₃×#3131	1	1	yes	characteristic
261	C₂	C₂₃×#3142	1	1	yes	characteristic
262	C₂	C₂₃×#3232	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
85. #29221	C₂₃	direct (N × H)
186. C₂₃	#29221	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×#2859	2
C₄×C₂₃×#125	2
C₂₃×#2100	2
C₂₃×#2099	2
C₂₃×#1991	2
C₂₃×#1989	2
C₂₃×#1835	2
C₄×C₂₃×C₄⋊C₈	2
#29221	23

Subgroups — C23×#29221 #19176847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29221 #19176847