Subgroups — C₂₃×#17430 #19165056

Order 5888.

884 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2984	C₂	1	2	yes	maximal
3	C₂₃×#1470	C₂	1	6	yes	maximal
4	C₂₃×#1467	C₂	1	3	yes	maximal
5	C₂₃×#1461	C₂	1	3	yes	maximal
6	C₂₃×#557	—	2	2	no	—
7	C₂₃×#557	—	2	6	no	—
8	C₂×C₂₃×C₄⋊Q₈	C₄	1	3	yes	—
9	C₂×C₂₃×#125	C₄	1	3	yes	—
10	C₂×C₂₃×#125	C₂²	1	3	yes	—
11	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#446	—	2	6	no	—
16	C₈×Q₈×C₂₃	—	2	2	no	—
17	C₂₃×#432	C₂²	1	6	yes	—
18	C₂₃×#431	C₂²	1	6	yes	—
19	C₂₃×#324	C₂²	1	1	yes	characteristic
20	C₂₃×#324	C₂²	1	3	yes	—
21	C₂₃×#324	C₂²	1	6	yes	—
22	C₂₃×#324	C₂²	1	6	yes	—
23	C₂²×Q₈×C₂₃	C₂×C₄	1	3	yes	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	—	2	6	no	—
26	C₂₃×C₄⋊Q₈	—	2	6	no	—
27	C₂₃×#125	—	2	6	no	—
28	C₂₃×#125	—	2	6	no	—
29	C₂₃×#125	—	2	6	no	—
30	C₂₃×#125	—	2	6	no	—
31	C₄×Q₈×C₂₃	—	2	3	no	—
32	C₄×Q₈×C₂₃	—	2	1	no	—
33	C₄×Q₈×C₂₃	—	2	3	no	—
34	C₄×Q₈×C₂₃	—	2	1	no	—
35	C₄×Q₈×C₂₃	—	2	6	no	—
36	C₄×Q₈×C₂₃	—	2	6	no	—
37	C₄×Q₈×C₂₃	D₄	1	2	yes	—
38	C₄×Q₈×C₂₃	D₄	1	6	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
47	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
49	C₂₃×C₄⋊C₈	—	2	6	no	—
50	C₂₃×C₄⋊C₈	—	2	6	no	—
51	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
52	C₂₃×C₂²⋊C₈	C₂³	1	6	yes	—
53	C₂₃×#97	—	2	6	no	—
54	C₄×C₈×C₂₃	—	2	6	no	—
55	C₂×Q₈×C₂₃	—	2	4	no	—
56	C₂×Q₈×C₂₃	—	2	6	no	—
57	C₂×Q₈×C₂₃	—	2	6	no	—
58	C₂×Q₈×C₂₃	—	4	3	no	—
59	C₂×Q₈×C₂₃	C₂²⋊C₄	1	3	yes	—
60	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	6	no	—
70	C₄⋊C₄×C₂₃	—	2	6	no	—
71	C₄⋊C₄×C₂₃	—	2	3	no	—
72	C₄⋊C₄×C₂₃	—	2	3	no	—
73	C₄⋊C₄×C₂₃	—	2	3	no	—
74	C₄⋊C₄×C₂₃	—	2	3	no	—
75	C₄⋊C₄×C₂₃	—	2	12	no	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	2	6	no	—
78	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
79	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
80	C₄²×C₂₃	—	2	3	no	—
81	C₄²×C₂₃	—	2	3	no	—
82	C₄²×C₂₃	—	2	3	no	—
83	C₄²×C₂₃	—	2	3	no	—
84	C₄²×C₂₃	C₂×D₄	1	6	yes	—
85	C₄²×C₂₃	C₂×D₄	1	6	yes	—
86	#17430	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	6	no	—
89	Q₈×C₂₃	—	4	12	no	—
90	Q₈×C₂₃	#101	1	4	yes	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	3	no	—
93	C₂×C₄×C₂₃	—	2	3	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	6	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	3	no	—
98	C₂×C₄×C₂₃	—	2	3	no	—
99	C₂×C₄×C₂₃	—	2	3	no	—
100	C₂×C₄×C₂₃	—	2	3	no	—
101	C₂×C₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	4	1	no	—
103	C₂×C₄×C₂₃	—	4	3	no	—
104	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
105	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
106	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₂₃	#131	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
109	C₈×C₂₃	—	4	2	no	—
110	C₈×C₂₃	—	4	6	no	—
111	C₂×#557	C₂×C₂₃	1	1	yes	characteristic
112	#2984	C₂×C₂₃	1	2	yes	—
113	#1470	C₂×C₂₃	1	6	yes	—
114	#1467	C₂×C₂₃	1	3	yes	—
115	#1461	C₂×C₂₃	1	3	yes	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
120	C₂²×C₂₃	#568	1	1	yes	characteristic
121	C₄×C₂₃	—	2	6	no	—
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	3	no	—
125	C₄×C₂₃	—	4	3	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	—	4	3	no	—
128	C₄×C₂₃	C₂×#101	1	6	yes	—
129	#557	—	2	2	no	—
130	#557	—	2	6	no	—
131	C₂×C₄⋊Q₈	C₄×C₂₃	1	3	yes	—
132	C₂×#125	C₄×C₂₃	1	3	yes	—
133	C₂×#125	C₂²×C₂₃	1	3	yes	—
134	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
137	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
138	#446	—	2	6	no	—
139	C₈×Q₈	—	2	2	no	—
140	#432	C₂²×C₂₃	1	6	yes	—
141	#431	C₂²×C₂₃	1	6	yes	—
142	#324	C₂²×C₂₃	1	1	yes	characteristic
143	#324	C₂²×C₂₃	1	3	yes	—
144	#324	C₂²×C₂₃	1	6	yes	—
145	#324	C₂²×C₂₃	1	6	yes	—
146	C₂×C₂₃	—	2	1	no	—
147	C₂×C₂₃	—	2	1	no	—
148	C₂×C₂₃	#2880	1	1	yes	characteristic
149	C₂×C₂₃	#2884	1	1	yes	characteristic
150	C₂×C₂₃	C₂²×#101	1	1	yes	characteristic
151	C₂²×Q₈	C₂×C₄×C₂₃	1	3	yes	—
152	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	2	6	no	—
154	C₄⋊Q₈	—	2	6	no	—
155	#125	—	2	6	no	—
156	#125	—	2	6	no	—
157	#125	—	2	6	no	—
158	#125	—	2	6	no	—
159	C₄×Q₈	—	2	3	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	—	2	3	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	—	2	6	no	—
164	C₄×Q₈	—	2	6	no	—
165	C₄×Q₈	D₄×C₂₃	1	2	yes	—
166	C₄×Q₈	D₄×C₂₃	1	6	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
168	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
169	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
170	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
172	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
174	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
175	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
176	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
177	C₄⋊C₈	—	2	6	no	—
178	C₄⋊C₈	—	2	6	no	—
179	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
180	C₂²⋊C₈	C₂³×C₂₃	1	6	yes	—
181	#97	—	2	6	no	—
182	C₄×C₈	—	2	6	no	—
183	C₂₃	#17430	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	6	no	—
186	C₂×Q₈	—	2	6	no	—
187	C₂×Q₈	—	4	3	no	—
188	C₂×Q₈	C₂²⋊C₄×C₂₃	1	3	yes	—
189	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
191	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
196	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	6	no	—
199	C₄⋊C₄	—	2	6	no	—
200	C₄⋊C₄	—	2	3	no	—
201	C₄⋊C₄	—	2	3	no	—
202	C₄⋊C₄	—	2	3	no	—
203	C₄⋊C₄	—	2	3	no	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	6	no	—
207	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
208	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
209	C₄²	—	2	3	no	—
210	C₄²	—	2	3	no	—
211	C₄²	—	2	3	no	—
212	C₄²	—	2	3	no	—
213	C₄²	C₂×D₄×C₂₃	1	6	yes	—
214	C₄²	C₂×D₄×C₂₃	1	6	yes	—
215	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
216	Q₈	—	2	6	no	—
217	Q₈	—	4	12	no	—
218	Q₈	C₂₃×#101	1	4	yes	—
219	C₂×C₄	—	2	3	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	3	no	—
225	C₂×C₄	—	2	3	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	3	no	—
228	C₂×C₄	—	2	3	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
233	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
235	C₂×C₄	C₂₃×#131	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
237	C₈	—	4	2	no	—
238	C₈	—	4	6	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
242	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
243	C₂²	C₂₃×#568	1	1	yes	characteristic
244	C₄	—	2	6	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	3	no	—
248	C₄	—	4	3	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	3	no	—
251	C₄	C₂×C₂₃×#101	1	6	yes	—
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	C₂₃×#2880	1	1	yes	characteristic
255	C₂	C₂₃×#2884	1	1	yes	characteristic
256	C₂	C₂²×C₂₃×#101	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
86. #17430	C₂₃	direct (N × H)
183. C₂₃	#17430	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×#557	2
C₂₃×#2984	2
C₂₃×#1470	2
C₂₃×#1467	2
C₂₃×#1461	2
#17430	23

Subgroups — C23×#17430 #19165056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17430 #19165056