Subgroups — C₂₃×#36584 #19184210

Order 5888.

1590 conjugacy classes of subgroups.

The subgroup lattice has 228 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	3	yes	maximal
2	C₂₃×#2571	C₂	1	6	yes	maximal
3	C₂₃×#2509	C₂	1	12	yes	maximal
4	C₂₃×#2351	C₂	1	4	yes	maximal
5	Q₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₄×C₂₃×#125	C₂	1	3	yes	maximal
7	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#125	C₂²	1	3	yes	—
9	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#400	C₂²	1	24	yes	—
13	C₂₃×#389	C₂²	1	12	yes	—
14	C₂₃×#389	C₂²	1	12	yes	—
15	C₂₃×#389	C₂²	1	12	yes	—
16	C₂₃×#389	C₂²	1	12	yes	—
17	C₂₃×#389	C₂²	1	6	yes	—
18	C₂₃×#387	C₂²	1	12	yes	—
19	C₂₃×#384	C₂²	1	6	yes	—
20	C₂₃×#382	C₂²	1	12	yes	—
21	C₂₃×#382	C₂²	1	1	yes	characteristic
22	C₂₃×#382	C₂²	1	6	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
27	C₄³×C₂₃	C₂²	1	3	yes	—
28	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#125	—	2	12	no	—
30	C₄×Q₈×C₂₃	—	2	8	no	—
31	C₄×Q₈×C₂₃	—	2	16	no	—
32	C₄×Q₈×C₂₃	Q₈	1	8	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#95	C₂³	1	4	yes	—
48	C₂₃×#95	C₂³	1	6	yes	—
49	C₂₃×#95	C₂³	1	12	yes	—
50	C₂₃×#95	C₂³	1	12	yes	—
51	C₂₃×#95	C₂³	1	24	yes	—
52	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
53	C₂×Q₈×C₂₃	#41	1	8	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
57	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	12	no	—
62	C₄⋊C₄×C₂₃	—	2	24	no	—
63	C₄⋊C₄×C₂₃	—	2	24	no	—
64	C₄⋊C₄×C₂₃	—	2	12	no	—
65	C₄⋊C₄×C₂₃	—	2	4	no	—
66	C₄⋊C₄×C₂₃	—	4	12	no	—
67	C₄⋊C₄×C₂₃	—	4	12	no	—
68	C₄⋊C₄×C₂₃	C₂×Q₈	1	12	yes	—
69	C₄²×C₂₃	—	2	12	no	—
70	C₄²×C₂₃	—	2	24	no	—
71	C₄²×C₂₃	—	2	6	no	—
72	C₄²×C₂₃	—	4	6	no	—
73	C₄²×C₂₃	C₂×Q₈	1	12	yes	—
74	C₄²×C₂₃	#41	1	4	yes	—
75	#36584	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
77	Q₈×C₂₃	#125	1	16	yes	complemented
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	6	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	12	no	—
83	C₂×C₄×C₂₃	—	2	6	no	—
84	C₂×C₄×C₂₃	—	2	12	no	—
85	C₂×C₄×C₂₃	—	2	6	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	6	no	—
88	C₂×C₄×C₂₃	—	2	12	no	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
91	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
92	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
94	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
95	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
96	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
97	#2859	C₂×C₂₃	1	3	yes	—
98	#2571	C₂×C₂₃	1	6	yes	—
99	#2509	C₂×C₂₃	1	12	yes	—
100	#2351	C₂×C₂₃	1	4	yes	—
101	Q₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
102	C₄×#125	C₂×C₂₃	1	3	yes	—
103	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
104	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
105	C₂²×C₂₃	C₂²×#41	1	2	yes	—
106	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
107	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
108	C₂²×C₂₃	#585	1	2	yes	—
109	C₄×C₂₃	—	2	4	no	—
110	C₄×C₂₃	—	2	6	no	—
111	C₄×C₂₃	—	4	12	no	—
112	C₄×C₂₃	—	4	4	no	—
113	C₄×C₂₃	—	4	6	no	—
114	C₄×C₂₃	C₂×#125	1	12	yes	—
115	C₄×C₂₃	#557	1	4	yes	—
116	C₂×#125	C₂²×C₂₃	1	3	yes	—
117	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
118	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
120	#400	C₂²×C₂₃	1	24	yes	—
121	#389	C₂²×C₂₃	1	6	yes	—
122	#389	C₂²×C₂₃	1	12	yes	—
123	#389	C₂²×C₂₃	1	12	yes	—
124	#389	C₂²×C₂₃	1	12	yes	—
125	#389	C₂²×C₂₃	1	12	yes	—
126	#387	C₂²×C₂₃	1	12	yes	—
127	#384	C₂²×C₂₃	1	6	yes	—
128	#382	C₂²×C₂₃	1	1	yes	characteristic
129	#382	C₂²×C₂₃	1	6	yes	—
130	#382	C₂²×C₂₃	1	12	yes	—
131	C₄×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
132	C₄×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
133	C₄×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
134	C₄×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
135	C₄³	C₂²×C₂₃	1	3	yes	—
136	C₂×C₂₃	C₂²×#125	1	1	yes	characteristic
137	C₂×C₂₃	C₂×#557	1	1	yes	characteristic
138	C₂×C₂₃	#3480	1	2	yes	—
139	C₂×C₂₃	#3504	1	1	yes	characteristic
140	C₂×C₂₃	#3518	1	2	yes	—
141	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
142	#125	—	2	12	no	—
143	C₄×Q₈	—	2	16	no	—
144	C₄×Q₈	—	2	8	no	—
145	C₄×Q₈	Q₈×C₂₃	1	8	yes	—
146	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
147	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
148	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
149	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
150	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
152	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
153	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
154	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
156	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
157	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
158	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
159	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
160	#95	C₂³×C₂₃	1	4	yes	—
161	#95	C₂³×C₂₃	1	6	yes	—
162	#95	C₂³×C₂₃	1	12	yes	—
163	#95	C₂³×C₂₃	1	12	yes	—
164	#95	C₂³×C₂₃	1	24	yes	—
165	C₂₃	#36584	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
167	C₂×Q₈	#41×C₂₃	1	8	yes	—
168	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
169	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
171	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
172	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
174	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	12	no	—
176	C₄⋊C₄	—	2	24	no	—
177	C₄⋊C₄	—	2	24	no	—
178	C₄⋊C₄	—	2	12	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	12	no	—
181	C₄⋊C₄	—	4	12	no	—
182	C₄⋊C₄	C₂×Q₈×C₂₃	1	12	yes	—
183	C₄²	—	2	12	no	—
184	C₄²	—	2	24	no	—
185	C₄²	—	2	6	no	—
186	C₄²	—	4	6	no	—
187	C₄²	C₂×Q₈×C₂₃	1	12	yes	—
188	C₄²	#41×C₂₃	1	4	yes	—
189	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
190	Q₈	C₂₃×#125	1	16	yes	complemented
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
204	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
205	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
206	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
207	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
208	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
209	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
210	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
211	C₂²	C₂²×#41×C₂₃	1	2	yes	—
212	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
213	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂²	C₂₃×#585	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	6	no	—
217	C₄	—	4	12	no	—
218	C₄	—	4	4	no	—
219	C₄	—	4	6	no	—
220	C₄	C₂×C₂₃×#125	1	12	yes	—
221	C₄	C₂₃×#557	1	4	yes	—
222	C₂	C₂²×C₂₃×#125	1	1	yes	characteristic
223	C₂	C₂×C₂₃×#557	1	1	yes	characteristic
224	C₂	C₂₃×#3480	1	2	yes	—
225	C₂	C₂₃×#3504	1	1	yes	characteristic
226	C₂	C₂₃×#3518	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
75. #36584	C₂₃	direct (N × H)
77. Q₈×C₂₃	#125	semidirect
165. C₂₃	#36584	direct (N × H)
190. Q₈	C₂₃×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2859	2
C₂₃×#2571	2
C₂₃×#2509	2
C₂₃×#2351	2
Q₈×C₄⋊C₄×C₂₃	2
C₄×C₂₃×#125	2
C₄²×Q₈×C₂₃	2
#36584	23

Subgroups — C23×#36584 #19184210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36584 #19184210