Subgroups — C₂₃×#33318 #19180944

Order 5888.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#501	C₂	1	2	yes	maximal
2	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2660	C₂	1	2	yes	maximal
5	C₂₃×#1847	C₂	1	4	yes	maximal
6	C₂₃×#1846	C₂	1	2	yes	maximal
7	C₂₃×#1844	C₂	1	2	yes	maximal
8	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂₃×#501	—	2	8	no	—
12	C₂₃×#499	—	2	4	no	—
13	C₂₃×#498	—	2	4	no	—
14	C₂×C₂₃×#107	C₂²	1	4	yes	—
15	C₂×C₂₃×#107	C₂²	1	2	yes	—
16	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	C₂²	1	4	yes	—
19	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#106	C₂²	1	2	yes	—
21	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#398	—	2	4	no	—
26	C₂₃×#395	—	2	4	no	—
27	C₂₃×#384	C₂²	1	4	yes	—
28	C₂₃×#382	C₂²	1	4	yes	—
29	C₂₃×#376	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊Q₈	—	2	4	no	—
35	C₂₃×C₄⋊Q₈	—	2	4	no	—
36	C₂₃×#125	—	2	4	no	—
37	C₂₃×#125	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×#107	—	2	4	no	—
48	C₂₃×#107	—	2	8	no	—
49	C₂₃×#106	—	2	4	no	—
50	C₂₃×#106	—	2	4	no	—
51	C₂₃×#106	—	2	2	no	—
52	C₂₃×#106	—	2	2	no	—
53	C₂₃×#106	—	2	8	no	—
54	C₂₃×#97	D₄	1	4	yes	—
55	C₂₃×#97	Q₈	1	8	yes	complemented
56	C₄×C₈×C₂₃	D₄	1	4	yes	—
57	C₄×C₈×C₂₃	Q₈	1	8	yes	complemented
58	C₂₃×#95	—	2	8	no	—
59	C₂₃×#95	D₄	1	4	yes	—
60	C₂×Q₈×C₂₃	—	4	4	no	—
61	C₂×Q₈×C₂₃	—	8	4	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	4	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	4	4	no	—
82	C₄⋊C₄×C₂₃	—	4	4	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₄⋊C₄×C₂₃	—	8	8	no	—
85	C₄²×C₂₃	C₂×D₄	1	2	yes	—
86	C₄²×C₂₃	C₂×D₄	1	2	yes	—
87	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
88	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
89	#33318	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	8	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	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	—	4	8	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	#127	1	4	yes	—
101	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
102	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
106	C₈×C₂₃	C₄⋊Q₈	1	4	yes	—
107	C₈×C₂₃	C₄⋊Q₈	1	4	yes	—
108	C₈×C₂₃	C₄⋊Q₈	1	8	yes	—
109	C₂×#501	C₂×C₂₃	1	2	yes	—
110	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
111	C₂×#498	C₂×C₂₃	1	1	yes	characteristic
112	#2660	C₂×C₂₃	1	2	yes	—
113	#1847	C₂×C₂₃	1	4	yes	—
114	#1846	C₂×C₂₃	1	2	yes	—
115	#1844	C₂×C₂₃	1	2	yes	—
116	#1726	C₂×C₂₃	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
119	C₂²×C₂₃	#577	1	4	yes	—
120	C₄×C₂₃	—	2	4	no	—
121	C₄×C₂₃	—	2	8	no	—
122	C₄×C₂₃	—	8	2	no	—
123	C₄×C₂₃	—	8	2	no	—
124	C₄×C₂₃	—	8	4	no	—
125	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
129	C₂×#125	C₂²×C₂₃	1	2	yes	—
130	#501	—	2	8	no	—
131	#499	—	2	4	no	—
132	#498	—	2	4	no	—
133	C₂×#107	C₂²×C₂₃	1	4	yes	—
134	C₂×#107	C₂²×C₂₃	1	2	yes	—
135	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×#106	C₂²×C₂₃	1	2	yes	—
137	C₂×#106	C₂²×C₂₃	1	4	yes	—
138	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×#106	C₂²×C₂₃	1	2	yes	—
140	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#97	C₂²×C₂₃	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
144	#398	—	2	4	no	—
145	#395	—	2	4	no	—
146	#384	C₂²×C₂₃	1	4	yes	—
147	#382	C₂²×C₂₃	1	4	yes	—
148	#376	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₂₃	#2868	1	1	yes	characteristic
150	C₂×C₂₃	#3154	1	2	yes	—
151	C₂×C₂₃	#3163	1	4	yes	—
152	C₂²×Q₈	—	2	2	no	—
153	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
156	C₄⋊Q₈	—	2	4	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	#125	—	2	4	no	—
159	#125	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
169	#107	—	2	4	no	—
170	#107	—	2	8	no	—
171	#106	—	2	4	no	—
172	#106	—	2	4	no	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	#106	—	2	8	no	—
176	#97	D₄×C₂₃	1	4	yes	—
177	#97	Q₈×C₂₃	1	8	yes	complemented
178	C₄×C₈	D₄×C₂₃	1	4	yes	—
179	C₄×C₈	Q₈×C₂₃	1	8	yes	complemented
180	#95	—	2	8	no	—
181	#95	D₄×C₂₃	1	4	yes	—
182	C₂₃	#33318	1	1	yes	characteristic, direct product factor, Sylow
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	8	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
190	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
194	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
195	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
196	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
197	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	8	8	no	—
208	C₄²	C₂×D₄×C₂₃	1	2	yes	—
209	C₄²	C₂×D₄×C₂₃	1	2	yes	—
210	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
211	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
212	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
213	Q₈	—	8	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₂₃×#127	1	4	yes	—
223	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
224	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
227	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
228	C₈	C₂₃×C₄⋊Q₈	1	4	yes	—
229	C₈	C₂₃×C₄⋊Q₈	1	4	yes	—
230	C₈	C₂₃×C₄⋊Q₈	1	8	yes	—
231	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
232	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
233	C₂²	C₂₃×#577	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	8	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
240	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
241	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
242	C₂	C₂₃×#2868	1	1	yes	characteristic
243	C₂	C₂₃×#3154	1	2	yes	—
244	C₂	C₂₃×#3163	1	4	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 ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
55. C₂₃×#97	Q₈	semidirect
57. C₄×C₈×C₂₃	Q₈	semidirect
89. #33318	C₂₃	direct (N × H)
177. #97	Q₈×C₂₃	semidirect
179. C₄×C₈	Q₈×C₂₃	semidirect
182. C₂₃	#33318	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×#501	2
C₂×C₂₃×#499	2
C₂×C₂₃×#498	2
C₂₃×#2660	2
C₂₃×#1847	2
C₂₃×#1846	2
C₂₃×#1844	2
C₂₃×#1726	2
#33318	23

Subgroups — C23×#33318 #19180944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33318 #19180944