Subgroups — C₂₃×#3082 #15670701

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#544	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#519	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#479	C₂	1	2	yes	maximal, complemented
4	C₂₃×#479	C₂	1	2	yes	maximal, complemented
5	C₂₃×#477	C₂	1	2	yes	maximal, complemented
6	C₂₃×#475	C₂	1	2	yes	maximal, complemented
7	C₂₃×#428	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#424	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#419	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	—	2	1	no	—
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂₃×#125	—	2	1	no	—
18	C₂₃×#125	C₂²	1	2	yes	—
19	C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₂₃	—	2	1	no	—
21	C₄×Q₈×C₂₃	—	2	1	no	—
22	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#117	—	2	2	no	—
25	C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#107	C₂²	1	1	yes	characteristic
31	C₂₃×#107	C₂²	1	2	yes	—
32	C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂₃×#106	C₂²	1	1	yes	characteristic
34	C₂₃×#106	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×#103	C₂²	1	2	yes	—
40	C₂₃×#103	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	2	2	no	—
43	C₂×Q₈×C₂₃	—	2	2	no	—
44	C₂×Q₈×C₂₃	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	M₁₆×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₂₃	—	2	4	no	—
77	Q₈×C₂₃	—	2	2	no	—
78	Q₈×C₂₃	—	8	2	no	—
79	Q₈×C₂₃	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	8	1	no	—
88	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	2	2	no	—
97	#3082	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
101	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₂₃	—	2	4	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	4	1	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	1	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
111	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
112	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
113	#544	C₂×C₂₃	1	1	yes	characteristic
114	#519	C₂×C₂₃	1	1	yes	characteristic
115	#479	C₂×C₂₃	1	2	yes	complemented
116	#479	C₂×C₂₃	1	2	yes	complemented
117	#477	C₂×C₂₃	1	2	yes	complemented
118	#475	C₂×C₂₃	1	2	yes	complemented
119	#428	C₂×C₂₃	1	1	yes	characteristic
120	#427	C₂×C₂₃	1	1	yes	characteristic
121	#424	C₂×C₂₃	1	1	yes	characteristic
122	#419	C₂×C₂₃	1	1	yes	characteristic
123	C₂×#103	C₂×C₂₃	1	1	yes	characteristic
124	C₂×C₂₃	—	2	2	no	—
125	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₂₃	#577	1	1	yes	characteristic
127	C₂×C₂₃	#578	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	1	no	—
132	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
133	#125	—	2	1	no	—
134	#125	C₂²×C₂₃	1	2	yes	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
139	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
140	#117	—	2	2	no	—
141	#117	C₂²×C₂₃	1	1	yes	characteristic
142	#117	C₂²×C₂₃	1	1	yes	characteristic
143	#117	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	#107	C₂²×C₂₃	1	1	yes	characteristic
146	#107	C₂²×C₂₃	1	1	yes	characteristic
147	#107	C₂²×C₂₃	1	2	yes	—
148	#106	C₂²×C₂₃	1	1	yes	characteristic
149	#106	C₂²×C₂₃	1	1	yes	characteristic
150	#106	C₂²×C₂₃	1	2	yes	—
151	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	#103	C₂²×C₂₃	1	2	yes	—
154	#103	C₂²×C₂₃	1	2	yes	—
155	#103	C₂²×C₂₃	1	2	yes	—
156	#103	C₂²×C₂₃	1	2	yes	—
157	C₂₃	#3082	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	Q₈×C₂₃	1	4	yes	complemented
162	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₂₃	1	2	yes	—
170	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
178	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	1	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₂³×C₂₃	1	2	yes	—
192	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	—	8	2	no	—
196	Q₈	C₂×Q₈×C₂₃	1	4	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
209	C₂×C₄	#41×C₂₃	1	2	yes	—
210	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
217	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
227	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
228	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂	C₂₃×#577	1	1	yes	characteristic
232	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#479	C₂	semidirect
4. C₂₃×#479	C₂	semidirect
5. C₂₃×#477	C₂	semidirect
6. C₂₃×#475	C₂	semidirect
44. C₂×Q₈×C₂₃	Q₈	semidirect
97. #3082	C₂₃	direct (N × H)
115. #479	C₂×C₂₃	semidirect
116. #479	C₂×C₂₃	semidirect
117. #477	C₂×C₂₃	semidirect
118. #475	C₂×C₂₃	semidirect
157. C₂₃	#3082	direct (N × H)
161. C₂×Q₈	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#544	2
C₂₃×#519	2
C₂₃×#479	2
C₂₃×#477	2
C₂₃×#475	2
C₂₃×#428	2
C₂₃×#427	2
C₂₃×#424	2
C₂₃×#419	2
C₂×C₂₃×#103	2
#3082	23

Subgroups — C23×#3082 #15670701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3082 #15670701