Subgroups — C₂₃×#3368 #15670987

Order 2944.

382 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#490	C₂	1	2	yes	maximal, complemented
4	C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#478	C₂	1	2	yes	maximal, complemented
6	C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#460	C₂	1	2	yes	maximal, complemented
8	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	—	2	1	no	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂₃×#127	C₂²	1	2	yes	—
19	C₂₃×#125	—	2	2	no	—
20	C₂₃×#125	C₂²	1	2	yes	—
21	C₂₃×#121	—	2	1	no	—
22	C₂₃×#121	—	2	2	no	—
23	C₄×Q₈×C₂₃	—	2	1	no	—
24	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#106	C₂²	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂₃×#102	C₂²	1	2	yes	—
36	C₂₃×#102	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	#41×C₂₃	—	4	4	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	SD₁₆×C₂₃	—	2	2	no	—
54	SD₁₆×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	2	1	no	—
72	C₄²×C₂₃	C₂³	1	2	yes	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	1	no	—
76	Q₈×C₂₃	—	2	2	no	—
77	Q₈×C₂₃	—	2	1	no	—
78	Q₈×C₂₃	—	4	2	no	—
79	Q₈×C₂₃	#41	1	2	yes	complemented
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	2	1	no	—
83	D₄×C₂₃	—	2	2	no	—
84	D₄×C₂₃	—	4	4	no	—
85	D₄×C₂₃	—	4	2	no	—
86	D₄×C₂₃	—	8	2	no	—
87	D₄×C₂₃	—	8	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	2	1	no	—
102	#3368	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	1	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	1	no	—
109	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	1	no	—
117	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
118	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
119	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
120	#557	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#550	C₂×C₂₃	1	1	yes	characteristic
122	#490	C₂×C₂₃	1	2	yes	complemented
123	#486	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#478	C₂×C₂₃	1	2	yes	complemented
125	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#460	C₂×C₂₃	1	2	yes	complemented
127	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#440	C₂×C₂₃	1	2	yes	complemented
130	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₂×C₂₃	—	8	2	no	—
132	C₂×C₂₃	—	8	1	no	—
133	C₂×C₂₃	#548	1	1	yes	characteristic
134	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₂₃	#577	1	1	yes	characteristic
136	C₂×#41	—	2	1	no	—
137	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
141	#127	C₂²×C₂₃	1	1	yes	characteristic
142	#127	C₂²×C₂₃	1	2	yes	—
143	#125	—	2	2	no	—
144	#125	C₂²×C₂₃	1	2	yes	—
145	#121	—	2	1	no	—
146	#121	—	2	2	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
150	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₄×D₄	C₂²×C₂₃	1	2	yes	—
153	#107	C₂²×C₂₃	1	2	yes	—
154	#106	C₂²×C₂₃	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
157	#103	C₂²×C₂₃	1	1	yes	characteristic
158	#102	C₂²×C₂₃	1	1	yes	characteristic
159	#102	C₂²×C₂₃	1	2	yes	—
160	#102	C₂²×C₂₃	1	2	yes	—
161	#102	C₂²×C₂₃	1	2	yes	—
162	#102	C₂²×C₂₃	1	2	yes	—
163	#97	C₂²×C₂₃	1	2	yes	—
164	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	C₂₃	#3368	1	1	yes	characteristic, direct product factor, Sylow
166	#41	—	4	4	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×D₄	C₂³×C₂₃	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	SD₁₆	—	2	2	no	—
179	SD₁₆	—	2	4	no	—
180	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₂₃	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
189	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
190	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂³×C₂₃	1	1	yes	characteristic
198	C₄²	C₂³×C₂₃	1	2	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	4	2	no	—
204	Q₈	#41×C₂₃	1	2	yes	complemented
205	D₄	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	8	2	no	—
212	D₄	—	8	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	1	no	—
233	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
242	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
243	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
244	C₂	—	8	2	no	—
245	C₂	—	8	1	no	—
246	C₂	C₂₃×#548	1	1	yes	characteristic
247	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#557	C₂	semidirect
3. C₂₃×#490	C₂	semidirect
4. C₂₃×#486	C₂	semidirect
5. C₂₃×#478	C₂	semidirect
6. C₂₃×#474	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#460	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#440	C₂	semidirect
11. C₄×SD₁₆×C₂₃	C₂	semidirect
79. Q₈×C₂₃	#41	semidirect
102. #3368	C₂₃	direct (N × H)
120. #557	C₂×C₂₃	semidirect
122. #490	C₂×C₂₃	semidirect
123. #486	C₂×C₂₃	semidirect
124. #478	C₂×C₂₃	semidirect
125. #474	C₂×C₂₃	semidirect
126. #460	C₂×C₂₃	semidirect
127. #460	C₂×C₂₃	semidirect
128. #446	C₂×C₂₃	semidirect
129. #440	C₂×C₂₃	semidirect
130. C₄×SD₁₆	C₂×C₂₃	semidirect
165. C₂₃	#3368	direct (N × H)
204. Q₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#557	2
C₂₃×#550	2
C₂₃×#490	2
C₂₃×#486	2
C₂₃×#478	2
C₂₃×#474	2
C₂₃×#460	2
C₂₃×#446	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
#3368	23

Subgroups — C23×#3368 #15670987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3368 #15670987