Subgroups — C₂₃×#3384 #15671003

Order 2944.

374 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#501	C₂	1	2	yes	maximal, complemented
4	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#478	C₂	1	2	yes	maximal, complemented
6	C₂₃×#476	C₂	1	2	yes	maximal, complemented
7	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#442	C₂	1	2	yes	maximal
11	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	—	2	1	no	—
13	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#125	C₂²	1	2	yes	—
18	C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₂₃	C₂²	1	2	yes	—
25	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#106	C₂²	1	2	yes	—
31	C₂₃×#106	C₂²	1	2	yes	—
32	C₂₃×#106	C₂²	1	2	yes	—
33	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂₃×#102	C₂²	1	1	yes	characteristic
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	C₂×Q₈×C₂₃	—	2	2	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	—	4	4	no	—
45	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	D₈×C₂₃	Q₈	1	4	yes	complemented
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	C₂³	1	2	yes	—
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	1	no	—
75	Q₈×C₂₃	—	2	2	no	—
76	Q₈×C₂₃	—	4	4	no	—
77	Q₈×C₂₃	—	4	2	no	—
78	Q₈×C₂₃	—	8	1	no	—
79	D₄×C₂₃	—	2	1	no	—
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	C₂×Q₈	1	2	yes	—
82	D₄×C₂₃	C₂×Q₈	1	2	yes	complemented
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₃	—	2	1	no	—
97	C₈×C₂₃	—	2	2	no	—
98	C₈×C₂₃	C₂×Q₈	1	2	yes	complemented
99	#3384	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	4	1	no	—
103	C₂²×C₂₃	—	4	1	no	—
104	C₂²×C₂₃	—	4	1	no	—
105	C₂²×C₂₃	—	4	1	no	—
106	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
116	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
117	#554	C₂×C₂₃	1	1	yes	characteristic, complemented
118	D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#501	C₂×C₂₃	1	2	yes	complemented
120	#498	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#478	C₂×C₂₃	1	2	yes	complemented
122	#476	C₂×C₂₃	1	2	yes	complemented
123	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#442	C₂×C₂₃	1	2	yes	—
127	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₂₃	—	4	2	no	—
129	C₂×C₂₃	—	4	2	no	—
130	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
131	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₂₃	#577	1	1	yes	characteristic
133	C₂²×Q₈	—	2	1	no	—
134	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
138	#125	C₂²×C₂₃	1	2	yes	—
139	C₂²⋊Q₈	—	2	1	no	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₂₃	1	2	yes	—
146	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₄×D₄	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	#107	C₂²×C₂₃	1	1	yes	characteristic
150	#107	C₂²×C₂₃	1	2	yes	—
151	#106	C₂²×C₂₃	1	2	yes	—
152	#106	C₂²×C₂₃	1	2	yes	—
153	#106	C₂²×C₂₃	1	2	yes	—
154	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
156	#102	C₂²×C₂₃	1	1	yes	characteristic
157	#102	C₂²×C₂₃	1	1	yes	characteristic
158	#102	C₂²×C₂₃	1	2	yes	—
159	#102	C₂²×C₂₃	1	2	yes	—
160	#97	C₂²×C₂₃	1	2	yes	—
161	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂₃	#3384	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	D₈	Q₈×C₂₃	1	4	yes	complemented
175	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₂₃	1	2	yes	—
177	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
185	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
188	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	C₂³×C₂₃	1	1	yes	characteristic
194	C₄²	C₂³×C₂₃	1	2	yes	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	2	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	C₂×Q₈×C₂₃	1	2	yes	—
204	D₄	C₂×Q₈×C₂₃	1	2	yes	complemented
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	C₂×Q₈×C₂₃	1	2	yes	complemented
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
236	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
237	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	4	2	no	—
240	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
241	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	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	C₂×D₈×C₂₃	C₂²
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×#554	C₂	semidirect
2. D₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#501	C₂	semidirect
4. C₂₃×#498	C₂	semidirect
5. C₂₃×#478	C₂	semidirect
6. C₂₃×#476	C₂	semidirect
7. C₂₃×#476	C₂	semidirect
8. C₂₃×#474	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
52. D₈×C₂₃	Q₈	semidirect
82. D₄×C₂₃	C₂×Q₈	semidirect
98. C₈×C₂₃	C₂×Q₈	semidirect
99. #3384	C₂₃	direct (N × H)
117. #554	C₂×C₂₃	semidirect
118. D₄×Q₈	C₂×C₂₃	semidirect
119. #501	C₂×C₂₃	semidirect
120. #498	C₂×C₂₃	semidirect
121. #478	C₂×C₂₃	semidirect
122. #476	C₂×C₂₃	semidirect
123. #476	C₂×C₂₃	semidirect
124. #474	C₂×C₂₃	semidirect
125. #446	C₂×C₂₃	semidirect
162. C₂₃	#3384	direct (N × H)
174. D₈	Q₈×C₂₃	semidirect
204. D₄	C₂×Q₈×C₂₃	semidirect
220. C₈	C₂×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×#554	2
D₄×Q₈×C₂₃	2
C₂₃×#501	2
C₂₃×#498	2
C₂₃×#478	2
C₂₃×#476	2
C₂₃×#474	2
C₂₃×#446	2
C₂₃×#442	2
C₄×D₈×C₂₃	2
#3384	23

Subgroups — C23×#3384 #15671003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3384 #15671003