Subgroups — C₂₃×#3299 #15670918

Order 2944.

388 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	C₂₃×#554	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#484	C₂	1	2	yes	maximal, complemented
5	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#467	C₂	1	2	yes	maximal, complemented
7	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#451	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂₃×#103	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	—	2	1	no	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×#125	—	2	1	no	—
20	C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₃	—	2	1	no	—
27	C₄×D₄×C₂₃	—	2	1	no	—
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#107	C₂²	1	1	yes	characteristic
31	C₂₃×#107	C₂²	1	2	yes	—
32	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂₃×#103	C₂²	1	2	yes	—
35	C₂₃×#103	C₂²	1	2	yes	—
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	1	yes	characteristic
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×C₂²⋊C₈	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	4	no	—
44	C₂×Q₈×C₂₃	—	4	2	no	—
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	Q₁₆×C₂₃	—	2	2	no	—
53	Q₁₆×C₂₃	—	4	2	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	2	1	no	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂³×C₂₃	C₂×D₄	1	2	yes	—
77	Q₈×C₂₃	—	2	1	no	—
78	Q₈×C₂₃	—	2	1	no	—
79	Q₈×C₂₃	—	2	2	no	—
80	Q₈×C₂₃	—	2	1	no	—
81	Q₈×C₂₃	—	2	2	no	—
82	Q₈×C₂₃	—	4	4	no	—
83	Q₈×C₂₃	—	8	1	no	—
84	Q₈×C₂₃	#41	1	2	yes	—
85	D₄×C₂₃	Q₁₆	1	4	yes	complemented
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₃	—	2	1	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	2	2	no	—
103	#3299	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	C₂×Q₁₆	1	4	yes	—
107	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	—	4	1	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
117	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
119	#554	C₂×C₂₃	1	1	yes	characteristic
120	D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
121	#487	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#484	C₂×C₂₃	1	2	yes	complemented
123	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#467	C₂×C₂₃	1	2	yes	complemented
125	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#451	C₂×C₂₃	1	2	yes	complemented
127	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
129	C₂×#103	C₂×C₂₃	1	2	yes	complemented
130	C₂×C₂₃	—	2	4	no	—
131	C₂×C₂₃	#546	1	1	yes	characteristic
132	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂×C₂₃	#577	1	1	yes	characteristic
134	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
135	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
137	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
141	#125	—	2	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
145	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
146	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₄×D₄	—	2	1	no	—
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	#107	C₂²×C₂₃	1	1	yes	characteristic
153	#107	C₂²×C₂₃	1	2	yes	—
154	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
155	#103	C₂²×C₂₃	1	1	yes	characteristic
156	#103	C₂²×C₂₃	1	2	yes	—
157	#103	C₂²×C₂₃	1	2	yes	—
158	#103	C₂²×C₂₃	1	2	yes	—
159	#103	C₂²×C₂₃	1	1	yes	characteristic
160	#103	C₂²×C₂₃	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
162	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂₃	#3299	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
169	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
170	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₂₃	1	2	yes	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
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	2	no	—
195	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂³×C₂₃	1	1	yes	characteristic
199	C₂³	C₂×D₄×C₂₃	1	2	yes	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	8	1	no	—
207	Q₈	#41×C₂₃	1	2	yes	—
208	D₄	Q₁₆×C₂₃	1	4	yes	complemented
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
221	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	C₂×Q₁₆×C₂₃	1	4	yes	—
229	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
239	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
240	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂	—	2	4	no	—
242	C₂	C₂₃×#546	1	1	yes	characteristic
243	C₂	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
244	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×D₄×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#487	C₂	semidirect
4. C₂₃×#484	C₂	semidirect
5. C₂₃×#477	C₂	semidirect
6. C₂₃×#467	C₂	semidirect
7. C₂₃×#462	C₂	semidirect
8. C₂₃×#451	C₂	semidirect
9. C₄×Q₁₆×C₂₃	C₂	semidirect
11. C₂×C₂₃×#103	C₂	semidirect
85. D₄×C₂₃	Q₁₆	semidirect
103. #3299	C₂₃	direct (N × H)
121. #487	C₂×C₂₃	semidirect
122. #484	C₂×C₂₃	semidirect
123. #477	C₂×C₂₃	semidirect
124. #467	C₂×C₂₃	semidirect
125. #462	C₂×C₂₃	semidirect
126. #451	C₂×C₂₃	semidirect
127. C₄×Q₁₆	C₂×C₂₃	semidirect
129. C₂×#103	C₂×C₂₃	semidirect
163. C₂₃	#3299	direct (N × H)
208. D₄	Q₁₆×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×#487	2
C₂₃×#484	2
C₂₃×#477	2
C₂₃×#467	2
C₂₃×#462	2
C₂₃×#451	2
C₄×Q₁₆×C₂₃	2
C₈×D₄×C₂₃	2
C₂×C₂₃×#103	2
#3299	23

Subgroups — C23×#3299 #15670918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3299 #15670918