Subgroups — C₂₃×#3278 #15670897

Order 2944.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	2	yes	maximal, complemented
2	C₂₃×#550	C₂	1	1	yes	maximal, characteristic
3	D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#468	C₂	1	2	yes	maximal, complemented
6	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#452	C₂	1	2	yes	maximal, complemented
9	C₂₃×#448	C₂	1	2	yes	maximal, complemented
10	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#137	—	2	8	no	—
15	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
21	C₂₃×C₄⋊Q₈	—	2	1	no	—
22	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂₃×#127	—	2	1	no	—
24	C₂₃×#127	C₂²	1	1	yes	characteristic
25	C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂₃×#121	—	2	2	no	—
28	C₂₃×#121	C₂²	1	2	yes	—
29	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₂₃	—	2	1	no	—
32	C₄×D₄×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#107	C₂²	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂₃×#103	C₂²	1	1	yes	characteristic
37	C₂₃×#103	C₂²	1	1	yes	characteristic
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×#102	C₂²	1	2	yes	—
40	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
42	#41×C₂₃	—	2	4	no	—
43	#41×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	2	2	no	—
52	C₂×D₄×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
59	Q₁₆×C₂₃	—	2	4	no	—
60	Q₁₆×C₂₃	D₄	1	4	yes	complemented
61	SD₁₆×C₂₃	—	2	4	no	—
62	SD₁₆×C₂₃	—	2	4	no	—
63	SD₁₆×C₂₃	—	4	2	no	—
64	M₁₆×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂²⋊C₄×C₂₃	—	2	2	no	—
78	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
79	C₄²×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	C₂×D₄	1	2	yes	—
84	Q₈×C₂₃	—	2	1	no	—
85	Q₈×C₂₃	—	2	1	no	—
86	Q₈×C₂₃	—	2	2	no	—
87	Q₈×C₂₃	—	2	2	no	—
88	Q₈×C₂₃	—	4	4	no	—
89	Q₈×C₂₃	C₂×D₄	1	2	yes	—
90	Q₈×C₂₃	C₂×D₄	1	2	yes	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	2	2	no	—
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	8	1	no	—
97	D₄×C₂₃	—	8	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	—	2	1	no	—
113	C₈×C₂₃	C₂×D₄	1	2	yes	—
114	#3278	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	#137	1	2	yes	complemented
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×#137	C₂×C₂₃	1	2	yes	complemented
133	#550	C₂×C₂₃	1	1	yes	characteristic
134	D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#492	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#468	C₂×C₂₃	1	2	yes	complemented
137	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#452	C₂×C₂₃	1	2	yes	complemented
140	#448	C₂×C₂₃	1	2	yes	complemented
141	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×C₂₃	—	4	2	no	—
144	C₂×C₂₃	—	8	2	no	—
145	C₂×C₂₃	D₄²	1	1	yes	characteristic
146	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
147	C₂×C₂₃	#577	1	1	yes	characteristic
148	C₂×#41	C₂²×C₂₃	1	2	yes	—
149	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
150	#137	—	2	8	no	—
151	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
152	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
154	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
156	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
157	C₄⋊Q₈	—	2	1	no	—
158	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
159	#127	—	2	1	no	—
160	#127	C₂²×C₂₃	1	1	yes	characteristic
161	C₂²⋊Q₈	—	2	2	no	—
162	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
163	#121	—	2	2	no	—
164	#121	C₂²×C₂₃	1	2	yes	—
165	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
170	#107	C₂²×C₂₃	1	1	yes	characteristic
171	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
172	#103	C₂²×C₂₃	1	1	yes	characteristic
173	#103	C₂²×C₂₃	1	1	yes	characteristic
174	#103	C₂²×C₂₃	1	2	yes	—
175	#102	C₂²×C₂₃	1	2	yes	—
176	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
177	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
178	C₂₃	#3278	1	1	yes	characteristic, direct product factor, Sylow
179	#41	—	2	4	no	—
180	#41	—	4	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
185	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
186	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
192	C₂×D₄	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
196	Q₁₆	—	2	4	no	—
197	Q₁₆	D₄×C₂₃	1	4	yes	complemented
198	SD₁₆	—	2	4	no	—
199	SD₁₆	—	2	4	no	—
200	SD₁₆	—	4	2	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₂₃	1	2	yes	—
204	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂³×C₂₃	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	C₂×D₄×C₂₃	1	2	yes	—
221	Q₈	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	2	2	no	—
225	Q₈	—	4	4	no	—
226	Q₈	C₂×D₄×C₂₃	1	2	yes	—
227	Q₈	C₂×D₄×C₂₃	1	2	yes	—
228	D₄	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×D₄×C₂₃	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	2	no	—
265	C₄	C₂₃×#137	1	2	yes	complemented
266	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
267	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂	—	4	2	no	—
269	C₂	—	8	2	no	—
270	C₂	D₄²×C₂₃	1	1	yes	characteristic
271	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
272	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×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#137	C₂	semidirect
3. D₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#492	C₂	semidirect
5. C₂₃×#468	C₂	semidirect
6. C₂₃×#462	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#452	C₂	semidirect
9. C₂₃×#448	C₂	semidirect
10. C₄×Q₁₆×C₂₃	C₂	semidirect
11. C₂₃×#436	C₂	semidirect
60. Q₁₆×C₂₃	D₄	semidirect
114. #3278	C₂₃	direct (N × H)
129. C₄×C₂₃	#137	semidirect
132. C₂×#137	C₂×C₂₃	semidirect
134. D₄×Q₈	C₂×C₂₃	semidirect
135. #492	C₂×C₂₃	semidirect
136. #468	C₂×C₂₃	semidirect
137. #462	C₂×C₂₃	semidirect
138. #460	C₂×C₂₃	semidirect
139. #452	C₂×C₂₃	semidirect
140. #448	C₂×C₂₃	semidirect
141. C₄×Q₁₆	C₂×C₂₃	semidirect
142. #436	C₂×C₂₃	semidirect
178. C₂₃	#3278	direct (N × H)
197. Q₁₆	D₄×C₂₃	semidirect
265. C₄	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#137	2
C₂₃×#550	2
D₄×Q₈×C₂₃	2
C₂₃×#492	2
C₂₃×#468	2
C₂₃×#462	2
C₂₃×#460	2
C₂₃×#452	2
C₂₃×#448	2
C₄×Q₁₆×C₂₃	2
C₂₃×#436	2
#3278	23

Subgroups — C23×#3278 #15670897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3278 #15670897