Subgroups — C₂×C₂₃×#3293 #19208730

Order 5888.

2598 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3293	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#473	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#456	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#443	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	4	yes	—
8	C₂₃×#578	—	2	16	no	—
9	C₂₃×#578	C₂²	1	32	yes	complemented
10	C₂×C₂₃×#137	—	2	8	no	—
11	C₂×C₂₃×#137	C₂²	1	8	yes	—
12	C₂×C₂₃×#135	—	2	4	no	—
13	C₂×C₂₃×#135	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₂₃	—	2	4	no	—
15	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
16	C₂×C₂₃×#131	C₂²	1	4	yes	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂₃×#494	C₂²	1	8	yes	complemented
19	C₂₃×#473	C₂²	1	32	yes	complemented
20	C₂₃×#456	C₂²	1	32	yes	complemented
21	C₂₃×#443	C₂²	1	16	yes	complemented
22	C₂×C₂₃×#108	C₂²	1	2	yes	—
23	C₂×C₂₃×#104	C₂²	1	4	yes	—
24	C₂×C₂₃×#101	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×2₋¹⁺⁴	—	2	16	no	—
27	C₂₃×2₋¹⁺⁴	—	4	4	no	—
28	C₂₃×2₋¹⁺⁴	C₂³	1	16	yes	—
29	C₂×#41×C₂₃	—	2	8	no	—
30	C₂×#41×C₂₃	—	4	4	no	—
31	C₂×#41×C₂₃	C₂³	1	4	yes	—
32	C₂²×Q₈×C₂₃	—	2	8	no	—
33	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
34	C₂₃×#137	—	2	16	no	—
35	C₂₃×#137	—	2	64	no	—
36	C₂₃×#137	C₂³	1	32	yes	—
37	C₂₃×#135	—	2	32	no	—
38	C₂₃×#135	D₄	1	8	yes	—
39	C₂₃×#135	C₂³	1	8	yes	—
40	C₂×Q₁₆×C₂₃	—	2	8	no	—
41	C₂×Q₁₆×C₂₃	—	2	32	no	—
42	C₂×Q₁₆×C₂₃	—	2	4	no	—
43	C₂×Q₁₆×C₂₃	—	2	4	no	—
44	C₂×Q₁₆×C₂₃	—	2	8	no	—
45	C₂×Q₁₆×C₂₃	D₄	1	2	yes	—
46	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
47	C₂×SD₁₆×C₂₃	—	2	8	no	—
48	C₂×SD₁₆×C₂₃	D₄	1	4	yes	—
49	C₂×D₈×C₂₃	D₄	1	2	yes	—
50	C₂₃×#131	—	2	8	no	—
51	C₂₃×#131	C₂³	1	16	yes	—
52	C₂×M₁₆×C₂₃	—	2	4	no	—
53	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
54	C₂²×C₈×C₂₃	—	2	4	no	—
55	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂₃×C₄⋊Q₈	—	4	2	no	—
57	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
58	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#108	C₂³	1	8	yes	—
61	C₂₃×#104	C₂³	1	16	yes	—
62	C₂₃×#101	C₂³	1	16	yes	—
63	C₄×C₈×C₂₃	C₂³	1	4	yes	—
64	#41×C₂₃	—	2	4	no	—
65	#41×C₂₃	—	2	4	no	—
66	#41×C₂₃	—	2	32	no	—
67	#41×C₂₃	—	4	16	no	—
68	#41×C₂₃	—	4	16	no	—
69	#41×C₂₃	—	4	16	no	—
70	#41×C₂₃	C₂×D₄	1	8	yes	—
71	#41×C₂₃	C₂⁴	1	8	yes	—
72	C₂×Q₈×C₂₃	—	2	4	no	—
73	C₂×Q₈×C₂₃	—	2	32	no	—
74	C₂×Q₈×C₂₃	—	2	8	no	—
75	C₂×Q₈×C₂₃	—	2	8	no	—
76	C₂×Q₈×C₂₃	—	4	4	no	—
77	C₂×Q₈×C₂₃	—	4	16	no	—
78	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
79	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
80	C₂×D₄×C₂₃	—	2	4	no	—
81	C₂×D₄×C₂₃	—	4	4	no	—
82	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₂₃	—	2	1	no	—
84	C₂²×C₄×C₂₃	—	2	4	no	—
85	C₂²×C₄×C₂₃	—	2	4	no	—
86	C₂²×C₄×C₂₃	—	4	8	no	—
87	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₂₃	—	2	32	no	—
89	Q₁₆×C₂₃	—	2	16	no	—
90	Q₁₆×C₂₃	—	2	16	no	—
91	Q₁₆×C₂₃	C₂×D₄	1	8	yes	—
92	SD₁₆×C₂₃	—	2	32	no	—
93	SD₁₆×C₂₃	C₂×D₄	1	16	yes	—
94	D₈×C₂₃	C₂×D₄	1	8	yes	—
95	M₁₆×C₂₃	—	2	4	no	—
96	M₁₆×C₂₃	—	2	16	no	—
97	M₁₆×C₂₃	C₂⁴	1	8	yes	—
98	C₂×C₈×C₂₃	—	2	4	no	—
99	C₂×C₈×C₂₃	—	2	16	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
104	C₄⋊C₄×C₂₃	—	4	8	no	—
105	C₄²×C₂₃	—	2	1	no	—
106	C₄²×C₂₃	C₂⁴	1	2	yes	—
107	C₂×#3293	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₃	—	2	4	no	—
109	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
110	Q₈×C₂₃	—	2	4	no	—
111	Q₈×C₂₃	—	2	16	no	—
112	Q₈×C₂₃	—	4	16	no	—
113	Q₈×C₂₃	—	4	8	no	—
114	Q₈×C₂₃	C₂²×D₄	1	8	yes	—
115	D₄×C₂₃	—	2	8	no	—
116	D₄×C₂₃	—	2	8	no	—
117	D₄×C₂₃	—	2	4	no	—
118	D₄×C₂₃	—	4	16	no	—
119	D₄×C₂₃	C₂²×D₄	1	8	yes	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	8	no	—
124	C₂×C₄×C₂₃	—	2	8	no	—
125	C₂×C₄×C₂₃	—	2	8	no	—
126	C₂×C₄×C₂₃	—	4	32	no	—
127	C₂×C₄×C₂₃	—	4	4	no	—
128	C₂×C₄×C₂₃	—	4	4	no	—
129	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
133	C₈×C₂₃	—	2	8	no	—
134	C₈×C₂₃	C₂²×D₄	1	4	yes	—
135	C₈×C₂₃	C₂²×D₄	1	4	yes	—
136	C₂×#578	C₂×C₂₃	1	4	yes	complemented
137	#3293	C₂×C₂₃	1	16	yes	direct product factor
138	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#473	C₂×C₂₃	1	4	yes	complemented
140	C₂×#456	C₂×C₂₃	1	4	yes	complemented
141	C₂×#443	C₂×C₂₃	1	2	yes	complemented
142	C₂²×C₂₃	—	2	2	no	—
143	C₂²×C₂₃	—	2	8	no	—
144	C₂²×C₂₃	—	4	8	no	—
145	C₂²×C₂₃	D₄²	1	1	yes	characteristic
146	C₂²×C₂₃	D₄²	1	1	yes	characteristic
147	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₂₃	—	2	4	no	—
149	C₄×C₂₃	—	2	8	no	—
150	C₄×C₂₃	—	4	8	no	—
151	C₄×C₂₃	D₄²	1	2	yes	—
152	C₄×C₂₃	C₂³×D₄	1	2	yes	—
153	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	4	yes	—
154	#578	—	2	16	no	—
155	#578	C₂²×C₂₃	1	32	yes	complemented
156	C₂×#137	—	2	8	no	—
157	C₂×#137	C₂²×C₂₃	1	8	yes	—
158	C₂×#135	—	2	4	no	—
159	C₂×#135	C₂²×C₂₃	1	2	yes	—
160	C₂²×Q₁₆	—	2	4	no	—
161	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
162	C₂×#131	C₂²×C₂₃	1	4	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
164	#494	C₂²×C₂₃	1	8	yes	complemented
165	#473	C₂²×C₂₃	1	32	yes	complemented
166	#456	C₂²×C₂₃	1	32	yes	complemented
167	#443	C₂²×C₂₃	1	16	yes	complemented
168	C₂×#108	C₂²×C₂₃	1	2	yes	—
169	C₂×#104	C₂²×C₂₃	1	4	yes	—
170	C₂×#101	C₂²×C₂₃	1	4	yes	—
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₂₃	—	2	1	no	—
173	C₂×C₂₃	—	2	1	no	—
174	C₂×C₂₃	—	4	8	no	—
175	C₂×C₂₃	#3293	1	2	yes	direct product factor
176	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
177	2₋¹⁺⁴	—	2	16	no	—
178	2₋¹⁺⁴	—	4	4	no	—
179	2₋¹⁺⁴	C₂³×C₂₃	1	16	yes	—
180	C₂×#41	—	2	8	no	—
181	C₂×#41	—	4	4	no	—
182	C₂×#41	C₂³×C₂₃	1	4	yes	—
183	C₂²×Q₈	—	2	8	no	—
184	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
185	#137	—	2	16	no	—
186	#137	—	2	64	no	—
187	#137	C₂³×C₂₃	1	32	yes	—
188	#135	—	2	32	no	—
189	#135	D₄×C₂₃	1	8	yes	—
190	#135	C₂³×C₂₃	1	8	yes	—
191	C₂×Q₁₆	—	2	8	no	—
192	C₂×Q₁₆	—	2	32	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	2	4	no	—
195	C₂×Q₁₆	—	2	8	no	—
196	C₂×Q₁₆	D₄×C₂₃	1	2	yes	—
197	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
198	C₂×SD₁₆	—	2	8	no	—
199	C₂×SD₁₆	D₄×C₂₃	1	4	yes	—
200	C₂×D₈	D₄×C₂₃	1	2	yes	—
201	#131	—	2	8	no	—
202	#131	C₂³×C₂₃	1	16	yes	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
207	C₄⋊Q₈	—	4	2	no	—
208	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
209	C₂×C₄⋊C₄	—	4	2	no	—
210	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
211	#108	C₂³×C₂₃	1	8	yes	—
212	#104	C₂³×C₂₃	1	16	yes	—
213	#101	C₂³×C₂₃	1	16	yes	—
214	C₄×C₈	C₂³×C₂₃	1	4	yes	—
215	C₂₃	C₂×#3293	1	1	yes	characteristic, direct product factor, Sylow
216	#41	—	2	4	no	—
217	#41	—	2	4	no	—
218	#41	—	2	32	no	—
219	#41	—	4	16	no	—
220	#41	—	4	16	no	—
221	#41	—	4	16	no	—
222	#41	C₂×D₄×C₂₃	1	8	yes	—
223	#41	C₂⁴×C₂₃	1	8	yes	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	32	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	16	no	—
230	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
231	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	8	no	—
239	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
240	Q₁₆	—	2	32	no	—
241	Q₁₆	—	2	16	no	—
242	Q₁₆	—	2	16	no	—
243	Q₁₆	C₂×D₄×C₂₃	1	8	yes	—
244	SD₁₆	—	2	32	no	—
245	SD₁₆	C₂×D₄×C₂₃	1	16	yes	—
246	D₈	C₂×D₄×C₂₃	1	8	yes	—
247	M₁₆	—	2	4	no	—
248	M₁₆	—	2	16	no	—
249	M₁₆	C₂⁴×C₂₃	1	8	yes	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	16	no	—
252	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
253	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
255	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂⁴×C₂₃	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
261	Q₈	—	2	4	no	—
262	Q₈	—	2	16	no	—
263	Q₈	—	4	16	no	—
264	Q₈	—	4	8	no	—
265	Q₈	C₂²×D₄×C₂₃	1	8	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	8	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	16	no	—
270	D₄	C₂²×D₄×C₂₃	1	8	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	32	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
282	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
284	C₈	—	2	8	no	—
285	C₈	C₂²×D₄×C₂₃	1	4	yes	—
286	C₈	C₂²×D₄×C₂₃	1	4	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	8	no	—
290	C₂²	D₄²×C₂₃	1	1	yes	characteristic
291	C₂²	D₄²×C₂₃	1	1	yes	characteristic
292	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
293	C₄	—	2	4	no	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	D₄²×C₂₃	1	2	yes	—
297	C₄	C₂³×D₄×C₂₃	1	2	yes	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	8	no	—
301	C₂	C₂₃×#3293	1	2	yes	direct product factor
302	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#578	C₂	semidirect
2. C₂₃×#3293	C₂	direct (N × H)
3. C₂×C₂₃×#494	C₂	semidirect
4. C₂×C₂₃×#473	C₂	semidirect
5. C₂×C₂₃×#456	C₂	semidirect
6. C₂×C₂₃×#443	C₂	semidirect
9. C₂₃×#578	C₂²	semidirect
18. C₂₃×#494	C₂²	semidirect
19. C₂₃×#473	C₂²	semidirect
20. C₂₃×#456	C₂²	semidirect
21. C₂₃×#443	C₂²	semidirect
107. C₂×#3293	C₂₃	direct (N × H)
136. C₂×#578	C₂×C₂₃	semidirect
137. #3293	C₂×C₂₃	direct (N × H)
138. C₂×#494	C₂×C₂₃	semidirect
139. C₂×#473	C₂×C₂₃	semidirect
140. C₂×#456	C₂×C₂₃	semidirect
141. C₂×#443	C₂×C₂₃	semidirect
155. #578	C₂²×C₂₃	semidirect
164. #494	C₂²×C₂₃	semidirect
165. #473	C₂²×C₂₃	semidirect
166. #456	C₂²×C₂₃	semidirect
167. #443	C₂²×C₂₃	semidirect
175. C₂×C₂₃	#3293	direct (N × H)
215. C₂₃	C₂×#3293	direct (N × H)
301. C₂	C₂₃×#3293	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#578	2
C₂₃×#3293	2
C₂×C₂₃×#494	2
C₂×C₂₃×#473	2
C₂×C₂₃×#456	2
C₂×C₂₃×#443	2
C₂×#3293	23

Subgroups — C2×C23×#3293 #19208730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3293 #19208730