Subgroups — C₂×C₂₃×#1361 #19159553

Order 5888.

518 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1361	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₂₃	—	2	1	no	—
6	C₂²×D₈×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×#127	—	2	1	no	—
8	C₂²×C₁₆×C₂₃	—	2	1	no	—
9	C₂₃×#493	—	2	1	no	—
10	C₂₃×#493	—	2	1	no	—
11	C₂₃×#493	C₄	1	2	yes	—
12	C₂₃×#493	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₁₆	—	2	1	no	—
16	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
17	C₂₃×#335	—	2	1	no	—
18	C₂₃×#335	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₂₃	—	2	2	no	—
20	C₂²×D₄×C₂₃	—	4	1	no	—
21	C₂²×D₄×C₂₃	—	4	1	no	—
22	C₂×D₈×C₂₃	—	2	2	no	—
23	C₂×D₈×C₂₃	—	2	2	no	—
24	C₂×D₈×C₂₃	—	2	2	no	—
25	C₂×D₈×C₂₃	—	4	4	no	—
26	C₂×D₈×C₂₃	—	4	2	no	—
27	C₂×D₈×C₂₃	—	4	2	no	—
28	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₂₃	—	4	1	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂₃×#127	—	2	2	no	—
33	C₂₃×#127	—	2	2	no	—
34	C₂₃×#127	—	4	1	no	—
35	C₂₃×#127	—	4	1	no	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂×C₁₆×C₂₃	—	4	2	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	—	4	1	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂⁴×C₂₃	—	4	2	no	—
46	C₂×D₄×C₂₃	—	2	4	no	—
47	C₂×D₄×C₂₃	—	4	4	no	—
48	C₂×D₄×C₂₃	—	4	4	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	—	4	2	no	—
51	C₂×D₄×C₂₃	—	4	2	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	8	4	no	—
55	C₂×D₄×C₂₃	—	8	2	no	—
56	C₂×D₄×C₂₃	—	8	2	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
60	D₈×C₂₃	—	4	4	no	—
61	D₈×C₂₃	—	4	4	no	—
62	D₈×C₂₃	—	4	4	no	—
63	D₈×C₂₃	—	4	2	no	—
64	D₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	8	2	no	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	D₈	1	2	yes	—
71	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₂₃	D₈	1	1	yes	characteristic
76	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₂₃	—	4	2	no	—
79	C₂×#1361	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	—	4	4	no	—
81	C₂³×C₂₃	—	8	4	no	—
82	C₂³×C₂₃	—	8	4	no	—
83	C₂³×C₂₃	—	8	4	no	—
84	C₂³×C₂₃	#102	1	1	yes	characteristic
85	D₄×C₂₃	—	4	4	no	—
86	D₄×C₂₃	—	4	4	no	—
87	D₄×C₂₃	—	8	4	no	—
88	D₄×C₂₃	—	8	4	no	—
89	D₄×C₂₃	—	8	4	no	—
90	D₄×C₂₃	—	8	4	no	—
91	D₄×C₂₃	—	8	2	no	—
92	D₄×C₂₃	—	8	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	#100	1	2	yes	—
95	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#102	1	2	yes	—
98	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
100	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₈×C₂₃	—	2	1	no	—
105	C₈×C₂₃	—	2	1	no	—
106	C₈×C₂₃	—	8	2	no	—
107	C₈×C₂₃	#102	1	2	yes	—
108	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
109	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×#493	C₂×C₂₃	1	1	yes	characteristic
111	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
113	#1361	C₂×C₂₃	1	4	yes	direct product factor
114	C₂²×C₂₃	—	8	4	no	—
115	C₂²×C₂₃	—	8	4	no	—
116	C₂²×C₂₃	—	8	4	no	—
117	C₂²×C₂₃	—	16	8	no	—
118	C₂²×C₂₃	#331	1	2	yes	—
119	C₂²×C₂₃	#357	1	2	yes	—
120	C₂²×C₂₃	#361	1	2	yes	—
121	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	#331	1	2	yes	—
125	C₄×C₂₃	#357	1	2	yes	—
126	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
127	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
128	C₄×C₂₃	C₂×D₁₆	1	1	yes	characteristic
129	C₄×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
130	C₂²×D₈	—	2	1	no	—
131	C₂²×D₈	C₄×C₂₃	1	2	yes	—
132	C₂×#127	—	2	1	no	—
133	C₂²×C₁₆	—	2	1	no	—
134	#493	—	2	1	no	—
135	#493	—	2	1	no	—
136	#493	C₄×C₂₃	1	2	yes	—
137	#493	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄⋊C₈	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₄⋊C₁₆	—	2	1	no	—
141	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
142	#335	—	2	1	no	—
143	#335	C₂²×C₂₃	1	2	yes	complemented
144	C₂×C₂₃	—	16	4	no	—
145	C₂×C₂₃	#1361	1	4	yes	direct product factor
146	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
147	C₂×C₂₃	C₂×#357	1	1	yes	characteristic
148	C₂×C₂₃	C₂×#361	1	1	yes	characteristic
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	4	1	no	—
151	C₂²×D₄	—	4	1	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	4	4	no	—
156	C₂×D₈	—	4	2	no	—
157	C₂×D₈	—	4	2	no	—
158	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
159	C₂²×C₈	—	4	1	no	—
160	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
162	#127	—	2	2	no	—
163	#127	—	2	2	no	—
164	#127	—	4	1	no	—
165	#127	—	4	1	no	—
166	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	4	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	4	1	no	—
171	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
175	C₂₃	C₂×#1361	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	4	2	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	8	4	no	—
186	C₂×D₄	—	8	2	no	—
187	C₂×D₄	—	8	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
191	D₈	—	4	4	no	—
192	D₈	—	4	4	no	—
193	D₈	—	4	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	8	2	no	—
199	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₈	D₈×C₂₃	1	2	yes	—
202	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₄²	D₈×C₂₃	1	1	yes	characteristic
207	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	C₂₃×#102	1	1	yes	characteristic
215	D₄	—	4	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	8	4	no	—
218	D₄	—	8	4	no	—
219	D₄	—	8	4	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	C₂₃×#100	1	2	yes	—
225	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
226	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
227	C₂×C₄	C₂₃×#102	1	2	yes	—
228	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
229	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
230	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
233	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	C₂₃×#102	1	2	yes	—
238	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
239	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
240	C₂²	—	8	4	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	16	8	no	—
244	C₂²	C₂₃×#331	1	2	yes	—
245	C₂²	C₂₃×#357	1	2	yes	—
246	C₂²	C₂₃×#361	1	2	yes	—
247	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	C₂₃×#331	1	2	yes	—
251	C₄	C₂₃×#357	1	2	yes	—
252	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
253	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
254	C₄	C₂×C₂₃×D₁₆	1	1	yes	characteristic
255	C₄	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
256	C₂	—	16	4	no	—
257	C₂	C₂₃×#1361	1	4	yes	direct product factor
258	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
259	C₂	C₂×C₂₃×#357	1	1	yes	characteristic
260	C₂	C₂×C₂₃×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₂₃×#335	C₂	semidirect
4. C₂₃×#1361	C₂	direct (N × H)
16. C₂₃×C₄⋊C₁₆	C₂²	semidirect
18. C₂₃×#335	C₂²	semidirect
79. C₂×#1361	C₂₃	direct (N × H)
111. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
112. C₂×#335	C₂×C₂₃	semidirect
113. #1361	C₂×C₂₃	direct (N × H)
141. C₄⋊C₁₆	C₂²×C₂₃	semidirect
143. #335	C₂²×C₂₃	semidirect
145. C₂×C₂₃	#1361	direct (N × H)
175. C₂₃	C₂×#1361	direct (N × H)
257. C₂	C₂₃×#1361	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×#493	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₂₃×#335	2
C₂₃×#1361	2
C₂×#1361	23

Subgroups — C2×C23×#1361 #19159553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1361 #19159553