Subgroups — C₂×C₂₃×#1336 #19159229

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1336	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₂₃×#135	C₄	1	1	yes	characteristic
6	C₂×C₂₃×#131	—	2	1	no	—
7	C₂×C₂₃×M₃₂	—	2	1	no	—
8	C₂₃×#443	—	2	1	no	—
9	C₂₃×#443	—	2	1	no	—
10	C₂₃×#443	C₄	1	2	yes	—
11	C₂₃×#443	C₂²	1	2	yes	—
12	C₂×C₂₃×#108	C₄	1	1	yes	characteristic
13	C₂×C₂₃×#104	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#364	—	2	1	no	—
17	C₂₃×#364	C₂²	1	2	yes	complemented
18	C₂₃×#322	—	2	1	no	—
19	C₂₃×#322	C₂²	1	2	yes	complemented
20	C₂×#41×C₂₃	—	2	1	no	—
21	C₂₃×#135	—	2	1	no	—
22	C₂₃×#135	—	2	1	no	—
23	C₂₃×#135	C₈	1	2	yes	complemented
24	C₂₃×#135	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₂₃	C₈	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₂₃	—	2	1	no	—
27	C₂×D₈×C₂₃	C₈	1	1	yes	characteristic, complemented
28	C₂₃×#131	—	2	2	no	—
29	C₂₃×#131	—	2	2	no	—
30	C₂₃×#131	—	4	1	no	—
31	C₂₃×#131	—	4	1	no	—
32	C₂×M₁₆×C₂₃	—	2	1	no	—
33	C₂×M₁₆×C₂₃	—	4	1	no	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	—	4	1	no	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂₃×M₃₂	—	2	2	no	—
40	C₂₃×M₃₂	—	4	1	no	—
41	C₂×C₁₆×C₂₃	—	4	1	no	—
42	C₂₃×#108	—	2	1	no	—
43	C₂₃×#108	C₂×C₄	1	2	yes	—
44	C₂₃×#104	—	2	2	no	—
45	C₂₃×#104	—	2	2	no	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	#41×C₂₃	—	2	2	no	—
53	#41×C₂₃	—	2	2	no	—
54	#41×C₂₃	—	4	1	no	—
55	#41×C₂₃	—	4	1	no	—
56	C₂×Q₈×C₂₃	—	2	1	no	—
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	4	1	no	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	Q₁₆×C₂₃	—	2	1	no	—
63	Q₁₆×C₂₃	C₂×C₈	1	2	yes	complemented
64	SD₁₆×C₂₃	—	2	2	no	—
65	SD₁₆×C₂₃	—	2	2	no	—
66	D₈×C₂₃	—	2	1	no	—
67	D₈×C₂₃	C₂×C₈	1	2	yes	complemented
68	M₁₆×C₂₃	—	2	2	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
82	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
84	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
85	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
86	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₃	—	2	1	no	—
90	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₂₃	—	4	2	no	—
93	C₂×#1336	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	4	1	no	—
95	C₂³×C₂₃	#104	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	2	no	—
97	Q₈×C₂₃	—	4	1	no	—
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#104	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₈×C₂₃	—	4	4	no	—
115	C₈×C₂₃	—	4	2	no	—
116	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₈×C₂₃	#102	1	1	yes	characteristic
119	C₈×C₂₃	#102	1	1	yes	characteristic
120	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
121	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
123	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
125	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×#322	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#1336	C₂×C₂₃	1	4	yes	direct product factor
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	8	2	no	—
131	C₂²×C₂₃	D₄⋊C₈	1	1	yes	characteristic
132	C₂²×C₂₃	D₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	4	2	no	—
137	C₄×C₂₃	D₄⋊C₈	1	1	yes	characteristic
138	C₄×C₂₃	D₄⋊C₈	1	1	yes	characteristic
139	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
141	C₂×#135	C₄×C₂₃	1	1	yes	characteristic
142	C₂×#131	—	2	1	no	—
143	C₂×M₃₂	—	2	1	no	—
144	#443	—	2	1	no	—
145	#443	—	2	1	no	—
146	#443	C₄×C₂₃	1	2	yes	—
147	#443	C₂²×C₂₃	1	2	yes	—
148	C₂×#108	C₄×C₂₃	1	1	yes	characteristic
149	C₂×#104	—	2	1	no	—
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	#364	—	2	1	no	—
153	#364	C₂²×C₂₃	1	2	yes	complemented
154	#322	—	2	1	no	—
155	#322	C₂²×C₂₃	1	2	yes	complemented
156	C₂×C₂₃	—	2	1	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	8	2	no	—
159	C₂×C₂₃	#1336	1	2	yes	direct product factor
160	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
161	C₂×#41	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	C₈×C₂₃	1	2	yes	complemented
165	#135	C₂×C₄×C₂₃	1	2	yes	—
166	C₂×Q₁₆	C₈×C₂₃	1	1	yes	characteristic, complemented
167	C₂×SD₁₆	—	2	1	no	—
168	C₂×D₈	C₈×C₂₃	1	1	yes	characteristic, complemented
169	#131	—	2	2	no	—
170	#131	—	2	2	no	—
171	#131	—	4	1	no	—
172	#131	—	4	1	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	4	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	4	1	no	—
177	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
180	M₃₂	—	2	2	no	—
181	M₃₂	—	4	1	no	—
182	C₂×C₁₆	—	4	1	no	—
183	#108	—	2	1	no	—
184	#108	C₂×C₄×C₂₃	1	2	yes	—
185	#104	—	2	2	no	—
186	#104	—	2	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
191	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂₃	C₂×#1336	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×C₈×C₂₃	1	2	yes	complemented
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×C₈×C₂₃	1	2	yes	complemented
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
222	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
223	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
224	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
225	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
226	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
227	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
228	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₁₆	—	4	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	C₂₃×#104	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	Q₈	—	4	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	4	2	no	—
257	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
258	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
259	C₈	C₂₃×#102	1	1	yes	characteristic
260	C₈	C₂₃×#102	1	1	yes	characteristic
261	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
262	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
263	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
264	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	C₂₃×D₄⋊C₈	1	1	yes	characteristic
269	C₂²	C₂₃×D₄⋊C₈	1	1	yes	characteristic
270	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	C₂₃×D₄⋊C₈	1	1	yes	characteristic
275	C₄	C₂₃×D₄⋊C₈	1	1	yes	characteristic
276	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
277	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	8	2	no	—
281	C₂	C₂₃×#1336	1	2	yes	direct product factor
282	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	D₄⋊C₈
prime radical	C₂×C₂₃×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₈×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×#364	C₂	semidirect
3. C₂×C₂₃×#322	C₂	semidirect
4. C₂₃×#1336	C₂	direct (N × H)
17. C₂₃×#364	C₂²	semidirect
19. C₂₃×#322	C₂²	semidirect
23. C₂₃×#135	C₈	semidirect
25. C₂×Q₁₆×C₂₃	C₈	semidirect
27. C₂×D₈×C₂₃	C₈	semidirect
63. Q₁₆×C₂₃	C₂×C₈	semidirect
67. D₈×C₂₃	C₂×C₈	semidirect
93. C₂×#1336	C₂₃	direct (N × H)
125. C₂×#364	C₂×C₂₃	semidirect
126. C₂×#322	C₂×C₂₃	semidirect
127. #1336	C₂×C₂₃	direct (N × H)
153. #364	C₂²×C₂₃	semidirect
155. #322	C₂²×C₂₃	semidirect
159. C₂×C₂₃	#1336	direct (N × H)
164. #135	C₈×C₂₃	semidirect
166. C₂×Q₁₆	C₈×C₂₃	semidirect
168. C₂×D₈	C₈×C₂₃	semidirect
193. C₂₃	C₂×#1336	direct (N × H)
205. Q₁₆	C₂×C₈×C₂₃	semidirect
209. D₈	C₂×C₈×C₂₃	semidirect
281. C₂	C₂₃×#1336	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×#443	2
C₂×C₂₃×#364	2
C₂×C₂₃×#322	2
C₂₃×#1336	2
C₂×#1336	23

Subgroups — C2×C23×#1336 #19159229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1336 #19159229