Subgroups — C₂×C₂₃×#2179 #19181088

Order 5888.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#443	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2179	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#359	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#135	C₂²	1	2	yes	—
8	C₂×C₂₃×#131	—	2	2	no	—
9	C₂₃×#508	C₄	1	4	yes	complemented
10	C₂₃×#508	C₂²	1	4	yes	—
11	C₂×C₂₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₂₃×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₂₃×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#443	—	2	4	no	—
17	C₂₃×#443	C₂²	1	8	yes	complemented
18	C₂×C₂₃×#108	C₂²	1	2	yes	—
19	C₂×C₂₃×#104	—	2	2	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#368	C₂²	1	4	yes	complemented
22	C₂₃×#359	C₂²	1	8	yes	complemented
23	C₂₃×#346	C₂²	1	4	yes	complemented
24	C₂×#41×C₂₃	—	2	2	no	—
25	C₂₃×#135	—	2	2	no	—
26	C₂₃×#135	—	2	2	no	—
27	C₂₃×#135	C₂×C₄	1	4	yes	—
28	C₂₃×#135	C₂³	1	4	yes	—
29	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×SD₁₆×C₂₃	—	2	2	no	—
31	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂₃×#131	—	2	8	no	—
33	C₂₃×#131	—	4	4	no	—
34	C₂×M₁₆×C₂₃	—	2	2	no	—
35	C₂×M₁₆×C₂₃	—	4	2	no	—
36	C₂²×C₈×C₂₃	—	4	2	no	—
37	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂₃×Q₃₂	C₂×C₄	1	4	yes	complemented
41	C₂₃×SD₃₂	C₂×C₄	1	8	yes	complemented
42	C₂₃×D₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₆×C₂₃	—	2	1	no	—
44	C₂×C₁₆×C₂₃	—	2	1	no	—
45	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
48	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
49	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
50	C₂₃×#108	—	2	2	no	—
51	C₂₃×#108	C₂³	1	4	yes	—
52	C₂₃×#104	—	2	8	no	—
53	C₄×C₈×C₂₃	D₄	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	#41×C₂₃	—	2	4	no	—
56	#41×C₂₃	—	2	4	no	—
57	#41×C₂₃	—	4	2	no	—
58	#41×C₂₃	—	4	2	no	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₂₃	—	2	2	no	—
66	Q₁₆×C₂₃	C₂²×C₄	1	4	yes	—
67	SD₁₆×C₂₃	—	2	4	no	—
68	SD₁₆×C₂₃	—	2	4	no	—
69	D₈×C₂₃	—	2	2	no	—
70	D₈×C₂₃	C₂²×C₄	1	4	yes	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	8	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	4	8	no	—
76	C₂×C₈×C₂₃	D₈	1	2	yes	—
77	C₂×C₈×C₂₃	D₈	1	2	yes	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₂₃	—	2	2	no	—
88	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×#2179	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	#135	1	1	yes	characteristic
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	2	4	no	—
96	D₄×C₂₃	—	4	4	no	—
97	D₄×C₂₃	—	4	4	no	—
98	D₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₂₃	—	4	4	no	—
111	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
113	C₈×C₂₃	C₂×D₈	1	4	yes	—
114	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂×#508	C₂×C₂₃	1	1	yes	characteristic
117	C₂×#443	C₂×C₂₃	1	2	yes	complemented
118	#2179	C₂×C₂₃	1	8	yes	direct product factor
119	C₂×#368	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×#359	C₂×C₂₃	1	2	yes	complemented
121	C₂×#346	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	4	4	no	—
124	C₂²×C₂₃	—	8	4	no	—
125	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
126	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
127	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	4	4	no	—
130	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
131	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
132	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
134	C₂×#135	C₂²×C₂₃	1	2	yes	—
135	C₂×#131	—	2	2	no	—
136	#508	C₄×C₂₃	1	4	yes	complemented
137	#508	C₂²×C₂₃	1	4	yes	—
138	C₂×Q₃₂	C₄×C₂₃	1	1	yes	characteristic, complemented
139	C₂×SD₃₂	C₄×C₂₃	1	2	yes	complemented
140	C₂×D₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
141	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
143	#443	—	2	4	no	—
144	#443	C₂²×C₂₃	1	8	yes	complemented
145	C₂×#108	C₂²×C₂₃	1	2	yes	—
146	C₂×#104	—	2	2	no	—
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	#368	C₂²×C₂₃	1	4	yes	complemented
149	#359	C₂²×C₂₃	1	8	yes	complemented
150	#346	C₂²×C₂₃	1	4	yes	complemented
151	C₂×C₂₃	—	2	1	no	—
152	C₂×C₂₃	—	2	1	no	—
153	C₂×C₂₃	—	8	4	no	—
154	C₂×C₂₃	#2179	1	2	yes	direct product factor
155	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×#41	—	2	2	no	—
157	#135	—	2	2	no	—
158	#135	—	2	2	no	—
159	#135	C₂×C₄×C₂₃	1	4	yes	—
160	#135	C₂³×C₂₃	1	4	yes	—
161	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
164	#131	—	2	8	no	—
165	#131	—	4	4	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	4	2	no	—
168	C₂²×C₈	—	4	2	no	—
169	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
172	Q₃₂	C₂×C₄×C₂₃	1	4	yes	complemented
173	SD₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
174	D₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
182	#108	—	2	2	no	—
183	#108	C₂³×C₂₃	1	4	yes	—
184	#104	—	2	8	no	—
185	C₄×C₈	D₄×C₂₃	1	2	yes	—
186	C₄×C₈	C₂³×C₂₃	1	2	yes	—
187	C₂₃	C₂×#2179	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	2	4	no	—
189	#41	—	2	4	no	—
190	#41	—	4	2	no	—
191	#41	—	4	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₂²×C₄×C₂₃	1	4	yes	—
200	SD₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×C₄×C₂₃	1	4	yes	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	8	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	8	no	—
209	C₂×C₈	D₈×C₂₃	1	2	yes	—
210	C₂×C₈	D₈×C₂₃	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
215	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
216	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
217	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	C₂×D₄×C₂₃	1	2	yes	—
220	C₁₆	—	2	2	no	—
221	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
222	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂₃×#135	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
242	C₈	—	4	4	no	—
243	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
244	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
245	C₈	C₂×D₈×C₂₃	1	4	yes	—
246	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
247	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
252	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
253	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
257	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
258	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
259	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	8	4	no	—
263	C₂	C₂₃×#2179	1	2	yes	direct product factor
264	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₄×D₈
prime radical	C₂×C₂₃×#508	C₂
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×#443	C₂	semidirect
3. C₂₃×#2179	C₂	direct (N × H)
4. C₂×C₂₃×#368	C₂	semidirect
5. C₂×C₂₃×#359	C₂	semidirect
6. C₂×C₂₃×#346	C₂	semidirect
9. C₂₃×#508	C₄	semidirect
11. C₂×C₂₃×Q₃₂	C₄	semidirect
12. C₂×C₂₃×SD₃₂	C₄	semidirect
13. C₂×C₂₃×D₁₆	C₄	semidirect
17. C₂₃×#443	C₂²	semidirect
21. C₂₃×#368	C₂²	semidirect
22. C₂₃×#359	C₂²	semidirect
23. C₂₃×#346	C₂²	semidirect
40. C₂₃×Q₃₂	C₂×C₄	semidirect
41. C₂₃×SD₃₂	C₂×C₄	semidirect
42. C₂₃×D₁₆	C₂×C₄	semidirect
90. C₂×#2179	C₂₃	direct (N × H)
117. C₂×#443	C₂×C₂₃	semidirect
118. #2179	C₂×C₂₃	direct (N × H)
119. C₂×#368	C₂×C₂₃	semidirect
120. C₂×#359	C₂×C₂₃	semidirect
121. C₂×#346	C₂×C₂₃	semidirect
136. #508	C₄×C₂₃	semidirect
138. C₂×Q₃₂	C₄×C₂₃	semidirect
139. C₂×SD₃₂	C₄×C₂₃	semidirect
140. C₂×D₁₆	C₄×C₂₃	semidirect
144. #443	C₂²×C₂₃	semidirect
148. #368	C₂²×C₂₃	semidirect
149. #359	C₂²×C₂₃	semidirect
150. #346	C₂²×C₂₃	semidirect
154. C₂×C₂₃	#2179	direct (N × H)
172. Q₃₂	C₂×C₄×C₂₃	semidirect
173. SD₃₂	C₂×C₄×C₂₃	semidirect
174. D₁₆	C₂×C₄×C₂₃	semidirect
187. C₂₃	C₂×#2179	direct (N × H)
263. C₂	C₂₃×#2179	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×#508	2
C₂×C₂₃×#443	2
C₂₃×#2179	2
C₂×C₂₃×#368	2
C₂×C₂₃×#359	2
C₂×C₂₃×#346	2
C₂×#2179	23

Subgroups — C2×C23×#2179 #19181088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2179 #19181088