Subgroups — C₂³×C₂₃×#99 #19207692

Order 5888.

7902 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	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂₃×#99	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₂₃	C₄	1	1	yes	characteristic, complemented
5	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×D₄×C₂₃	—	2	14	no	—
7	C₂³×D₄×C₂₃	—	2	2	no	—
8	C₂³×D₄×C₂₃	C₄	1	14	yes	complemented
9	C₂³×D₄×C₂₃	C₂²	1	14	yes	—
10	C₂⁴×C₄×C₂₃	—	2	2	no	—
11	C₂⁴×C₄×C₂₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₂²⋊C₄×C₂₃	—	2	14	no	—
13	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	28	yes	complemented
14	C₂×C₂₃×#99	C₂²	1	112	yes	direct product factor
15	C₂⁵×C₂₃	—	2	7	no	—
16	C₂⁵×C₂₃	—	2	1	no	—
17	C₂⁵×C₂₃	—	4	8	no	—
18	C₂⁵×C₂₃	—	4	8	no	—
19	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂⁵×C₂₃	C₂×C₄	1	7	yes	—
21	C₂⁵×C₂₃	C₂×C₄	1	14	yes	complemented
22	C₂⁵×C₂₃	D₄	1	2	yes	—
23	C₂⁵×C₂₃	D₄	1	14	yes	—
24	C₂⁵×C₂₃	C₂³	1	7	yes	—
25	C₂²×D₄×C₂₃	—	2	28	no	—
26	C₂²×D₄×C₂₃	—	2	84	no	—
27	C₂²×D₄×C₂₃	—	2	28	no	—
28	C₂²×D₄×C₂₃	—	2	84	no	—
29	C₂²×D₄×C₂₃	C₂×C₄	1	84	yes	complemented
30	C₂²×D₄×C₂₃	C₂³	1	28	yes	—
31	C₂³×C₄×C₂₃	—	2	7	no	—
32	C₂³×C₄×C₂₃	—	2	1	no	—
33	C₂³×C₄×C₂₃	—	2	28	no	—
34	C₂³×C₄×C₂₃	—	4	16	no	—
35	C₂³×C₄×C₂₃	C₂×C₄	1	14	yes	complemented
36	C₂×C₂²⋊C₄×C₂₃	—	2	84	no	—
37	C₂×C₂²⋊C₄×C₂₃	C₂³	1	56	yes	complemented
38	C₂₃×#99	C₂³	1	64	yes	direct product factor
39	C₂⁴×C₂₃	—	2	16	no	—
40	C₂⁴×C₂₃	—	2	8	no	—
41	C₂⁴×C₂₃	—	2	42	no	—
42	C₂⁴×C₂₃	—	2	14	no	—
43	C₂⁴×C₂₃	—	4	28	no	—
44	C₂⁴×C₂₃	—	4	56	no	—
45	C₂⁴×C₂₃	—	4	28	no	—
46	C₂⁴×C₂₃	—	4	56	no	—
47	C₂⁴×C₂₃	—	4	56	no	—
48	C₂⁴×C₂₃	—	4	8	no	—
49	C₂⁴×C₂₃	C₂²⋊C₄	1	7	yes	—
50	C₂⁴×C₂₃	C₂²⋊C₄	1	14	yes	—
51	C₂⁴×C₂₃	C₂²⋊C₄	1	42	yes	—
52	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂⁴×C₂₃	C₂²×C₄	1	7	yes	—
54	C₂⁴×C₂₃	C₂²×C₄	1	21	yes	—
55	C₂⁴×C₂₃	C₂²×C₄	1	28	yes	complemented
56	C₂⁴×C₂₃	C₂×D₄	1	14	yes	—
57	C₂⁴×C₂₃	C₂×D₄	1	42	yes	—
58	C₂⁴×C₂₃	C₂⁴	1	7	yes	—
59	C₂×D₄×C₂₃	—	2	168	no	—
60	C₂×D₄×C₂₃	—	2	168	no	—
61	C₂×D₄×C₂₃	—	2	56	no	—
62	C₂×D₄×C₂₃	—	2	56	no	—
63	C₂×D₄×C₂₃	C₂²×C₄	1	56	yes	complemented
64	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₂₃	—	2	42	no	—
66	C₂²×C₄×C₂₃	—	2	14	no	—
67	C₂²×C₄×C₂₃	—	2	56	no	—
68	C₂²×C₄×C₂₃	—	4	112	no	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	28	yes	complemented
70	C₂²⋊C₄×C₂₃	—	2	56	no	—
71	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	complemented
72	C₂³×#99	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	56	no	—
74	C₂³×C₂₃	—	2	28	no	—
75	C₂³×C₂₃	—	2	28	no	—
76	C₂³×C₂₃	—	2	56	no	—
77	C₂³×C₂₃	—	2	28	no	—
78	C₂³×C₂₃	—	2	28	no	—
79	C₂³×C₂₃	—	4	84	no	—
80	C₂³×C₂₃	—	4	42	no	—
81	C₂³×C₂₃	—	4	84	no	—
82	C₂³×C₂₃	—	4	14	no	—
83	C₂³×C₂₃	—	4	56	no	—
84	C₂³×C₂₃	—	4	168	no	—
85	C₂³×C₂₃	—	4	56	no	—
86	C₂³×C₂₃	#99	1	8	yes	direct product factor
87	C₂³×C₂₃	C₂×C₂²⋊C₄	1	21	yes	—
88	C₂³×C₂₃	C₂×C₂²⋊C₄	1	42	yes	—
89	C₂³×C₂₃	C₂×C₂²⋊C₄	1	42	yes	—
90	C₂³×C₂₃	C₂×C₂²⋊C₄	1	7	yes	—
91	C₂³×C₂₃	C₂³×C₄	1	7	yes	—
92	C₂³×C₂₃	C₂³×C₄	1	7	yes	—
93	C₂³×C₂₃	C₂³×C₄	1	8	yes	complemented
94	C₂³×C₂₃	C₂²×D₄	1	14	yes	—
95	C₂³×C₂₃	C₂²×D₄	1	14	yes	—
96	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
97	D₄×C₂₃	—	2	112	no	—
98	D₄×C₂₃	—	2	16	no	—
99	C₂×C₄×C₂₃	—	2	28	no	—
100	C₂×C₄×C₂₃	—	2	28	no	—
101	C₂×C₄×C₂₃	—	2	16	no	—
102	C₂×C₄×C₂₃	—	4	112	no	—
103	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	complemented
104	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
106	C₂²×#99	C₂×C₂₃	1	28	yes	direct product factor
107	C₂²×C₂₃	—	2	28	no	—
108	C₂²×C₂₃	—	2	42	no	—
109	C₂²×C₂₃	—	2	14	no	—
110	C₂²×C₂₃	—	2	84	no	—
111	C₂²×C₂₃	—	2	8	no	—
112	C₂²×C₂₃	—	4	42	no	—
113	C₂²×C₂₃	—	4	7	no	—
114	C₂²×C₂₃	—	4	14	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	—	4	8	no	—
117	C₂²×C₂₃	—	4	56	no	—
118	C₂²×C₂₃	—	4	56	no	—
119	C₂²×C₂₃	C₂×#99	1	28	yes	direct product factor
120	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	7	yes	—
121	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	14	yes	—
122	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	7	yes	—
123	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
124	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
125	C₄×C₂₃	—	2	8	no	—
126	C₄×C₂₃	—	4	16	no	—
127	C₂⁶	C₄×C₂₃	1	1	yes	characteristic, complemented
128	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
129	C₂³×D₄	—	2	2	no	—
130	C₂³×D₄	—	2	14	no	—
131	C₂³×D₄	C₄×C₂₃	1	14	yes	complemented
132	C₂³×D₄	C₂²×C₂₃	1	14	yes	—
133	C₂⁴×C₄	—	2	2	no	—
134	C₂⁴×C₄	C₄×C₂₃	1	1	yes	characteristic, complemented
135	C₂²×C₂²⋊C₄	—	2	14	no	—
136	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	28	yes	complemented
137	C₂×#99	C₂²×C₂₃	1	112	yes	direct product factor
138	C₂×C₂₃	—	2	2	no	—
139	C₂×C₂₃	—	2	7	no	—
140	C₂×C₂₃	—	2	1	no	—
141	C₂×C₂₃	—	2	14	no	—
142	C₂×C₂₃	—	4	8	no	—
143	C₂×C₂₃	C₂²×#99	1	14	yes	direct product factor
144	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
145	C₂⁵	—	2	7	no	—
146	C₂⁵	—	2	1	no	—
147	C₂⁵	—	4	8	no	—
148	C₂⁵	—	4	8	no	—
149	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂⁵	C₂×C₄×C₂₃	1	7	yes	—
151	C₂⁵	C₂×C₄×C₂₃	1	14	yes	complemented
152	C₂⁵	D₄×C₂₃	1	2	yes	—
153	C₂⁵	D₄×C₂₃	1	14	yes	—
154	C₂⁵	C₂³×C₂₃	1	7	yes	—
155	C₂²×D₄	—	2	28	no	—
156	C₂²×D₄	—	2	84	no	—
157	C₂²×D₄	—	2	28	no	—
158	C₂²×D₄	—	2	84	no	—
159	C₂²×D₄	C₂×C₄×C₂₃	1	84	yes	complemented
160	C₂²×D₄	C₂³×C₂₃	1	28	yes	—
161	C₂³×C₄	—	2	7	no	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	—	2	28	no	—
164	C₂³×C₄	—	4	16	no	—
165	C₂³×C₄	C₂×C₄×C₂₃	1	14	yes	complemented
166	C₂×C₂²⋊C₄	—	2	84	no	—
167	C₂×C₂²⋊C₄	C₂³×C₂₃	1	56	yes	complemented
168	#99	C₂³×C₂₃	1	64	yes	direct product factor
169	C₂₃	C₂³×#99	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	—	2	16	no	—
171	C₂⁴	—	2	8	no	—
172	C₂⁴	—	2	42	no	—
173	C₂⁴	—	2	14	no	—
174	C₂⁴	—	4	28	no	—
175	C₂⁴	—	4	56	no	—
176	C₂⁴	—	4	28	no	—
177	C₂⁴	—	4	56	no	—
178	C₂⁴	—	4	56	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	C₂²⋊C₄×C₂₃	1	7	yes	—
181	C₂⁴	C₂²⋊C₄×C₂₃	1	14	yes	—
182	C₂⁴	C₂²⋊C₄×C₂₃	1	42	yes	—
183	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
184	C₂⁴	C₂²×C₄×C₂₃	1	7	yes	—
185	C₂⁴	C₂²×C₄×C₂₃	1	21	yes	—
186	C₂⁴	C₂²×C₄×C₂₃	1	28	yes	complemented
187	C₂⁴	C₂×D₄×C₂₃	1	14	yes	—
188	C₂⁴	C₂×D₄×C₂₃	1	42	yes	—
189	C₂⁴	C₂⁴×C₂₃	1	7	yes	—
190	C₂×D₄	—	2	168	no	—
191	C₂×D₄	—	2	168	no	—
192	C₂×D₄	—	2	56	no	—
193	C₂×D₄	—	2	56	no	—
194	C₂×D₄	C₂²×C₄×C₂₃	1	56	yes	complemented
195	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
196	C₂²×C₄	—	2	42	no	—
197	C₂²×C₄	—	2	14	no	—
198	C₂²×C₄	—	2	56	no	—
199	C₂²×C₄	—	4	112	no	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	28	yes	complemented
201	C₂²⋊C₄	—	2	56	no	—
202	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	complemented
203	C₂³	—	2	56	no	—
204	C₂³	—	2	28	no	—
205	C₂³	—	2	28	no	—
206	C₂³	—	2	56	no	—
207	C₂³	—	2	28	no	—
208	C₂³	—	2	28	no	—
209	C₂³	—	4	84	no	—
210	C₂³	—	4	42	no	—
211	C₂³	—	4	84	no	—
212	C₂³	—	4	14	no	—
213	C₂³	—	4	56	no	—
214	C₂³	—	4	168	no	—
215	C₂³	—	4	56	no	—
216	C₂³	C₂₃×#99	1	8	yes	direct product factor
217	C₂³	C₂×C₂²⋊C₄×C₂₃	1	21	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₂₃	1	42	yes	—
219	C₂³	C₂×C₂²⋊C₄×C₂₃	1	42	yes	—
220	C₂³	C₂×C₂²⋊C₄×C₂₃	1	7	yes	—
221	C₂³	C₂³×C₄×C₂₃	1	7	yes	—
222	C₂³	C₂³×C₄×C₂₃	1	7	yes	—
223	C₂³	C₂³×C₄×C₂₃	1	8	yes	complemented
224	C₂³	C₂²×D₄×C₂₃	1	14	yes	—
225	C₂³	C₂²×D₄×C₂₃	1	14	yes	—
226	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
227	D₄	—	2	16	no	—
228	D₄	—	2	112	no	—
229	C₂×C₄	—	2	28	no	—
230	C₂×C₄	—	2	28	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	4	112	no	—
233	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	complemented
234	C₂²	—	2	28	no	—
235	C₂²	—	2	42	no	—
236	C₂²	—	2	14	no	—
237	C₂²	—	2	84	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	42	no	—
240	C₂²	—	4	7	no	—
241	C₂²	—	4	14	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	56	no	—
245	C₂²	—	4	56	no	—
246	C₂²	C₂×C₂₃×#99	1	28	yes	direct product factor
247	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	7	yes	—
248	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	14	yes	—
249	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	7	yes	—
250	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
251	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	4	16	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	7	no	—
257	C₂	—	2	14	no	—
258	C₂	—	4	8	no	—
259	C₂	C₂²×C₂₃×#99	1	14	yes	direct product factor
260	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂²⋊C₄
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁴×D₄×C₂₃	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂⁶×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#99	C₂	direct (N × H)
4. C₂⁶×C₂₃	C₄	semidirect
8. C₂³×D₄×C₂₃	C₄	semidirect
11. C₂⁴×C₄×C₂₃	C₄	semidirect
13. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
14. C₂×C₂₃×#99	C₂²	direct (N × H)
21. C₂⁵×C₂₃	C₂×C₄	semidirect
29. C₂²×D₄×C₂₃	C₂×C₄	semidirect
35. C₂³×C₄×C₂₃	C₂×C₄	semidirect
37. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
38. C₂₃×#99	C₂³	direct (N × H)
55. C₂⁴×C₂₃	C₂²×C₄	semidirect
63. C₂×D₄×C₂₃	C₂²×C₄	semidirect
69. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
71. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
72. C₂³×#99	C₂₃	direct (N × H)
86. C₂³×C₂₃	#99	direct (N × H)
93. C₂³×C₂₃	C₂³×C₄	semidirect
103. C₂×C₄×C₂₃	C₂³×C₄	semidirect
105. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
106. C₂²×#99	C₂×C₂₃	direct (N × H)
119. C₂²×C₂₃	C₂×#99	direct (N × H)
127. C₂⁶	C₄×C₂₃	semidirect
131. C₂³×D₄	C₄×C₂₃	semidirect
134. C₂⁴×C₄	C₄×C₂₃	semidirect
136. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
137. C₂×#99	C₂²×C₂₃	direct (N × H)
143. C₂×C₂₃	C₂²×#99	direct (N × H)
151. C₂⁵	C₂×C₄×C₂₃	semidirect
159. C₂²×D₄	C₂×C₄×C₂₃	semidirect
165. C₂³×C₄	C₂×C₄×C₂₃	semidirect
167. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
168. #99	C₂³×C₂₃	direct (N × H)
169. C₂₃	C₂³×#99	direct (N × H)
186. C₂⁴	C₂²×C₄×C₂₃	semidirect
194. C₂×D₄	C₂²×C₄×C₂₃	semidirect
200. C₂²×C₄	C₂²×C₄×C₂₃	semidirect
202. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
216. C₂³	C₂₃×#99	direct (N × H)
223. C₂³	C₂³×C₄×C₂₃	semidirect
233. C₂×C₄	C₂³×C₄×C₂₃	semidirect
246. C₂²	C₂×C₂₃×#99	direct (N × H)
259. C₂	C₂²×C₂₃×#99	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂²×C₂₃×#99	2
C₂³×#99	23

Subgroups — C23×C23×#99 #19207692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×#99 #19207692