Subgroups — C₂×C₂₃×#2298 #19181651

Order 5888.

2902 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	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2298	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#388	C₂	1	4	yes	maximal
4	C₂×C₂₃×#387	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
7	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₄	1	16	yes	complemented
11	C₂×C₂₃×#117	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
17	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#388	C₂²	1	32	yes	—
22	C₂₃×#387	C₂²	1	32	yes	complemented
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	16	yes	complemented
24	C₄×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
25	C₂₃×#376	C₂²	1	8	yes	complemented
26	C₂×C₂₃×#95	C₂²	1	4	yes	—
27	C₂×C₂₃×#95	C₂²	1	4	yes	—
28	C₂×C₂₃×#95	C₂²	1	4	yes	—
29	C₄³×C₂₃	C₂²	1	8	yes	complemented
30	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₂₃	—	2	8	no	—
32	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#117	C₂×C₄	1	64	yes	complemented
41	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
46	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
47	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
48	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
49	C₂×C₄²×C₂₃	—	2	16	no	—
50	C₂×C₄²×C₂₃	—	2	8	no	—
51	C₂×C₄²×C₂₃	—	2	8	no	—
52	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
53	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
55	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
56	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
57	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
59	C₂₃×#95	C₂³	1	16	yes	—
60	C₂₃×#95	C₂³	1	16	yes	—
61	C₂₃×#95	C₂³	1	16	yes	—
62	C₂⁴×C₂₃	—	2	8	no	—
63	C₂⁴×C₂₃	—	2	4	no	—
64	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
66	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	—	2	8	no	—
68	C₂²×C₄×C₂₃	—	2	8	no	—
69	C₂²×C₄×C₂₃	—	2	8	no	—
70	C₂²×C₄×C₂₃	—	2	8	no	—
71	C₂²×C₄×C₂₃	—	2	4	no	—
72	C₂²×C₄×C₂₃	—	2	32	no	—
73	C₂²×C₄×C₂₃	—	2	16	no	—
74	C₂²×C₄×C₂₃	—	2	8	no	—
75	C₂²×C₄×C₂₃	—	2	8	no	—
76	C₂²×C₄×C₂₃	—	2	8	no	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
81	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₂₃	#41	1	16	yes	—
83	C₂²×C₄×C₂₃	#41	1	8	yes	—
84	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
90	C₄⋊C₄×C₂₃	—	2	32	no	—
91	C₄⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
92	C₂²⋊C₄×C₂₃	—	2	32	no	—
93	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
94	C₄²×C₂₃	—	2	32	no	—
95	C₄²×C₂₃	—	2	16	no	—
96	C₄²×C₂₃	—	2	16	no	—
97	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
98	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
99	C₂×#2298	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	8	no	—
101	C₂³×C₂₃	—	2	16	no	—
102	C₂³×C₂₃	—	2	8	no	—
103	C₂³×C₂₃	—	2	4	no	—
104	C₂³×C₂₃	—	4	16	no	—
105	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
106	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
107	C₂³×C₂₃	C₂×#41	1	4	yes	—
108	C₂³×C₂₃	C₂×#41	1	8	yes	—
109	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
110	C₂×C₄×C₂₃	—	2	8	no	—
111	C₂×C₄×C₂₃	—	2	16	no	—
112	C₂×C₄×C₂₃	—	2	16	no	—
113	C₂×C₄×C₂₃	—	2	32	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	2	8	no	—
116	C₂×C₄×C₂₃	—	2	8	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	32	no	—
120	C₂×C₄×C₂₃	—	2	8	no	—
121	C₂×C₄×C₂₃	—	2	8	no	—
122	C₂×C₄×C₂₃	—	2	8	no	—
123	C₂×C₄×C₂₃	—	4	16	no	—
124	C₂×C₄×C₂₃	#117	1	16	yes	—
125	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
126	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
127	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
128	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
129	C₂×C₄×C₂₃	C₂×#41	1	16	yes	—
130	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
131	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic
132	#2298	C₂×C₂₃	1	16	yes	direct product factor
133	C₂×#388	C₂×C₂₃	1	4	yes	—
134	C₂×#387	C₂×C₂₃	1	4	yes	complemented
135	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
136	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
137	C₂×#376	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂²×C₂₃	—	2	4	no	—
140	C₂²×C₂₃	—	2	8	no	—
141	C₂²×C₂₃	—	4	8	no	—
142	C₂²×C₂₃	—	4	16	no	—
143	C₂²×C₂₃	C₂×#117	1	4	yes	—
144	C₂²×C₂₃	C₄×#41	1	8	yes	—
145	C₂²×C₂₃	#529	1	16	yes	—
146	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
147	C₂²×C₂₃	C₂²×#41	1	2	yes	—
148	C₂²×C₂₃	C₂²×#41	1	4	yes	—
149	C₄×C₂₃	—	2	4	no	—
150	C₄×C₂₃	—	2	8	no	—
151	C₄×C₂₃	—	2	8	no	—
152	C₄×C₂₃	—	2	16	no	—
153	C₄×C₂₃	—	4	8	no	—
154	C₄×C₂₃	C₂×#117	1	8	yes	—
155	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×#117	C₄×C₂₃	1	16	yes	complemented
157	C₂×#117	C₂²×C₂₃	1	8	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
162	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
163	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
164	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
165	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
166	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
167	#388	C₂²×C₂₃	1	32	yes	—
168	#387	C₂²×C₂₃	1	32	yes	complemented
169	C₄×C₄⋊C₄	C₂²×C₂₃	1	16	yes	complemented
170	C₄×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
171	#376	C₂²×C₂₃	1	8	yes	complemented
172	C₂×#95	C₂²×C₂₃	1	4	yes	—
173	C₂×#95	C₂²×C₂₃	1	4	yes	—
174	C₂×#95	C₂²×C₂₃	1	4	yes	—
175	C₄³	C₂²×C₂₃	1	8	yes	complemented
176	C₂×C₂₃	—	4	4	no	—
177	C₂×C₂₃	#2298	1	8	yes	direct product factor
178	C₂×C₂₃	C₂²×#117	1	1	yes	characteristic
179	C₂×C₂₃	C₂×C₄×#41	1	2	yes	—
180	C₂×C₂₃	C₂×#529	1	4	yes	—
181	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
184	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
185	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
187	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
189	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
190	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
191	#117	C₂×C₄×C₂₃	1	64	yes	complemented
192	C₂×C₄⋊C₄	—	2	16	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
194	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
195	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
196	C₂×C₂²⋊C₄	—	2	16	no	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
199	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
200	C₂×C₄²	—	2	16	no	—
201	C₂×C₄²	—	2	8	no	—
202	C₂×C₄²	—	2	8	no	—
203	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
204	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
205	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
206	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
207	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
208	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
209	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
210	#95	C₂³×C₂₃	1	16	yes	—
211	#95	C₂³×C₂₃	1	16	yes	—
212	#95	C₂³×C₂₃	1	16	yes	—
213	C₂₃	C₂×#2298	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	—	2	8	no	—
215	C₂⁴	—	2	4	no	—
216	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
217	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
218	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	32	no	—
225	C₂²×C₄	—	2	16	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
230	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
231	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
233	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
234	C₂²×C₄	#41×C₂₃	1	16	yes	—
235	C₂²×C₄	#41×C₂₃	1	8	yes	—
236	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
237	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
241	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
242	C₄⋊C₄	—	2	32	no	—
243	C₄⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
244	C₂²⋊C₄	—	2	32	no	—
245	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
246	C₄²	—	2	32	no	—
247	C₄²	—	2	16	no	—
248	C₄²	—	2	16	no	—
249	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
250	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	8	no	—
254	C₂³	—	2	16	no	—
255	C₂³	—	4	16	no	—
256	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
257	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
258	C₂³	C₂×#41×C₂₃	1	4	yes	—
259	C₂³	C₂×#41×C₂₃	1	8	yes	—
260	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	16	no	—
268	C₂×C₄	—	2	16	no	—
269	C₂×C₄	—	2	32	no	—
270	C₂×C₄	—	2	32	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	C₂₃×#117	1	16	yes	—
276	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
278	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
279	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
280	C₂×C₄	C₂×#41×C₂₃	1	16	yes	—
281	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	4	8	no	—
285	C₂²	—	4	16	no	—
286	C₂²	C₂×C₂₃×#117	1	4	yes	—
287	C₂²	C₄×#41×C₂₃	1	8	yes	—
288	C₂²	C₂₃×#529	1	16	yes	—
289	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
290	C₂²	C₂²×#41×C₂₃	1	2	yes	—
291	C₂²	C₂²×#41×C₂₃	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	8	no	—
294	C₄	—	2	8	no	—
295	C₄	—	2	16	no	—
296	C₄	—	4	8	no	—
297	C₄	C₂×C₂₃×#117	1	8	yes	—
298	C₂	—	4	4	no	—
299	C₂	C₂₃×#2298	1	8	yes	direct product factor
300	C₂	C₂²×C₂₃×#117	1	1	yes	characteristic
301	C₂	C₂×C₄×#41×C₂₃	1	2	yes	—
302	C₂	C₂×C₂₃×#529	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2298	C₂	direct (N × H)
4. C₂×C₂₃×#387	C₂	semidirect
5. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
7. C₂×C₂₃×#376	C₂	semidirect
8. C₂×C₄³×C₂₃	C₂	semidirect
10. C₂×C₂₃×#117	C₄	semidirect
22. C₂₃×#387	C₂²	semidirect
23. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
25. C₂₃×#376	C₂²	semidirect
29. C₄³×C₂₃	C₂²	semidirect
40. C₂₃×#117	C₂×C₄	semidirect
99. C₂×#2298	C₂₃	direct (N × H)
132. #2298	C₂×C₂₃	direct (N × H)
134. C₂×#387	C₂×C₂₃	semidirect
135. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
137. C₂×#376	C₂×C₂₃	semidirect
138. C₂×C₄³	C₂×C₂₃	semidirect
156. C₂×#117	C₄×C₂₃	semidirect
168. #387	C₂²×C₂₃	semidirect
169. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
171. #376	C₂²×C₂₃	semidirect
175. C₄³	C₂²×C₂₃	semidirect
177. C₂×C₂₃	#2298	direct (N × H)
191. #117	C₂×C₄×C₂₃	semidirect
213. C₂₃	C₂×#2298	direct (N × H)
299. C₂	C₂₃×#2298	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#117	2
C₂₃×#2298	2
C₂×C₂₃×#388	2
C₂×C₂₃×#387	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#376	2
C₂×C₄³×C₂₃	2
C₂×#2298	23

Subgroups — C2×C23×#2298 #19181651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2298 #19181651