Subgroups — C₂²×C₂₃×#397 #19182163

Order 5888.

5230 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂³×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#397	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₂₃	—	2	8	no	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	complemented
14	C₂²×C₂²⋊C₄×C₂₃	—	2	16	no	—
15	C₂²×C₂²⋊C₄×C₂₃	—	2	8	no	—
16	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	24	yes	—
17	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
18	C₂₃×#397	C₂²	1	64	yes	direct product factor
19	C₂×C₂₃×#95	C₂²	1	24	yes	complemented
20	C₂×C₂₃×#95	C₂²	1	12	yes	complemented
21	C₂⁵×C₂₃	—	2	4	no	—
22	C₂⁵×C₂₃	—	2	4	no	—
23	C₂⁵×C₂₃	—	2	8	no	—
24	C₂⁵×C₂₃	—	2	8	no	—
25	C₂⁵×C₂₃	D₄	1	8	yes	—
26	C₂⁵×C₂₃	C₂³	1	6	yes	—
27	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	—	2	8	no	—
29	C₂³×C₄×C₂₃	—	2	8	no	—
30	C₂³×C₄×C₂₃	—	2	8	no	—
31	C₂³×C₄×C₂₃	—	2	4	no	—
32	C₂³×C₄×C₂₃	—	2	8	no	—
33	C₂³×C₄×C₂₃	—	2	8	no	—
34	C₂³×C₄×C₂₃	—	2	8	no	—
35	C₂³×C₄×C₂₃	—	2	8	no	—
36	C₂³×C₄×C₂₃	—	2	8	no	—
37	C₂³×C₄×C₂₃	D₄	1	8	yes	—
38	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
39	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
40	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
41	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	complemented
44	C₂×C₂²⋊C₄×C₂₃	—	2	96	no	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	48	no	—
46	C₂×C₂²⋊C₄×C₂₃	C₂³	1	32	yes	—
47	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
48	C₂₃×#95	C₂³	1	32	yes	complemented
49	C₂₃×#95	C₂³	1	16	yes	complemented
50	C₂⁴×C₂₃	—	2	32	no	—
51	C₂⁴×C₂₃	—	2	64	no	—
52	C₂⁴×C₂₃	—	2	12	no	—
53	C₂⁴×C₂₃	—	2	12	no	—
54	C₂⁴×C₂₃	—	2	24	no	—
55	C₂⁴×C₂₃	—	2	24	no	—
56	C₂⁴×C₂₃	—	4	16	no	—
57	C₂⁴×C₂₃	—	4	16	no	—
58	C₂⁴×C₂₃	—	4	32	no	—
59	C₂⁴×C₂₃	C₂×D₄	1	24	yes	—
60	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
61	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
62	C₂⁴×C₂₃	#41	1	4	yes	—
63	C₂⁴×C₂₃	#41	1	8	yes	—
64	C₂⁴×C₂₃	#41	1	8	yes	—
65	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
66	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
67	C₂²×C₄×C₂₃	—	2	32	no	—
68	C₂²×C₄×C₂₃	—	2	24	no	—
69	C₂²×C₄×C₂₃	—	2	24	no	—
70	C₂²×C₄×C₂₃	—	2	24	no	—
71	C₂²×C₄×C₂₃	—	2	12	no	—
72	C₂²×C₄×C₂₃	—	2	24	no	—
73	C₂²×C₄×C₂₃	—	2	24	no	—
74	C₂²×C₄×C₂₃	—	2	24	no	—
75	C₂²×C₄×C₂₃	—	2	24	no	—
76	C₂²×C₄×C₂₃	—	2	24	no	—
77	C₂²×C₄×C₂₃	—	4	32	no	—
78	C₂²×C₄×C₂₃	—	4	32	no	—
79	C₂²×C₄×C₂₃	—	4	16	no	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
81	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
85	C₄⋊C₄×C₂₃	—	2	32	no	—
86	C₂²⋊C₄×C₂₃	—	2	64	no	—
87	C₂²⋊C₄×C₂₃	—	2	32	no	—
88	C₂²×#397	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	2	48	no	—
90	C₂³×C₂₃	—	2	96	no	—
91	C₂³×C₂₃	—	2	4	no	—
92	C₂³×C₂₃	—	2	4	no	—
93	C₂³×C₂₃	—	2	8	no	—
94	C₂³×C₂₃	—	2	8	no	—
95	C₂³×C₂₃	—	4	24	no	—
96	C₂³×C₂₃	—	4	24	no	—
97	C₂³×C₂₃	—	4	48	no	—
98	C₂³×C₂₃	—	4	128	no	—
99	C₂³×C₂₃	#121	1	16	yes	—
100	C₂³×C₂₃	#123	1	16	yes	—
101	C₂³×C₂₃	#123	1	32	yes	—
102	C₂³×C₂₃	#124	1	16	yes	—
103	C₂³×C₂₃	#126	1	32	yes	—
104	C₂³×C₂₃	C₂²×D₄	1	8	yes	—
105	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
106	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
107	C₂³×C₂₃	C₂×#41	1	6	yes	—
108	C₂³×C₂₃	C₂×#41	1	12	yes	—
109	C₂³×C₂₃	C₂×#41	1	12	yes	—
110	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
111	C₂×C₄×C₂₃	—	2	48	no	—
112	C₂×C₄×C₂₃	—	2	8	no	—
113	C₂×C₄×C₂₃	—	2	8	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	8	no	—
117	C₂×C₄×C₂₃	—	2	8	no	—
118	C₂×C₄×C₂₃	—	2	8	no	—
119	C₂×C₄×C₂₃	—	2	8	no	—
120	C₂×C₄×C₂₃	—	2	8	no	—
121	C₂×C₄×C₂₃	—	4	48	no	—
122	C₂×C₄×C₂₃	—	4	48	no	—
123	C₂×C₄×C₂₃	—	4	24	no	—
124	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
125	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
127	C₂³×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
128	C₂×#397	C₂×C₂₃	1	24	yes	direct product factor
129	C₂²×#95	C₂×C₂₃	1	2	yes	complemented
130	C₂²×#95	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₂²×C₂₃	—	2	8	no	—
132	C₂²×C₂₃	—	2	16	no	—
133	C₂²×C₂₃	—	4	96	no	—
134	C₂²×C₂₃	—	4	4	no	—
135	C₂²×C₂₃	—	4	4	no	—
136	C₂²×C₂₃	—	4	8	no	—
137	C₂²×C₂₃	#397	1	64	yes	direct product factor
138	C₂²×C₂₃	C₂×#121	1	12	yes	—
139	C₂²×C₂₃	C₂×#123	1	12	yes	—
140	C₂²×C₂₃	C₂×#123	1	24	yes	—
141	C₂²×C₂₃	C₂×#124	1	12	yes	—
142	C₂²×C₂₃	C₂×#126	1	24	yes	—
143	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
144	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
145	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
146	C₂²×C₂₃	C₂²×#41	1	2	yes	—
147	C₂²×C₂₃	C₂²×#41	1	2	yes	—
148	C₄×C₂₃	—	2	8	no	—
149	C₄×C₂₃	—	4	8	no	—
150	C₄×C₂₃	—	4	8	no	—
151	C₄×C₂₃	—	4	4	no	—
152	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
153	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
154	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
155	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	—	2	8	no	—
158	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	complemented
159	C₂²×C₂²⋊C₄	—	2	16	no	—
160	C₂²×C₂²⋊C₄	—	2	8	no	—
161	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	24	yes	—
162	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
163	#397	C₂²×C₂₃	1	64	yes	direct product factor
164	C₂×#95	C₂²×C₂₃	1	24	yes	complemented
165	C₂×#95	C₂²×C₂₃	1	12	yes	complemented
166	C₂×C₂₃	—	4	8	no	—
167	C₂×C₂₃	C₂×#397	1	24	yes	direct product factor
168	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
169	C₂×C₂₃	C₂²×#123	1	1	yes	characteristic
170	C₂×C₂₃	C₂²×#123	1	2	yes	—
171	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
172	C₂×C₂₃	C₂²×#126	1	2	yes	—
173	C₂⁵	—	2	4	no	—
174	C₂⁵	—	2	4	no	—
175	C₂⁵	—	2	8	no	—
176	C₂⁵	—	2	8	no	—
177	C₂⁵	D₄×C₂₃	1	8	yes	—
178	C₂⁵	C₂³×C₂₃	1	6	yes	—
179	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
180	C₂³×C₄	—	2	8	no	—
181	C₂³×C₄	—	2	8	no	—
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	—	2	8	no	—
187	C₂³×C₄	—	2	8	no	—
188	C₂³×C₄	—	2	8	no	—
189	C₂³×C₄	D₄×C₂₃	1	8	yes	—
190	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
191	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
192	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
193	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
194	C₂×C₄⋊C₄	—	2	48	no	—
195	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	complemented
196	C₂×C₂²⋊C₄	—	2	96	no	—
197	C₂×C₂²⋊C₄	—	2	48	no	—
198	C₂×C₂²⋊C₄	C₂³×C₂₃	1	32	yes	—
199	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
200	#95	C₂³×C₂₃	1	32	yes	complemented
201	#95	C₂³×C₂₃	1	16	yes	complemented
202	C₂₃	C₂²×#397	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	—	2	32	no	—
204	C₂⁴	—	2	64	no	—
205	C₂⁴	—	2	12	no	—
206	C₂⁴	—	2	12	no	—
207	C₂⁴	—	2	24	no	—
208	C₂⁴	—	2	24	no	—
209	C₂⁴	—	4	16	no	—
210	C₂⁴	—	4	16	no	—
211	C₂⁴	—	4	32	no	—
212	C₂⁴	C₂×D₄×C₂₃	1	24	yes	—
213	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
214	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
215	C₂⁴	#41×C₂₃	1	4	yes	—
216	C₂⁴	#41×C₂₃	1	8	yes	—
217	C₂⁴	#41×C₂₃	1	8	yes	—
218	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
219	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
220	C₂²×C₄	—	2	32	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	24	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	—	2	24	no	—
226	C₂²×C₄	—	2	24	no	—
227	C₂²×C₄	—	2	24	no	—
228	C₂²×C₄	—	2	24	no	—
229	C₂²×C₄	—	2	24	no	—
230	C₂²×C₄	—	4	32	no	—
231	C₂²×C₄	—	4	32	no	—
232	C₂²×C₄	—	4	16	no	—
233	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
234	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
235	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
237	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
238	C₄⋊C₄	—	2	32	no	—
239	C₂²⋊C₄	—	2	64	no	—
240	C₂²⋊C₄	—	2	32	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	2	8	no	—
245	C₂³	—	2	48	no	—
246	C₂³	—	2	96	no	—
247	C₂³	—	4	128	no	—
248	C₂³	—	4	24	no	—
249	C₂³	—	4	24	no	—
250	C₂³	—	4	48	no	—
251	C₂³	C₂₃×#121	1	16	yes	—
252	C₂³	C₂₃×#123	1	16	yes	—
253	C₂³	C₂₃×#123	1	32	yes	—
254	C₂³	C₂₃×#124	1	16	yes	—
255	C₂³	C₂₃×#126	1	32	yes	—
256	C₂³	C₂²×D₄×C₂₃	1	8	yes	—
257	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
258	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
259	C₂³	C₂×#41×C₂₃	1	6	yes	—
260	C₂³	C₂×#41×C₂₃	1	12	yes	—
261	C₂³	C₂×#41×C₂₃	1	12	yes	—
262	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	48	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	4	48	no	—
274	C₂×C₄	—	4	48	no	—
275	C₂×C₄	—	4	24	no	—
276	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	96	no	—
283	C₂²	C₂₃×#397	1	64	yes	direct product factor
284	C₂²	C₂×C₂₃×#121	1	12	yes	—
285	C₂²	C₂×C₂₃×#123	1	12	yes	—
286	C₂²	C₂×C₂₃×#123	1	24	yes	—
287	C₂²	C₂×C₂₃×#124	1	12	yes	—
288	C₂²	C₂×C₂₃×#126	1	24	yes	—
289	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
290	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
291	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
292	C₂²	C₂²×#41×C₂₃	1	2	yes	—
293	C₂²	C₂²×#41×C₂₃	1	2	yes	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	4	no	—
298	C₂	—	4	8	no	—
299	C₂	C₂×C₂₃×#397	1	24	yes	direct product factor
300	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
301	C₂	C₂²×C₂₃×#123	1	1	yes	characteristic
302	C₂	C₂²×C₂₃×#123	1	2	yes	—
303	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
304	C₂	C₂²×C₂₃×#126	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
4. C₂×C₂₃×#397	C₂	direct (N × H)
5. C₂²×C₂₃×#95	C₂	semidirect
6. C₂²×C₂₃×#95	C₂	semidirect
13. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
18. C₂₃×#397	C₂²	direct (N × H)
19. C₂×C₂₃×#95	C₂²	semidirect
20. C₂×C₂₃×#95	C₂²	semidirect
43. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
48. C₂₃×#95	C₂³	semidirect
49. C₂₃×#95	C₂³	semidirect
88. C₂²×#397	C₂₃	direct (N × H)
125. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
128. C₂×#397	C₂×C₂₃	direct (N × H)
129. C₂²×#95	C₂×C₂₃	semidirect
130. C₂²×#95	C₂×C₂₃	semidirect
137. C₂²×C₂₃	#397	direct (N × H)
158. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
163. #397	C₂²×C₂₃	direct (N × H)
164. C₂×#95	C₂²×C₂₃	semidirect
165. C₂×#95	C₂²×C₂₃	semidirect
167. C₂×C₂₃	C₂×#397	direct (N × H)
195. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
200. #95	C₂³×C₂₃	semidirect
201. #95	C₂³×C₂₃	semidirect
202. C₂₃	C₂²×#397	direct (N × H)
283. C₂²	C₂₃×#397	direct (N × H)
299. C₂	C₂×C₂₃×#397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#397	2
C₂²×C₂₃×#95	2
C₂²×#397	23

Subgroups — C22×C23×#397 #19182163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#397 #19182163