Subgroups — C₂×C₂₃×#1631 #19166131

Order 5888.

1322 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1631	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₂₃	—	2	1	no	—
10	C₂²×SD₁₆×C₂₃	—	2	2	no	—
11	C₂₃×#550	—	2	2	no	—
12	C₂₃×#550	—	2	2	no	—
13	C₂₃×#550	—	2	2	no	—
14	C₂₃×#550	C₂²	1	4	yes	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#127	—	2	1	no	—
17	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#121	—	2	2	no	—
19	C₂×C₂₃×#121	—	2	1	no	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#492	—	2	2	no	—
24	C₂₃×#492	C₂²	1	4	yes	complemented
25	C₂₃×#477	—	2	2	no	—
26	C₂₃×#477	C₂²	1	4	yes	complemented
27	C₂₃×#476	—	2	2	no	—
28	C₂₃×#476	C₂²	1	4	yes	complemented
29	C₂×C₂₃×#107	—	2	1	no	—
30	C₂×C₂₃×#106	—	2	1	no	—
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	—	2	1	no	—
34	C₂×C₂₃×#102	—	2	1	no	—
35	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#331	C₂²	1	4	yes	complemented
37	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
38	C₂₃×D₄⋊C₈	C₂²	1	4	yes	complemented
39	C₂×#41×C₂₃	—	2	4	no	—
40	C₂×#41×C₂₃	—	2	4	no	—
41	C₂×#41×C₂₃	—	4	4	no	—
42	C₂×#41×C₂₃	—	4	4	no	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	—	2	2	no	—
46	C₂²×D₄×C₂₃	—	2	2	no	—
47	C₂²×D₄×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₂₃	D₄	1	2	yes	—
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	—	2	1	no	—
53	C₂×SD₁₆×C₂₃	—	2	8	no	—
54	C₂×SD₁₆×C₂₃	—	4	8	no	—
55	C₂²×C₈×C₂₃	—	2	1	no	—
56	C₂²×C₈×C₂₃	—	2	1	no	—
57	C₂²×C₈×C₂₃	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×#127	—	2	4	no	—
63	C₂₃×#127	—	2	4	no	—
64	C₂₃×#127	—	2	1	no	—
65	C₂₃×#127	—	2	1	no	—
66	C₂₃×#127	D₄	1	2	yes	—
67	C₂₃×#127	C₂³	1	2	yes	—
68	C₂₃×#121	—	2	4	no	—
69	C₂₃×#121	—	2	8	no	—
70	C₂₃×#121	—	2	8	no	—
71	C₂₃×#121	—	2	4	no	—
72	C₄×Q₈×C₂₃	—	2	1	no	—
73	C₄×Q₈×C₂₃	—	2	1	no	—
74	C₄×Q₈×C₂₃	D₄	1	2	yes	—
75	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
76	C₄×D₄×C₂₃	—	2	4	no	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	—	2	1	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	D₄	1	2	yes	—
81	C₄×D₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×#107	—	2	4	no	—
92	C₂₃×#106	—	2	4	no	—
93	C₂₃×C₄⋊C₈	—	2	1	no	—
94	C₂₃×C₄⋊C₈	—	2	1	no	—
95	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
96	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
97	C₂₃×#103	—	2	4	no	—
98	C₂₃×#102	—	2	4	no	—
99	C₄×C₈×C₂₃	—	2	1	no	—
100	C₄×C₈×C₂₃	C₂³	1	2	yes	—
101	C₂⁴×C₂₃	—	2	1	no	—
102	C₂⁴×C₂₃	—	2	2	no	—
103	#41×C₂₃	—	4	8	no	—
104	#41×C₂₃	—	4	16	no	—
105	#41×C₂₃	—	4	8	no	—
106	C₂×Q₈×C₂₃	—	2	1	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	4	4	no	—
110	C₂×Q₈×C₂₃	—	4	1	no	—
111	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	complemented
112	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
113	C₂×D₄×C₂₃	—	2	8	no	—
114	C₂×D₄×C₂₃	—	2	8	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	2	no	—
117	C₂×D₄×C₂₃	—	2	2	no	—
118	C₂×D₄×C₂₃	—	2	2	no	—
119	C₂×D₄×C₂₃	—	2	2	no	—
120	C₂×D₄×C₂₃	—	2	2	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	4	8	no	—
124	C₂×D₄×C₂₃	—	4	4	no	—
125	C₂×D₄×C₂₃	—	4	2	no	—
126	C₂×D₄×C₂₃	—	4	2	no	—
127	C₂×D₄×C₂₃	—	4	2	no	—
128	C₂×D₄×C₂₃	—	4	2	no	—
129	C₂×D₄×C₂₃	—	8	4	no	—
130	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	8	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	4	8	no	—
139	C₂²×C₄×C₂₃	—	4	2	no	—
140	C₂²×C₄×C₂₃	—	4	2	no	—
141	C₂²×C₄×C₂₃	—	4	1	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	SD₁₆×C₂₃	—	4	16	no	—
146	C₂×C₈×C₂₃	—	2	4	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	4	2	no	—
151	C₂×C₈×C₂₃	—	4	2	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	4	1	no	—
159	C₄⋊C₄×C₂₃	—	4	1	no	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₂₃	—	2	8	no	—
163	C₂²⋊C₄×C₂₃	—	2	2	no	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₄²×C₂₃	—	2	2	no	—
166	C₄²×C₂₃	—	2	2	no	—
167	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
171	C₂×#1631	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₂₃	—	2	2	no	—
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	—	4	4	no	—
175	C₂³×C₂₃	—	4	2	no	—
176	C₂³×C₂₃	—	4	2	no	—
177	C₂³×C₂₃	—	4	4	no	—
178	C₂³×C₂₃	—	4	4	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	#120	1	1	yes	characteristic
181	Q₈×C₂₃	—	2	2	no	—
182	Q₈×C₂₃	—	2	4	no	—
183	Q₈×C₂₃	—	4	4	no	—
184	Q₈×C₂₃	—	8	2	no	—
185	Q₈×C₂₃	C₂×SD₁₆	1	4	yes	complemented
186	D₄×C₂₃	—	2	2	no	—
187	D₄×C₂₃	—	2	2	no	—
188	D₄×C₂₃	—	4	4	no	—
189	D₄×C₂₃	—	4	16	no	—
190	D₄×C₂₃	—	4	2	no	—
191	D₄×C₂₃	—	4	2	no	—
192	D₄×C₂₃	—	4	2	no	—
193	D₄×C₂₃	—	4	2	no	—
194	D₄×C₂₃	—	4	4	no	—
195	D₄×C₂₃	—	4	2	no	—
196	D₄×C₂₃	—	4	4	no	—
197	D₄×C₂₃	—	8	2	no	—
198	D₄×C₂₃	—	8	8	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	4	16	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	4	2	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	8	2	no	—
212	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
213	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
214	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
216	C₂×C₄×C₂₃	#135	1	2	yes	—
217	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄×C₂₃	#137	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₂₃	—	2	4	no	—
223	C₈×C₂₃	—	4	2	no	—
224	C₈×C₂₃	—	4	2	no	—
225	C₂×#550	C₂×C₂₃	1	1	yes	characteristic
226	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#1631	C₂×C₂₃	1	8	yes	direct product factor
230	C₂×#331	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂²×C₂₃	—	4	4	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	—	4	2	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	4	no	—
238	C₂²×C₂₃	—	4	2	no	—
239	C₂²×C₂₃	—	8	4	no	—
240	C₂²×C₂₃	—	8	8	no	—
241	C₂²×C₂₃	#450	1	2	yes	—
242	C₂²×C₂₃	#452	1	2	yes	—
243	C₂²×C₂₃	#453	1	2	yes	—
244	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
245	C₄×C₂₃	—	2	4	no	—
246	C₄×C₂₃	—	2	1	no	—
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	4	2	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	—	8	2	no	—
251	C₄×C₂₃	#450	1	2	yes	—
252	C₄×C₂₃	#452	1	2	yes	—
253	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
254	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
255	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
257	C₂²×#41	—	2	1	no	—
258	C₂²×SD₁₆	—	2	2	no	—
259	#550	—	2	2	no	—
260	#550	—	2	2	no	—
261	#550	—	2	2	no	—
262	#550	C₂²×C₂₃	1	4	yes	—
263	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×#127	—	2	1	no	—
265	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#121	—	2	2	no	—
267	C₂×#121	—	2	1	no	—
268	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×C₄×D₄	—	2	1	no	—
270	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
271	#492	—	2	2	no	—
272	#492	C₂²×C₂₃	1	4	yes	complemented
273	#477	—	2	2	no	—
274	#477	C₂²×C₂₃	1	4	yes	complemented
275	#476	—	2	2	no	—
276	#476	C₂²×C₂₃	1	4	yes	complemented
277	C₂×#107	—	2	1	no	—
278	C₂×#106	—	2	1	no	—
279	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×#103	—	2	1	no	—
282	C₂×#102	—	2	1	no	—
283	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
284	#331	C₂²×C₂₃	1	4	yes	complemented
285	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
286	D₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
287	C₂×C₂₃	—	8	2	no	—
288	C₂×C₂₃	—	8	4	no	—
289	C₂×C₂₃	#1631	1	4	yes	direct product factor
290	C₂×C₂₃	C₂×#450	1	1	yes	characteristic
291	C₂×C₂₃	C₂×#452	1	1	yes	characteristic
292	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
293	C₂×#41	—	2	4	no	—
294	C₂×#41	—	2	4	no	—
295	C₂×#41	—	4	4	no	—
296	C₂×#41	—	4	4	no	—
297	C₂²×Q₈	—	2	1	no	—
298	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	D₄×C₂₃	1	2	yes	—
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	—	2	1	no	—
307	C₂×SD₁₆	—	2	8	no	—
308	C₂×SD₁₆	—	4	8	no	—
309	C₂²×C₈	—	2	1	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
316	#127	—	2	4	no	—
317	#127	—	2	4	no	—
318	#127	—	2	1	no	—
319	#127	—	2	1	no	—
320	#127	D₄×C₂₃	1	2	yes	—
321	#127	C₂³×C₂₃	1	2	yes	—
322	#121	—	2	4	no	—
323	#121	—	2	8	no	—
324	#121	—	2	8	no	—
325	#121	—	2	4	no	—
326	C₄×Q₈	—	2	1	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	D₄×C₂₃	1	2	yes	—
329	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	D₄×C₂₃	1	2	yes	—
335	C₄×D₄	C₂³×C₂₃	1	2	yes	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
340	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	1	no	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	#107	—	2	4	no	—
346	#106	—	2	4	no	—
347	C₄⋊C₈	—	2	1	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
350	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
351	#103	—	2	4	no	—
352	#102	—	2	4	no	—
353	C₄×C₈	—	2	1	no	—
354	C₄×C₈	C₂³×C₂₃	1	2	yes	—
355	C₂₃	C₂×#1631	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	2	no	—
358	#41	—	4	8	no	—
359	#41	—	4	16	no	—
360	#41	—	4	8	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	—	4	1	no	—
366	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	complemented
367	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	4	8	no	—
379	C₂×D₄	—	4	4	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	8	4	no	—
385	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
386	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	4	8	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	SD₁₆	—	4	16	no	—
401	C₂×C₈	—	2	4	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	4	2	no	—
406	C₂×C₈	—	4	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	4	1	no	—
414	C₄⋊C₄	—	4	1	no	—
415	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
416	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	4	4	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	C₂₃×#120	1	1	yes	characteristic
435	Q₈	—	2	2	no	—
436	Q₈	—	2	4	no	—
437	Q₈	—	4	4	no	—
438	Q₈	—	8	2	no	—
439	Q₈	C₂×SD₁₆×C₂₃	1	4	yes	complemented
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	4	no	—
443	D₄	—	4	16	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	4	no	—
451	D₄	—	8	2	no	—
452	D₄	—	8	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
467	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
468	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
469	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
470	C₂×C₄	C₂₃×#135	1	2	yes	—
471	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
472	C₂×C₄	C₂₃×#137	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	8	4	no	—
486	C₂²	—	8	8	no	—
487	C₂²	C₂₃×#450	1	2	yes	—
488	C₂²	C₂₃×#452	1	2	yes	—
489	C₂²	C₂₃×#453	1	2	yes	—
490	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	8	2	no	—
497	C₄	C₂₃×#450	1	2	yes	—
498	C₄	C₂₃×#452	1	2	yes	—
499	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
500	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
501	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
502	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
503	C₂	—	8	2	no	—
504	C₂	—	8	4	no	—
505	C₂	C₂₃×#1631	1	4	yes	direct product factor
506	C₂	C₂×C₂₃×#450	1	1	yes	characteristic
507	C₂	C₂×C₂₃×#452	1	1	yes	characteristic
508	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#492	C₂	semidirect
3. C₂×C₂₃×#477	C₂	semidirect
4. C₂×C₂₃×#476	C₂	semidirect
5. C₂₃×#1631	C₂	direct (N × H)
6. C₂×C₂₃×#331	C₂	semidirect
7. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
8. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
24. C₂₃×#492	C₂²	semidirect
26. C₂₃×#477	C₂²	semidirect
28. C₂₃×#476	C₂²	semidirect
36. C₂₃×#331	C₂²	semidirect
37. C₂₃×Q₈⋊C₈	C₂²	semidirect
38. C₂₃×D₄⋊C₈	C₂²	semidirect
111. C₂×Q₈×C₂₃	SD₁₆	semidirect
171. C₂×#1631	C₂₃	direct (N × H)
185. Q₈×C₂₃	C₂×SD₁₆	semidirect
226. C₂×#492	C₂×C₂₃	semidirect
227. C₂×#477	C₂×C₂₃	semidirect
228. C₂×#476	C₂×C₂₃	semidirect
229. #1631	C₂×C₂₃	direct (N × H)
230. C₂×#331	C₂×C₂₃	semidirect
231. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
232. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
272. #492	C₂²×C₂₃	semidirect
274. #477	C₂²×C₂₃	semidirect
276. #476	C₂²×C₂₃	semidirect
284. #331	C₂²×C₂₃	semidirect
285. Q₈⋊C₈	C₂²×C₂₃	semidirect
286. D₄⋊C₈	C₂²×C₂₃	semidirect
289. C₂×C₂₃	#1631	direct (N × H)
355. C₂₃	C₂×#1631	direct (N × H)
366. C₂×Q₈	SD₁₆×C₂₃	semidirect
439. Q₈	C₂×SD₁₆×C₂₃	semidirect
505. C₂	C₂₃×#1631	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×#550	2
C₂×C₂₃×#492	2
C₂×C₂₃×#477	2
C₂×C₂₃×#476	2
C₂₃×#1631	2
C₂×C₂₃×#331	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×C₂₃×D₄⋊C₈	2
C₂×#1631	23

Subgroups — C2×C23×#1631 #19166131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1631 #19166131