Subgroups — C₂×C₂₃×#1626 #19166116

Order 5888.

1030 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1626	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₂₃	—	2	1	no	—
10	C₂²×SD₁₆×C₂₃	—	2	1	no	—
11	C₂₃×#557	—	2	2	no	—
12	C₂₃×#557	—	2	2	no	—
13	C₂₃×#557	—	2	2	no	—
14	C₂₃×#557	C₂²	1	4	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#125	—	2	2	no	—
19	C₂×C₂₃×#125	—	2	1	no	—
20	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#486	—	2	2	no	—
25	C₂₃×#486	C₂²	1	4	yes	—
26	C₂₃×#460	—	2	2	no	—
27	C₂₃×#460	—	2	2	no	—
28	C₂₃×#460	C₂²	1	4	yes	—
29	C₂₃×#460	C₂²	1	4	yes	—
30	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	—	2	1	no	—
33	C₂×C₂₃×#102	—	2	1	no	—
34	C₂×C₂₃×#102	—	2	2	no	—
35	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#332	C₂²	1	4	yes	complemented
37	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
38	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
39	C₂²×Q₈×C₂₃	—	2	1	no	—
40	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₂₃	—	2	1	no	—
43	C₂²×D₄×C₂₃	—	2	1	no	—
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂×SD₁₆×C₂₃	—	2	4	no	—
46	C₂×SD₁₆×C₂₃	—	2	4	no	—
47	C₂×SD₁₆×C₂₃	—	4	4	no	—
48	C₂×SD₁₆×C₂₃	—	4	4	no	—
49	C₂²×C₈×C₂₃	—	2	1	no	—
50	C₂²×C₈×C₂₃	—	2	1	no	—
51	C₂²×C₈×C₂₃	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	—	2	4	no	—
53	C₂₃×C₄⋊Q₈	—	2	4	no	—
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
57	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₂₃×#127	—	2	1	no	—
59	C₂₃×#127	—	2	1	no	—
60	C₂₃×#127	—	2	1	no	—
61	C₂₃×#127	C₂³	1	2	yes	—
62	C₂₃×#125	—	2	8	no	—
63	C₂₃×#125	—	2	4	no	—
64	C₂₃×#125	—	2	8	no	—
65	C₂₃×#125	—	2	4	no	—
66	C₄×Q₈×C₂₃	—	2	4	no	—
67	C₄×Q₈×C₂₃	—	2	4	no	—
68	C₄×Q₈×C₂₃	—	2	4	no	—
69	C₄×Q₈×C₂₃	—	2	4	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	—	2	1	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	D₄	1	2	yes	—
76	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
77	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
87	C₂×C₄²×C₂₃	—	2	2	no	—
88	C₂×C₄²×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊C₈	—	2	1	no	—
92	C₂₃×C₄⋊C₈	—	2	1	no	—
93	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
94	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
95	C₂₃×#102	—	2	4	no	—
96	C₂₃×#102	—	2	4	no	—
97	C₂₃×#102	—	2	8	no	—
98	C₄×C₈×C₂₃	—	2	1	no	—
99	C₄×C₈×C₂₃	C₂³	1	2	yes	—
100	C₂⁴×C₂₃	—	4	1	no	—
101	C₂×Q₈×C₂₃	—	2	1	no	—
102	C₂×Q₈×C₂₃	—	2	1	no	—
103	C₂×Q₈×C₂₃	—	2	2	no	—
104	C₂×Q₈×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	—	2	2	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	4	2	no	—
109	C₂×Q₈×C₂₃	—	4	2	no	—
110	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	complemented
111	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
112	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
113	C₂×D₄×C₂₃	—	2	2	no	—
114	C₂×D₄×C₂₃	—	2	2	no	—
115	C₂×D₄×C₂₃	—	2	2	no	—
116	C₂×D₄×C₂₃	—	4	4	no	—
117	C₂×D₄×C₂₃	—	4	1	no	—
118	C₂×D₄×C₂₃	—	4	1	no	—
119	C₂×D₄×C₂₃	—	4	1	no	—
120	C₂×D₄×C₂₃	—	4	2	no	—
121	C₂×D₄×C₂₃	—	4	2	no	—
122	C₂×D₄×C₂₃	—	4	2	no	—
123	C₂×D₄×C₂₃	—	4	2	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	SD₁₆×C₂₃	—	4	8	no	—
133	SD₁₆×C₂₃	—	4	8	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	4	2	no	—
138	C₂×C₈×C₂₃	—	4	2	no	—
139	C₂×C₈×C₂₃	—	4	2	no	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	8	no	—
142	C₄⋊C₄×C₂₃	—	2	8	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	2	1	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
154	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
155	C₄²×C₂₃	—	2	8	no	—
156	C₄²×C₂₃	—	2	2	no	—
157	C₄²×C₂₃	—	2	2	no	—
158	C₄²×C₂₃	—	2	2	no	—
159	C₄²×C₂₃	—	2	2	no	—
160	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
164	C₂×#1626	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₂₃	—	4	2	no	—
166	C₂³×C₂₃	—	8	2	no	—
167	C₂³×C₂₃	—	8	2	no	—
168	C₂³×C₂₃	—	8	2	no	—
169	C₂³×C₂₃	#120	1	1	yes	characteristic
170	Q₈×C₂₃	—	2	2	no	—
171	Q₈×C₂₃	—	2	4	no	—
172	Q₈×C₂₃	—	2	2	no	—
173	Q₈×C₂₃	—	2	2	no	—
174	Q₈×C₂₃	—	4	2	no	—
175	Q₈×C₂₃	—	4	2	no	—
176	Q₈×C₂₃	—	4	2	no	—
177	Q₈×C₂₃	—	4	4	no	—
178	Q₈×C₂₃	C₂×SD₁₆	1	4	yes	complemented
179	D₄×C₂₃	—	4	4	no	—
180	D₄×C₂₃	—	4	2	no	—
181	D₄×C₂₃	—	4	2	no	—
182	D₄×C₂₃	—	4	2	no	—
183	D₄×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	8	2	no	—
185	D₄×C₂₃	—	8	2	no	—
186	D₄×C₂₃	—	8	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	4	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
199	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
202	C₂×C₄×C₂₃	#135	1	2	yes	—
203	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄×C₂₃	#137	1	2	yes	—
205	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₈×C₂₃	—	4	2	no	—
209	C₈×C₂₃	—	4	2	no	—
210	C₈×C₂₃	—	4	2	no	—
211	C₂×#557	C₂×C₂₃	1	1	yes	characteristic, complemented
212	C₂×#486	C₂×C₂₃	1	1	yes	characteristic
213	C₂×#460	C₂×C₂₃	1	1	yes	characteristic
214	C₂×#460	C₂×C₂₃	1	1	yes	characteristic
215	#1626	C₂×C₂₃	1	8	yes	direct product factor
216	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
218	C₂×Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
219	C₂²×C₂₃	—	8	2	no	—
220	C₂²×C₂₃	—	8	2	no	—
221	C₂²×C₂₃	—	8	2	no	—
222	C₂²×C₂₃	—	16	4	no	—
223	C₂²×C₂₃	#448	1	2	yes	—
224	C₂²×C₂₃	#449	1	2	yes	—
225	C₂²×C₂₃	#454	1	2	yes	—
226	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
227	C₄×C₂₃	—	2	4	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	2	1	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	4	4	no	—
232	C₄×C₂₃	—	4	2	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	#448	1	2	yes	—
235	C₄×C₂₃	#449	1	2	yes	—
236	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
237	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
238	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
240	C₂²×SD₁₆	—	2	1	no	—
241	C₂²×SD₁₆	—	2	1	no	—
242	#557	—	2	2	no	—
243	#557	—	2	2	no	—
244	#557	—	2	2	no	—
245	#557	C₂²×C₂₃	1	4	yes	complemented
246	C₂×C₄⋊Q₈	—	2	1	no	—
247	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×#125	—	2	2	no	—
250	C₂×#125	—	2	1	no	—
251	C₂×C₄×Q₈	—	2	1	no	—
252	C₂×C₄×Q₈	—	2	1	no	—
253	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	#486	—	2	2	no	—
256	#486	C₂²×C₂₃	1	4	yes	—
257	#460	—	2	2	no	—
258	#460	—	2	2	no	—
259	#460	C₂²×C₂₃	1	4	yes	—
260	#460	C₂²×C₂₃	1	4	yes	—
261	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×#102	—	2	2	no	—
264	C₂×#102	—	2	1	no	—
265	C₂×#102	—	2	1	no	—
266	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
267	#332	C₂²×C₂₃	1	4	yes	complemented
268	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
269	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
270	C₂×C₂₃	—	16	2	no	—
271	C₂×C₂₃	#1626	1	4	yes	direct product factor
272	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
273	C₂×C₂₃	C₂×#449	1	1	yes	characteristic
274	C₂×C₂₃	C₂×#454	1	1	yes	characteristic
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
277	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂×SD₁₆	—	2	4	no	—
282	C₂×SD₁₆	—	2	4	no	—
283	C₂×SD₁₆	—	4	4	no	—
284	C₂×SD₁₆	—	4	4	no	—
285	C₂²×C₈	—	2	1	no	—
286	C₂²×C₈	—	2	1	no	—
287	C₂²×C₈	—	2	1	no	—
288	C₄⋊Q₈	—	2	4	no	—
289	C₄⋊Q₈	—	2	4	no	—
290	C₄⋊Q₈	—	2	1	no	—
291	C₄⋊Q₈	—	2	1	no	—
292	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
293	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
294	#127	—	2	1	no	—
295	#127	—	2	1	no	—
296	#127	—	2	1	no	—
297	#127	C₂³×C₂₃	1	2	yes	—
298	#125	—	2	8	no	—
299	#125	—	2	4	no	—
300	#125	—	2	8	no	—
301	#125	—	2	4	no	—
302	C₄×Q₈	—	2	4	no	—
303	C₄×Q₈	—	2	4	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×Q₈	—	2	4	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	—	2	1	no	—
310	C₄×Q₈	D₄×C₂₃	1	2	yes	—
311	C₄×Q₈	D₄×C₂₃	1	2	yes	—
312	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
313	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
321	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
330	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
331	#102	—	2	4	no	—
332	#102	—	2	4	no	—
333	#102	—	2	8	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	C₂³×C₂₃	1	2	yes	—
336	C₂₃	C₂×#1626	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	—	4	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂×Q₈	—	4	2	no	—
347	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	complemented
348	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
349	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	4	4	no	—
354	C₂×D₄	—	4	1	no	—
355	C₂×D₄	—	4	1	no	—
356	C₂×D₄	—	4	1	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
369	SD₁₆	—	4	8	no	—
370	SD₁₆	—	4	8	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	4	2	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	4	2	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
390	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
391	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
392	C₄²	—	2	8	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
401	C₂³	—	4	2	no	—
402	C₂³	—	8	2	no	—
403	C₂³	—	8	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	C₂₃×#120	1	1	yes	characteristic
406	Q₈	—	2	2	no	—
407	Q₈	—	2	4	no	—
408	Q₈	—	2	2	no	—
409	Q₈	—	2	2	no	—
410	Q₈	—	4	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	4	no	—
414	Q₈	C₂×SD₁₆×C₂₃	1	4	yes	complemented
415	D₄	—	4	4	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	8	2	no	—
421	D₄	—	8	2	no	—
422	D₄	—	8	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
435	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
436	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
437	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
438	C₂×C₄	C₂₃×#135	1	2	yes	—
439	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
440	C₂×C₄	C₂₃×#137	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
444	C₈	—	4	2	no	—
445	C₈	—	4	2	no	—
446	C₈	—	4	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	16	4	no	—
451	C₂²	C₂₃×#448	1	2	yes	—
452	C₂²	C₂₃×#449	1	2	yes	—
453	C₂²	C₂₃×#454	1	2	yes	—
454	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
455	C₄	—	2	4	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	4	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	C₂₃×#448	1	2	yes	—
463	C₄	C₂₃×#449	1	2	yes	—
464	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
465	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
466	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
467	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
468	C₂	—	16	2	no	—
469	C₂	C₂₃×#1626	1	4	yes	direct product factor
470	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
471	C₂	C₂×C₂₃×#449	1	1	yes	characteristic
472	C₂	C₂×C₂₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
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
1. C₂×C₂₃×#557	C₂	semidirect
5. C₂₃×#1626	C₂	direct (N × H)
6. C₂×C₂₃×#332	C₂	semidirect
7. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
8. C₂×C₂₃×Q₈⋊C₈	C₂	semidirect
14. C₂₃×#557	C₂²	semidirect
36. C₂₃×#332	C₂²	semidirect
37. C₂₃×Q₈⋊C₈	C₂²	semidirect
38. C₂₃×Q₈⋊C₈	C₂²	semidirect
110. C₂×Q₈×C₂₃	SD₁₆	semidirect
164. C₂×#1626	C₂₃	direct (N × H)
178. Q₈×C₂₃	C₂×SD₁₆	semidirect
211. C₂×#557	C₂×C₂₃	semidirect
215. #1626	C₂×C₂₃	direct (N × H)
216. C₂×#332	C₂×C₂₃	semidirect
217. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
218. C₂×Q₈⋊C₈	C₂×C₂₃	semidirect
245. #557	C₂²×C₂₃	semidirect
267. #332	C₂²×C₂₃	semidirect
268. Q₈⋊C₈	C₂²×C₂₃	semidirect
269. Q₈⋊C₈	C₂²×C₂₃	semidirect
271. C₂×C₂₃	#1626	direct (N × H)
336. C₂₃	C₂×#1626	direct (N × H)
347. C₂×Q₈	SD₁₆×C₂₃	semidirect
414. Q₈	C₂×SD₁₆×C₂₃	semidirect
469. C₂	C₂₃×#1626	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×#557	2
C₂×C₂₃×#486	2
C₂×C₂₃×#460	2
C₂₃×#1626	2
C₂×C₂₃×#332	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×#1626	23

Subgroups — C2×C23×#1626 #19166116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1626 #19166116