Subgroups — C₂×C₂₃×#1647 #19166121

Order 5888.

1238 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1647	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×D₄⋊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	1	no	—
11	C₂₃×#547	—	2	2	no	—
12	C₂₃×#547	—	2	2	no	—
13	C₂₃×#547	—	2	2	no	—
14	C₂₃×#547	C₂²	1	4	yes	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#123	—	2	2	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×#121	—	2	1	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₂₃×#500	—	2	2	no	—
24	C₂₃×#500	C₂²	1	4	yes	complemented
25	C₂₃×#474	—	2	2	no	—
26	C₂₃×#474	C₂²	1	4	yes	complemented
27	C₂₃×#461	—	2	2	no	—
28	C₂₃×#461	C₂²	1	4	yes	complemented
29	C₂×C₂₃×#107	—	2	1	no	—
30	C₂×C₂₃×#107	—	2	2	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×#332	C₂²	1	4	yes	complemented
37	C₂₃×D₄⋊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	—	2	1	no	—
45	C₂²×D₄×C₂₃	—	2	2	no	—
46	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	—	2	2	no	—
49	C₂³×C₄×C₂₃	—	2	1	no	—
50	C₂³×C₄×C₂₃	—	2	1	no	—
51	C₂×SD₁₆×C₂₃	—	2	4	no	—
52	C₂×SD₁₆×C₂₃	—	4	4	no	—
53	C₂²×C₈×C₂₃	—	2	1	no	—
54	C₂²×C₈×C₂₃	—	2	1	no	—
55	C₂²×C₈×C₂₃	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	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	D₄	1	2	yes	—
62	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₂₃×#123	—	2	8	no	—
65	C₂₃×#123	—	2	8	no	—
66	C₂₃×C₂²⋊Q₈	—	2	4	no	—
67	C₂₃×C₂²⋊Q₈	—	2	4	no	—
68	C₂₃×#121	—	2	4	no	—
69	C₂₃×#121	—	2	4	no	—
70	C₄×D₄×C₂₃	—	2	1	no	—
71	C₄×D₄×C₂₃	—	2	1	no	—
72	C₄×D₄×C₂₃	—	2	1	no	—
73	C₄×D₄×C₂₃	—	2	1	no	—
74	C₄×D₄×C₂₃	D₄	1	2	yes	—
75	C₄×D₄×C₂₃	D₄	1	2	yes	—
76	C₄×D₄×C₂₃	C₂³	1	2	yes	—
77	C₄×D₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	4	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×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×#107	—	2	4	no	—
92	C₂₃×#107	—	2	8	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	1	no	—
103	#41×C₂₃	—	4	8	no	—
104	#41×C₂₃	—	4	16	no	—
105	#41×C₂₃	—	4	8	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	4	1	no	—
110	C₂×Q₈×C₂₃	—	4	2	no	—
111	C₂×Q₈×C₂₃	—	4	2	no	—
112	C₂×Q₈×C₂₃	—	4	2	no	—
113	C₂×Q₈×C₂₃	—	4	2	no	—
114	C₂×D₄×C₂₃	—	2	8	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	1	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	—	4	8	no	—
121	C₂×D₄×C₂₃	D₈	1	4	yes	—
122	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₂₃	—	2	8	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	2	no	—
132	C₂²×C₄×C₂₃	—	4	8	no	—
133	C₂²×C₄×C₂₃	—	4	2	no	—
134	C₂²×C₄×C₂₃	—	4	2	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	2	no	—
137	C₂²×C₄×C₂₃	—	4	1	no	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	SD₁₆×C₂₃	—	4	8	no	—
142	C₂×C₈×C₂₃	—	2	4	no	—
143	C₂×C₈×C₂₃	—	2	2	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	4	2	no	—
147	C₂×C₈×C₂₃	—	4	2	no	—
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	4	1	no	—
157	C₄⋊C₄×C₂₃	—	4	1	no	—
158	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²⋊C₄×C₂₃	—	2	8	no	—
162	C₂²⋊C₄×C₂₃	—	2	2	no	—
163	C₂²⋊C₄×C₂₃	—	2	2	no	—
164	C₂²⋊C₄×C₂₃	—	2	2	no	—
165	C₂²⋊C₄×C₂₃	—	2	2	no	—
166	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
170	C₂×#1647	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	2	2	no	—
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	—	4	2	no	—
176	C₂³×C₂₃	—	4	2	no	—
177	C₂³×C₂₃	—	4	2	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	#120	1	1	yes	characteristic
180	Q₈×C₂₃	—	4	2	no	—
181	Q₈×C₂₃	—	4	2	no	—
182	Q₈×C₂₃	—	4	4	no	—
183	Q₈×C₂₃	—	4	2	no	—
184	Q₈×C₂₃	—	4	2	no	—
185	Q₈×C₂₃	—	8	2	no	—
186	D₄×C₂₃	—	2	2	no	—
187	D₄×C₂₃	—	2	2	no	—
188	D₄×C₂₃	—	2	2	no	—
189	D₄×C₂₃	—	2	4	no	—
190	D₄×C₂₃	—	4	16	no	—
191	D₄×C₂₃	—	4	2	no	—
192	D₄×C₂₃	C₂×D₈	1	4	yes	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	4	16	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	4	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	2	no	—
208	C₂×C₄×C₂₃	—	8	2	no	—
209	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
210	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
211	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
213	C₂×C₄×C₂₃	#135	1	2	yes	—
214	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
215	C₂×C₄×C₂₃	#137	1	2	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₈×C₂₃	—	2	4	no	—
220	C₈×C₂₃	—	4	2	no	—
221	C₈×C₂₃	—	4	2	no	—
222	C₂×#547	C₂×C₂₃	1	1	yes	characteristic
223	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#1647	C₂×C₂₃	1	8	yes	direct product factor
227	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂²×C₂₃	—	2	4	no	—
231	C₂²×C₂₃	—	4	8	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	—	4	2	no	—
236	C₂²×C₂₃	—	4	2	no	—
237	C₂²×C₂₃	—	8	4	no	—
238	C₂²×C₂₃	#447	1	2	yes	—
239	C₂²×C₂₃	#452	1	2	yes	—
240	C₂²×C₂₃	#456	1	2	yes	—
241	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
242	C₄×C₂₃	—	2	1	no	—
243	C₄×C₂₃	—	2	1	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	4	4	no	—
246	C₄×C₂₃	—	4	2	no	—
247	C₄×C₂₃	—	8	2	no	—
248	C₄×C₂₃	#447	1	2	yes	—
249	C₄×C₂₃	#452	1	2	yes	—
250	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
251	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
252	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
254	C₂²×#41	—	2	1	no	—
255	C₂²×SD₁₆	—	2	1	no	—
256	#547	—	2	2	no	—
257	#547	—	2	2	no	—
258	#547	—	2	2	no	—
259	#547	C₂²×C₂₃	1	4	yes	—
260	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#123	—	2	2	no	—
263	C₂×C₂²⋊Q₈	—	2	1	no	—
264	C₂×#121	—	2	1	no	—
265	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	—	2	1	no	—
268	#500	—	2	2	no	—
269	#500	C₂²×C₂₃	1	4	yes	complemented
270	#474	—	2	2	no	—
271	#474	C₂²×C₂₃	1	4	yes	complemented
272	#461	—	2	2	no	—
273	#461	C₂²×C₂₃	1	4	yes	complemented
274	C₂×#107	—	2	2	no	—
275	C₂×#107	—	2	1	no	—
276	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
278	C₂×#103	—	2	1	no	—
279	C₂×#102	—	2	1	no	—
280	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
281	#332	C₂²×C₂₃	1	4	yes	complemented
282	D₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
283	D₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
284	C₂×C₂₃	—	4	4	no	—
285	C₂×C₂₃	—	8	2	no	—
286	C₂×C₂₃	#1647	1	4	yes	direct product factor
287	C₂×C₂₃	C₂×#447	1	1	yes	characteristic
288	C₂×C₂₃	C₂×#452	1	1	yes	characteristic
289	C₂×C₂₃	C₂×#456	1	1	yes	characteristic
290	C₂×#41	—	2	4	no	—
291	C₂×#41	—	2	4	no	—
292	C₂×#41	—	4	4	no	—
293	C₂×#41	—	4	4	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
298	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	—	2	1	no	—
301	C₂³×C₄	—	2	1	no	—
302	C₂×SD₁₆	—	2	4	no	—
303	C₂×SD₁₆	—	4	4	no	—
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	2	1	no	—
306	C₂²×C₈	—	2	1	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	—	2	1	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
313	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
314	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
315	#123	—	2	8	no	—
316	#123	—	2	8	no	—
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	#121	—	2	4	no	—
320	#121	—	2	4	no	—
321	C₄×D₄	—	2	1	no	—
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	D₄×C₂₃	1	2	yes	—
326	C₄×D₄	D₄×C₂₃	1	2	yes	—
327	C₄×D₄	C₂³×C₂₃	1	2	yes	—
328	C₄×D₄	C₂³×C₂₃	1	2	yes	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	4	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
336	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
337	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
342	#107	—	2	4	no	—
343	#107	—	2	8	no	—
344	C₄⋊C₈	—	2	1	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
347	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
348	#103	—	2	4	no	—
349	#102	—	2	4	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	C₂³×C₂₃	1	2	yes	—
352	C₂₃	C₂×#1647	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	—	2	1	no	—
354	C₂⁴	—	2	1	no	—
355	#41	—	4	8	no	—
356	#41	—	4	16	no	—
357	#41	—	4	8	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	4	1	no	—
362	C₂×Q₈	—	4	2	no	—
363	C₂×Q₈	—	4	2	no	—
364	C₂×Q₈	—	4	2	no	—
365	C₂×Q₈	—	4	2	no	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	4	8	no	—
373	C₂×D₄	D₈×C₂₃	1	4	yes	—
374	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	8	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	1	no	—
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	SD₁₆	—	4	8	no	—
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	4	2	no	—
399	C₂×C₈	—	4	2	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	4	1	no	—
409	C₄⋊C₄	—	4	1	no	—
410	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
411	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
412	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
419	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
420	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
421	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
422	C₂³	—	2	4	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	C₂₃×#120	1	1	yes	characteristic
431	Q₈	—	4	2	no	—
432	Q₈	—	4	2	no	—
433	Q₈	—	4	4	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	8	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	—	2	4	no	—
441	D₄	—	4	16	no	—
442	D₄	—	4	2	no	—
443	D₄	C₂×D₈×C₂₃	1	4	yes	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	4	16	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	8	2	no	—
460	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
461	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
462	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
463	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
464	C₂×C₄	C₂₃×#135	1	2	yes	—
465	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
466	C₂×C₄	C₂₃×#137	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	—	4	2	no	—
472	C₈	—	4	2	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	4	8	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	8	4	no	—
481	C₂²	C₂₃×#447	1	2	yes	—
482	C₂²	C₂₃×#452	1	2	yes	—
483	C₂²	C₂₃×#456	1	2	yes	—
484	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	2	no	—
490	C₄	—	8	2	no	—
491	C₄	C₂₃×#447	1	2	yes	—
492	C₄	C₂₃×#452	1	2	yes	—
493	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
494	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
495	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
496	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
497	C₂	—	4	4	no	—
498	C₂	—	8	2	no	—
499	C₂	C₂₃×#1647	1	4	yes	direct product factor
500	C₂	C₂×C₂₃×#447	1	1	yes	characteristic
501	C₂	C₂×C₂₃×#452	1	1	yes	characteristic
502	C₂	C₂×C₂₃×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#547	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×#500	C₂	semidirect
3. C₂×C₂₃×#474	C₂	semidirect
4. C₂×C₂₃×#461	C₂	semidirect
5. C₂₃×#1647	C₂	direct (N × H)
6. C₂×C₂₃×#332	C₂	semidirect
7. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
8. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
24. C₂₃×#500	C₂²	semidirect
26. C₂₃×#474	C₂²	semidirect
28. C₂₃×#461	C₂²	semidirect
36. C₂₃×#332	C₂²	semidirect
37. C₂₃×D₄⋊C₈	C₂²	semidirect
38. C₂₃×D₄⋊C₈	C₂²	semidirect
170. C₂×#1647	C₂₃	direct (N × H)
223. C₂×#500	C₂×C₂₃	semidirect
224. C₂×#474	C₂×C₂₃	semidirect
225. C₂×#461	C₂×C₂₃	semidirect
226. #1647	C₂×C₂₃	direct (N × H)
227. C₂×#332	C₂×C₂₃	semidirect
228. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
229. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
269. #500	C₂²×C₂₃	semidirect
271. #474	C₂²×C₂₃	semidirect
273. #461	C₂²×C₂₃	semidirect
281. #332	C₂²×C₂₃	semidirect
282. D₄⋊C₈	C₂²×C₂₃	semidirect
283. D₄⋊C₈	C₂²×C₂₃	semidirect
286. C₂×C₂₃	#1647	direct (N × H)
352. C₂₃	C₂×#1647	direct (N × H)
499. C₂	C₂₃×#1647	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×#547	2
C₂×C₂₃×#500	2
C₂×C₂₃×#474	2
C₂×C₂₃×#461	2
C₂₃×#1647	2
C₂×C₂₃×#332	2
C₂×C₂₃×D₄⋊C₈	2
C₂×#1647	23

Subgroups — C2×C23×#1647 #19166121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1647 #19166121