Subgroups — C₂×C₂₃×#3208 #19208401

Order 5888.

2822 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3208	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#492	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#486	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#460	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#450	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₂₃	—	2	1	no	—
10	C₂³×D₄×C₂₃	—	2	1	no	—
11	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
13	C₂₃×#533	—	2	4	no	—
14	C₂₃×#533	C₂²	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₂₃×#127	—	2	1	no	—
18	C₂×C₂₃×#127	—	2	1	no	—
19	C₂×C₂₃×#127	—	2	4	no	—
20	C₂×C₂₃×#127	C₂²	1	8	yes	—
21	C₂×C₂₃×#127	C₂²	1	2	yes	—
22	C₂×C₂₃×#127	C₂²	1	2	yes	—
23	C₂×C₂₃×#125	—	2	1	no	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
27	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
28	C₂×C₂₃×#117	—	2	1	no	—
29	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#492	C₂²	1	16	yes	complemented
31	C₂₃×#486	C₂²	1	16	yes	complemented
32	C₂₃×#460	C₂²	1	32	yes	complemented
33	C₂₃×#450	C₂²	1	32	yes	complemented
34	C₂₃×#431	C₂²	1	8	yes	complemented
35	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂×C₂₃×#102	C₂²	1	8	yes	complemented
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
39	C₂⁵×C₂₃	—	2	2	no	—
40	C₂²×Q₈×C₂₃	—	2	1	no	—
41	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×D₄×C₂₃	—	2	8	no	—
43	C₂²×D₄×C₂₃	—	2	2	no	—
44	C₂²×D₄×C₂₃	—	2	2	no	—
45	C₂²×D₄×C₂₃	—	2	2	no	—
46	C₂²×D₄×C₂₃	—	2	8	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	4	no	—
49	C₂²×D₄×C₂₃	—	2	2	no	—
50	C₂²×D₄×C₂₃	—	4	8	no	—
51	C₂²×D₄×C₂₃	—	4	8	no	—
52	C₂²×D₄×C₂₃	—	4	8	no	—
53	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
54	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
55	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×SD₁₆×C₂₃	—	4	16	no	—
59	C₂×SD₁₆×C₂₃	C₂³	1	32	yes	complemented
60	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
61	C₂₃×C₄⋊Q₈	—	2	2	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
64	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
65	C₂₃×#127	—	2	8	no	—
66	C₂₃×#127	—	2	4	no	—
67	C₂₃×#127	—	2	8	no	—
68	C₂₃×#127	—	2	4	no	—
69	C₂₃×#127	C₂³	1	8	yes	—
70	C₂₃×#127	C₂³	1	8	yes	—
71	C₂₃×#125	—	2	8	no	—
72	C₂₃×C₂²⋊Q₈	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
75	C₄×Q₈×C₂₃	—	2	8	no	—
76	C₄×Q₈×C₂₃	—	2	4	no	—
77	C₄×Q₈×C₂₃	C₂³	1	8	yes	complemented
78	C₂₃×#117	—	2	8	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄²×C₂₃	—	2	2	no	—
87	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	D₄	1	4	yes	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
93	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
94	C₂₃×#102	C₂³	1	32	yes	complemented
95	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
96	C₄×C₈×C₂₃	C₂³	1	8	yes	complemented
97	C₂⁴×C₂₃	—	2	8	no	—
98	C₂⁴×C₂₃	—	2	4	no	—
99	C₂⁴×C₂₃	—	4	8	no	—
100	C₂⁴×C₂₃	—	4	8	no	—
101	C₂⁴×C₂₃	—	4	8	no	—
102	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₂₃	—	2	2	no	—
104	C₂×Q₈×C₂₃	—	2	4	no	—
105	C₂×Q₈×C₂₃	—	2	8	no	—
106	C₂×Q₈×C₂₃	—	4	4	no	—
107	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
108	C₂×D₄×C₂₃	—	2	8	no	—
109	C₂×D₄×C₂₃	—	2	4	no	—
110	C₂×D₄×C₂₃	—	2	8	no	—
111	C₂×D₄×C₂₃	—	2	8	no	—
112	C₂×D₄×C₂₃	—	2	8	no	—
113	C₂×D₄×C₂₃	—	2	8	no	—
114	C₂×D₄×C₂₃	—	2	16	no	—
115	C₂×D₄×C₂₃	—	4	16	no	—
116	C₂×D₄×C₂₃	—	4	8	no	—
117	C₂×D₄×C₂₃	—	4	8	no	—
118	C₂×D₄×C₂₃	—	4	16	no	—
119	C₂×D₄×C₂₃	—	4	32	no	—
120	C₂×D₄×C₂₃	—	4	8	no	—
121	C₂×D₄×C₂₃	—	4	8	no	—
122	C₂×D₄×C₂₃	—	4	32	no	—
123	C₂×D₄×C₂₃	—	4	8	no	—
124	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
125	C₂²×C₄×C₂₃	—	2	4	no	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	complemented
130	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₂₃	—	4	32	no	—
141	C₂×C₈×C₂₃	—	2	8	no	—
142	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
143	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	4	no	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	8	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
150	C₂²⋊C₄×C₂₃	—	2	8	no	—
151	C₄²×C₂₃	—	2	8	no	—
152	C₄²×C₂₃	C₂×D₄	1	4	yes	—
153	C₄²×C₂₃	C₂×D₄	1	2	yes	—
154	C₄²×C₂₃	C₂×D₄	1	4	yes	—
155	C₄²×C₂₃	C₂×D₄	1	2	yes	—
156	C₄²×C₂₃	C₂⁴	1	2	yes	—
157	C₄²×C₂₃	C₂⁴	1	2	yes	—
158	C₂×#3208	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₂₃	—	2	2	no	—
160	C₂³×C₂₃	—	2	2	no	—
161	C₂³×C₂₃	—	2	8	no	—
162	C₂³×C₂₃	—	4	16	no	—
163	C₂³×C₂₃	—	4	16	no	—
164	C₂³×C₂₃	—	4	8	no	—
165	C₂³×C₂₃	—	4	16	no	—
166	C₂³×C₂₃	—	4	8	no	—
167	C₂³×C₂₃	—	4	8	no	—
168	C₂³×C₂₃	—	8	32	no	—
169	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
170	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
171	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₂₃	—	2	8	no	—
173	Q₈×C₂₃	—	4	4	no	—
174	Q₈×C₂₃	—	4	8	no	—
175	D₄×C₂₃	—	2	16	no	—
176	D₄×C₂₃	—	4	16	no	—
177	D₄×C₂₃	—	4	16	no	—
178	D₄×C₂₃	—	4	8	no	—
179	D₄×C₂₃	—	4	16	no	—
180	D₄×C₂₃	—	4	16	no	—
181	D₄×C₂₃	—	4	16	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	4	no	—
185	C₂×C₄×C₂₃	—	2	8	no	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	4	4	no	—
190	C₂×C₄×C₂₃	—	4	4	no	—
191	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
192	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
193	C₂×C₄×C₂₃	C₂×SD₁₆	1	16	yes	complemented
194	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
195	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
202	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
203	C₈×C₂₃	—	2	8	no	—
204	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
205	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic
206	#3208	C₂×C₂₃	1	16	yes	direct product factor
207	C₂×#492	C₂×C₂₃	1	2	yes	complemented
208	C₂×#486	C₂×C₂₃	1	2	yes	complemented
209	C₂×#460	C₂×C₂₃	1	4	yes	complemented
210	C₂×#450	C₂×C₂₃	1	4	yes	complemented
211	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
212	C₂²×C₂₃	—	2	2	no	—
213	C₂²×C₂₃	—	2	8	no	—
214	C₂²×C₂₃	—	2	2	no	—
215	C₂²×C₂₃	—	4	8	no	—
216	C₂²×C₂₃	—	4	8	no	—
217	C₂²×C₂₃	—	4	8	no	—
218	C₂²×C₂₃	—	8	32	no	—
219	C₂²×C₂₃	—	8	16	no	—
220	C₂²×C₂₃	#535	1	2	yes	—
221	C₂²×C₂₃	C₂²×SD₁₆	1	4	yes	—
222	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
223	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
224	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
225	C₄×C₂₃	—	2	2	no	—
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	4	4	no	—
228	C₄×C₂₃	—	4	4	no	—
229	C₄×C₂₃	#535	1	2	yes	—
230	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
231	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
232	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
233	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
234	C₂³×D₄	—	2	1	no	—
235	C₂³×D₄	—	2	1	no	—
236	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
237	C₂²×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
238	#533	—	2	4	no	—
239	#533	C₂²×C₂₃	1	8	yes	complemented
240	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
241	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	C₂×#127	—	2	1	no	—
243	C₂×#127	—	2	1	no	—
244	C₂×#127	—	2	4	no	—
245	C₂×#127	C₂²×C₂₃	1	2	yes	—
246	C₂×#127	C₂²×C₂₃	1	8	yes	—
247	C₂×#127	C₂²×C₂₃	1	2	yes	—
248	C₂×#125	—	2	1	no	—
249	C₂×C₂²⋊Q₈	—	2	1	no	—
250	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄×Q₈	—	2	1	no	—
252	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
253	C₂×#117	—	2	1	no	—
254	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
255	#492	C₂²×C₂₃	1	16	yes	complemented
256	#486	C₂²×C₂₃	1	16	yes	complemented
257	#460	C₂²×C₂₃	1	32	yes	complemented
258	#450	C₂²×C₂₃	1	32	yes	complemented
259	#431	C₂²×C₂₃	1	8	yes	complemented
260	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
261	C₂×#102	C₂²×C₂₃	1	8	yes	complemented
262	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
263	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
264	C₂×C₂₃	—	2	4	no	—
265	C₂×C₂₃	—	8	8	no	—
266	C₂×C₂₃	#3208	1	4	yes	direct product factor
267	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
268	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
269	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂⁵	—	2	2	no	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	4	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	2	8	no	—
277	C₂²×D₄	—	2	2	no	—
278	C₂²×D₄	—	2	2	no	—
279	C₂²×D₄	—	2	8	no	—
280	C₂²×D₄	—	2	2	no	—
281	C₂²×D₄	—	4	8	no	—
282	C₂²×D₄	—	4	8	no	—
283	C₂²×D₄	—	4	8	no	—
284	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
285	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
286	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
289	C₂×SD₁₆	—	4	16	no	—
290	C₂×SD₁₆	C₂³×C₂₃	1	32	yes	complemented
291	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
295	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
296	#127	—	2	8	no	—
297	#127	—	2	4	no	—
298	#127	—	2	8	no	—
299	#127	—	2	4	no	—
300	#127	C₂³×C₂₃	1	8	yes	—
301	#127	C₂³×C₂₃	1	8	yes	—
302	#125	—	2	8	no	—
303	C₂²⋊Q₈	—	2	8	no	—
304	C₂²⋊Q₈	—	2	4	no	—
305	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
306	C₄×Q₈	—	2	8	no	—
307	C₄×Q₈	—	2	4	no	—
308	C₄×Q₈	C₂³×C₂₃	1	8	yes	complemented
309	#117	—	2	8	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
319	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄²	D₄×C₂₃	1	4	yes	—
321	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
324	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
325	#102	C₂³×C₂₃	1	32	yes	complemented
326	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
327	C₄×C₈	C₂³×C₂₃	1	8	yes	complemented
328	C₂₃	C₂×#3208	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁴	—	2	8	no	—
330	C₂⁴	—	2	4	no	—
331	C₂⁴	—	4	8	no	—
332	C₂⁴	—	4	8	no	—
333	C₂⁴	—	4	8	no	—
334	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	8	no	—
338	C₂×Q₈	—	4	4	no	—
339	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
340	C₂×D₄	—	2	8	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	8	no	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	16	no	—
347	C₂×D₄	—	4	16	no	—
348	C₂×D₄	—	4	8	no	—
349	C₂×D₄	—	4	8	no	—
350	C₂×D₄	—	4	32	no	—
351	C₂×D₄	—	4	16	no	—
352	C₂×D₄	—	4	8	no	—
353	C₂×D₄	—	4	8	no	—
354	C₂×D₄	—	4	32	no	—
355	C₂×D₄	—	4	8	no	—
356	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	complemented
362	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
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	4	yes	—
369	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
372	SD₁₆	—	4	32	no	—
373	C₂×C₈	—	2	8	no	—
374	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
382	C₂²⋊C₄	—	2	8	no	—
383	C₄²	—	2	8	no	—
384	C₄²	C₂×D₄×C₂₃	1	4	yes	—
385	C₄²	C₂×D₄×C₂₃	1	2	yes	—
386	C₄²	C₂×D₄×C₂₃	1	4	yes	—
387	C₄²	C₂×D₄×C₂₃	1	2	yes	—
388	C₄²	C₂⁴×C₂₃	1	2	yes	—
389	C₄²	C₂⁴×C₂₃	1	2	yes	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	8	no	—
393	C₂³	—	4	16	no	—
394	C₂³	—	4	16	no	—
395	C₂³	—	4	16	no	—
396	C₂³	—	4	8	no	—
397	C₂³	—	4	8	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	8	32	no	—
400	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
401	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
402	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
403	Q₈	—	2	8	no	—
404	Q₈	—	4	8	no	—
405	Q₈	—	4	4	no	—
406	D₄	—	2	16	no	—
407	D₄	—	4	16	no	—
408	D₄	—	4	16	no	—
409	D₄	—	4	16	no	—
410	D₄	—	4	16	no	—
411	D₄	—	4	16	no	—
412	D₄	—	4	8	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
423	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
424	C₂×C₄	C₂×SD₁₆×C₂₃	1	16	yes	complemented
425	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
427	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
428	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
430	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
433	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
434	C₈	—	2	8	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	8	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	8	no	—
439	C₂²	—	4	8	no	—
440	C₂²	—	4	8	no	—
441	C₂²	—	8	32	no	—
442	C₂²	—	8	16	no	—
443	C₂²	C₂₃×#535	1	2	yes	—
444	C₂²	C₂²×SD₁₆×C₂₃	1	4	yes	—
445	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
446	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
447	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	4	no	—
452	C₄	C₂₃×#535	1	2	yes	—
453	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
454	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
455	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
456	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
457	C₂	—	2	4	no	—
458	C₂	—	8	8	no	—
459	C₂	C₂₃×#3208	1	4	yes	direct product factor
460	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
461	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
462	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3208	C₂	direct (N × H)
4. C₂×C₂₃×#492	C₂	semidirect
5. C₂×C₂₃×#486	C₂	semidirect
6. C₂×C₂₃×#460	C₂	semidirect
7. C₂×C₂₃×#450	C₂	semidirect
8. C₂×C₂₃×#431	C₂	semidirect
12. C₂²×SD₁₆×C₂₃	C₂²	semidirect
14. C₂₃×#533	C₂²	semidirect
15. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
27. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
30. C₂₃×#492	C₂²	semidirect
31. C₂₃×#486	C₂²	semidirect
32. C₂₃×#460	C₂²	semidirect
33. C₂₃×#450	C₂²	semidirect
34. C₂₃×#431	C₂²	semidirect
35. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
36. C₂×C₂₃×#102	C₂²	semidirect
38. C₂×C₄×C₈×C₂₃	C₂²	semidirect
59. C₂×SD₁₆×C₂₃	C₂³	semidirect
63. C₂₃×C₄⋊Q₈	C₂³	semidirect
64. C₂₃×C₄⋊Q₈	C₂³	semidirect
77. C₄×Q₈×C₂₃	C₂³	semidirect
93. C₂₃×C₄⋊C₈	C₂³	semidirect
94. C₂₃×#102	C₂³	semidirect
96. C₄×C₈×C₂₃	C₂³	semidirect
129. C₂²×C₄×C₂₃	SD₁₆	semidirect
158. C₂×#3208	C₂₃	direct (N × H)
193. C₂×C₄×C₂₃	C₂×SD₁₆	semidirect
204. C₂×#533	C₂×C₂₃	semidirect
206. #3208	C₂×C₂₃	direct (N × H)
207. C₂×#492	C₂×C₂₃	semidirect
208. C₂×#486	C₂×C₂₃	semidirect
209. C₂×#460	C₂×C₂₃	semidirect
210. C₂×#450	C₂×C₂₃	semidirect
211. C₂×#431	C₂×C₂₃	semidirect
237. C₂²×SD₁₆	C₂²×C₂₃	semidirect
239. #533	C₂²×C₂₃	semidirect
240. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
241. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
252. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
255. #492	C₂²×C₂₃	semidirect
256. #486	C₂²×C₂₃	semidirect
257. #460	C₂²×C₂₃	semidirect
258. #450	C₂²×C₂₃	semidirect
259. #431	C₂²×C₂₃	semidirect
260. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
261. C₂×#102	C₂²×C₂₃	semidirect
263. C₂×C₄×C₈	C₂²×C₂₃	semidirect
266. C₂×C₂₃	#3208	direct (N × H)
290. C₂×SD₁₆	C₂³×C₂₃	semidirect
294. C₄⋊Q₈	C₂³×C₂₃	semidirect
295. C₄⋊Q₈	C₂³×C₂₃	semidirect
308. C₄×Q₈	C₂³×C₂₃	semidirect
324. C₄⋊C₈	C₂³×C₂₃	semidirect
325. #102	C₂³×C₂₃	semidirect
327. C₄×C₈	C₂³×C₂₃	semidirect
328. C₂₃	C₂×#3208	direct (N × H)
361. C₂²×C₄	SD₁₆×C₂₃	semidirect
424. C₂×C₄	C₂×SD₁₆×C₂₃	semidirect
459. C₂	C₂₃×#3208	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×#533	2
C₂²×C₂₃×#127	2
C₂₃×#3208	2
C₂×C₂₃×#492	2
C₂×C₂₃×#486	2
C₂×C₂₃×#460	2
C₂×C₂₃×#450	2
C₂×C₂₃×#431	2
C₂×#3208	23

Subgroups — C2×C23×#3208 #19208401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3208 #19208401