Subgroups — C₂×C₂₃×#3298 #19208742

Order 5888.

2374 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3298	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#483	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#465	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#448	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
18	D₄×Q₈×C₂₃	—	2	4	no	—
19	D₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
20	C₂₃×#547	—	2	4	no	—
21	C₂₃×#547	C₂²	1	8	yes	—
22	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#123	—	2	2	no	—
26	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
28	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₂×C₂₃×#121	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	—	2	1	no	—
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₂₃×#487	C₂²	1	8	yes	complemented
36	C₂₃×#483	C₂²	1	16	yes	complemented
37	C₂₃×#475	C₂²	1	8	yes	complemented
38	C₂₃×#465	C₂²	1	16	yes	complemented
39	C₂₃×#461	C₂²	1	8	yes	complemented
40	C₂₃×#448	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
42	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
43	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#106	C₂²	1	2	yes	—
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#103	C₂²	1	2	yes	—
48	C₂×C₂₃×#103	C₂²	1	2	yes	—
49	C₂×C₂₃×#103	C₂²	1	2	yes	—
50	C₂×C₂₃×#103	C₂²	1	2	yes	—
51	C₂×C₂₃×#103	C₂²	1	16	yes	complemented
52	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×#41×C₂₃	—	2	8	no	—
56	C₂×#41×C₂₃	—	4	8	no	—
57	C₂²×Q₈×C₂₃	—	2	4	no	—
58	C₂²×Q₈×C₂₃	—	2	2	no	—
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	—	2	8	no	—
61	C₂²×Q₈×C₂₃	—	4	4	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	—	2	2	no	—
66	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	—	2	2	no	—
69	C₂³×C₄×C₂₃	—	2	2	no	—
70	C₂³×C₄×C₂₃	—	2	1	no	—
71	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₂₃	—	2	4	no	—
73	C₂×SD₁₆×C₂₃	—	4	4	no	—
74	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
76	C₂²×C₈×C₂₃	—	4	4	no	—
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
81	C₂₃×C₄⋊Q₈	—	2	8	no	—
82	C₂₃×C₄⋊Q₈	—	2	2	no	—
83	C₂₃×C₄⋊Q₈	—	2	2	no	—
84	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
85	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
86	C₂₃×#123	—	2	16	no	—
87	C₂₃×C₂²⋊Q₈	—	2	4	no	—
88	C₂₃×C₂²⋊Q₈	—	2	8	no	—
89	C₂₃×C₂²⋊Q₈	—	2	16	no	—
90	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
91	C₂₃×#121	—	2	4	no	—
92	C₂₃×#121	C₂³	1	8	yes	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
95	C₄×D₄×C₂₃	—	2	8	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	D₄	1	4	yes	—
98	C₄×D₄×C₂₃	C₂³	1	4	yes	—
99	C₄×D₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂₃×#106	C₂³	1	4	yes	—
119	C₂₃×#106	C₂³	1	8	yes	—
120	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
121	C₂₃×#103	C₂³	1	4	yes	—
122	C₂₃×#103	C₂³	1	8	yes	—
123	C₂₃×#103	C₂³	1	8	yes	—
124	C₂₃×#103	C₂³	1	8	yes	—
125	C₂₃×#103	C₂³	1	8	yes	—
126	C₂₃×#102	C₂³	1	4	yes	—
127	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₄×C₈×C₂₃	C₂³	1	4	yes	—
129	C₂⁴×C₂₃	—	2	1	no	—
130	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
131	#41×C₂₃	—	4	32	no	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	8	no	—
134	C₂×Q₈×C₂₃	—	2	2	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	16	no	—
138	C₂×Q₈×C₂₃	—	2	4	no	—
139	C₂×Q₈×C₂₃	—	2	4	no	—
140	C₂×Q₈×C₂₃	—	4	8	no	—
141	C₂×Q₈×C₂₃	—	4	8	no	—
142	C₂×Q₈×C₂₃	—	4	16	no	—
143	C₂×Q₈×C₂₃	—	4	4	no	—
144	C₂×Q₈×C₂₃	#41	1	4	yes	complemented
145	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
147	C₂×D₄×C₂₃	—	2	8	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	4	8	no	—
152	C₂×D₄×C₂₃	SD₁₆	1	8	yes	complemented
153	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
154	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	8	no	—
159	C₂²×C₄×C₂₃	—	2	8	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	2	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	2	no	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	4	8	no	—
166	C₂²×C₄×C₂₃	—	4	8	no	—
167	C₂²×C₄×C₂₃	—	4	4	no	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	SD₁₆×C₂₃	—	2	8	no	—
175	SD₁₆×C₂₃	—	4	8	no	—
176	C₂×C₈×C₂₃	—	2	2	no	—
177	C₂×C₈×C₂₃	—	2	2	no	—
178	C₂×C₈×C₂₃	—	2	4	no	—
179	C₂×C₈×C₂₃	—	4	8	no	—
180	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
181	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
182	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	8	no	—
186	C₄⋊C₄×C₂₃	—	2	4	no	—
187	C₄⋊C₄×C₂₃	—	2	4	no	—
188	C₄⋊C₄×C₂₃	—	2	4	no	—
189	C₄⋊C₄×C₂₃	—	2	8	no	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
196	C₂²⋊C₄×C₂₃	—	2	8	no	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	2	8	no	—
199	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
200	C₄²×C₂₃	—	2	4	no	—
201	C₄²×C₂₃	C₂×D₄	1	2	yes	—
202	C₄²×C₂₃	C₂⁴	1	2	yes	—
203	C₂×#3298	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₂₃	—	2	4	no	—
205	C₂³×C₂₃	—	2	4	no	—
206	C₂³×C₂₃	—	2	2	no	—
207	C₂³×C₂₃	—	4	2	no	—
208	C₂³×C₂₃	—	4	2	no	—
209	C₂³×C₂₃	—	4	2	no	—
210	C₂³×C₂₃	C₂×SD₁₆	1	8	yes	—
211	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
212	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	4	no	—
214	Q₈×C₂₃	—	2	2	no	—
215	Q₈×C₂₃	—	2	4	no	—
216	Q₈×C₂₃	—	2	4	no	—
217	Q₈×C₂₃	—	4	16	no	—
218	Q₈×C₂₃	—	4	4	no	—
219	Q₈×C₂₃	—	4	8	no	—
220	Q₈×C₂₃	C₂×#41	1	4	yes	complemented
221	D₄×C₂₃	—	2	2	no	—
222	D₄×C₂₃	—	2	2	no	—
223	D₄×C₂₃	—	4	16	no	—
224	D₄×C₂₃	—	4	2	no	—
225	D₄×C₂₃	C₂×SD₁₆	1	16	yes	complemented
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	8	no	—
231	C₂×C₄×C₂₃	—	2	4	no	—
232	C₂×C₄×C₂₃	—	2	4	no	—
233	C₂×C₄×C₂₃	—	2	2	no	—
234	C₂×C₄×C₂₃	—	2	4	no	—
235	C₂×C₄×C₂₃	—	2	2	no	—
236	C₂×C₄×C₂₃	—	2	4	no	—
237	C₂×C₄×C₂₃	—	4	16	no	—
238	C₂×C₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	4	no	—
241	C₂×C₄×C₂₃	—	4	16	no	—
242	C₂×C₄×C₂₃	—	4	2	no	—
243	C₂×C₄×C₂₃	—	4	8	no	—
244	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
247	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
250	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
251	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
252	C₈×C₂₃	—	2	2	no	—
253	C₈×C₂₃	—	2	2	no	—
254	C₈×C₂₃	—	2	4	no	—
255	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
256	C₂×#547	C₂×C₂₃	1	1	yes	characteristic
257	#3298	C₂×C₂₃	1	16	yes	direct product factor
258	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂×#483	C₂×C₂₃	1	2	yes	complemented
260	C₂×#475	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂×#465	C₂×C₂₃	1	2	yes	complemented
262	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
263	C₂×#448	C₂×C₂₃	1	2	yes	complemented
264	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
267	C₂²×C₂₃	—	2	4	no	—
268	C₂²×C₂₃	—	2	16	no	—
269	C₂²×C₂₃	—	2	4	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	4	2	no	—
272	C₂²×C₂₃	—	4	2	no	—
273	C₂²×C₂₃	—	8	4	no	—
274	C₂²×C₂₃	#546	1	2	yes	—
275	C₂²×C₂₃	C₂²×SD₁₆	1	8	yes	—
276	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
277	C₂²×C₂₃	#578	1	2	yes	—
278	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
279	C₄×C₂₃	—	2	4	no	—
280	C₄×C₂₃	—	2	4	no	—
281	C₄×C₂₃	—	2	2	no	—
282	C₄×C₂₃	—	4	4	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	4	no	—
285	C₄×C₂₃	—	4	2	no	—
286	C₄×C₂₃	#546	1	2	yes	—
287	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
288	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
289	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂²×#41	—	2	1	no	—
291	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
292	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
293	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
294	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
295	D₄×Q₈	—	2	4	no	—
296	D₄×Q₈	C₂²×C₂₃	1	8	yes	complemented
297	#547	—	2	4	no	—
298	#547	C₂²×C₂₃	1	8	yes	—
299	C₂×C₄⋊Q₈	—	2	1	no	—
300	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×#123	—	2	2	no	—
303	C₂×C₂²⋊Q₈	—	2	1	no	—
304	C₂×C₂²⋊Q₈	—	2	2	no	—
305	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
306	C₂×#121	C₂²×C₂₃	1	2	yes	—
307	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄×D₄	—	2	1	no	—
309	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
311	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
312	#487	C₂²×C₂₃	1	8	yes	complemented
313	#483	C₂²×C₂₃	1	16	yes	complemented
314	#475	C₂²×C₂₃	1	8	yes	complemented
315	#465	C₂²×C₂₃	1	16	yes	complemented
316	#461	C₂²×C₂₃	1	8	yes	complemented
317	#448	C₂²×C₂₃	1	16	yes	complemented
318	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
319	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
320	C₂×#106	C₂²×C₂₃	1	2	yes	—
321	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#103	C₂²×C₂₃	1	2	yes	—
324	C₂×#103	C₂²×C₂₃	1	2	yes	—
325	C₂×#103	C₂²×C₂₃	1	2	yes	—
326	C₂×#103	C₂²×C₂₃	1	2	yes	—
327	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#103	C₂²×C₂₃	1	16	yes	complemented
329	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
331	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×C₂₃	—	2	8	no	—
333	C₂×C₂₃	—	8	2	no	—
334	C₂×C₂₃	#3298	1	4	yes	direct product factor
335	C₂×C₂₃	C₂×#546	1	1	yes	characteristic
336	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
337	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
338	C₂×#41	—	2	8	no	—
339	C₂×#41	—	4	8	no	—
340	C₂²×Q₈	—	2	2	no	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×Q₈	—	2	4	no	—
343	C₂²×Q₈	—	2	8	no	—
344	C₂²×Q₈	—	4	4	no	—
345	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
346	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
347	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
350	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
351	C₂³×C₄	—	2	2	no	—
352	C₂³×C₄	—	2	2	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
355	C₂×SD₁₆	—	2	4	no	—
356	C₂×SD₁₆	—	4	4	no	—
357	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
359	C₂²×C₈	—	4	4	no	—
360	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
362	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
364	C₄⋊Q₈	—	2	8	no	—
365	C₄⋊Q₈	—	2	2	no	—
366	C₄⋊Q₈	—	2	2	no	—
367	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
368	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
369	#123	—	2	16	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	—	2	8	no	—
372	C₂²⋊Q₈	—	2	16	no	—
373	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
374	#121	—	2	4	no	—
375	#121	C₂³×C₂₃	1	8	yes	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
378	C₄×D₄	—	2	8	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	D₄×C₂₃	1	4	yes	—
381	C₄×D₄	C₂³×C₂₃	1	4	yes	—
382	C₄×D₄	C₂³×C₂₃	1	4	yes	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	4	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
394	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
401	#106	C₂³×C₂₃	1	4	yes	—
402	#106	C₂³×C₂₃	1	8	yes	—
403	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
404	#103	C₂³×C₂₃	1	4	yes	—
405	#103	C₂³×C₂₃	1	8	yes	—
406	#103	C₂³×C₂₃	1	8	yes	—
407	#103	C₂³×C₂₃	1	8	yes	—
408	#103	C₂³×C₂₃	1	8	yes	—
409	#102	C₂³×C₂₃	1	4	yes	—
410	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
411	C₄×C₈	C₂³×C₂₃	1	4	yes	—
412	C₂₃	C₂×#3298	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	—	2	1	no	—
414	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
415	#41	—	4	32	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	16	no	—
422	C₂×Q₈	—	2	4	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	4	8	no	—
425	C₂×Q₈	—	4	8	no	—
426	C₂×Q₈	—	4	16	no	—
427	C₂×Q₈	—	4	4	no	—
428	C₂×Q₈	#41×C₂₃	1	4	yes	complemented
429	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
430	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	4	8	no	—
436	C₂×D₄	SD₁₆×C₂₃	1	8	yes	complemented
437	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
438	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	—	4	8	no	—
451	C₂²×C₄	—	4	4	no	—
452	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
458	SD₁₆	—	2	8	no	—
459	SD₁₆	—	4	8	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	4	no	—
463	C₂×C₈	—	4	8	no	—
464	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
465	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
466	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	8	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
478	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
484	C₄²	—	2	4	no	—
485	C₄²	C₂×D₄×C₂₃	1	2	yes	—
486	C₄²	C₂⁴×C₂₃	1	2	yes	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	2	no	—
493	C₂³	C₂×SD₁₆×C₂₃	1	8	yes	—
494	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
495	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
496	Q₈	—	2	2	no	—
497	Q₈	—	2	4	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	2	4	no	—
500	Q₈	—	4	16	no	—
501	Q₈	—	4	8	no	—
502	Q₈	—	4	4	no	—
503	Q₈	C₂×#41×C₂₃	1	4	yes	complemented
504	D₄	—	2	2	no	—
505	D₄	—	2	2	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	2	no	—
508	D₄	C₂×SD₁₆×C₂₃	1	16	yes	complemented
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	16	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	16	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
528	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
532	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
533	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
534	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
535	C₈	—	2	2	no	—
536	C₈	—	2	2	no	—
537	C₈	—	2	4	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	16	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	8	4	no	—
545	C₂²	C₂₃×#546	1	2	yes	—
546	C₂²	C₂²×SD₁₆×C₂₃	1	8	yes	—
547	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
548	C₂²	C₂₃×#578	1	2	yes	—
549	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	C₂₃×#546	1	2	yes	—
558	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
559	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
560	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
561	C₂	—	2	8	no	—
562	C₂	—	8	2	no	—
563	C₂	C₂₃×#3298	1	4	yes	direct product factor
564	C₂	C₂×C₂₃×#546	1	1	yes	characteristic
565	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
566	C₂	C₂×C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#3298	C₂	direct (N × H)
4. C₂×C₂₃×#487	C₂	semidirect
5. C₂×C₂₃×#483	C₂	semidirect
6. C₂×C₂₃×#475	C₂	semidirect
7. C₂×C₂₃×#465	C₂	semidirect
8. C₂×C₂₃×#461	C₂	semidirect
9. C₂×C₂₃×#448	C₂	semidirect
10. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂²×C₂₃×#103	C₂	semidirect
19. D₄×Q₈×C₂₃	C₂²	semidirect
35. C₂₃×#487	C₂²	semidirect
36. C₂₃×#483	C₂²	semidirect
37. C₂₃×#475	C₂²	semidirect
38. C₂₃×#465	C₂²	semidirect
39. C₂₃×#461	C₂²	semidirect
40. C₂₃×#448	C₂²	semidirect
41. C₄×SD₁₆×C₂₃	C₂²	semidirect
42. C₈×D₄×C₂₃	C₂²	semidirect
51. C₂×C₂₃×#103	C₂²	semidirect
144. C₂×Q₈×C₂₃	#41	semidirect
152. C₂×D₄×C₂₃	SD₁₆	semidirect
203. C₂×#3298	C₂₃	direct (N × H)
220. Q₈×C₂₃	C₂×#41	semidirect
225. D₄×C₂₃	C₂×SD₁₆	semidirect
255. C₂×D₄×Q₈	C₂×C₂₃	semidirect
257. #3298	C₂×C₂₃	direct (N × H)
258. C₂×#487	C₂×C₂₃	semidirect
259. C₂×#483	C₂×C₂₃	semidirect
260. C₂×#475	C₂×C₂₃	semidirect
261. C₂×#465	C₂×C₂₃	semidirect
262. C₂×#461	C₂×C₂₃	semidirect
263. C₂×#448	C₂×C₂₃	semidirect
264. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
265. C₂×C₈×D₄	C₂×C₂₃	semidirect
266. C₂²×#103	C₂×C₂₃	semidirect
296. D₄×Q₈	C₂²×C₂₃	semidirect
312. #487	C₂²×C₂₃	semidirect
313. #483	C₂²×C₂₃	semidirect
314. #475	C₂²×C₂₃	semidirect
315. #465	C₂²×C₂₃	semidirect
316. #461	C₂²×C₂₃	semidirect
317. #448	C₂²×C₂₃	semidirect
318. C₄×SD₁₆	C₂²×C₂₃	semidirect
319. C₈×D₄	C₂²×C₂₃	semidirect
328. C₂×#103	C₂²×C₂₃	semidirect
334. C₂×C₂₃	#3298	direct (N × H)
412. C₂₃	C₂×#3298	direct (N × H)
428. C₂×Q₈	#41×C₂₃	semidirect
436. C₂×D₄	SD₁₆×C₂₃	semidirect
503. Q₈	C₂×#41×C₂₃	semidirect
508. D₄	C₂×SD₁₆×C₂₃	semidirect
563. C₂	C₂₃×#3298	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×D₄×Q₈×C₂₃	2
C₂×C₂₃×#547	2
C₂₃×#3298	2
C₂×C₂₃×#487	2
C₂×C₂₃×#483	2
C₂×C₂₃×#475	2
C₂×C₂₃×#465	2
C₂×C₂₃×#461	2
C₂×C₂₃×#448	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂²×C₂₃×#103	2
C₂×#3298	23

Subgroups — C2×C23×#3298 #19208742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3298 #19208742