Subgroups — C₂×C₂₃×#3500 #19210432

Order 5888.

11534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3500	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₂₃×#548	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#546	C₂	1	12	yes	maximal, complemented
6	C₂×C₂₃×#536	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#535	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×#532	C₂	1	3	yes	maximal, complemented
9	C₂²×C₂₃×#124	C₂	1	3	yes	maximal, complemented
10	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×2₋¹⁺⁴	—	2	16	no	—
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	16	yes	complemented
13	C₂×C₂₃×H(5,𝔽₂)	—	2	16	no	—
14	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	16	yes	complemented
15	C₂²×#41×C₂₃	C₂²	1	3	yes	—
16	C₂²×#41×C₂₃	C₂²	1	2	yes	—
17	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×#41×C₂₃	C₂²	1	3	yes	complemented
19	C₂²×#41×C₂₃	C₂²	1	6	yes	complemented
20	C₂³×Q₈×C₂₃	C₂²	1	3	yes	complemented
21	C₂³×Q₈×C₂₃	C₂²	1	2	yes	complemented
22	C₂³×D₄×C₂₃	C₂²	1	3	yes	—
23	C₂³×D₄×C₂₃	C₂²	1	6	yes	—
24	C₂₃×#548	C₂²	1	64	yes	complemented
25	C₂₃×#546	C₂²	1	192	yes	complemented
26	C₂₃×#536	C₂²	1	48	yes	complemented
27	C₂₃×#535	C₂²	1	48	yes	complemented
28	C₂₃×#532	C₂²	1	48	yes	complemented
29	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
30	C₂×C₂₃×#127	C₂²	1	3	yes	complemented
31	C₂×C₂₃×#124	C₂²	1	3	yes	complemented
32	C₂×C₂₃×#124	C₂²	1	6	yes	complemented
33	C₂×C₂₃×#124	C₂²	1	12	yes	complemented
34	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₂₃×#124	C₂²	1	48	yes	complemented
36	C₂×C₂₃×#123	C₂²	1	12	yes	complemented
37	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	complemented
38	C₂×C₂₃×#121	C₂²	1	12	yes	complemented
39	C₂×C₂₃×#121	C₂²	1	12	yes	complemented
40	C₂×C₂₃×#120	C₂²	1	12	yes	complemented
41	C₄×#41×C₂₃	C₂²	1	16	yes	complemented
42	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
43	C₂×C₄×D₄×C₂₃	C₂²	1	6	yes	complemented
44	C₂×C₄×D₄×C₂₃	C₂²	1	12	yes	complemented
45	C₂×C₂₃×#117	C₂²	1	3	yes	complemented
46	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	complemented
47	C₂²×C₄²×C₂₃	C₂²	1	3	yes	complemented
48	C₂⁵×C₂₃	C₂³	1	6	yes	—
49	C₂₃×2₋¹⁺⁴	—	2	128	no	—
50	C₂₃×H(5,𝔽₂)	—	2	128	no	—
51	C₂×#41×C₂₃	—	2	16	no	—
52	C₂×#41×C₂₃	—	2	48	no	—
53	C₂×#41×C₂₃	—	2	24	no	—
54	C₂×#41×C₂₃	—	2	24	no	—
55	C₂×#41×C₂₃	D₄	1	16	yes	complemented
56	C₂×#41×C₂₃	C₂³	1	16	yes	—
57	C₂×#41×C₂₃	C₂³	1	24	yes	—
58	C₂×#41×C₂₃	C₂³	1	8	yes	—
59	C₂×#41×C₂₃	C₂³	1	24	yes	complemented
60	C₂×#41×C₂₃	C₂³	1	48	yes	complemented
61	C₂²×Q₈×C₂₃	—	2	16	no	—
62	C₂²×Q₈×C₂₃	—	2	24	no	—
63	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₂₃	C₂³	1	16	yes	complemented
68	C₂²×Q₈×C₂₃	C₂³	1	24	yes	complemented
69	C₂²×D₄×C₂₃	—	2	48	no	—
70	C₂²×D₄×C₂₃	—	2	24	no	—
71	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
72	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
73	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
74	C₂²×D₄×C₂₃	C₂³	1	24	yes	—
75	C₂²×D₄×C₂₃	C₂³	1	48	yes	—
76	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
77	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
79	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
80	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
81	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
82	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
83	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
84	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
85	C₂₃×C₄⋊Q₈	C₂³	1	24	yes	complemented
86	C₂₃×#127	C₂³	1	24	yes	complemented
87	C₂₃×#124	C₂³	1	96	yes	complemented
88	C₂₃×#124	C₂³	1	24	yes	complemented
89	C₂₃×#124	C₂³	1	48	yes	complemented
90	C₂₃×#124	C₂³	1	8	yes	complemented
91	C₂₃×#123	C₂³	1	96	yes	complemented
92	C₂₃×C₂²⋊Q₈	C₂³	1	96	yes	complemented
93	C₂₃×#121	C₂³	1	96	yes	complemented
94	C₂₃×#121	C₂³	1	96	yes	complemented
95	C₂₃×#120	C₂³	1	96	yes	complemented
96	C₄×Q₈×C₂₃	C₂³	1	16	yes	complemented
97	C₄×D₄×C₂₃	C₂³	1	96	yes	complemented
98	C₄×D₄×C₂₃	C₂³	1	48	yes	complemented
99	C₂₃×#117	C₂³	1	24	yes	complemented
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
104	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	96	yes	complemented
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
109	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	24	yes	complemented
111	C₂⁴×C₂₃	—	2	24	no	—
112	C₂⁴×C₂₃	—	4	24	no	—
113	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
114	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
115	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
116	C₂⁴×C₂₃	C₂⁴	1	24	yes	—
117	#41×C₂₃	—	2	64	no	—
118	#41×C₂₃	—	2	192	no	—
119	#41×C₂₃	—	2	96	no	—
120	#41×C₂₃	—	2	96	no	—
121	#41×C₂₃	C₂×D₄	1	64	yes	complemented
122	C₂×Q₈×C₂₃	—	2	64	no	—
123	C₂×Q₈×C₂₃	—	2	24	no	—
124	C₂×Q₈×C₂₃	—	2	12	no	—
125	C₂×Q₈×C₂₃	—	2	96	no	—
126	C₂×Q₈×C₂₃	—	2	4	no	—
127	C₂×Q₈×C₂₃	C₂×D₄	1	8	yes	—
128	C₂×Q₈×C₂₃	C₂⁴	1	24	yes	—
129	C₂×Q₈×C₂₃	C₂⁴	1	12	yes	—
130	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
131	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
132	C₂×D₄×C₂₃	—	2	24	no	—
133	C₂×D₄×C₂₃	—	2	24	no	—
134	C₂×D₄×C₂₃	—	2	192	no	—
135	C₂×D₄×C₂₃	—	2	24	no	—
136	C₂×D₄×C₂₃	—	2	12	no	—
137	C₂×D₄×C₂₃	—	2	96	no	—
138	C₂×D₄×C₂₃	—	2	4	no	—
139	C₂×D₄×C₂₃	—	4	24	no	—
140	C₂×D₄×C₂₃	C₂×D₄	1	24	yes	—
141	C₂×D₄×C₂₃	C₂⁴	1	24	yes	—
142	C₂×D₄×C₂₃	C₂⁴	1	24	yes	—
143	C₂×D₄×C₂₃	C₂⁴	1	24	yes	—
144	C₂×D₄×C₂₃	C₂⁴	1	12	yes	—
145	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
146	C₂×D₄×C₂₃	C₂⁴	1	48	yes	—
147	C₂×D₄×C₂₃	C₂⁴	1	12	yes	—
148	C₂²×C₄×C₂₃	—	2	24	no	—
149	C₂²×C₄×C₂₃	—	2	24	no	—
150	C₂²×C₄×C₂₃	—	2	24	no	—
151	C₂²×C₄×C₂₃	—	4	12	no	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
166	C₂²⋊C₄×C₂₃	C₂⁴	1	48	yes	—
167	C₂²⋊C₄×C₂₃	C₂⁴	1	96	yes	—
168	C₂²⋊C₄×C₂₃	C₂⁴	1	24	yes	—
169	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
170	C₄²×C₂₃	C₂⁴	1	4	yes	—
171	C₄²×C₂₃	C₂⁴	1	12	yes	—
172	C₄²×C₂₃	C₂⁴	1	24	yes	—
173	C₂×#3500	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₂₃	—	2	12	no	—
175	C₂³×C₂₃	—	2	4	no	—
176	C₂³×C₂₃	—	2	12	no	—
177	C₂³×C₂₃	—	2	8	no	—
178	C₂³×C₂₃	—	2	24	no	—
179	C₂³×C₂₃	—	2	48	no	—
180	C₂³×C₂₃	—	4	48	no	—
181	C₂³×C₂₃	—	8	48	no	—
182	C₂³×C₂₃	C₂²×D₄	1	12	yes	—
183	C₂³×C₂₃	C₂⁵	1	12	yes	—
184	C₂³×C₂₃	C₂⁵	1	4	yes	—
185	C₂³×C₂₃	C₂⁵	1	6	yes	—
186	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
187	Q₈×C₂₃	—	2	48	no	—
188	Q₈×C₂₃	—	2	24	no	—
189	Q₈×C₂₃	—	2	8	no	—
190	Q₈×C₂₃	C₂²×D₄	1	16	yes	—
191	D₄×C₂₃	—	2	48	no	—
192	D₄×C₂₃	—	2	48	no	—
193	D₄×C₂₃	—	2	48	no	—
194	D₄×C₂₃	—	2	24	no	—
195	D₄×C₂₃	—	2	8	no	—
196	D₄×C₂₃	—	4	48	no	—
197	D₄×C₂₃	C₂²×D₄	1	48	yes	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	12	no	—
200	C₂×C₄×C₂₃	—	2	48	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	12	no	—
203	C₂×C₄×C₂₃	—	2	48	no	—
204	C₂×C₄×C₂₃	—	2	48	no	—
205	C₂×C₄×C₂₃	—	4	24	no	—
206	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₂₃	C₂²×D₄	1	48	yes	—
209	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
210	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
211	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
212	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
213	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
214	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
215	C₂²×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#3500	C₂×C₂₃	1	32	yes	direct product factor
218	C₂×#548	C₂×C₂₃	1	4	yes	complemented
219	C₂×#546	C₂×C₂₃	1	12	yes	complemented
220	C₂×#536	C₂×C₂₃	1	3	yes	complemented
221	C₂×#535	C₂×C₂₃	1	3	yes	complemented
222	C₂×#532	C₂×C₂₃	1	3	yes	complemented
223	C₂²×#124	C₂×C₂₃	1	3	yes	complemented
224	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂²×C₂₃	—	2	12	no	—
226	C₂²×C₂₃	—	2	24	no	—
227	C₂²×C₂₃	—	2	4	no	—
228	C₂²×C₂₃	—	2	12	no	—
229	C₂²×C₂₃	—	2	8	no	—
230	C₂²×C₂₃	—	2	24	no	—
231	C₂²×C₂₃	—	4	8	no	—
232	C₂²×C₂₃	—	4	24	no	—
233	C₂²×C₂₃	—	8	48	no	—
234	C₂²×C₂₃	C₂³×D₄	1	12	yes	—
235	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
236	C₂²×C₂₃	#585	1	4	yes	—
237	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
238	C₄×C₂₃	—	2	4	no	—
239	C₄×C₂₃	—	2	12	no	—
240	C₄×C₂₃	—	2	4	no	—
241	C₄×C₂₃	—	2	12	no	—
242	C₄×C₂₃	C₂³×D₄	1	12	yes	—
243	C₄×C₂₃	C₂³×D₄	1	4	yes	—
244	C₂×2₋¹⁺⁴	—	2	16	no	—
245	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	16	yes	complemented
246	C₂×H(5,𝔽₂)	—	2	16	no	—
247	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	16	yes	complemented
248	C₂²×#41	C₂²×C₂₃	1	3	yes	—
249	C₂²×#41	C₂²×C₂₃	1	2	yes	—
250	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
251	C₂²×#41	C₂²×C₂₃	1	3	yes	complemented
252	C₂²×#41	C₂²×C₂₃	1	6	yes	complemented
253	C₂³×Q₈	C₂²×C₂₃	1	3	yes	complemented
254	C₂³×Q₈	C₂²×C₂₃	1	2	yes	complemented
255	C₂³×D₄	C₂²×C₂₃	1	3	yes	—
256	C₂³×D₄	C₂²×C₂₃	1	6	yes	—
257	#548	C₂²×C₂₃	1	64	yes	complemented
258	#546	C₂²×C₂₃	1	192	yes	complemented
259	#536	C₂²×C₂₃	1	48	yes	complemented
260	#535	C₂²×C₂₃	1	48	yes	complemented
261	#532	C₂²×C₂₃	1	48	yes	complemented
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	complemented
263	C₂×#127	C₂²×C₂₃	1	3	yes	complemented
264	C₂×#124	C₂²×C₂₃	1	3	yes	complemented
265	C₂×#124	C₂²×C₂₃	1	6	yes	complemented
266	C₂×#124	C₂²×C₂₃	1	12	yes	complemented
267	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
268	C₂×#124	C₂²×C₂₃	1	48	yes	complemented
269	C₂×#123	C₂²×C₂₃	1	12	yes	complemented
270	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	complemented
271	C₂×#121	C₂²×C₂₃	1	12	yes	complemented
272	C₂×#121	C₂²×C₂₃	1	12	yes	complemented
273	C₂×#120	C₂²×C₂₃	1	12	yes	complemented
274	C₄×#41	C₂²×C₂₃	1	16	yes	complemented
275	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
276	C₂×C₄×D₄	C₂²×C₂₃	1	6	yes	complemented
277	C₂×C₄×D₄	C₂²×C₂₃	1	12	yes	complemented
278	C₂×#117	C₂²×C₂₃	1	3	yes	complemented
279	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	complemented
280	C₂²×C₄²	C₂²×C₂₃	1	3	yes	complemented
281	C₂×C₂₃	—	2	12	no	—
282	C₂×C₂₃	—	4	4	no	—
283	C₂×C₂₃	—	4	12	no	—
284	C₂×C₂₃	#3500	1	4	yes	direct product factor
285	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
286	C₂×C₂₃	C₂×#585	1	2	yes	—
287	C₂⁵	C₂³×C₂₃	1	6	yes	—
288	2₋¹⁺⁴	—	2	128	no	—
289	H(5,𝔽₂)	—	2	128	no	—
290	C₂×#41	—	2	16	no	—
291	C₂×#41	—	2	48	no	—
292	C₂×#41	—	2	24	no	—
293	C₂×#41	—	2	24	no	—
294	C₂×#41	D₄×C₂₃	1	16	yes	complemented
295	C₂×#41	C₂³×C₂₃	1	24	yes	—
296	C₂×#41	C₂³×C₂₃	1	16	yes	—
297	C₂×#41	C₂³×C₂₃	1	8	yes	—
298	C₂×#41	C₂³×C₂₃	1	24	yes	complemented
299	C₂×#41	C₂³×C₂₃	1	48	yes	complemented
300	C₂²×Q₈	—	2	16	no	—
301	C₂²×Q₈	—	2	24	no	—
302	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
303	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
304	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
305	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
306	C₂²×Q₈	C₂³×C₂₃	1	24	yes	complemented
307	C₂²×Q₈	C₂³×C₂₃	1	16	yes	complemented
308	C₂²×D₄	—	2	48	no	—
309	C₂²×D₄	—	2	24	no	—
310	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
311	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
312	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
313	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
314	C₂²×D₄	C₂³×C₂₃	1	24	yes	—
315	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₂²×D₄	C₂³×C₂₃	1	48	yes	—
317	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
318	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
319	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
320	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
321	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
322	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
323	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
324	C₄⋊Q₈	C₂³×C₂₃	1	24	yes	complemented
325	#127	C₂³×C₂₃	1	24	yes	complemented
326	#124	C₂³×C₂₃	1	24	yes	complemented
327	#124	C₂³×C₂₃	1	48	yes	complemented
328	#124	C₂³×C₂₃	1	96	yes	complemented
329	#124	C₂³×C₂₃	1	8	yes	complemented
330	#123	C₂³×C₂₃	1	96	yes	complemented
331	C₂²⋊Q₈	C₂³×C₂₃	1	96	yes	complemented
332	#121	C₂³×C₂₃	1	96	yes	complemented
333	#121	C₂³×C₂₃	1	96	yes	complemented
334	#120	C₂³×C₂₃	1	96	yes	complemented
335	C₄×Q₈	C₂³×C₂₃	1	16	yes	complemented
336	C₄×D₄	C₂³×C₂₃	1	48	yes	complemented
337	C₄×D₄	C₂³×C₂₃	1	96	yes	complemented
338	#117	C₂³×C₂₃	1	24	yes	complemented
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₂₃	1	96	yes	complemented
346	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
347	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
348	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₂₃	1	24	yes	complemented
350	C₂₃	C₂×#3500	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	24	no	—
352	C₂⁴	—	4	24	no	—
353	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
354	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
355	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
356	C₂⁴	C₂⁴×C₂₃	1	24	yes	—
357	#41	—	2	64	no	—
358	#41	—	2	192	no	—
359	#41	—	2	96	no	—
360	#41	—	2	96	no	—
361	#41	C₂×D₄×C₂₃	1	64	yes	complemented
362	C₂×Q₈	—	2	64	no	—
363	C₂×Q₈	—	2	24	no	—
364	C₂×Q₈	—	2	12	no	—
365	C₂×Q₈	—	2	96	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	C₂×D₄×C₂₃	1	8	yes	—
368	C₂×Q₈	C₂⁴×C₂₃	1	24	yes	—
369	C₂×Q₈	C₂⁴×C₂₃	1	12	yes	—
370	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
371	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
372	C₂×D₄	—	2	24	no	—
373	C₂×D₄	—	2	24	no	—
374	C₂×D₄	—	2	192	no	—
375	C₂×D₄	—	2	24	no	—
376	C₂×D₄	—	2	12	no	—
377	C₂×D₄	—	2	96	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	4	24	no	—
380	C₂×D₄	C₂×D₄×C₂₃	1	24	yes	—
381	C₂×D₄	C₂⁴×C₂₃	1	24	yes	—
382	C₂×D₄	C₂⁴×C₂₃	1	24	yes	—
383	C₂×D₄	C₂⁴×C₂₃	1	24	yes	—
384	C₂×D₄	C₂⁴×C₂₃	1	12	yes	—
385	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
386	C₂×D₄	C₂⁴×C₂₃	1	48	yes	—
387	C₂×D₄	C₂⁴×C₂₃	1	12	yes	—
388	C₂²×C₄	—	2	24	no	—
389	C₂²×C₄	—	2	24	no	—
390	C₂²×C₄	—	2	24	no	—
391	C₂²×C₄	—	4	12	no	—
392	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
393	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
395	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
396	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
400	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
401	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
402	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
403	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
404	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
405	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
406	C₂²⋊C₄	C₂⁴×C₂₃	1	48	yes	—
407	C₂²⋊C₄	C₂⁴×C₂₃	1	96	yes	—
408	C₂²⋊C₄	C₂⁴×C₂₃	1	24	yes	—
409	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
410	C₄²	C₂⁴×C₂₃	1	4	yes	—
411	C₄²	C₂⁴×C₂₃	1	12	yes	—
412	C₄²	C₂⁴×C₂₃	1	24	yes	—
413	C₂³	—	2	12	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	12	no	—
416	C₂³	—	2	8	no	—
417	C₂³	—	2	24	no	—
418	C₂³	—	2	48	no	—
419	C₂³	—	4	48	no	—
420	C₂³	—	8	48	no	—
421	C₂³	C₂²×D₄×C₂₃	1	12	yes	—
422	C₂³	C₂⁵×C₂₃	1	12	yes	—
423	C₂³	C₂⁵×C₂₃	1	4	yes	—
424	C₂³	C₂⁵×C₂₃	1	6	yes	—
425	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
426	Q₈	—	2	48	no	—
427	Q₈	—	2	24	no	—
428	Q₈	—	2	8	no	—
429	Q₈	C₂²×D₄×C₂₃	1	16	yes	—
430	D₄	—	2	48	no	—
431	D₄	—	2	48	no	—
432	D₄	—	2	48	no	—
433	D₄	—	2	24	no	—
434	D₄	—	2	8	no	—
435	D₄	—	4	48	no	—
436	D₄	C₂²×D₄×C₂₃	1	48	yes	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	12	no	—
439	C₂×C₄	—	2	48	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	12	no	—
442	C₂×C₄	—	2	48	no	—
443	C₂×C₄	—	2	48	no	—
444	C₂×C₄	—	4	24	no	—
445	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
446	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
447	C₂×C₄	C₂²×D₄×C₂₃	1	48	yes	—
448	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
449	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
450	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
451	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
452	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
453	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
454	C₂²	—	2	24	no	—
455	C₂²	—	2	12	no	—
456	C₂²	—	2	4	no	—
457	C₂²	—	2	12	no	—
458	C₂²	—	2	8	no	—
459	C₂²	—	2	24	no	—
460	C₂²	—	4	24	no	—
461	C₂²	—	4	8	no	—
462	C₂²	—	8	48	no	—
463	C₂²	C₂³×D₄×C₂₃	1	12	yes	—
464	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
465	C₂²	C₂₃×#585	1	4	yes	—
466	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
467	C₄	—	2	4	no	—
468	C₄	—	2	12	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	12	no	—
471	C₄	C₂³×D₄×C₂₃	1	12	yes	—
472	C₄	C₂³×D₄×C₂₃	1	4	yes	—
473	C₂	—	2	12	no	—
474	C₂	—	4	12	no	—
475	C₂	—	4	4	no	—
476	C₂	C₂₃×#3500	1	4	yes	direct product factor
477	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
478	C₂	C₂×C₂₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×2₋¹⁺⁴	C₂	semidirect
2. C₂²×C₂₃×H(5,𝔽₂)	C₂	semidirect
3. C₂₃×#3500	C₂	direct (N × H)
4. C₂×C₂₃×#548	C₂	semidirect
5. C₂×C₂₃×#546	C₂	semidirect
6. C₂×C₂₃×#536	C₂	semidirect
7. C₂×C₂₃×#535	C₂	semidirect
8. C₂×C₂₃×#532	C₂	semidirect
9. C₂²×C₂₃×#124	C₂	semidirect
10. C₂×C₄×#41×C₂₃	C₂	semidirect
12. C₂×C₂₃×2₋¹⁺⁴	C₂²	semidirect
14. C₂×C₂₃×H(5,𝔽₂)	C₂²	semidirect
18. C₂²×#41×C₂₃	C₂²	semidirect
19. C₂²×#41×C₂₃	C₂²	semidirect
20. C₂³×Q₈×C₂₃	C₂²	semidirect
21. C₂³×Q₈×C₂₃	C₂²	semidirect
24. C₂₃×#548	C₂²	semidirect
25. C₂₃×#546	C₂²	semidirect
26. C₂₃×#536	C₂²	semidirect
27. C₂₃×#535	C₂²	semidirect
28. C₂₃×#532	C₂²	semidirect
29. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
30. C₂×C₂₃×#127	C₂²	semidirect
31. C₂×C₂₃×#124	C₂²	semidirect
32. C₂×C₂₃×#124	C₂²	semidirect
33. C₂×C₂₃×#124	C₂²	semidirect
34. C₂×C₂₃×#124	C₂²	semidirect
35. C₂×C₂₃×#124	C₂²	semidirect
36. C₂×C₂₃×#123	C₂²	semidirect
37. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
38. C₂×C₂₃×#121	C₂²	semidirect
39. C₂×C₂₃×#121	C₂²	semidirect
40. C₂×C₂₃×#120	C₂²	semidirect
41. C₄×#41×C₂₃	C₂²	semidirect
42. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
43. C₂×C₄×D₄×C₂₃	C₂²	semidirect
44. C₂×C₄×D₄×C₂₃	C₂²	semidirect
45. C₂×C₂₃×#117	C₂²	semidirect
46. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
47. C₂²×C₄²×C₂₃	C₂²	semidirect
55. C₂×#41×C₂₃	D₄	semidirect
59. C₂×#41×C₂₃	C₂³	semidirect
60. C₂×#41×C₂₃	C₂³	semidirect
67. C₂²×Q₈×C₂₃	C₂³	semidirect
68. C₂²×Q₈×C₂₃	C₂³	semidirect
85. C₂₃×C₄⋊Q₈	C₂³	semidirect
86. C₂₃×#127	C₂³	semidirect
87. C₂₃×#124	C₂³	semidirect
88. C₂₃×#124	C₂³	semidirect
89. C₂₃×#124	C₂³	semidirect
90. C₂₃×#124	C₂³	semidirect
91. C₂₃×#123	C₂³	semidirect
92. C₂₃×C₂²⋊Q₈	C₂³	semidirect
93. C₂₃×#121	C₂³	semidirect
94. C₂₃×#121	C₂³	semidirect
95. C₂₃×#120	C₂³	semidirect
96. C₄×Q₈×C₂₃	C₂³	semidirect
97. C₄×D₄×C₂₃	C₂³	semidirect
98. C₄×D₄×C₂₃	C₂³	semidirect
99. C₂₃×#117	C₂³	semidirect
106. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
110. C₂×C₄²×C₂₃	C₂³	semidirect
121. #41×C₂₃	C₂×D₄	semidirect
173. C₂×#3500	C₂₃	direct (N × H)
215. C₂²×2₋¹⁺⁴	C₂×C₂₃	semidirect
216. C₂²×H(5,𝔽₂)	C₂×C₂₃	semidirect
217. #3500	C₂×C₂₃	direct (N × H)
218. C₂×#548	C₂×C₂₃	semidirect
219. C₂×#546	C₂×C₂₃	semidirect
220. C₂×#536	C₂×C₂₃	semidirect
221. C₂×#535	C₂×C₂₃	semidirect
222. C₂×#532	C₂×C₂₃	semidirect
223. C₂²×#124	C₂×C₂₃	semidirect
224. C₂×C₄×#41	C₂×C₂₃	semidirect
245. C₂×2₋¹⁺⁴	C₂²×C₂₃	semidirect
247. C₂×H(5,𝔽₂)	C₂²×C₂₃	semidirect
251. C₂²×#41	C₂²×C₂₃	semidirect
252. C₂²×#41	C₂²×C₂₃	semidirect
253. C₂³×Q₈	C₂²×C₂₃	semidirect
254. C₂³×Q₈	C₂²×C₂₃	semidirect
257. #548	C₂²×C₂₃	semidirect
258. #546	C₂²×C₂₃	semidirect
259. #536	C₂²×C₂₃	semidirect
260. #535	C₂²×C₂₃	semidirect
261. #532	C₂²×C₂₃	semidirect
262. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
263. C₂×#127	C₂²×C₂₃	semidirect
264. C₂×#124	C₂²×C₂₃	semidirect
265. C₂×#124	C₂²×C₂₃	semidirect
266. C₂×#124	C₂²×C₂₃	semidirect
267. C₂×#124	C₂²×C₂₃	semidirect
268. C₂×#124	C₂²×C₂₃	semidirect
269. C₂×#123	C₂²×C₂₃	semidirect
270. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
271. C₂×#121	C₂²×C₂₃	semidirect
272. C₂×#121	C₂²×C₂₃	semidirect
273. C₂×#120	C₂²×C₂₃	semidirect
274. C₄×#41	C₂²×C₂₃	semidirect
275. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
276. C₂×C₄×D₄	C₂²×C₂₃	semidirect
277. C₂×C₄×D₄	C₂²×C₂₃	semidirect
278. C₂×#117	C₂²×C₂₃	semidirect
279. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
280. C₂²×C₄²	C₂²×C₂₃	semidirect
284. C₂×C₂₃	#3500	direct (N × H)
294. C₂×#41	D₄×C₂₃	semidirect
298. C₂×#41	C₂³×C₂₃	semidirect
299. C₂×#41	C₂³×C₂₃	semidirect
306. C₂²×Q₈	C₂³×C₂₃	semidirect
307. C₂²×Q₈	C₂³×C₂₃	semidirect
324. C₄⋊Q₈	C₂³×C₂₃	semidirect
325. #127	C₂³×C₂₃	semidirect
326. #124	C₂³×C₂₃	semidirect
327. #124	C₂³×C₂₃	semidirect
328. #124	C₂³×C₂₃	semidirect
329. #124	C₂³×C₂₃	semidirect
330. #123	C₂³×C₂₃	semidirect
331. C₂²⋊Q₈	C₂³×C₂₃	semidirect
332. #121	C₂³×C₂₃	semidirect
333. #121	C₂³×C₂₃	semidirect
334. #120	C₂³×C₂₃	semidirect
335. C₄×Q₈	C₂³×C₂₃	semidirect
336. C₄×D₄	C₂³×C₂₃	semidirect
337. C₄×D₄	C₂³×C₂₃	semidirect
338. #117	C₂³×C₂₃	semidirect
345. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
349. C₂×C₄²	C₂³×C₂₃	semidirect
350. C₂₃	C₂×#3500	direct (N × H)
361. #41	C₂×D₄×C₂₃	semidirect
476. C₂	C₂₃×#3500	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂₃×2₋¹⁺⁴	2
C₂²×C₂₃×H(5,𝔽₂)	2
C₂₃×#3500	2
C₂×C₂₃×#548	2
C₂×C₂₃×#546	2
C₂×C₂₃×#536	2
C₂×C₂₃×#535	2
C₂×C₂₃×#532	2
C₂²×C₂₃×#124	2
C₂×C₄×#41×C₂₃	2
C₂×#3500	23

Subgroups — C2×C23×#3500 #19210432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3500 #19210432