Subgroups — C₂×C₂₃×#3523 #19210413

Order 5888.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3523	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#557	C₂	1	3	yes	maximal, complemented
4	C₂×C₂₃×#556	C₂	1	3	yes	maximal, complemented
5	C₂×C₂₃×#552	C₂	1	6	yes	maximal, complemented
6	C₂×D₄×Q₈×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#541	C₂	1	6	yes	maximal, complemented
9	C₂×C₂₃×#533	C₂	1	3	yes	maximal, complemented
10	C₂×C₂₃×#529	C₂	1	3	yes	maximal
11	C₂×C₂₃×#519	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#557	C₂²	1	16	yes	complemented
16	C₂₃×#557	C₂²	1	48	yes	complemented
17	C₂₃×#556	C₂²	1	48	yes	complemented
18	C₂₃×#552	C₂²	1	96	yes	complemented
19	D₄×Q₈×C₂₃	C₂²	1	32	yes	complemented
20	D₄×Q₈×C₂₃	C₂²	1	16	yes	complemented
21	C₂₃×#541	C₂²	1	96	yes	complemented
22	C₂₃×#533	C₂²	1	48	yes	complemented
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	6	yes	—
26	C₂₃×#529	C₂²	1	48	yes	—
27	C₂×C₂₃×#126	C₂²	1	6	yes	—
28	C₂×C₂₃×#126	C₂²	1	12	yes	—
29	C₂×C₂₃×#125	C₂²	1	3	yes	—
30	C₂×C₂₃×#125	C₂²	1	6	yes	—
31	C₂×C₂₃×#125	C₂²	1	12	yes	—
32	C₂×C₂₃×#125	C₂²	1	6	yes	—
33	C₂×C₂₃×#124	C₂²	1	3	yes	—
34	C₂×C₂₃×#123	C₂²	1	6	yes	—
35	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	3	yes	—
36	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	6	yes	—
37	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	6	yes	—
38	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	6	yes	—
39	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
40	C₂₃×#519	C₂²	1	32	yes	complemented
41	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
42	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
43	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
44	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
45	C₂×C₄×Q₈×C₂₃	C₂²	1	6	yes	—
46	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
47	C₂×C₄×Q₈×C₂₃	C₂²	1	6	yes	—
48	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
49	C₂×C₄×Q₈×C₂₃	C₂²	1	6	yes	—
50	C₂×C₄×Q₈×C₂₃	C₂²	1	16	yes	complemented
51	C₂×C₄×D₄×C₂₃	C₂²	1	6	yes	—
52	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
53	C₂×C₂₃×#117	C₂²	1	6	yes	—
54	C₂×C₂₃×#117	C₂²	1	6	yes	—
55	C₂×C₂₃×#117	C₂²	1	6	yes	—
56	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
57	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
58	C₂²×Q₈×C₂₃	—	2	8	no	—
59	C₂²×Q₈×C₂₃	—	2	16	no	—
60	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
61	C₂²×Q₈×C₂₃	C₂³	1	16	yes	—
62	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
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×D₄×C₂₃	C₂³	1	2	yes	—
67	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
69	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	24	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	48	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	24	yes	—
75	C₂₃×#126	C₂³	1	96	yes	—
76	C₂₃×#126	C₂³	1	48	yes	—
77	C₂₃×#125	C₂³	1	96	yes	—
78	C₂₃×#125	C₂³	1	48	yes	—
79	C₂₃×#125	C₂³	1	24	yes	—
80	C₂₃×#125	C₂³	1	48	yes	—
81	C₂₃×#124	C₂³	1	24	yes	—
82	C₂₃×#123	C₂³	1	48	yes	—
83	C₂₃×C₂²⋊Q₈	C₂³	1	48	yes	—
84	C₂₃×C₂²⋊Q₈	C₂³	1	48	yes	—
85	C₂₃×C₂²⋊Q₈	C₂³	1	48	yes	—
86	C₂₃×C₂²⋊Q₈	C₂³	1	24	yes	—
87	C₂₃×#121	C₂³	1	8	yes	—
88	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
89	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
90	C₄×Q₈×C₂₃	C₂³	1	48	yes	—
91	C₄×Q₈×C₂₃	C₂³	1	24	yes	—
92	C₄×Q₈×C₂₃	C₂³	1	48	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	24	yes	—
94	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	24	yes	—
96	C₄×Q₈×C₂₃	C₂³	1	48	yes	—
97	C₄×D₄×C₂₃	C₂³	1	48	yes	—
98	C₄×D₄×C₂₃	C₂³	1	24	yes	—
99	C₂₃×#117	C₂³	1	48	yes	—
100	C₂₃×#117	C₂³	1	48	yes	—
101	C₂₃×#117	C₂³	1	48	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
115	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
120	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
121	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
122	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
123	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
125	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
128	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
129	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
130	C₂×Q₈×C₂₃	—	2	32	no	—
131	C₂×Q₈×C₂₃	—	2	64	no	—
132	C₂×Q₈×C₂₃	—	2	12	no	—
133	C₂×Q₈×C₂₃	—	2	24	no	—
134	C₂×Q₈×C₂₃	—	2	12	no	—
135	C₂×Q₈×C₂₃	#41	1	8	yes	—
136	C₂×Q₈×C₂₃	C₂⁴	1	12	yes	—
137	C₂×Q₈×C₂₃	C₂⁴	1	24	yes	—
138	C₂×Q₈×C₂₃	C₂⁴	1	12	yes	—
139	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
140	C₂×D₄×C₂₃	—	2	8	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
143	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₂₃	—	2	12	no	—
145	C₂²×C₄×C₂₃	—	2	12	no	—
146	C₂²×C₄×C₂₃	—	2	24	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
157	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
161	C₄⋊C₄×C₂₃	C₂⁴	1	12	yes	—
162	C₄⋊C₄×C₂₃	C₂⁴	1	12	yes	—
163	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
166	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
167	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
168	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
169	C₄⋊C₄×C₂₃	C₂⁴	1	12	yes	—
170	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
171	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
172	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
173	C₄⋊C₄×C₂₃	C₂⁴	1	24	yes	—
174	C₂²⋊C₄×C₂₃	C₂⁴	1	8	yes	—
175	C₂²⋊C₄×C₂₃	C₂⁴	1	24	yes	—
176	C₂²⋊C₄×C₂₃	C₂⁴	1	24	yes	—
177	C₂²⋊C₄×C₂₃	C₂⁴	1	24	yes	—
178	C₂²⋊C₄×C₂₃	C₂⁴	1	24	yes	—
179	C₄²×C₂₃	C₂⁴	1	12	yes	—
180	C₄²×C₂₃	C₂⁴	1	12	yes	—
181	C₄²×C₂₃	C₂⁴	1	12	yes	—
182	C₄²×C₂₃	C₂⁴	1	12	yes	—
183	C₄²×C₂₃	C₂⁴	1	48	yes	—
184	C₄²×C₂₃	C₂⁴	1	24	yes	—
185	C₂×#3523	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	2	2	no	—
188	C₂³×C₂₃	—	2	4	no	—
189	C₂³×C₂₃	—	2	4	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
192	C₂³×C₂₃	C₂⁵	1	4	yes	—
193	C₂³×C₂₃	C₂⁵	1	2	yes	—
194	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
195	Q₈×C₂₃	—	2	24	no	—
196	Q₈×C₂₃	—	2	48	no	—
197	Q₈×C₂₃	—	2	24	no	—
198	Q₈×C₂₃	C₂×#41	1	16	yes	—
199	D₄×C₂₃	—	2	16	no	—
200	D₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	6	no	—
202	C₂×C₄×C₂₃	—	2	12	no	—
203	C₂×C₄×C₂₃	—	2	24	no	—
204	C₂×C₄×C₂₃	—	2	6	no	—
205	C₂×C₄×C₂₃	—	2	6	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	12	no	—
209	C₂×C₄×C₂₃	—	2	24	no	—
210	C₂×C₄×C₂₃	—	2	48	no	—
211	C₂×C₄×C₂₃	—	2	8	no	—
212	C₂×C₄×C₂₃	C₂×#41	1	12	yes	—
213	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
214	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
215	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
216	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
217	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
218	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
219	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
220	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
222	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
223	#3523	C₂×C₂₃	1	32	yes	direct product factor
224	C₂×#557	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂×#557	C₂×C₂₃	1	3	yes	complemented
226	C₂×#556	C₂×C₂₃	1	3	yes	complemented
227	C₂×#552	C₂×C₂₃	1	6	yes	complemented
228	C₂×D₄×Q₈	C₂×C₂₃	1	2	yes	complemented
229	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂×#541	C₂×C₂₃	1	6	yes	complemented
231	C₂×#533	C₂×C₂₃	1	3	yes	complemented
232	C₂×#529	C₂×C₂₃	1	3	yes	—
233	C₂×#519	C₂×C₂₃	1	2	yes	complemented
234	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	2	no	—
237	C₂²×C₂₃	—	2	8	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	4	8	no	—
242	C₂²×C₂₃	C₂²×#41	1	2	yes	—
243	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
244	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
245	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
246	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
247	C₄×C₂₃	—	2	6	no	—
248	C₄×C₂₃	—	2	12	no	—
249	C₄×C₂₃	—	2	6	no	—
250	C₄×C₂₃	—	2	6	no	—
251	C₄×C₂₃	—	2	2	no	—
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	—	2	12	no	—
254	C₄×C₂₃	C₂²×#41	1	12	yes	—
255	C₄×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
256	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
257	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
258	#557	C₂²×C₂₃	1	16	yes	complemented
259	#557	C₂²×C₂₃	1	48	yes	complemented
260	#556	C₂²×C₂₃	1	48	yes	complemented
261	#552	C₂²×C₂₃	1	96	yes	complemented
262	D₄×Q₈	C₂²×C₂₃	1	32	yes	complemented
263	D₄×Q₈	C₂²×C₂₃	1	16	yes	complemented
264	#541	C₂²×C₂₃	1	96	yes	complemented
265	#533	C₂²×C₂₃	1	48	yes	complemented
266	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
268	C₂×C₄⋊Q₈	C₂²×C₂₃	1	6	yes	—
269	#529	C₂²×C₂₃	1	48	yes	—
270	C₂×#126	C₂²×C₂₃	1	6	yes	—
271	C₂×#126	C₂²×C₂₃	1	12	yes	—
272	C₂×#125	C₂²×C₂₃	1	3	yes	—
273	C₂×#125	C₂²×C₂₃	1	6	yes	—
274	C₂×#125	C₂²×C₂₃	1	12	yes	—
275	C₂×#125	C₂²×C₂₃	1	6	yes	—
276	C₂×#124	C₂²×C₂₃	1	3	yes	—
277	C₂×#123	C₂²×C₂₃	1	6	yes	—
278	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	3	yes	—
279	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	6	yes	—
280	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	6	yes	—
281	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	6	yes	—
282	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
283	#519	C₂²×C₂₃	1	32	yes	complemented
284	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
285	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
286	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
288	C₂×C₄×Q₈	C₂²×C₂₃	1	6	yes	—
289	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
290	C₂×C₄×Q₈	C₂²×C₂₃	1	6	yes	—
291	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
292	C₂×C₄×Q₈	C₂²×C₂₃	1	6	yes	—
293	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	complemented
294	C₂×C₄×D₄	C₂²×C₂₃	1	6	yes	—
295	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
296	C₂×#117	C₂²×C₂₃	1	6	yes	—
297	C₂×#117	C₂²×C₂₃	1	6	yes	—
298	C₂×#117	C₂²×C₂₃	1	6	yes	—
299	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
300	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
301	C₂×C₂₃	—	2	4	no	—
302	C₂×C₂₃	—	4	4	no	—
303	C₂×C₂₃	#3523	1	4	yes	direct product factor
304	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
305	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
306	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂²×Q₈	—	2	8	no	—
308	C₂²×Q₈	—	2	16	no	—
309	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
310	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
311	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
312	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
313	C₂²×Q₈	C₂³×C₂₃	1	16	yes	—
314	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
316	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
318	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
319	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
320	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₂₃	1	24	yes	—
322	C₄⋊Q₈	C₂³×C₂₃	1	24	yes	—
323	C₄⋊Q₈	C₂³×C₂₃	1	48	yes	—
324	#126	C₂³×C₂₃	1	48	yes	—
325	#126	C₂³×C₂₃	1	96	yes	—
326	#125	C₂³×C₂₃	1	24	yes	—
327	#125	C₂³×C₂₃	1	48	yes	—
328	#125	C₂³×C₂₃	1	96	yes	—
329	#125	C₂³×C₂₃	1	48	yes	—
330	#124	C₂³×C₂₃	1	24	yes	—
331	#123	C₂³×C₂₃	1	48	yes	—
332	C₂²⋊Q₈	C₂³×C₂₃	1	24	yes	—
333	C₂²⋊Q₈	C₂³×C₂₃	1	48	yes	—
334	C₂²⋊Q₈	C₂³×C₂₃	1	48	yes	—
335	C₂²⋊Q₈	C₂³×C₂₃	1	48	yes	—
336	#121	C₂³×C₂₃	1	8	yes	—
337	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
338	C₄×Q₈	C₂³×C₂₃	1	24	yes	—
339	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
340	C₄×Q₈	C₂³×C₂₃	1	24	yes	—
341	C₄×Q₈	C₂³×C₂₃	1	48	yes	—
342	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
343	C₄×Q₈	C₂³×C₂₃	1	48	yes	—
344	C₄×Q₈	C₂³×C₂₃	1	24	yes	—
345	C₄×Q₈	C₂³×C₂₃	1	48	yes	—
346	C₄×D₄	C₂³×C₂₃	1	48	yes	—
347	C₄×D₄	C₂³×C₂₃	1	24	yes	—
348	#117	C₂³×C₂₃	1	48	yes	—
349	#117	C₂³×C₂₃	1	48	yes	—
350	#117	C₂³×C₂₃	1	48	yes	—
351	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
352	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
353	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
354	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
356	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
358	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
359	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
360	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
361	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
362	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
364	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
370	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
371	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
372	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
373	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
374	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
375	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
376	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
377	C₂₃	C₂×#3523	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
379	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
380	C₂×Q₈	—	2	32	no	—
381	C₂×Q₈	—	2	64	no	—
382	C₂×Q₈	—	2	12	no	—
383	C₂×Q₈	—	2	24	no	—
384	C₂×Q₈	—	2	12	no	—
385	C₂×Q₈	#41×C₂₃	1	8	yes	—
386	C₂×Q₈	C₂⁴×C₂₃	1	12	yes	—
387	C₂×Q₈	C₂⁴×C₂₃	1	12	yes	—
388	C₂×Q₈	C₂⁴×C₂₃	1	24	yes	—
389	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
390	C₂×D₄	—	2	8	no	—
391	C₂×D₄	—	2	4	no	—
392	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
393	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
394	C₂²×C₄	—	2	12	no	—
395	C₂²×C₄	—	2	12	no	—
396	C₂²×C₄	—	2	24	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
400	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
402	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
404	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
405	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
406	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
407	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
408	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
409	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
410	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
411	C₄⋊C₄	C₂⁴×C₂₃	1	12	yes	—
412	C₄⋊C₄	C₂⁴×C₂₃	1	12	yes	—
413	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
414	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
415	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
416	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
417	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
418	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
419	C₄⋊C₄	C₂⁴×C₂₃	1	12	yes	—
420	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
421	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
422	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
423	C₄⋊C₄	C₂⁴×C₂₃	1	24	yes	—
424	C₂²⋊C₄	C₂⁴×C₂₃	1	8	yes	—
425	C₂²⋊C₄	C₂⁴×C₂₃	1	24	yes	—
426	C₂²⋊C₄	C₂⁴×C₂₃	1	24	yes	—
427	C₂²⋊C₄	C₂⁴×C₂₃	1	24	yes	—
428	C₂²⋊C₄	C₂⁴×C₂₃	1	24	yes	—
429	C₄²	C₂⁴×C₂₃	1	12	yes	—
430	C₄²	C₂⁴×C₂₃	1	12	yes	—
431	C₄²	C₂⁴×C₂₃	1	12	yes	—
432	C₄²	C₂⁴×C₂₃	1	12	yes	—
433	C₄²	C₂⁴×C₂₃	1	48	yes	—
434	C₄²	C₂⁴×C₂₃	1	24	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	4	no	—
440	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
441	C₂³	C₂⁵×C₂₃	1	4	yes	—
442	C₂³	C₂⁵×C₂₃	1	2	yes	—
443	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
444	Q₈	—	2	24	no	—
445	Q₈	—	2	48	no	—
446	Q₈	—	2	24	no	—
447	Q₈	C₂×#41×C₂₃	1	16	yes	—
448	D₄	—	2	16	no	—
449	D₄	—	2	8	no	—
450	C₂×C₄	—	2	6	no	—
451	C₂×C₄	—	2	12	no	—
452	C₂×C₄	—	2	24	no	—
453	C₂×C₄	—	2	6	no	—
454	C₂×C₄	—	2	6	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	12	no	—
458	C₂×C₄	—	2	24	no	—
459	C₂×C₄	—	2	48	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	C₂×#41×C₂₃	1	12	yes	—
462	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
463	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
464	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
465	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
466	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
467	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
468	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
469	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
470	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
471	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	8	no	—
479	C₂²	C₂²×#41×C₂₃	1	2	yes	—
480	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
481	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
482	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
483	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
484	C₄	—	2	6	no	—
485	C₄	—	2	12	no	—
486	C₄	—	2	6	no	—
487	C₄	—	2	6	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	4	no	—
490	C₄	—	2	12	no	—
491	C₄	C₂²×#41×C₂₃	1	12	yes	—
492	C₄	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
493	C₂	—	2	4	no	—
494	C₂	—	4	4	no	—
495	C₂	C₂₃×#3523	1	4	yes	direct product factor
496	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic
497	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
498	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
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. C₂₃×#3523	C₂	direct (N × H)
2. C₂×C₂₃×#557	C₂	semidirect
3. C₂×C₂₃×#557	C₂	semidirect
4. C₂×C₂₃×#556	C₂	semidirect
5. C₂×C₂₃×#552	C₂	semidirect
6. C₂×D₄×Q₈×C₂₃	C₂	semidirect
7. C₂×D₄×Q₈×C₂₃	C₂	semidirect
8. C₂×C₂₃×#541	C₂	semidirect
9. C₂×C₂₃×#533	C₂	semidirect
11. C₂×C₂₃×#519	C₂	semidirect
12. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
15. C₂₃×#557	C₂²	semidirect
16. C₂₃×#557	C₂²	semidirect
17. C₂₃×#556	C₂²	semidirect
18. C₂₃×#552	C₂²	semidirect
19. D₄×Q₈×C₂₃	C₂²	semidirect
20. D₄×Q₈×C₂₃	C₂²	semidirect
21. C₂₃×#541	C₂²	semidirect
22. C₂₃×#533	C₂²	semidirect
40. C₂₃×#519	C₂²	semidirect
50. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
185. C₂×#3523	C₂₃	direct (N × H)
223. #3523	C₂×C₂₃	direct (N × H)
224. C₂×#557	C₂×C₂₃	semidirect
225. C₂×#557	C₂×C₂₃	semidirect
226. C₂×#556	C₂×C₂₃	semidirect
227. C₂×#552	C₂×C₂₃	semidirect
228. C₂×D₄×Q₈	C₂×C₂₃	semidirect
229. C₂×D₄×Q₈	C₂×C₂₃	semidirect
230. C₂×#541	C₂×C₂₃	semidirect
231. C₂×#533	C₂×C₂₃	semidirect
233. C₂×#519	C₂×C₂₃	semidirect
234. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
258. #557	C₂²×C₂₃	semidirect
259. #557	C₂²×C₂₃	semidirect
260. #556	C₂²×C₂₃	semidirect
261. #552	C₂²×C₂₃	semidirect
262. D₄×Q₈	C₂²×C₂₃	semidirect
263. D₄×Q₈	C₂²×C₂₃	semidirect
264. #541	C₂²×C₂₃	semidirect
265. #533	C₂²×C₂₃	semidirect
283. #519	C₂²×C₂₃	semidirect
293. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
303. C₂×C₂₃	#3523	direct (N × H)
377. C₂₃	C₂×#3523	direct (N × H)
495. C₂	C₂₃×#3523	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3523	2
C₂×C₂₃×#557	2
C₂×C₂₃×#556	2
C₂×C₂₃×#552	2
C₂×D₄×Q₈×C₂₃	2
C₂×C₂₃×#541	2
C₂×C₂₃×#533	2
C₂×C₂₃×#529	2
C₂×C₂₃×#519	2
C₂²×C₄×Q₈×C₂₃	2
C₂×#3523	23

Subgroups — C2×C23×#3523 #19210413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3523 #19210413