Subgroups — C₂×C₂₃×#2437 #19182273

Order 5888.

4926 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂₃×#2437	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#396	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#393	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#385	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#385	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#380	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#379	C₂	1	1	yes	maximal, characteristic
11	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊Q₈	—	2	16	no	—
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	16	yes	complemented
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₂²⋊C₄×C₂₃	—	2	16	no	—
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
25	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#399	C₂²	1	8	yes	complemented
33	C₂₃×#396	C₂²	1	16	yes	complemented
34	C₂₃×#393	C₂²	1	8	yes	complemented
35	C₂₃×#385	C₂²	1	16	yes	complemented
36	C₂₃×#385	C₂²	1	16	yes	complemented
37	C₂₃×#380	C₂²	1	16	yes	complemented
38	C₂₃×#379	C₂²	1	8	yes	—
39	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
41	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
42	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic, complemented
43	C₂⁵×C₂₃	—	2	8	no	—
44	C₂⁵×C₂₃	—	2	8	no	—
45	C₂⁵×C₂₃	—	4	2	no	—
46	C₂⁵×C₂₃	—	4	2	no	—
47	C₂⁵×C₂₃	Q₈	1	8	yes	complemented
48	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
49	C₂⁵×C₂₃	C₂³	1	4	yes	—
50	C₂⁵×C₂₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₂₃	—	2	8	no	—
52	C₂²×Q₈×C₂₃	—	4	2	no	—
53	C₂²×Q₈×C₂₃	C₂³	1	4	yes	complemented
54	C₂³×C₄×C₂₃	—	2	8	no	—
55	C₂³×C₄×C₂₃	—	2	8	no	—
56	C₂³×C₄×C₂₃	—	2	8	no	—
57	C₂³×C₄×C₂₃	—	2	8	no	—
58	C₂³×C₄×C₂₃	—	4	4	no	—
59	C₂³×C₄×C₂₃	—	4	4	no	—
60	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
67	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
68	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
69	C₂₃×C₂²⋊Q₈	—	2	64	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
74	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	complemented
80	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	64	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₂×C₂²⋊C₄×C₂₃	D₄	1	16	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
97	C₂₃×#95	C₂³	1	4	yes	complemented
98	C₂₃×#95	C₂³	1	8	yes	complemented
99	C₂₃×#95	C₂³	1	8	yes	complemented
100	C₂₃×#95	C₂³	1	4	yes	complemented
101	C₂⁴×C₂₃	—	2	4	no	—
102	C₂⁴×C₂₃	—	2	4	no	—
103	C₂⁴×C₂₃	—	2	4	no	—
104	C₂⁴×C₂₃	—	2	4	no	—
105	C₂⁴×C₂₃	—	2	64	no	—
106	C₂⁴×C₂₃	—	2	16	no	—
107	C₂⁴×C₂₃	—	2	4	no	—
108	C₂⁴×C₂₃	—	2	16	no	—
109	C₂⁴×C₂₃	—	2	4	no	—
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	4	16	no	—
113	C₂⁴×C₂₃	—	4	4	no	—
114	C₂⁴×C₂₃	—	4	4	no	—
115	C₂⁴×C₂₃	—	4	16	no	—
116	C₂⁴×C₂₃	—	4	32	no	—
117	C₂⁴×C₂₃	—	4	16	no	—
118	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
119	C₂⁴×C₂₃	C₂×Q₈	1	4	yes	—
120	C₂⁴×C₂₃	C₂×Q₈	1	8	yes	—
121	C₂⁴×C₂₃	C₂×Q₈	1	16	yes	complemented
122	C₂⁴×C₂₃	#41	1	8	yes	—
123	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
124	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
125	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
126	C₂×Q₈×C₂₃	—	2	16	no	—
127	C₂×Q₈×C₂₃	—	4	4	no	—
128	C₂×Q₈×C₂₃	—	4	16	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	4	no	—
138	C₂²×C₄×C₂₃	—	2	16	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	16	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	2	2	no	—
145	C₂²×C₄×C₂₃	—	2	2	no	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	—	2	16	no	—
149	C₂²×C₄×C₂₃	—	2	16	no	—
150	C₂²×C₄×C₂₃	—	4	8	no	—
151	C₂²×C₄×C₂₃	—	4	8	no	—
152	C₂²×C₄×C₂₃	—	4	32	no	—
153	C₂²×C₄×C₂₃	—	4	32	no	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
163	C₄⋊C₄×C₂₃	—	2	16	no	—
164	C₄⋊C₄×C₂₃	—	2	16	no	—
165	C₄⋊C₄×C₂₃	—	2	16	no	—
166	C₄⋊C₄×C₂₃	—	4	8	no	—
167	C₄⋊C₄×C₂₃	—	4	4	no	—
168	C₂²⋊C₄×C₂₃	—	2	16	no	—
169	C₂²⋊C₄×C₂₃	—	2	16	no	—
170	C₂²⋊C₄×C₂₃	—	2	16	no	—
171	C₂²⋊C₄×C₂₃	—	2	16	no	—
172	C₂²⋊C₄×C₂₃	—	2	16	no	—
173	C₂²⋊C₄×C₂₃	—	4	4	no	—
174	C₂²⋊C₄×C₂₃	—	4	4	no	—
175	C₂²⋊C₄×C₂₃	C₂×D₄	1	32	yes	—
176	C₂×#2437	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	16	no	—
178	C₂³×C₂₃	—	2	16	no	—
179	C₂³×C₂₃	—	2	16	no	—
180	C₂³×C₂₃	—	2	16	no	—
181	C₂³×C₂₃	—	2	16	no	—
182	C₂³×C₂₃	—	2	16	no	—
183	C₂³×C₂₃	—	2	4	no	—
184	C₂³×C₂₃	—	2	4	no	—
185	C₂³×C₂₃	—	2	4	no	—
186	C₂³×C₂₃	—	2	4	no	—
187	C₂³×C₂₃	—	2	16	no	—
188	C₂³×C₂₃	—	2	64	no	—
189	C₂³×C₂₃	—	2	8	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	2	4	no	—
192	C₂³×C₂₃	—	2	2	no	—
193	C₂³×C₂₃	—	2	2	no	—
194	C₂³×C₂₃	—	4	16	no	—
195	C₂³×C₂₃	—	4	128	no	—
196	C₂³×C₂₃	—	4	16	no	—
197	C₂³×C₂₃	—	4	32	no	—
198	C₂³×C₂₃	—	4	16	no	—
199	C₂³×C₂₃	—	4	4	no	—
200	C₂³×C₂₃	—	4	8	no	—
201	C₂³×C₂₃	—	4	4	no	—
202	C₂³×C₂₃	C₂²⋊Q₈	1	32	yes	complemented
203	C₂³×C₂₃	C₂²×D₄	1	8	yes	—
204	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
205	C₂³×C₂₃	C₂²×Q₈	1	4	yes	—
206	C₂³×C₂₃	C₂²×Q₈	1	8	yes	—
207	C₂³×C₂₃	C₂²×Q₈	1	2	yes	—
208	C₂³×C₂₃	C₂×#41	1	8	yes	—
209	C₂³×C₂₃	C₂×#41	1	4	yes	—
210	C₂³×C₂₃	H(5,𝔽₂)	1	2	yes	—
211	C₂³×C₂₃	H(5,𝔽₂)	1	2	yes	—
212	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
213	Q₈×C₂₃	—	4	16	no	—
214	C₂×C₄×C₂₃	—	2	16	no	—
215	C₂×C₄×C₂₃	—	2	16	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	4	no	—
221	C₂×C₄×C₂₃	—	2	4	no	—
222	C₂×C₄×C₂₃	—	2	4	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	2	2	no	—
231	C₂×C₄×C₂₃	—	2	4	no	—
232	C₂×C₄×C₂₃	—	2	4	no	—
233	C₂×C₄×C₂₃	—	4	32	no	—
234	C₂×C₄×C₂₃	—	4	32	no	—
235	C₂×C₄×C₂₃	—	4	8	no	—
236	C₂×C₄×C₂₃	—	4	4	no	—
237	C₂×C₄×C₂₃	—	4	8	no	—
238	C₂×C₄×C₂₃	—	4	8	no	—
239	C₂×C₄×C₂₃	—	4	4	no	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
241	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
242	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	complemented
243	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
244	#2437	C₂×C₂₃	1	16	yes	direct product factor
245	C₂×#399	C₂×C₂₃	1	1	yes	characteristic, complemented
246	C₂×#396	C₂×C₂₃	1	2	yes	complemented
247	C₂×#393	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#385	C₂×C₂₃	1	2	yes	complemented
249	C₂×#385	C₂×C₂₃	1	2	yes	complemented
250	C₂×#380	C₂×C₂₃	1	2	yes	complemented
251	C₂×#379	C₂×C₂₃	1	1	yes	characteristic
252	C₂²×C₂₃	—	2	8	no	—
253	C₂²×C₂₃	—	2	8	no	—
254	C₂²×C₂₃	—	2	64	no	—
255	C₂²×C₂₃	—	2	8	no	—
256	C₂²×C₂₃	—	2	8	no	—
257	C₂²×C₂₃	—	2	8	no	—
258	C₂²×C₂₃	—	2	8	no	—
259	C₂²×C₂₃	—	2	8	no	—
260	C₂²×C₂₃	—	2	4	no	—
261	C₂²×C₂₃	—	4	64	no	—
262	C₂²×C₂₃	—	4	16	no	—
263	C₂²×C₂₃	—	4	2	no	—
264	C₂²×C₂₃	—	4	4	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	8	yes	—
267	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	16	yes	complemented
268	C₂²×C₂₃	#543	1	4	yes	—
269	C₂²×C₂₃	D₄²	1	4	yes	—
270	C₂²×C₂₃	#546	1	8	yes	—
271	C₂²×C₂₃	#551	1	4	yes	—
272	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
273	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
274	C₂²×C₂₃	C₂²×#41	1	2	yes	—
275	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
277	C₄×C₂₃	—	2	8	no	—
278	C₄×C₂₃	—	2	8	no	—
279	C₄×C₂₃	—	4	4	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	4	no	—
282	C₄×C₂₃	—	4	4	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
285	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
286	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
287	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
288	C₂×C₂²⋊Q₈	—	2	16	no	—
289	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	16	yes	complemented
290	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
291	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
292	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
293	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
294	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
295	C₂²×C₂²⋊C₄	—	2	16	no	—
296	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
297	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
298	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
299	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
300	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
301	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
302	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
305	#399	C₂²×C₂₃	1	8	yes	complemented
306	#396	C₂²×C₂₃	1	16	yes	complemented
307	#393	C₂²×C₂₃	1	8	yes	complemented
308	#385	C₂²×C₂₃	1	16	yes	complemented
309	#385	C₂²×C₂₃	1	16	yes	complemented
310	#380	C₂²×C₂₃	1	16	yes	complemented
311	#379	C₂²×C₂₃	1	8	yes	—
312	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic, complemented
313	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
314	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
315	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic, complemented
316	C₂×C₂₃	—	2	16	no	—
317	C₂×C₂₃	—	4	4	no	—
318	C₂×C₂₃	#2437	1	8	yes	direct product factor
319	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	2	yes	—
320	C₂×C₂₃	C₂×#543	1	1	yes	characteristic
321	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
322	C₂×C₂₃	C₂×#546	1	2	yes	—
323	C₂×C₂₃	C₂×#551	1	1	yes	characteristic
324	C₂⁵	—	2	8	no	—
325	C₂⁵	—	2	8	no	—
326	C₂⁵	—	4	2	no	—
327	C₂⁵	—	4	2	no	—
328	C₂⁵	Q₈×C₂₃	1	8	yes	complemented
329	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
330	C₂⁵	C₂³×C₂₃	1	4	yes	—
331	C₂⁵	C₂³×C₂₃	1	2	yes	—
332	C₂²×Q₈	—	2	8	no	—
333	C₂²×Q₈	—	4	2	no	—
334	C₂²×Q₈	C₂³×C₂₃	1	4	yes	complemented
335	C₂³×C₄	—	2	8	no	—
336	C₂³×C₄	—	2	8	no	—
337	C₂³×C₄	—	2	8	no	—
338	C₂³×C₄	—	2	8	no	—
339	C₂³×C₄	—	4	4	no	—
340	C₂³×C₄	—	4	4	no	—
341	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
342	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
345	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
346	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
347	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
348	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
349	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
350	C₂²⋊Q₈	—	2	64	no	—
351	C₂×C₄⋊C₄	—	2	8	no	—
352	C₂×C₄⋊C₄	—	2	8	no	—
353	C₂×C₄⋊C₄	—	2	8	no	—
354	C₂×C₄⋊C₄	—	4	4	no	—
355	C₂×C₄⋊C₄	—	4	2	no	—
356	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
358	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
359	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
360	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	complemented
361	C₂×C₂²⋊C₄	—	2	8	no	—
362	C₂×C₂²⋊C₄	—	2	8	no	—
363	C₂×C₂²⋊C₄	—	2	8	no	—
364	C₂×C₂²⋊C₄	—	2	8	no	—
365	C₂×C₂²⋊C₄	—	2	8	no	—
366	C₂×C₂²⋊C₄	—	2	64	no	—
367	C₂×C₂²⋊C₄	—	4	2	no	—
368	C₂×C₂²⋊C₄	—	4	2	no	—
369	C₂×C₂²⋊C₄	D₄×C₂₃	1	16	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
378	#95	C₂³×C₂₃	1	4	yes	complemented
379	#95	C₂³×C₂₃	1	8	yes	complemented
380	#95	C₂³×C₂₃	1	8	yes	complemented
381	#95	C₂³×C₂₃	1	4	yes	complemented
382	C₂₃	C₂×#2437	1	1	yes	characteristic, direct product factor, Sylow
383	C₂⁴	—	2	4	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	2	4	no	—
387	C₂⁴	—	2	64	no	—
388	C₂⁴	—	2	16	no	—
389	C₂⁴	—	2	4	no	—
390	C₂⁴	—	2	16	no	—
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	—	4	16	no	—
395	C₂⁴	—	4	4	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	—	4	16	no	—
398	C₂⁴	—	4	16	no	—
399	C₂⁴	—	4	32	no	—
400	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
401	C₂⁴	C₂×Q₈×C₂₃	1	4	yes	—
402	C₂⁴	C₂×Q₈×C₂₃	1	8	yes	—
403	C₂⁴	C₂×Q₈×C₂₃	1	16	yes	complemented
404	C₂⁴	#41×C₂₃	1	8	yes	—
405	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
406	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
407	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
408	C₂×Q₈	—	2	16	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	16	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	16	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	16	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	16	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	32	no	—
435	C₂²×C₄	—	4	32	no	—
436	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
438	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
445	C₄⋊C₄	—	2	16	no	—
446	C₄⋊C₄	—	2	16	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	—	4	8	no	—
449	C₄⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	—	2	16	no	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	—	2	16	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	—	4	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₂₃	1	32	yes	—
458	C₂³	—	2	16	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	16	no	—
464	C₂³	—	2	16	no	—
465	C₂³	—	2	16	no	—
466	C₂³	—	2	16	no	—
467	C₂³	—	2	16	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	64	no	—
474	C₂³	—	2	8	no	—
475	C₂³	—	4	16	no	—
476	C₂³	—	4	128	no	—
477	C₂³	—	4	16	no	—
478	C₂³	—	4	32	no	—
479	C₂³	—	4	16	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	8	no	—
483	C₂³	C₂₃×C₂²⋊Q₈	1	32	yes	complemented
484	C₂³	C₂²×D₄×C₂₃	1	8	yes	—
485	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
486	C₂³	C₂²×Q₈×C₂₃	1	4	yes	—
487	C₂³	C₂²×Q₈×C₂₃	1	8	yes	—
488	C₂³	C₂²×Q₈×C₂₃	1	2	yes	—
489	C₂³	C₂×#41×C₂₃	1	8	yes	—
490	C₂³	C₂×#41×C₂₃	1	4	yes	—
491	C₂³	C₂₃×H(5,𝔽₂)	1	2	yes	—
492	C₂³	C₂₃×H(5,𝔽₂)	1	2	yes	—
493	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
494	Q₈	—	4	16	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	16	no	—
500	C₂×C₄	—	2	16	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	4	32	no	—
515	C₂×C₄	—	4	32	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
522	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
523	C₂²	—	2	8	no	—
524	C₂²	—	2	8	no	—
525	C₂²	—	2	64	no	—
526	C₂²	—	2	8	no	—
527	C₂²	—	2	8	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	2	8	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	4	64	no	—
533	C₂²	—	4	16	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	2	no	—
537	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	8	yes	—
538	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	16	yes	complemented
539	C₂²	C₂₃×#543	1	4	yes	—
540	C₂²	D₄²×C₂₃	1	4	yes	—
541	C₂²	C₂₃×#546	1	8	yes	—
542	C₂²	C₂₃×#551	1	4	yes	—
543	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
544	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
545	C₂²	C₂²×#41×C₂₃	1	2	yes	—
546	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
547	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
548	C₄	—	2	8	no	—
549	C₄	—	2	8	no	—
550	C₄	—	4	4	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	2	no	—
555	C₂	—	2	16	no	—
556	C₂	—	4	4	no	—
557	C₂	C₂₃×#2437	1	8	yes	direct product factor
558	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	2	yes	—
559	C₂	C₂×C₂₃×#543	1	1	yes	characteristic
560	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic
561	C₂	C₂×C₂₃×#546	1	2	yes	—
562	C₂	C₂×C₂₃×#551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×C₂²⋊Q₈	C₂	semidirect
3. C₂₃×#2437	C₂	direct (N × H)
4. C₂×C₂₃×#399	C₂	semidirect
5. C₂×C₂₃×#396	C₂	semidirect
6. C₂×C₂₃×#393	C₂	semidirect
7. C₂×C₂₃×#385	C₂	semidirect
8. C₂×C₂₃×#385	C₂	semidirect
9. C₂×C₂₃×#380	C₂	semidirect
12. C₂³×Q₈×C₂₃	C₂²	semidirect
16. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
32. C₂₃×#399	C₂²	semidirect
33. C₂₃×#396	C₂²	semidirect
34. C₂₃×#393	C₂²	semidirect
35. C₂₃×#385	C₂²	semidirect
36. C₂₃×#385	C₂²	semidirect
37. C₂₃×#380	C₂²	semidirect
39. C₂×C₂₃×#95	C₂²	semidirect
40. C₂×C₂₃×#95	C₂²	semidirect
41. C₂×C₂₃×#95	C₂²	semidirect
42. C₂×C₂₃×#95	C₂²	semidirect
47. C₂⁵×C₂₃	Q₈	semidirect
53. C₂²×Q₈×C₂₃	C₂³	semidirect
75. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
76. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
77. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
78. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
79. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
97. C₂₃×#95	C₂³	semidirect
98. C₂₃×#95	C₂³	semidirect
99. C₂₃×#95	C₂³	semidirect
100. C₂₃×#95	C₂³	semidirect
121. C₂⁴×C₂₃	C₂×Q₈	semidirect
176. C₂×#2437	C₂₃	direct (N × H)
202. C₂³×C₂₃	C₂²⋊Q₈	semidirect
242. C₂²×C₂²⋊Q₈	C₂×C₂₃	semidirect
244. #2437	C₂×C₂₃	direct (N × H)
245. C₂×#399	C₂×C₂₃	semidirect
246. C₂×#396	C₂×C₂₃	semidirect
247. C₂×#393	C₂×C₂₃	semidirect
248. C₂×#385	C₂×C₂₃	semidirect
249. C₂×#385	C₂×C₂₃	semidirect
250. C₂×#380	C₂×C₂₃	semidirect
267. C₂²×C₂₃	C₂×C₂²⋊Q₈	semidirect
285. C₂³×Q₈	C₂²×C₂₃	semidirect
289. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
290. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
291. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
292. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
293. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
294. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
305. #399	C₂²×C₂₃	semidirect
306. #396	C₂²×C₂₃	semidirect
307. #393	C₂²×C₂₃	semidirect
308. #385	C₂²×C₂₃	semidirect
309. #385	C₂²×C₂₃	semidirect
310. #380	C₂²×C₂₃	semidirect
312. C₂×#95	C₂²×C₂₃	semidirect
313. C₂×#95	C₂²×C₂₃	semidirect
314. C₂×#95	C₂²×C₂₃	semidirect
315. C₂×#95	C₂²×C₂₃	semidirect
318. C₂×C₂₃	#2437	direct (N × H)
328. C₂⁵	Q₈×C₂₃	semidirect
334. C₂²×Q₈	C₂³×C₂₃	semidirect
356. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
357. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
358. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
359. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
360. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
378. #95	C₂³×C₂₃	semidirect
379. #95	C₂³×C₂₃	semidirect
380. #95	C₂³×C₂₃	semidirect
381. #95	C₂³×C₂₃	semidirect
382. C₂₃	C₂×#2437	direct (N × H)
403. C₂⁴	C₂×Q₈×C₂₃	semidirect
483. C₂³	C₂₃×C₂²⋊Q₈	semidirect
538. C₂²	C₂×C₂₃×C₂²⋊Q₈	semidirect
557. C₂	C₂₃×#2437	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×C₂²⋊Q₈	2
C₂³×C₂²⋊C₄×C₂₃	2
C₂₃×#2437	2
C₂×C₂₃×#399	2
C₂×C₂₃×#396	2
C₂×C₂₃×#393	2
C₂×C₂₃×#385	2
C₂×C₂₃×#380	2
C₂×C₂₃×#379	2
C₂×#2437	23

Subgroups — C2×C23×#2437 #19182273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2437 #19182273