Subgroups — C₂×D₄×C₂²⋊C₄×C₂₃ #19181724

Order 5888.

8774 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₂₃	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#390	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#386	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#379	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₂₃	C₂²	1	2	yes	—
11	C₂³×D₄×C₂₃	—	2	16	no	—
12	C₂³×D₄×C₂₃	C₄	1	16	yes	complemented
13	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₂₃	C₂²	1	8	yes	—
16	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	complemented
19	C₂×C₄×D₄×C₂₃	—	2	16	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	16	yes	complemented
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₂₃	—	2	16	no	—
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	complemented
27	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
28	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
30	C₂²×C₄²×C₂₃	C₂²	1	2	yes	complemented
31	C₂₃×#390	C₂²	1	16	yes	complemented
32	C₂₃×#386	C₂²	1	32	yes	complemented
33	C₂₃×#380	C₂²	1	8	yes	complemented
34	C₂₃×#379	C₂²	1	16	yes	complemented
35	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
36	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
37	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
38	C₂⁵×C₂₃	—	2	16	no	—
39	C₂⁵×C₂₃	—	2	8	no	—
40	C₂⁵×C₂₃	—	2	8	no	—
41	C₂⁵×C₂₃	—	4	8	no	—
42	C₂⁵×C₂₃	C₂×C₄	1	16	yes	complemented
43	C₂⁵×C₂₃	C₂³	1	2	yes	—
44	C₂⁵×C₂₃	C₂³	1	8	yes	—
45	C₂⁵×C₂₃	C₂³	1	2	yes	—
46	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₂₃	—	2	16	no	—
48	C₂²×D₄×C₂₃	—	2	128	no	—
49	C₂²×D₄×C₂₃	—	2	64	no	—
50	C₂²×D₄×C₂₃	—	2	8	no	—
51	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	—
52	C₂²×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
53	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
54	C₂²×D₄×C₂₃	C₂×C₄	1	64	yes	complemented
55	C₂²×D₄×C₂₃	D₄	1	16	yes	—
56	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
57	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₂₃	—	2	8	no	—
60	C₂³×C₄×C₂₃	—	2	16	no	—
61	C₂³×C₄×C₂₃	—	2	16	no	—
62	C₂³×C₄×C₂₃	—	4	8	no	—
63	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	complemented
64	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
68	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₂₃	C₂³	1	16	yes	complemented
71	C₄×D₄×C₂₃	—	2	64	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
74	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
76	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
77	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	64	no	—
79	C₂×C₂²⋊C₄×C₂₃	—	4	8	no	—
80	C₂×C₂²⋊C₄×C₂₃	D₄	1	8	yes	direct product factor
81	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	complemented
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	complemented
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	complemented
87	C₂×C₄²×C₂₃	—	2	8	no	—
88	C₂×C₄²×C₂₃	C₂³	1	8	yes	complemented
89	C₂₃×#95	C₂³	1	8	yes	complemented
90	C₂₃×#95	C₂³	1	8	yes	complemented
91	C₂⁴×C₂₃	—	2	4	no	—
92	C₂⁴×C₂₃	—	2	4	no	—
93	C₂⁴×C₂₃	—	2	4	no	—
94	C₂⁴×C₂₃	—	2	8	no	—
95	C₂⁴×C₂₃	—	2	4	no	—
96	C₂⁴×C₂₃	—	2	64	no	—
97	C₂⁴×C₂₃	—	2	64	no	—
98	C₂⁴×C₂₃	—	2	16	no	—
99	C₂⁴×C₂₃	—	2	4	no	—
100	C₂⁴×C₂₃	—	2	32	no	—
101	C₂⁴×C₂₃	—	2	8	no	—
102	C₂⁴×C₂₃	—	2	16	no	—
103	C₂⁴×C₂₃	—	2	4	no	—
104	C₂⁴×C₂₃	—	2	8	no	—
105	C₂⁴×C₂₃	—	4	32	no	—
106	C₂⁴×C₂₃	—	4	16	no	—
107	C₂⁴×C₂₃	—	4	64	no	—
108	C₂⁴×C₂₃	—	4	64	no	—
109	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
110	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
111	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
112	C₂⁴×C₂₃	C₂²×C₄	1	32	yes	complemented
113	C₂⁴×C₂₃	C₂×D₄	1	16	yes	—
114	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
115	C₂⁴×C₂₃	#41	1	4	yes	—
116	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
117	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
118	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
119	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×D₄×C₂₃	—	2	64	no	—
121	C₂×D₄×C₂₃	—	2	32	no	—
122	C₂×D₄×C₂₃	—	2	256	no	—
123	C₂×D₄×C₂₃	—	2	32	no	—
124	C₂×D₄×C₂₃	—	2	16	no	—
125	C₂×D₄×C₂₃	C₂²⋊C₄	1	64	yes	direct product factor
126	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
127	C₂×D₄×C₂₃	C₂²×C₄	1	32	yes	complemented
128	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	complemented
129	C₂×D₄×C₂₃	C₂×D₄	1	32	yes	—
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	32	no	—
134	C₂²×C₄×C₂₃	—	2	16	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	8	no	—
137	C₂²×C₄×C₂₃	—	2	8	no	—
138	C₂²×C₄×C₂₃	—	2	32	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	8	no	—
142	C₂²×C₄×C₂₃	—	2	32	no	—
143	C₂²×C₄×C₂₃	—	4	16	no	—
144	C₂²×C₄×C₂₃	—	4	64	no	—
145	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	complemented
148	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	complemented
150	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
155	C₄⋊C₄×C₂₃	—	2	16	no	—
156	C₂²⋊C₄×C₂₃	—	2	16	no	—
157	C₂²⋊C₄×C₂₃	—	2	32	no	—
158	C₂²⋊C₄×C₂₃	—	2	8	no	—
159	C₂²⋊C₄×C₂₃	—	2	32	no	—
160	C₂²⋊C₄×C₂₃	—	4	16	no	—
161	C₂²⋊C₄×C₂₃	C₂×D₄	1	16	yes	direct product factor
162	C₄²×C₂₃	—	2	16	no	—
163	C₂×D₄×C₂²⋊C₄	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₂₃	—	2	8	no	—
165	C₂³×C₂₃	—	2	16	no	—
166	C₂³×C₂₃	—	2	8	no	—
167	C₂³×C₂₃	—	2	16	no	—
168	C₂³×C₂₃	—	2	32	no	—
169	C₂³×C₂₃	—	2	16	no	—
170	C₂³×C₂₃	—	2	16	no	—
171	C₂³×C₂₃	—	2	32	no	—
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	—	2	4	no	—
175	C₂³×C₂₃	—	2	8	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	2	16	no	—
178	C₂³×C₂₃	—	2	64	no	—
179	C₂³×C₂₃	—	2	16	no	—
180	C₂³×C₂₃	—	2	64	no	—
181	C₂³×C₂₃	—	2	16	no	—
182	C₂³×C₂₃	—	2	4	no	—
183	C₂³×C₂₃	—	2	8	no	—
184	C₂³×C₂₃	—	2	4	no	—
185	C₂³×C₂₃	—	2	8	no	—
186	C₂³×C₂₃	—	4	32	no	—
187	C₂³×C₂₃	—	4	256	no	—
188	C₂³×C₂₃	—	4	64	no	—
189	C₂³×C₂₃	—	4	64	no	—
190	C₂³×C₂₃	—	4	16	no	—
191	C₂³×C₂₃	—	4	16	no	—
192	C₂³×C₂₃	C₂×C₂²⋊C₄	1	32	yes	complemented
193	C₂³×C₂₃	C₄×D₄	1	16	yes	complemented
194	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
195	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
196	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
197	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
198	C₂³×C₂₃	C₂²×D₄	1	16	yes	—
199	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
200	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
201	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
202	C₂³×C₂₃	C₂×#41	1	4	yes	—
203	C₂³×C₂₃	C₂×#41	1	2	yes	—
204	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
205	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
206	D₄×C₂₃	—	2	64	no	—
207	D₄×C₂₃	—	2	32	no	—
208	D₄×C₂₃	C₂×C₂²⋊C₄	1	64	yes	direct product factor
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	16	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	2	32	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	8	no	—
217	C₂×C₄×C₂₃	—	2	8	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	8	no	—
221	C₂×C₄×C₂₃	—	4	64	no	—
222	C₂×C₄×C₂₃	—	4	16	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	16	yes	complemented
225	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
226	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
227	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	complemented
229	C₂⁴×D₄	C₂×C₂₃	1	1	yes	characteristic
230	C₂²×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
231	C₂³×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
232	D₄×C₂²⋊C₄	C₂×C₂₃	1	16	yes	direct product factor
233	C₂×#390	C₂×C₂₃	1	2	yes	complemented
234	C₂×#386	C₂×C₂₃	1	4	yes	complemented
235	C₂×#380	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#379	C₂×C₂₃	1	2	yes	complemented
237	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
238	C₂²×C₂₃	—	2	32	no	—
239	C₂²×C₂₃	—	2	64	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	2	8	no	—
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	—	2	8	no	—
244	C₂²×C₂₃	—	2	16	no	—
245	C₂²×C₂₃	—	2	8	no	—
246	C₂²×C₂₃	—	2	8	no	—
247	C₂²×C₂₃	—	2	16	no	—
248	C₂²×C₂₃	—	2	8	no	—
249	C₂²×C₂₃	—	2	8	no	—
250	C₂²×C₂₃	—	2	8	no	—
251	C₂²×C₂₃	—	4	128	no	—
252	C₂²×C₂₃	—	4	32	no	—
253	C₂²×C₂₃	—	4	8	no	—
254	C₂²×C₂₃	—	4	8	no	—
255	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
256	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
257	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
258	C₂²×C₂₃	C₂×C₄×D₄	1	8	yes	complemented
259	C₂²×C₂₃	#518	1	4	yes	—
260	C₂²×C₂₃	D₄²	1	8	yes	—
261	C₂²×C₂₃	#546	1	8	yes	—
262	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
263	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
264	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
265	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
266	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
267	C₄×C₂₃	—	2	4	no	—
268	C₄×C₂₃	—	2	8	no	—
269	C₄×C₂₃	—	4	8	no	—
270	C₄×C₂₃	—	4	4	no	—
271	C₄×C₂₃	C₂²×C₂²⋊C₄	1	8	yes	complemented
272	C₂⁶	C₂²×C₂₃	1	2	yes	—
273	C₂³×D₄	—	2	16	no	—
274	C₂³×D₄	C₄×C₂₃	1	16	yes	complemented
275	C₂³×D₄	C₂²×C₂₃	1	8	yes	—
276	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
277	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
278	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
279	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
280	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	complemented
281	C₂×C₄×D₄	—	2	16	no	—
282	C₂×C₄×D₄	C₂²×C₂₃	1	16	yes	complemented
283	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
284	C₂²×C₂²⋊C₄	—	2	16	no	—
285	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
287	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
288	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	complemented
289	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
290	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
291	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
292	C₂²×C₄²	C₂²×C₂₃	1	2	yes	complemented
293	#390	C₂²×C₂₃	1	16	yes	complemented
294	#386	C₂²×C₂₃	1	32	yes	complemented
295	#380	C₂²×C₂₃	1	8	yes	complemented
296	#379	C₂²×C₂₃	1	16	yes	complemented
297	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
298	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
299	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
300	C₂×C₂₃	—	2	8	no	—
301	C₂×C₂₃	—	2	16	no	—
302	C₂×C₂₃	—	4	8	no	—
303	C₂×C₂₃	D₄×C₂²⋊C₄	1	8	yes	direct product factor
304	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
306	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
307	C₂×C₂₃	C₂×D₄²	1	2	yes	—
308	C₂×C₂₃	C₂×#546	1	2	yes	—
309	C₂⁵	—	2	8	no	—
310	C₂⁵	—	2	16	no	—
311	C₂⁵	—	2	8	no	—
312	C₂⁵	—	4	8	no	—
313	C₂⁵	C₂×C₄×C₂₃	1	16	yes	complemented
314	C₂⁵	C₂³×C₂₃	1	2	yes	—
315	C₂⁵	C₂³×C₂₃	1	8	yes	—
316	C₂⁵	C₂³×C₂₃	1	2	yes	—
317	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
318	C₂²×D₄	—	2	16	no	—
319	C₂²×D₄	—	2	128	no	—
320	C₂²×D₄	—	2	64	no	—
321	C₂²×D₄	—	2	8	no	—
322	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	—
323	C₂²×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
324	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
325	C₂²×D₄	C₂×C₄×C₂₃	1	64	yes	complemented
326	C₂²×D₄	D₄×C₂₃	1	16	yes	—
327	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
328	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
329	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
330	C₂³×C₄	—	2	8	no	—
331	C₂³×C₄	—	2	16	no	—
332	C₂³×C₄	—	2	16	no	—
333	C₂³×C₄	—	4	8	no	—
334	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
335	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
336	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
337	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
339	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
340	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₂₃	1	16	yes	complemented
342	C₄×D₄	—	2	64	no	—
343	C₂×C₄⋊C₄	—	2	8	no	—
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
345	C₂×C₂²⋊C₄	—	2	8	no	—
346	C₂×C₂²⋊C₄	—	2	16	no	—
347	C₂×C₂²⋊C₄	—	2	4	no	—
348	C₂×C₂²⋊C₄	—	2	16	no	—
349	C₂×C₂²⋊C₄	—	2	64	no	—
350	C₂×C₂²⋊C₄	—	4	8	no	—
351	C₂×C₂²⋊C₄	D₄×C₂₃	1	8	yes	direct product factor
352	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	complemented
356	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	complemented
357	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	complemented
358	C₂×C₄²	—	2	8	no	—
359	C₂×C₄²	C₂³×C₂₃	1	8	yes	complemented
360	#95	C₂³×C₂₃	1	8	yes	complemented
361	#95	C₂³×C₂₃	1	8	yes	complemented
362	C₂₃	C₂×D₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	4	no	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	—	2	8	no	—
367	C₂⁴	—	2	4	no	—
368	C₂⁴	—	2	64	no	—
369	C₂⁴	—	2	64	no	—
370	C₂⁴	—	2	16	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	2	32	no	—
373	C₂⁴	—	2	8	no	—
374	C₂⁴	—	2	16	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	8	no	—
377	C₂⁴	—	4	32	no	—
378	C₂⁴	—	4	16	no	—
379	C₂⁴	—	4	64	no	—
380	C₂⁴	—	4	64	no	—
381	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
382	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
383	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
384	C₂⁴	C₂²×C₄×C₂₃	1	32	yes	complemented
385	C₂⁴	C₂×D₄×C₂₃	1	16	yes	—
386	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
387	C₂⁴	#41×C₂₃	1	4	yes	—
388	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
389	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
390	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
391	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
392	C₂×D₄	—	2	64	no	—
393	C₂×D₄	—	2	32	no	—
394	C₂×D₄	—	2	256	no	—
395	C₂×D₄	—	2	32	no	—
396	C₂×D₄	—	2	16	no	—
397	C₂×D₄	C₂²⋊C₄×C₂₃	1	64	yes	direct product factor
398	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
399	C₂×D₄	C₂²×C₄×C₂₃	1	32	yes	complemented
400	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	complemented
401	C₂×D₄	C₂×D₄×C₂₃	1	32	yes	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	32	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	32	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	32	no	—
415	C₂²×C₄	—	4	16	no	—
416	C₂²×C₄	—	4	64	no	—
417	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
418	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
419	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	complemented
420	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
421	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	complemented
422	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
424	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
427	C₄⋊C₄	—	2	16	no	—
428	C₂²⋊C₄	—	2	16	no	—
429	C₂²⋊C₄	—	2	32	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	32	no	—
432	C₂²⋊C₄	—	4	16	no	—
433	C₂²⋊C₄	C₂×D₄×C₂₃	1	16	yes	direct product factor
434	C₄²	—	2	16	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	8	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	2	16	no	—
444	C₂³	—	2	32	no	—
445	C₂³	—	2	16	no	—
446	C₂³	—	2	16	no	—
447	C₂³	—	2	32	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	8	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	8	no	—
452	C₂³	—	2	16	no	—
453	C₂³	—	2	64	no	—
454	C₂³	—	2	16	no	—
455	C₂³	—	2	64	no	—
456	C₂³	—	2	16	no	—
457	C₂³	—	4	32	no	—
458	C₂³	—	4	64	no	—
459	C₂³	—	4	64	no	—
460	C₂³	—	4	256	no	—
461	C₂³	—	4	16	no	—
462	C₂³	—	4	16	no	—
463	C₂³	C₂×C₂²⋊C₄×C₂₃	1	32	yes	complemented
464	C₂³	C₄×D₄×C₂₃	1	16	yes	complemented
465	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
466	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
467	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
468	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
469	C₂³	C₂²×D₄×C₂₃	1	16	yes	—
470	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
471	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
472	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
473	C₂³	C₂×#41×C₂₃	1	4	yes	—
474	C₂³	C₂×#41×C₂₃	1	2	yes	—
475	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
476	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
477	D₄	—	2	64	no	—
478	D₄	—	2	32	no	—
479	D₄	C₂×C₂²⋊C₄×C₂₃	1	64	yes	direct product factor
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	16	no	—
485	C₂×C₄	—	2	32	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	8	no	—
492	C₂×C₄	—	4	64	no	—
493	C₂×C₄	—	4	16	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	16	yes	complemented
496	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
497	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
498	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
499	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	complemented
500	C₂²	—	2	8	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	16	no	—
507	C₂²	—	2	32	no	—
508	C₂²	—	2	64	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	8	no	—
511	C₂²	—	2	8	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	4	128	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	4	32	no	—
517	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
518	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	16	yes	complemented
519	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
520	C₂²	C₂×C₄×D₄×C₂₃	1	8	yes	complemented
521	C₂²	C₂₃×#518	1	4	yes	—
522	C₂²	D₄²×C₂₃	1	8	yes	—
523	C₂²	C₂₃×#546	1	8	yes	—
524	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
525	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
526	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
527	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
528	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
529	C₄	—	2	8	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	8	no	—
532	C₄	—	4	4	no	—
533	C₄	C₂²×C₂²⋊C₄×C₂₃	1	8	yes	complemented
534	C₂	—	2	8	no	—
535	C₂	—	2	16	no	—
536	C₂	—	4	8	no	—
537	C₂	D₄×C₂²⋊C₄×C₂₃	1	8	yes	direct product factor
538	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
539	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
540	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
541	C₂	C₂×D₄²×C₂₃	1	2	yes	—
542	C₂	C₂×C₂₃×#546	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁴×D₄×C₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₂₃	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₂₃	C₂	semidirect
4. D₄×C₂²⋊C₄×C₂₃	C₂	direct (N × H)
5. C₂×C₂₃×#390	C₂	semidirect
6. C₂×C₂₃×#386	C₂	semidirect
7. C₂×C₂₃×#380	C₂	semidirect
8. C₂×C₂₃×#379	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
12. C₂³×D₄×C₂₃	C₄	semidirect
18. C₂⁴×C₄×C₂₃	C₂²	semidirect
20. C₂×C₄×D₄×C₂₃	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
28. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
30. C₂²×C₄²×C₂₃	C₂²	semidirect
31. C₂₃×#390	C₂²	semidirect
32. C₂₃×#386	C₂²	semidirect
33. C₂₃×#380	C₂²	semidirect
34. C₂₃×#379	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
36. C₂×C₂₃×#95	C₂²	semidirect
37. C₂×C₂₃×#95	C₂²	semidirect
42. C₂⁵×C₂₃	C₂×C₄	semidirect
52. C₂²×D₄×C₂₃	C₂×C₄	semidirect
53. C₂²×D₄×C₂₃	C₂×C₄	semidirect
54. C₂²×D₄×C₂₃	C₂×C₄	semidirect
63. C₂³×C₄×C₂₃	C₂×C₄	semidirect
70. C₂³×C₄×C₂₃	C₂³	semidirect
73. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
80. C₂×C₂²⋊C₄×C₂₃	D₄	direct (N × H)
84. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
85. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
86. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
88. C₂×C₄²×C₂₃	C₂³	semidirect
89. C₂₃×#95	C₂³	semidirect
90. C₂₃×#95	C₂³	semidirect
112. C₂⁴×C₂₃	C₂²×C₄	semidirect
125. C₂×D₄×C₂₃	C₂²⋊C₄	direct (N × H)
127. C₂×D₄×C₂₃	C₂²×C₄	semidirect
128. C₂×D₄×C₂₃	C₂²×C₄	semidirect
147. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
149. C₂²×C₄×C₂₃	C₂×D₄	semidirect
161. C₂²⋊C₄×C₂₃	C₂×D₄	direct (N × H)
163. C₂×D₄×C₂²⋊C₄	C₂₃	direct (N × H)
192. C₂³×C₂₃	C₂×C₂²⋊C₄	semidirect
193. C₂³×C₂₃	C₄×D₄	semidirect
208. D₄×C₂₃	C₂×C₂²⋊C₄	direct (N × H)
224. C₂×C₄×C₂₃	C₂×C₂²⋊C₄	semidirect
228. C₂×C₄×C₂₃	C₂²×D₄	semidirect
230. C₂²×C₄×D₄	C₂×C₂₃	semidirect
231. C₂³×C₂²⋊C₄	C₂×C₂₃	semidirect
232. D₄×C₂²⋊C₄	C₂×C₂₃	direct (N × H)
233. C₂×#390	C₂×C₂₃	semidirect
234. C₂×#386	C₂×C₂₃	semidirect
235. C₂×#380	C₂×C₂₃	semidirect
236. C₂×#379	C₂×C₂₃	semidirect
237. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
256. C₂²×C₂₃	C₂²×C₂²⋊C₄	semidirect
258. C₂²×C₂₃	C₂×C₄×D₄	semidirect
271. C₄×C₂₃	C₂²×C₂²⋊C₄	semidirect
274. C₂³×D₄	C₄×C₂₃	semidirect
280. C₂⁴×C₄	C₂²×C₂₃	semidirect
282. C₂×C₄×D₄	C₂²×C₂₃	semidirect
283. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
288. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
289. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
290. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
291. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
292. C₂²×C₄²	C₂²×C₂₃	semidirect
293. #390	C₂²×C₂₃	semidirect
294. #386	C₂²×C₂₃	semidirect
295. #380	C₂²×C₂₃	semidirect
296. #379	C₂²×C₂₃	semidirect
297. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
298. C₂×#95	C₂²×C₂₃	semidirect
299. C₂×#95	C₂²×C₂₃	semidirect
303. C₂×C₂₃	D₄×C₂²⋊C₄	direct (N × H)
313. C₂⁵	C₂×C₄×C₂₃	semidirect
323. C₂²×D₄	C₂×C₄×C₂₃	semidirect
324. C₂²×D₄	C₂×C₄×C₂₃	semidirect
325. C₂²×D₄	C₂×C₄×C₂₃	semidirect
334. C₂³×C₄	C₂×C₄×C₂₃	semidirect
341. C₂³×C₄	C₂³×C₂₃	semidirect
344. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
351. C₂×C₂²⋊C₄	D₄×C₂₃	direct (N × H)
355. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
356. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
357. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
359. C₂×C₄²	C₂³×C₂₃	semidirect
360. #95	C₂³×C₂₃	semidirect
361. #95	C₂³×C₂₃	semidirect
362. C₂₃	C₂×D₄×C₂²⋊C₄	direct (N × H)
384. C₂⁴	C₂²×C₄×C₂₃	semidirect
397. C₂×D₄	C₂²⋊C₄×C₂₃	direct (N × H)
399. C₂×D₄	C₂²×C₄×C₂₃	semidirect
400. C₂×D₄	C₂²×C₄×C₂₃	semidirect
419. C₂²×C₄	C₂²×C₄×C₂₃	semidirect
421. C₂²×C₄	C₂×D₄×C₂₃	semidirect
433. C₂²⋊C₄	C₂×D₄×C₂₃	direct (N × H)
463. C₂³	C₂×C₂²⋊C₄×C₂₃	semidirect
464. C₂³	C₄×D₄×C₂₃	semidirect
479. D₄	C₂×C₂²⋊C₄×C₂₃	direct (N × H)
495. C₂×C₄	C₂×C₂²⋊C₄×C₂₃	semidirect
499. C₂×C₄	C₂²×D₄×C₂₃	semidirect
518. C₂²	C₂²×C₂²⋊C₄×C₂₃	semidirect
520. C₂²	C₂×C₄×D₄×C₂₃	semidirect
533. C₄	C₂²×C₂²⋊C₄×C₂₃	semidirect
537. C₂	D₄×C₂²⋊C₄×C₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₂₃	2
C₂²×C₄×D₄×C₂₃	2
C₂³×C₂²⋊C₄×C₂₃	2
D₄×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#390	2
C₂×C₂₃×#386	2
C₂×C₂₃×#380	2
C₂×C₂₃×#379	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂×D₄×C₂²⋊C₄	23

Subgroups — C2×D4×C22⋊C4×C23 #19181724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₂²⋊C₄×C₂₃ #19181724