Subgroups — C₂×C₂₃×#2897 #19207710

Order 5888.

5226 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2897	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#417	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#418	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#416	C₂	1	3	yes	maximal, complemented
8	C₂²×C₂₃×#102	C₂	1	3	yes	maximal, complemented
9	C₂×C₂₃×H(5,𝔽₂)	—	2	8	no	—
10	C₂×C₂₃×H(5,𝔽₂)	—	2	4	no	—
11	C₂×C₂₃×H(5,𝔽₂)	C₄	1	16	yes	complemented
12	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	8	yes	—
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂²×#41×C₂₃	C₂²	1	3	yes	—
15	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₂₃	—	2	3	no	—
17	C₂³×D₄×C₂₃	C₂²	1	3	yes	—
18	C₂×C₂₃×#131	—	2	8	no	—
19	C₂×C₂₃×#131	C₂²	1	8	yes	complemented
20	C₂²×M₁₆×C₂₃	C₂²	1	3	yes	complemented
21	C₂³×C₈×C₂₃	C₂²	1	3	yes	complemented
22	C₂₃×#520	—	2	4	no	—
23	C₂₃×#520	C₂²	1	8	yes	complemented
24	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
25	C₂×C₄×D₄×C₂₃	—	2	3	no	—
26	C₂×C₂₃×#117	C₂²	1	3	yes	complemented
27	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	complemented
28	C₂₃×#418	C₂²	1	24	yes	complemented
29	C₂₃×#417	C₂²	1	24	yes	complemented
30	C₂₃×#416	C₂²	1	24	yes	complemented
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₂₃×#103	C₂²	1	3	yes	complemented
33	C₂×C₂₃×#102	C₂²	1	3	yes	complemented
34	C₂×C₂₃×#102	C₂²	1	6	yes	complemented
35	C₂×C₂₃×#102	C₂²	1	3	yes	complemented
36	C₂×C₂₃×#102	C₂²	1	24	yes	complemented
37	C₂⁵×C₂₃	—	2	3	no	—
38	C₂₃×H(5,𝔽₂)	—	2	64	no	—
39	C₂₃×H(5,𝔽₂)	—	2	32	no	—
40	C₂₃×H(5,𝔽₂)	C₂×C₄	1	64	yes	complemented
41	C₂×#41×C₂₃	—	2	16	no	—
42	C₂×#41×C₂₃	—	2	8	no	—
43	C₂×#41×C₂₃	—	2	12	no	—
44	C₂×#41×C₂₃	—	2	6	no	—
45	C₂×#41×C₂₃	C₂×C₄	1	8	yes	—
46	C₂×#41×C₂₃	C₂×C₄	1	24	yes	complemented
47	C₂×#41×C₂₃	D₄	1	8	yes	—
48	C₂×#41×C₂₃	D₄	1	4	yes	—
49	C₂×#41×C₂₃	C₂³	1	12	yes	—
50	C₂×#41×C₂₃	C₂³	1	4	yes	—
51	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	—	2	3	no	—
54	C₂²×D₄×C₂₃	—	2	48	no	—
55	C₂²×D₄×C₂₃	—	2	24	no	—
56	C₂²×D₄×C₂₃	—	2	3	no	—
57	C₂²×D₄×C₂₃	—	2	6	no	—
58	C₂²×D₄×C₂₃	—	2	12	no	—
59	C₂²×D₄×C₂₃	C₂×C₄	1	24	yes	complemented
60	C₂²×D₄×C₂₃	D₄	1	3	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
62	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
63	C₂³×C₄×C₂₃	—	2	3	no	—
64	C₂³×C₄×C₂₃	—	2	3	no	—
65	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
66	C₂₃×#131	—	2	32	no	—
67	C₂×M₁₆×C₂₃	—	2	12	no	—
68	C₂×M₁₆×C₂₃	C₂³	1	12	yes	complemented
69	C₂²×C₈×C₂₃	—	2	12	no	—
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	12	yes	complemented
73	C₄×Q₈×C₂₃	—	2	8	no	—
74	C₄×D₄×C₂₃	—	2	24	no	—
75	C₂₃×#117	—	2	6	no	—
76	C₂₃×#117	C₂³	1	12	yes	complemented
77	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	complemented
82	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
83	C₂×C₄²×C₂₃	—	2	3	no	—
84	C₂₃×#103	C₂³	1	4	yes	complemented
85	C₂₃×#103	C₂³	1	12	yes	complemented
86	C₂₃×#102	C₂³	1	12	yes	complemented
87	C₂₃×#102	C₂³	1	24	yes	complemented
88	C₂₃×#102	C₂³	1	12	yes	complemented
89	C₂⁴×C₂₃	—	2	3	no	—
90	C₂⁴×C₂₃	—	2	24	no	—
91	C₂⁴×C₂₃	—	2	12	no	—
92	C₂⁴×C₂₃	—	8	6	no	—
93	C₂⁴×C₂₃	—	8	6	no	—
94	C₂⁴×C₂₃	C₂×D₄	1	3	yes	—
95	#41×C₂₃	—	2	64	no	—
96	#41×C₂₃	—	2	48	no	—
97	#41×C₂₃	—	2	24	no	—
98	#41×C₂₃	C₂²⋊C₄	1	32	yes	—
99	#41×C₂₃	C₂²×C₄	1	16	yes	—
100	#41×C₂₃	C₂²×C₄	1	48	yes	complemented
101	#41×C₂₃	C₂×D₄	1	16	yes	—
102	C₂×Q₈×C₂₃	—	2	2	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	C₂²⋊C₄	1	8	yes	—
105	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
106	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
107	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
108	C₂×D₄×C₂₃	—	2	24	no	—
109	C₂×D₄×C₂₃	—	2	12	no	—
110	C₂×D₄×C₂₃	—	2	192	no	—
111	C₂×D₄×C₂₃	—	2	24	no	—
112	C₂×D₄×C₂₃	—	2	48	no	—
113	C₂×D₄×C₂₃	—	2	12	no	—
114	C₂×D₄×C₂₃	—	2	3	no	—
115	C₂×D₄×C₂₃	—	2	6	no	—
116	C₂×D₄×C₂₃	—	2	24	no	—
117	C₂×D₄×C₂₃	C₂²⋊C₄	1	24	yes	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	12	yes	—
119	C₂×D₄×C₂₃	C₂²×C₄	1	48	yes	complemented
120	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	—
121	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	24	no	—
124	C₂²×C₄×C₂₃	—	2	12	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	3	no	—
127	C₂²×C₄×C₂₃	—	2	12	no	—
128	C₂²×C₄×C₂₃	—	8	6	no	—
129	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
130	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
131	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
135	M₁₆×C₂₃	—	2	24	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	2	6	no	—
138	C₂×C₈×C₂₃	—	2	24	no	—
139	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
140	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
141	C₄⋊C₄×C₂₃	—	2	3	no	—
142	C₄⋊C₄×C₂₃	—	2	12	no	—
143	C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
145	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
146	C₂²⋊C₄×C₂₃	—	2	12	no	—
147	C₄²×C₂₃	—	2	12	no	—
148	C₂×#2897	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₂₃	—	2	6	no	—
150	C₂³×C₂₃	—	2	6	no	—
151	C₂³×C₂₃	—	2	12	no	—
152	C₂³×C₂₃	—	2	6	no	—
153	C₂³×C₂₃	—	2	48	no	—
154	C₂³×C₂₃	—	4	6	no	—
155	C₂³×C₂₃	—	4	6	no	—
156	C₂³×C₂₃	—	8	48	no	—
157	C₂³×C₂₃	—	8	12	no	—
158	C₂³×C₂₃	—	8	12	no	—
159	C₂³×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
160	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
161	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₂₃	—	2	4	no	—
163	Q₈×C₂₃	—	2	2	no	—
164	Q₈×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
165	Q₈×C₂₃	C₂³×C₄	1	4	yes	—
166	D₄×C₂₃	—	2	48	no	—
167	D₄×C₂₃	—	2	12	no	—
168	D₄×C₂₃	—	2	48	no	—
169	D₄×C₂₃	—	2	6	no	—
170	D₄×C₂₃	C₂×C₂²⋊C₄	1	48	yes	—
171	D₄×C₂₃	C₂³×C₄	1	12	yes	—
172	C₂×C₄×C₂₃	—	2	4	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	48	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	6	no	—
177	C₂×C₄×C₂₃	—	4	6	no	—
178	C₂×C₄×C₂₃	—	4	2	no	—
179	C₂×C₄×C₂₃	—	8	12	no	—
180	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
181	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	24	yes	—
182	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
183	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
184	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
185	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
188	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
189	C₈×C₂₃	—	2	2	no	—
190	C₈×C₂₃	—	2	6	no	—
191	C₂²×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic
192	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
193	C₂×#520	C₂×C₂₃	1	1	yes	characteristic, complemented
194	#2897	C₂×C₂₃	1	16	yes	direct product factor
195	C₂×#417	C₂×C₂₃	1	3	yes	complemented
196	C₂×#418	C₂×C₂₃	1	3	yes	complemented
197	C₂×#416	C₂×C₂₃	1	3	yes	complemented
198	C₂²×#102	C₂×C₂₃	1	3	yes	complemented
199	C₂²×C₂₃	—	2	24	no	—
200	C₂²×C₂₃	—	2	6	no	—
201	C₂²×C₂₃	—	2	6	no	—
202	C₂²×C₂₃	—	2	12	no	—
203	C₂²×C₂₃	—	4	24	no	—
204	C₂²×C₂₃	—	4	6	no	—
205	C₂²×C₂₃	—	4	6	no	—
206	C₂²×C₂₃	—	8	48	no	—
207	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	12	yes	—
208	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
209	C₂²×C₂₃	#576	1	2	yes	—
210	C₂²×C₂₃	#577	1	2	yes	—
211	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
212	C₄×C₂₃	—	2	4	no	—
213	C₄×C₂₃	—	4	6	no	—
214	C₄×C₂₃	—	4	2	no	—
215	C₄×C₂₃	C₂²×C₂²⋊C₄	1	12	yes	—
216	C₄×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
217	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
218	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
219	C₂×H(5,𝔽₂)	—	2	8	no	—
220	C₂×H(5,𝔽₂)	—	2	4	no	—
221	C₂×H(5,𝔽₂)	C₄×C₂₃	1	16	yes	complemented
222	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	8	yes	—
223	C₂²×#41	—	2	1	no	—
224	C₂²×#41	C₂²×C₂₃	1	3	yes	—
225	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
226	C₂³×D₄	—	2	3	no	—
227	C₂³×D₄	C₂²×C₂₃	1	3	yes	—
228	C₂×#131	—	2	8	no	—
229	C₂×#131	C₂²×C₂₃	1	8	yes	complemented
230	C₂²×M₁₆	C₂²×C₂₃	1	3	yes	complemented
231	C₂³×C₈	C₂²×C₂₃	1	3	yes	complemented
232	#520	—	2	4	no	—
233	#520	C₂²×C₂₃	1	8	yes	complemented
234	C₂×C₄×Q₈	—	2	1	no	—
235	C₂×C₄×D₄	—	2	3	no	—
236	C₂×#117	C₂²×C₂₃	1	3	yes	complemented
237	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	complemented
238	#418	C₂²×C₂₃	1	24	yes	complemented
239	#417	C₂²×C₂₃	1	24	yes	complemented
240	#416	C₂²×C₂₃	1	24	yes	complemented
241	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	C₂×#103	C₂²×C₂₃	1	3	yes	complemented
243	C₂×#102	C₂²×C₂₃	1	3	yes	complemented
244	C₂×#102	C₂²×C₂₃	1	6	yes	complemented
245	C₂×#102	C₂²×C₂₃	1	3	yes	complemented
246	C₂×#102	C₂²×C₂₃	1	24	yes	complemented
247	C₂×C₂₃	—	2	12	no	—
248	C₂×C₂₃	—	4	12	no	—
249	C₂×C₂₃	#2897	1	4	yes	direct product factor
250	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
252	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
253	C₂⁵	—	2	3	no	—
254	H(5,𝔽₂)	—	2	64	no	—
255	H(5,𝔽₂)	—	2	32	no	—
256	H(5,𝔽₂)	C₂×C₄×C₂₃	1	64	yes	complemented
257	C₂×#41	—	2	16	no	—
258	C₂×#41	—	2	8	no	—
259	C₂×#41	—	2	12	no	—
260	C₂×#41	—	2	6	no	—
261	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
262	C₂×#41	C₂×C₄×C₂₃	1	24	yes	complemented
263	C₂×#41	D₄×C₂₃	1	8	yes	—
264	C₂×#41	D₄×C₂₃	1	4	yes	—
265	C₂×#41	C₂³×C₂₃	1	12	yes	—
266	C₂×#41	C₂³×C₂₃	1	4	yes	—
267	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
268	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
269	C₂²×D₄	—	2	3	no	—
270	C₂²×D₄	—	2	48	no	—
271	C₂²×D₄	—	2	24	no	—
272	C₂²×D₄	—	2	3	no	—
273	C₂²×D₄	—	2	6	no	—
274	C₂²×D₄	—	2	12	no	—
275	C₂²×D₄	C₂×C₄×C₂₃	1	24	yes	complemented
276	C₂²×D₄	D₄×C₂₃	1	3	yes	—
277	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
278	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
279	C₂³×C₄	—	2	3	no	—
280	C₂³×C₄	—	2	3	no	—
281	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
282	#131	—	2	32	no	—
283	C₂×M₁₆	—	2	12	no	—
284	C₂×M₁₆	C₂³×C₂₃	1	12	yes	complemented
285	C₂²×C₈	—	2	12	no	—
286	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
288	C₂²×C₈	C₂³×C₂₃	1	12	yes	complemented
289	C₄×Q₈	—	2	8	no	—
290	C₄×D₄	—	2	24	no	—
291	#117	—	2	6	no	—
292	#117	C₂³×C₂₃	1	12	yes	complemented
293	C₂×C₄⋊C₄	—	2	3	no	—
294	C₂×C₄⋊C₄	—	2	6	no	—
295	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
296	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	complemented
298	C₂×C₂²⋊C₄	—	2	3	no	—
299	C₂×C₄²	—	2	3	no	—
300	#103	C₂³×C₂₃	1	4	yes	complemented
301	#103	C₂³×C₂₃	1	12	yes	complemented
302	#102	C₂³×C₂₃	1	12	yes	complemented
303	#102	C₂³×C₂₃	1	24	yes	complemented
304	#102	C₂³×C₂₃	1	12	yes	complemented
305	C₂₃	C₂×#2897	1	1	yes	characteristic, direct product factor, Sylow
306	C₂⁴	—	2	3	no	—
307	C₂⁴	—	2	24	no	—
308	C₂⁴	—	2	12	no	—
309	C₂⁴	—	8	6	no	—
310	C₂⁴	—	8	6	no	—
311	C₂⁴	C₂×D₄×C₂₃	1	3	yes	—
312	#41	—	2	64	no	—
313	#41	—	2	48	no	—
314	#41	—	2	24	no	—
315	#41	C₂²⋊C₄×C₂₃	1	32	yes	—
316	#41	C₂²×C₄×C₂₃	1	16	yes	—
317	#41	C₂²×C₄×C₂₃	1	48	yes	complemented
318	#41	C₂×D₄×C₂₃	1	16	yes	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	C₂²⋊C₄×C₂₃	1	8	yes	—
322	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
323	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
324	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
325	C₂×D₄	—	2	24	no	—
326	C₂×D₄	—	2	12	no	—
327	C₂×D₄	—	2	192	no	—
328	C₂×D₄	—	2	24	no	—
329	C₂×D₄	—	2	48	no	—
330	C₂×D₄	—	2	6	no	—
331	C₂×D₄	—	2	12	no	—
332	C₂×D₄	—	2	3	no	—
333	C₂×D₄	—	2	24	no	—
334	C₂×D₄	C₂²⋊C₄×C₂₃	1	24	yes	—
335	C₂×D₄	C₂²×C₄×C₂₃	1	12	yes	—
336	C₂×D₄	C₂²×C₄×C₂₃	1	48	yes	complemented
337	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	—
338	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	24	no	—
341	C₂²×C₄	—	2	12	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	3	no	—
344	C₂²×C₄	—	2	12	no	—
345	C₂²×C₄	—	8	6	no	—
346	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
347	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
348	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
349	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
350	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
352	M₁₆	—	2	24	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	6	no	—
355	C₂×C₈	—	2	24	no	—
356	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
357	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
358	C₄⋊C₄	—	2	3	no	—
359	C₄⋊C₄	—	2	12	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
362	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
363	C₂²⋊C₄	—	2	12	no	—
364	C₄²	—	2	12	no	—
365	C₂³	—	2	6	no	—
366	C₂³	—	2	6	no	—
367	C₂³	—	2	6	no	—
368	C₂³	—	2	12	no	—
369	C₂³	—	2	48	no	—
370	C₂³	—	4	6	no	—
371	C₂³	—	4	6	no	—
372	C₂³	—	8	12	no	—
373	C₂³	—	8	12	no	—
374	C₂³	—	8	48	no	—
375	C₂³	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
376	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
377	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
378	Q₈	—	2	2	no	—
379	Q₈	—	2	4	no	—
380	Q₈	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
381	Q₈	C₂³×C₄×C₂₃	1	4	yes	—
382	D₄	—	2	48	no	—
383	D₄	—	2	6	no	—
384	D₄	—	2	12	no	—
385	D₄	—	2	48	no	—
386	D₄	C₂×C₂²⋊C₄×C₂₃	1	48	yes	—
387	D₄	C₂³×C₄×C₂₃	1	12	yes	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	48	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	6	no	—
393	C₂×C₄	—	4	6	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	12	no	—
396	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
397	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	24	yes	—
398	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
399	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
400	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
401	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
402	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
403	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
404	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
405	C₈	—	2	2	no	—
406	C₈	—	2	6	no	—
407	C₂²	—	2	6	no	—
408	C₂²	—	2	6	no	—
409	C₂²	—	2	24	no	—
410	C₂²	—	2	12	no	—
411	C₂²	—	4	6	no	—
412	C₂²	—	4	6	no	—
413	C₂²	—	4	24	no	—
414	C₂²	—	8	48	no	—
415	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	12	yes	—
416	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
417	C₂²	C₂₃×#576	1	2	yes	—
418	C₂²	C₂₃×#577	1	2	yes	—
419	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
420	C₄	—	2	4	no	—
421	C₄	—	4	6	no	—
422	C₄	—	4	2	no	—
423	C₄	C₂²×C₂²⋊C₄×C₂₃	1	12	yes	—
424	C₄	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
425	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
426	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
427	C₂	—	2	12	no	—
428	C₂	—	4	12	no	—
429	C₂	C₂₃×#2897	1	4	yes	direct product factor
430	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
431	C₂	C₂×C₂₃×#576	1	1	yes	characteristic
432	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×#131	C₂	semidirect
3. C₂×C₂₃×#520	C₂	semidirect
4. C₂₃×#2897	C₂	direct (N × H)
5. C₂×C₂₃×#417	C₂	semidirect
6. C₂×C₂₃×#418	C₂	semidirect
7. C₂×C₂₃×#416	C₂	semidirect
8. C₂²×C₂₃×#102	C₂	semidirect
11. C₂×C₂₃×H(5,𝔽₂)	C₄	semidirect
19. C₂×C₂₃×#131	C₂²	semidirect
20. C₂²×M₁₆×C₂₃	C₂²	semidirect
21. C₂³×C₈×C₂₃	C₂²	semidirect
23. C₂₃×#520	C₂²	semidirect
26. C₂×C₂₃×#117	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
28. C₂₃×#418	C₂²	semidirect
29. C₂₃×#417	C₂²	semidirect
30. C₂₃×#416	C₂²	semidirect
31. C₂×C₂₃×#103	C₂²	semidirect
32. C₂×C₂₃×#103	C₂²	semidirect
33. C₂×C₂₃×#102	C₂²	semidirect
34. C₂×C₂₃×#102	C₂²	semidirect
35. C₂×C₂₃×#102	C₂²	semidirect
36. C₂×C₂₃×#102	C₂²	semidirect
40. C₂₃×H(5,𝔽₂)	C₂×C₄	semidirect
46. C₂×#41×C₂₃	C₂×C₄	semidirect
59. C₂²×D₄×C₂₃	C₂×C₄	semidirect
68. C₂×M₁₆×C₂₃	C₂³	semidirect
72. C₂²×C₈×C₂₃	C₂³	semidirect
76. C₂₃×#117	C₂³	semidirect
81. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
84. C₂₃×#103	C₂³	semidirect
85. C₂₃×#103	C₂³	semidirect
86. C₂₃×#102	C₂³	semidirect
87. C₂₃×#102	C₂³	semidirect
88. C₂₃×#102	C₂³	semidirect
100. #41×C₂₃	C₂²×C₄	semidirect
119. C₂×D₄×C₂₃	C₂²×C₄	semidirect
148. C₂×#2897	C₂₃	direct (N × H)
192. C₂²×#131	C₂×C₂₃	semidirect
193. C₂×#520	C₂×C₂₃	semidirect
194. #2897	C₂×C₂₃	direct (N × H)
195. C₂×#417	C₂×C₂₃	semidirect
196. C₂×#418	C₂×C₂₃	semidirect
197. C₂×#416	C₂×C₂₃	semidirect
198. C₂²×#102	C₂×C₂₃	semidirect
221. C₂×H(5,𝔽₂)	C₄×C₂₃	semidirect
229. C₂×#131	C₂²×C₂₃	semidirect
230. C₂²×M₁₆	C₂²×C₂₃	semidirect
231. C₂³×C₈	C₂²×C₂₃	semidirect
233. #520	C₂²×C₂₃	semidirect
236. C₂×#117	C₂²×C₂₃	semidirect
237. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
238. #418	C₂²×C₂₃	semidirect
239. #417	C₂²×C₂₃	semidirect
240. #416	C₂²×C₂₃	semidirect
241. C₂×#103	C₂²×C₂₃	semidirect
242. C₂×#103	C₂²×C₂₃	semidirect
243. C₂×#102	C₂²×C₂₃	semidirect
244. C₂×#102	C₂²×C₂₃	semidirect
245. C₂×#102	C₂²×C₂₃	semidirect
246. C₂×#102	C₂²×C₂₃	semidirect
249. C₂×C₂₃	#2897	direct (N × H)
256. H(5,𝔽₂)	C₂×C₄×C₂₃	semidirect
262. C₂×#41	C₂×C₄×C₂₃	semidirect
275. C₂²×D₄	C₂×C₄×C₂₃	semidirect
284. C₂×M₁₆	C₂³×C₂₃	semidirect
288. C₂²×C₈	C₂³×C₂₃	semidirect
292. #117	C₂³×C₂₃	semidirect
297. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
300. #103	C₂³×C₂₃	semidirect
301. #103	C₂³×C₂₃	semidirect
302. #102	C₂³×C₂₃	semidirect
303. #102	C₂³×C₂₃	semidirect
304. #102	C₂³×C₂₃	semidirect
305. C₂₃	C₂×#2897	direct (N × H)
317. #41	C₂²×C₄×C₂₃	semidirect
336. C₂×D₄	C₂²×C₄×C₂₃	semidirect
429. C₂	C₂₃×#2897	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×H(5,𝔽₂)	2
C₂²×C₂₃×#131	2
C₂×C₂₃×#520	2
C₂₃×#2897	2
C₂×C₂₃×#417	2
C₂×C₂₃×#418	2
C₂×C₂₃×#416	2
C₂²×C₂₃×#102	2
C₂×#2897	23

Subgroups — C2×C23×#2897 #19207710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2897 #19207710