Subgroups — C₂×C₂₃×#2357 #19181746

Order 5888.

3158 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2357	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#390	C₂	1	2	yes	maximal
4	C₂×C₂₃×#388	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#381	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₂₃	C₄	1	16	yes	complemented
13	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
18	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#390	C₂²	1	16	yes	—
24	C₂₃×#388	C₂²	1	32	yes	complemented
25	C₂₃×#382	C₂²	1	8	yes	complemented
26	C₂₃×#381	C₂²	1	16	yes	complemented
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	8	yes	complemented
28	C₄×C₄⋊C₄×C₂₃	C₂²	1	16	yes	complemented
29	C₄×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	complemented
30	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#95	C₂²	1	2	yes	—
32	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#95	C₂²	1	4	yes	—
34	C₂×C₂₃×#95	C₂²	1	2	yes	—
35	C₂⁵×C₂₃	C₂³	1	2	yes	—
36	C₂²×D₄×C₂₃	—	2	4	no	—
37	C₂²×D₄×C₂₃	—	2	4	no	—
38	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	—
39	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₂₃	—	4	8	no	—
41	C₂³×C₄×C₂₃	C₂×C₄	1	16	yes	—
42	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₄×D₄×C₂₃	C₂×C₄	1	64	yes	complemented
51	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
53	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
58	C₂×C₂²⋊C₄×C₂₃	—	4	8	no	—
59	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
60	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
61	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
62	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
63	C₂×C₄²×C₂₃	—	2	8	no	—
64	C₂×C₄²×C₂₃	—	4	8	no	—
65	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
66	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
67	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
68	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
69	C₂₃×#95	C₂³	1	4	yes	—
70	C₂₃×#95	C₂³	1	8	yes	—
71	C₂₃×#95	C₂³	1	4	yes	—
72	C₂₃×#95	C₂³	1	16	yes	—
73	C₂₃×#95	C₂³	1	8	yes	—
74	C₂⁴×C₂₃	—	2	8	no	—
75	C₂⁴×C₂₃	—	2	4	no	—
76	C₂⁴×C₂₃	—	2	4	no	—
77	C₂⁴×C₂₃	—	2	8	no	—
78	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
79	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
80	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
81	C₂×D₄×C₂₃	—	2	32	no	—
82	C₂×D₄×C₂₃	—	2	8	no	—
83	C₂×D₄×C₂₃	—	2	8	no	—
84	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
85	C₂²×C₄×C₂₃	—	2	4	no	—
86	C₂²×C₄×C₂₃	—	2	4	no	—
87	C₂²×C₄×C₂₃	—	2	4	no	—
88	C₂²×C₄×C₂₃	—	2	8	no	—
89	C₂²×C₄×C₂₃	—	2	4	no	—
90	C₂²×C₄×C₂₃	—	2	4	no	—
91	C₂²×C₄×C₂₃	—	2	8	no	—
92	C₂²×C₄×C₂₃	—	2	4	no	—
93	C₂²×C₄×C₂₃	—	2	8	no	—
94	C₂²×C₄×C₂₃	—	2	8	no	—
95	C₂²×C₄×C₂₃	—	2	4	no	—
96	C₂²×C₄×C₂₃	—	4	32	no	—
97	C₂²×C₄×C₂₃	—	4	16	no	—
98	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
101	C₂²×C₄×C₂₃	C₂²×C₄	1	32	yes	—
102	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₂₃	#41	1	8	yes	—
104	C₂²×C₄×C₂₃	#41	1	4	yes	—
105	C₂²×C₄×C₂₃	#41	1	4	yes	—
106	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
111	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
113	C₄⋊C₄×C₂₃	—	2	16	no	—
114	C₄⋊C₄×C₂₃	—	2	16	no	—
115	C₄⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
116	C₂²⋊C₄×C₂₃	—	2	32	no	—
117	C₂²⋊C₄×C₂₃	—	4	16	no	—
118	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
119	C₄²×C₂₃	—	2	16	no	—
120	C₄²×C₂₃	—	4	16	no	—
121	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
122	C₂×#2357	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₂₃	—	2	16	no	—
124	C₂³×C₂₃	—	2	16	no	—
125	C₂³×C₂₃	—	2	16	no	—
126	C₂³×C₂₃	—	2	8	no	—
127	C₂³×C₂₃	—	2	4	no	—
128	C₂³×C₂₃	—	2	4	no	—
129	C₂³×C₂₃	—	2	8	no	—
130	C₂³×C₂₃	—	4	16	no	—
131	C₂³×C₂₃	—	4	16	no	—
132	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
133	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
134	C₂³×C₂₃	C₂×#41	1	2	yes	—
135	C₂³×C₂₃	C₂×#41	1	2	yes	—
136	C₂³×C₂₃	C₂×#41	1	4	yes	—
137	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
138	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
139	D₄×C₂₃	—	2	32	no	—
140	C₂×C₄×C₂₃	—	2	8	no	—
141	C₂×C₄×C₂₃	—	2	8	no	—
142	C₂×C₄×C₂₃	—	2	16	no	—
143	C₂×C₄×C₂₃	—	2	16	no	—
144	C₂×C₄×C₂₃	—	2	4	no	—
145	C₂×C₄×C₂₃	—	2	4	no	—
146	C₂×C₄×C₂₃	—	2	4	no	—
147	C₂×C₄×C₂₃	—	2	8	no	—
148	C₂×C₄×C₂₃	—	2	4	no	—
149	C₂×C₄×C₂₃	—	2	4	no	—
150	C₂×C₄×C₂₃	—	2	8	no	—
151	C₂×C₄×C₂₃	—	2	4	no	—
152	C₂×C₄×C₂₃	—	2	8	no	—
153	C₂×C₄×C₂₃	—	2	8	no	—
154	C₂×C₄×C₂₃	—	2	4	no	—
155	C₂×C₄×C₂₃	—	4	32	no	—
156	C₂×C₄×C₂₃	—	4	8	no	—
157	C₂×C₄×C₂₃	—	4	16	no	—
158	C₂×C₄×C₂₃	#117	1	16	yes	—
159	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
160	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
161	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
162	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
163	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
164	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
165	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
166	C₂²×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
167	#2357	C₂×C₂₃	1	16	yes	direct product factor
168	C₂×#390	C₂×C₂₃	1	2	yes	—
169	C₂×#388	C₂×C₂₃	1	4	yes	complemented
170	C₂×#382	C₂×C₂₃	1	1	yes	characteristic, complemented
171	C₂×#381	C₂×C₂₃	1	2	yes	complemented
172	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
173	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
174	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
175	C₂²×C₂₃	—	2	8	no	—
176	C₂²×C₂₃	—	2	8	no	—
177	C₂²×C₂₃	—	2	8	no	—
178	C₂²×C₂₃	—	4	32	no	—
179	C₂²×C₂₃	—	4	8	no	—
180	C₂²×C₂₃	—	4	8	no	—
181	C₂²×C₂₃	C₂×#117	1	4	yes	—
182	C₂²×C₂₃	C₄×#41	1	4	yes	—
183	C₂²×C₂₃	#518	1	4	yes	—
184	C₂²×C₂₃	#553	1	8	yes	—
185	C₂²×C₂₃	#559	1	8	yes	—
186	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
187	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
188	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
189	C₂²×C₂₃	C₂²×#41	1	2	yes	—
190	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
191	C₄×C₂₃	—	2	4	no	—
192	C₄×C₂₃	—	2	4	no	—
193	C₄×C₂₃	—	2	8	no	—
194	C₄×C₂₃	—	2	8	no	—
195	C₄×C₂₃	—	4	4	no	—
196	C₄×C₂₃	—	4	8	no	—
197	C₄×C₂₃	C₂×#117	1	8	yes	—
198	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
199	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
200	C₂×C₄×D₄	C₄×C₂₃	1	16	yes	complemented
201	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
202	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
203	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
204	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
205	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
206	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
207	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
208	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
209	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
210	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
211	#390	C₂²×C₂₃	1	16	yes	—
212	#388	C₂²×C₂₃	1	32	yes	complemented
213	#382	C₂²×C₂₃	1	8	yes	complemented
214	#381	C₂²×C₂₃	1	16	yes	complemented
215	C₄×C₄⋊C₄	C₂²×C₂₃	1	8	yes	complemented
216	C₄×C₄⋊C₄	C₂²×C₂₃	1	16	yes	complemented
217	C₄×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	complemented
218	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
219	C₂×#95	C₂²×C₂₃	1	2	yes	—
220	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
221	C₂×#95	C₂²×C₂₃	1	4	yes	—
222	C₂×#95	C₂²×C₂₃	1	2	yes	—
223	C₂×C₂₃	—	4	8	no	—
224	C₂×C₂₃	#2357	1	8	yes	direct product factor
225	C₂×C₂₃	C₂²×#117	1	1	yes	characteristic
226	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
227	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
228	C₂×C₂₃	C₂×#553	1	2	yes	—
229	C₂×C₂₃	C₂×#559	1	2	yes	—
230	C₂⁵	C₂³×C₂₃	1	2	yes	—
231	C₂²×D₄	—	2	4	no	—
232	C₂²×D₄	—	2	4	no	—
233	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	—
234	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
235	C₂³×C₄	—	4	8	no	—
236	C₂³×C₄	C₂×C₄×C₂₃	1	16	yes	—
237	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
238	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
239	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
241	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
242	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
243	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
244	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
245	C₄×D₄	C₂×C₄×C₂₃	1	64	yes	complemented
246	C₂×C₄⋊C₄	—	2	8	no	—
247	C₂×C₄⋊C₄	—	2	8	no	—
248	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
249	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
250	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
251	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
252	C₂×C₂²⋊C₄	—	2	16	no	—
253	C₂×C₂²⋊C₄	—	4	8	no	—
254	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
255	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
256	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
258	C₂×C₄²	—	2	8	no	—
259	C₂×C₄²	—	4	8	no	—
260	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
261	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
262	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
263	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
264	#95	C₂³×C₂₃	1	4	yes	—
265	#95	C₂³×C₂₃	1	8	yes	—
266	#95	C₂³×C₂₃	1	4	yes	—
267	#95	C₂³×C₂₃	1	16	yes	—
268	#95	C₂³×C₂₃	1	8	yes	—
269	C₂₃	C₂×#2357	1	1	yes	characteristic, direct product factor, Sylow
270	C₂⁴	—	2	8	no	—
271	C₂⁴	—	2	4	no	—
272	C₂⁴	—	2	4	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
275	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
276	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₂×D₄	—	2	32	no	—
278	C₂×D₄	—	2	8	no	—
279	C₂×D₄	—	2	8	no	—
280	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	8	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	8	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	4	32	no	—
293	C₂²×C₄	—	4	16	no	—
294	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
295	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
296	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
297	C₂²×C₄	C₂²×C₄×C₂₃	1	32	yes	—
298	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
299	C₂²×C₄	#41×C₂₃	1	8	yes	—
300	C₂²×C₄	#41×C₂₃	1	4	yes	—
301	C₂²×C₄	#41×C₂₃	1	4	yes	—
302	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
303	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
304	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
305	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
306	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
307	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
308	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
309	C₄⋊C₄	—	2	16	no	—
310	C₄⋊C₄	—	2	16	no	—
311	C₄⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
312	C₂²⋊C₄	—	2	32	no	—
313	C₂²⋊C₄	—	4	16	no	—
314	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
315	C₄²	—	2	16	no	—
316	C₄²	—	4	16	no	—
317	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
318	C₂³	—	2	8	no	—
319	C₂³	—	2	4	no	—
320	C₂³	—	2	4	no	—
321	C₂³	—	2	8	no	—
322	C₂³	—	2	16	no	—
323	C₂³	—	2	16	no	—
324	C₂³	—	2	16	no	—
325	C₂³	—	4	16	no	—
326	C₂³	—	4	16	no	—
327	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
328	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
329	C₂³	C₂×#41×C₂₃	1	2	yes	—
330	C₂³	C₂×#41×C₂₃	1	4	yes	—
331	C₂³	C₂×#41×C₂₃	1	2	yes	—
332	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
333	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
334	D₄	—	2	32	no	—
335	C₂×C₄	—	2	16	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	16	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	4	32	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	16	no	—
353	C₂×C₄	C₂₃×#117	1	16	yes	—
354	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
355	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
356	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
357	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
358	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
359	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
360	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
361	C₂²	—	2	8	no	—
362	C₂²	—	2	8	no	—
363	C₂²	—	2	8	no	—
364	C₂²	—	4	8	no	—
365	C₂²	—	4	8	no	—
366	C₂²	—	4	32	no	—
367	C₂²	C₂×C₂₃×#117	1	4	yes	—
368	C₂²	C₄×#41×C₂₃	1	4	yes	—
369	C₂²	C₂₃×#518	1	4	yes	—
370	C₂²	C₂₃×#553	1	8	yes	—
371	C₂²	C₂₃×#559	1	8	yes	—
372	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
373	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
374	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
375	C₂²	C₂²×#41×C₂₃	1	2	yes	—
376	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
377	C₄	—	2	4	no	—
378	C₄	—	2	4	no	—
379	C₄	—	2	8	no	—
380	C₄	—	2	8	no	—
381	C₄	—	4	4	no	—
382	C₄	—	4	8	no	—
383	C₄	C₂×C₂₃×#117	1	8	yes	—
384	C₂	—	4	8	no	—
385	C₂	C₂₃×#2357	1	8	yes	direct product factor
386	C₂	C₂²×C₂₃×#117	1	1	yes	characteristic
387	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
388	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
389	C₂	C₂×C₂₃×#553	1	2	yes	—
390	C₂	C₂×C₂₃×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
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. C₂₃×#2357	C₂	direct (N × H)
4. C₂×C₂₃×#388	C₂	semidirect
5. C₂×C₂₃×#382	C₂	semidirect
6. C₂×C₂₃×#381	C₂	semidirect
7. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
12. C₂×C₄×D₄×C₂₃	C₄	semidirect
24. C₂₃×#388	C₂²	semidirect
25. C₂₃×#382	C₂²	semidirect
26. C₂₃×#381	C₂²	semidirect
27. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
28. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
29. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
50. C₄×D₄×C₂₃	C₂×C₄	semidirect
122. C₂×#2357	C₂₃	direct (N × H)
167. #2357	C₂×C₂₃	direct (N × H)
169. C₂×#388	C₂×C₂₃	semidirect
170. C₂×#382	C₂×C₂₃	semidirect
171. C₂×#381	C₂×C₂₃	semidirect
172. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
173. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
174. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
200. C₂×C₄×D₄	C₄×C₂₃	semidirect
212. #388	C₂²×C₂₃	semidirect
213. #382	C₂²×C₂₃	semidirect
214. #381	C₂²×C₂₃	semidirect
215. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
216. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
217. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
224. C₂×C₂₃	#2357	direct (N × H)
245. C₄×D₄	C₂×C₄×C₂₃	semidirect
269. C₂₃	C₂×#2357	direct (N × H)
385. C₂	C₂₃×#2357	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×C₄×D₄×C₂₃	2
C₂₃×#2357	2
C₂×C₂₃×#390	2
C₂×C₂₃×#388	2
C₂×C₂₃×#382	2
C₂×C₂₃×#381	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂×#2357	23

Subgroups — C2×C23×#2357 #19181746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2357 #19181746