Subgroups — C₂₃×#34195 #19181821

Order 5888.

2370 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2356	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2353	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2349	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2347	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2343	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2339	C₂	1	2	yes	maximal
8	C₂₃×#2320	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2318	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2283	C₂	1	2	yes	maximal, complemented
13	C₂×C₂₃×#383	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2275	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×#41×C₂₃	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂₃×#117	C₂²	1	2	yes	—
25	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#391	C₂²	1	4	yes	—
29	C₂₃×#389	C₂²	1	4	yes	—
30	C₂₃×#388	C₂²	1	8	yes	—
31	C₂₃×#387	C₂²	1	8	yes	—
32	C₂₃×#387	C₂²	1	8	yes	—
33	C₂₃×#386	C₂²	1	8	yes	—
34	C₂₃×#385	C₂²	1	8	yes	—
35	C₂₃×#383	C₂²	1	1	yes	characteristic
36	C₂₃×#383	C₂²	1	2	yes	—
37	C₂₃×#383	C₂²	1	4	yes	—
38	C₂₃×#383	C₂²	1	4	yes	—
39	C₂₃×#383	C₂²	1	4	yes	—
40	C₂₃×#383	C₂²	1	1	yes	characteristic
41	C₂₃×#382	C₂²	1	4	yes	—
42	C₂₃×#382	C₂²	1	2	yes	—
43	C₂₃×#382	C₂²	1	2	yes	—
44	C₂₃×#381	C₂²	1	2	yes	—
45	C₂₃×#381	C₂²	1	4	yes	—
46	C₂₃×#381	C₂²	1	2	yes	—
47	C₂₃×#380	C₂²	1	2	yes	—
48	C₂₃×#380	C₂²	1	2	yes	—
49	C₂₃×#380	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
55	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
56	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
58	C₂₃×#376	C₂²	1	2	yes	—
59	C₂₃×#376	C₂²	1	2	yes	—
60	C₂₃×#376	C₂²	1	4	yes	—
61	C₂×C₂₃×#95	C₂²	1	4	yes	—
62	C₂×C₂₃×#95	C₂²	1	4	yes	—
63	C₂×C₂₃×#95	C₂²	1	4	yes	—
64	C₂×#41×C₂₃	—	2	8	no	—
65	C₂×#41×C₂₃	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
72	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₂₃	—	2	16	no	—
79	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
83	C₂₃×#117	C₂×C₄	1	16	yes	complemented
84	C₂₃×#117	C₂×C₄	1	8	yes	complemented
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
99	C₂×C₄²×C₂₃	C₂×C₄	1	16	yes	complemented
100	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
108	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#95	C₂³	1	2	yes	—
111	C₂₃×#95	C₂³	1	2	yes	—
112	C₂₃×#95	C₂³	1	2	yes	—
113	C₂₃×#95	C₂³	1	4	yes	—
114	C₂₃×#95	C₂³	1	4	yes	—
115	C₂₃×#95	C₂³	1	4	yes	—
116	C₂₃×#95	C₂³	1	8	yes	—
117	C₂₃×#95	C₂³	1	8	yes	—
118	C₂₃×#95	C₂³	1	8	yes	—
119	C₂₃×#95	C₂³	1	2	yes	—
120	C₂₃×#95	C₂³	1	2	yes	—
121	C₂₃×#95	C₂³	1	4	yes	—
122	C₂₃×#95	C₂³	1	4	yes	—
123	C₂₃×#95	C₂³	1	8	yes	—
124	C₂₃×#95	C₂³	1	2	yes	—
125	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
126	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
127	#41×C₂₃	—	2	32	no	—
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₂₃	—	2	8	no	—
131	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
132	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
133	C₂×D₄×C₂₃	#41	1	8	yes	—
134	C₂²×C₄×C₂₃	—	2	8	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	16	no	—
137	C₂²×C₄×C₂₃	—	4	4	no	—
138	C₂²×C₄×C₂₃	—	4	2	no	—
139	C₂²×C₄×C₂₃	—	4	4	no	—
140	C₂²×C₄×C₂₃	—	4	2	no	—
141	C₂²×C₄×C₂₃	—	4	8	no	—
142	C₂²×C₄×C₂₃	—	4	8	no	—
143	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
144	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
147	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
161	C₄⋊C₄×C₂₃	—	2	8	no	—
162	C₄⋊C₄×C₂₃	—	4	4	no	—
163	C₄⋊C₄×C₂₃	—	4	8	no	—
164	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
168	C₂²⋊C₄×C₂₃	—	2	16	no	—
169	C₂²⋊C₄×C₂₃	—	4	4	no	—
170	C₂²⋊C₄×C₂₃	—	4	4	no	—
171	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
174	C₄²×C₂₃	—	2	8	no	—
175	C₄²×C₂₃	—	4	8	no	—
176	C₄²×C₂₃	—	4	4	no	—
177	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
178	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
179	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
180	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
181	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
182	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
183	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
184	#34195	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₂₃	—	2	4	no	—
186	C₂³×C₂₃	—	2	4	no	—
187	C₂³×C₂₃	—	2	2	no	—
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
191	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
192	C₂³×C₂₃	C₂×#41	1	8	yes	—
193	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
194	Q₈×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	2	16	no	—
196	D₄×C₂₃	#117	1	16	yes	—
197	C₂×C₄×C₂₃	—	2	16	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	8	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	4	no	—
211	C₂×C₄×C₂₃	—	2	8	no	—
212	C₂×C₄×C₂₃	—	2	8	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	4	8	no	—
217	C₂×C₄×C₂₃	—	4	8	no	—
218	C₂×C₄×C₂₃	—	4	4	no	—
219	C₂×C₄×C₂₃	—	4	4	no	—
220	C₂×C₄×C₂₃	—	8	16	no	—
221	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
227	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
228	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
229	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
230	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
231	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic
232	#2356	C₂×C₂₃	1	2	yes	complemented
233	#2353	C₂×C₂₃	1	4	yes	complemented
234	#2349	C₂×C₂₃	1	2	yes	complemented
235	#2347	C₂×C₂₃	1	2	yes	complemented
236	#2343	C₂×C₂₃	1	4	yes	complemented
237	#2339	C₂×C₂₃	1	2	yes	—
238	#2320	C₂×C₂₃	1	2	yes	complemented
239	#2318	C₂×C₂₃	1	4	yes	complemented
240	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#2283	C₂×C₂₃	1	2	yes	complemented
243	C₂×#383	C₂×C₂₃	1	2	yes	complemented
244	#2275	C₂×C₂₃	1	2	yes	complemented
245	C₂²×C₂₃	—	2	4	no	—
246	C₂²×C₂₃	—	2	4	no	—
247	C₂²×C₂₃	—	2	16	no	—
248	C₂²×C₂₃	—	2	2	no	—
249	C₂²×C₂₃	—	2	2	no	—
250	C₂²×C₂₃	—	2	2	no	—
251	C₂²×C₂₃	—	4	4	no	—
252	C₂²×C₂₃	C₂×#117	1	8	yes	—
253	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
254	C₂²×C₂₃	C₂²×#41	1	2	yes	—
255	C₂²×C₂₃	#585	1	2	yes	—
256	C₂²×C₂₃	#585	1	2	yes	—
257	C₄×C₂₃	—	2	2	no	—
258	C₄×C₂₃	—	2	4	no	—
259	C₄×C₂₃	—	2	2	no	—
260	C₄×C₂₃	—	2	2	no	—
261	C₄×C₂₃	—	2	4	no	—
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	2	4	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	4	4	no	—
266	C₄×C₂₃	—	4	8	no	—
267	C₄×C₂₃	—	4	4	no	—
268	C₄×C₂₃	C₂×#117	1	4	yes	—
269	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
270	C₄×#41	C₄×C₂₃	1	16	yes	complemented
271	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
274	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
275	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
277	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
278	C₂×#117	C₂²×C₂₃	1	2	yes	—
279	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
280	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
281	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
282	#391	C₂²×C₂₃	1	4	yes	—
283	#389	C₂²×C₂₃	1	4	yes	—
284	#388	C₂²×C₂₃	1	8	yes	—
285	#387	C₂²×C₂₃	1	8	yes	—
286	#387	C₂²×C₂₃	1	8	yes	—
287	#386	C₂²×C₂₃	1	8	yes	—
288	#385	C₂²×C₂₃	1	8	yes	—
289	#383	C₂²×C₂₃	1	4	yes	—
290	#383	C₂²×C₂₃	1	4	yes	—
291	#383	C₂²×C₂₃	1	4	yes	—
292	#383	C₂²×C₂₃	1	1	yes	characteristic
293	#383	C₂²×C₂₃	1	2	yes	—
294	#383	C₂²×C₂₃	1	1	yes	characteristic
295	#382	C₂²×C₂₃	1	4	yes	—
296	#382	C₂²×C₂₃	1	2	yes	—
297	#382	C₂²×C₂₃	1	2	yes	—
298	#381	C₂²×C₂₃	1	2	yes	—
299	#381	C₂²×C₂₃	1	4	yes	—
300	#381	C₂²×C₂₃	1	2	yes	—
301	#380	C₂²×C₂₃	1	2	yes	—
302	#380	C₂²×C₂₃	1	2	yes	—
303	#380	C₂²×C₂₃	1	4	yes	—
304	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
306	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
307	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
308	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
309	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
310	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
311	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
312	#376	C₂²×C₂₃	1	2	yes	—
313	#376	C₂²×C₂₃	1	2	yes	—
314	#376	C₂²×C₂₃	1	4	yes	—
315	C₂×#95	C₂²×C₂₃	1	4	yes	—
316	C₂×#95	C₂²×C₂₃	1	4	yes	—
317	C₂×#95	C₂²×C₂₃	1	4	yes	—
318	C₂×C₂₃	—	2	8	no	—
319	C₂×C₂₃	—	4	2	no	—
320	C₂×C₂₃	C₂²×#117	1	1	yes	characteristic
321	C₂×C₂₃	#3428	1	1	yes	characteristic
322	C₂×C₂₃	#3428	1	1	yes	characteristic
323	C₂×C₂₃	#3521	1	4	yes	—
324	C₂×#41	—	2	8	no	—
325	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
326	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
327	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
328	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
330	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
332	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
333	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
334	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
335	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
336	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
337	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
338	C₄×D₄	—	2	16	no	—
339	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
340	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
341	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
342	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
343	#117	C₂×C₄×C₂₃	1	16	yes	complemented
344	#117	C₂×C₄×C₂₃	1	8	yes	complemented
345	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
347	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
348	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
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	2	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
358	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
359	C₂×C₄²	C₂×C₄×C₂₃	1	16	yes	complemented
360	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
361	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
362	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
363	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
365	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
367	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
369	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
370	#95	C₂³×C₂₃	1	2	yes	—
371	#95	C₂³×C₂₃	1	2	yes	—
372	#95	C₂³×C₂₃	1	2	yes	—
373	#95	C₂³×C₂₃	1	4	yes	—
374	#95	C₂³×C₂₃	1	4	yes	—
375	#95	C₂³×C₂₃	1	4	yes	—
376	#95	C₂³×C₂₃	1	8	yes	—
377	#95	C₂³×C₂₃	1	8	yes	—
378	#95	C₂³×C₂₃	1	8	yes	—
379	#95	C₂³×C₂₃	1	2	yes	—
380	#95	C₂³×C₂₃	1	2	yes	—
381	#95	C₂³×C₂₃	1	4	yes	—
382	#95	C₂³×C₂₃	1	4	yes	—
383	#95	C₂³×C₂₃	1	8	yes	—
384	#95	C₂³×C₂₃	1	2	yes	—
385	C₂₃	#34195	1	1	yes	characteristic, direct product factor, Sylow
386	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
387	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
388	#41	—	2	32	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
393	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
394	C₂×D₄	#41×C₂₃	1	8	yes	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	—	4	8	no	—
404	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
405	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
406	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
407	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
408	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
409	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
412	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
419	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
420	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	4	4	no	—
424	C₄⋊C₄	—	4	8	no	—
425	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
427	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
428	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
434	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
435	C₄²	—	2	8	no	—
436	C₄²	—	4	8	no	—
437	C₄²	—	4	4	no	—
438	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
439	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
440	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
441	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
442	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
443	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
444	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
451	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
452	C₂³	C₂×#41×C₂₃	1	8	yes	—
453	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
454	Q₈	—	2	8	no	—
455	D₄	—	2	16	no	—
456	D₄	C₂₃×#117	1	16	yes	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	16	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	8	16	no	—
481	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
482	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
485	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
486	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
487	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
488	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
489	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
490	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	2	16	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	C₂×C₂₃×#117	1	8	yes	—
499	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
500	C₂²	C₂²×#41×C₂₃	1	2	yes	—
501	C₂²	C₂₃×#585	1	2	yes	—
502	C₂²	C₂₃×#585	1	2	yes	—
503	C₄	—	2	2	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	4	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	8	no	—
513	C₄	—	4	4	no	—
514	C₄	C₂×C₂₃×#117	1	4	yes	—
515	C₂	—	2	8	no	—
516	C₂	—	4	2	no	—
517	C₂	C₂²×C₂₃×#117	1	1	yes	characteristic
518	C₂	C₂₃×#3428	1	1	yes	characteristic
519	C₂	C₂₃×#3428	1	1	yes	characteristic
520	C₂	C₂₃×#3521	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂²×#41×C₂₃	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
2. C₂₃×#2356	C₂	semidirect
3. C₂₃×#2353	C₂	semidirect
4. C₂₃×#2349	C₂	semidirect
5. C₂₃×#2347	C₂	semidirect
6. C₂₃×#2343	C₂	semidirect
8. C₂₃×#2320	C₂	semidirect
9. C₂₃×#2318	C₂	semidirect
10. C₂₃×#2283	C₂	semidirect
11. C₂₃×#2283	C₂	semidirect
12. C₂₃×#2283	C₂	semidirect
13. C₂×C₂₃×#383	C₂	semidirect
14. C₂₃×#2275	C₂	semidirect
16. C₄×#41×C₂₃	C₄	semidirect
76. C₄×Q₈×C₂₃	C₂×C₄	semidirect
77. C₄×Q₈×C₂₃	C₂×C₄	semidirect
81. C₄×D₄×C₂₃	C₂×C₄	semidirect
82. C₄×D₄×C₂₃	C₂×C₄	semidirect
83. C₂₃×#117	C₂×C₄	semidirect
84. C₂₃×#117	C₂×C₄	semidirect
98. C₂×C₄²×C₂₃	C₂×C₄	semidirect
99. C₂×C₄²×C₂₃	C₂×C₄	semidirect
184. #34195	C₂₃	direct (N × H)
232. #2356	C₂×C₂₃	semidirect
233. #2353	C₂×C₂₃	semidirect
234. #2349	C₂×C₂₃	semidirect
235. #2347	C₂×C₂₃	semidirect
236. #2343	C₂×C₂₃	semidirect
238. #2320	C₂×C₂₃	semidirect
239. #2318	C₂×C₂₃	semidirect
240. #2283	C₂×C₂₃	semidirect
241. #2283	C₂×C₂₃	semidirect
242. #2283	C₂×C₂₃	semidirect
243. C₂×#383	C₂×C₂₃	semidirect
244. #2275	C₂×C₂₃	semidirect
270. C₄×#41	C₄×C₂₃	semidirect
336. C₄×Q₈	C₂×C₄×C₂₃	semidirect
337. C₄×Q₈	C₂×C₄×C₂₃	semidirect
341. C₄×D₄	C₂×C₄×C₂₃	semidirect
342. C₄×D₄	C₂×C₄×C₂₃	semidirect
343. #117	C₂×C₄×C₂₃	semidirect
344. #117	C₂×C₄×C₂₃	semidirect
358. C₂×C₄²	C₂×C₄×C₂₃	semidirect
359. C₂×C₄²	C₂×C₄×C₂₃	semidirect
385. C₂₃	#34195	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₂₃	2
C₂₃×#2356	2
C₂₃×#2353	2
C₂₃×#2349	2
C₂₃×#2347	2
C₂₃×#2343	2
C₂₃×#2339	2
C₂₃×#2320	2
C₂₃×#2318	2
C₂₃×#2283	2
C₂×C₂₃×#383	2
C₂₃×#2275	2
#34195	23

Subgroups — C23×#34195 #19181821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34195 #19181821