Subgroups — C₂₃×#3555 #15671174

Order 2944.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#563	C₂	1	4	yes	maximal, complemented
2	Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#557	C₂	1	2	yes	maximal, complemented
5	C₂₃×#556	C₂	1	2	yes	maximal, complemented
6	C₂₃×#556	C₂	1	2	yes	maximal, complemented
7	C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#552	C₂	1	2	yes	maximal, complemented
10	C₂₃×#544	C₂	1	2	yes	maximal, complemented
11	C₂₃×#542	C₂	1	2	yes	maximal
12	C₂₃×#541	C₂	1	4	yes	maximal, complemented
13	C₂₃×#541	C₂	1	2	yes	maximal, complemented
14	C₂₃×#541	C₂	1	2	yes	maximal, complemented
15	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂₃×#529	C₂	1	1	yes	maximal, characteristic
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
24	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
25	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂₃×#126	C₂²	1	4	yes	—
28	C₂₃×#126	C₂²	1	4	yes	—
29	C₂₃×#126	C₂²	1	4	yes	—
30	C₂₃×#126	C₂²	1	4	yes	—
31	C₂₃×#126	C₂²	1	4	yes	—
32	C₂₃×#126	C₂²	1	2	yes	—
33	C₂₃×#126	C₂²	1	4	yes	—
34	C₂₃×#125	C₂²	1	4	yes	—
35	C₂₃×#125	C₂²	1	2	yes	—
36	C₂₃×#125	C₂²	1	2	yes	—
37	C₂₃×#125	C₂²	1	1	yes	characteristic
38	C₂₃×#125	C₂²	1	4	yes	—
39	C₂₃×#125	C₂²	1	4	yes	—
40	C₂₃×#125	C₂²	1	2	yes	—
41	C₂₃×#125	C₂²	1	2	yes	—
42	C₂₃×#125	C₂²	1	4	yes	—
43	C₂₃×#125	C₂²	1	4	yes	—
44	C₂₃×#124	C₂²	1	1	yes	characteristic
45	C₂₃×#124	C₂²	1	2	yes	—
46	C₂₃×#124	C₂²	1	4	yes	—
47	C₂₃×#123	C₂²	1	4	yes	—
48	C₂₃×#123	C₂²	1	2	yes	—
49	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
50	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
51	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
52	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
54	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
55	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
56	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
57	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
58	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
59	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
60	C₂₃×#121	C₂²	1	1	yes	characteristic
61	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
62	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
63	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
64	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
65	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
66	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
67	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
68	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
69	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
70	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
71	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
72	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
73	C₄×D₄×C₂₃	C₂²	1	2	yes	—
74	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
75	C₄×D₄×C₂₃	C₂²	1	2	yes	—
76	C₂₃×#117	C₂²	1	2	yes	—
77	C₂₃×#117	C₂²	1	4	yes	—
78	C₂₃×#117	C₂²	1	2	yes	—
79	C₂₃×#117	C₂²	1	2	yes	—
80	C₂₃×#117	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
82	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
83	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
86	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
88	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
89	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
90	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₂₃	C₂³	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
95	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
97	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
101	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
107	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
113	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
117	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
118	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
119	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
120	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
124	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
125	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
126	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
130	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
137	C₄²×C₂₃	C₂³	1	1	yes	characteristic
138	C₄²×C₂₃	C₂³	1	1	yes	characteristic
139	C₄²×C₂₃	C₂³	1	1	yes	characteristic
140	C₄²×C₂₃	C₂³	1	1	yes	characteristic
141	C₄²×C₂₃	C₂³	1	4	yes	—
142	C₄²×C₂₃	C₂³	1	2	yes	—
143	C₄²×C₂₃	C₂³	1	2	yes	—
144	C₄²×C₂₃	C₂³	1	2	yes	—
145	C₄²×C₂₃	C₂³	1	4	yes	—
146	C₄²×C₂₃	C₂³	1	4	yes	—
147	C₄²×C₂₃	C₂³	1	4	yes	—
148	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
149	C₂³×C₂₃	C₂⁴	1	2	yes	—
150	Q₈×C₂₃	—	2	2	no	—
151	Q₈×C₂₃	—	2	2	no	—
152	Q₈×C₂₃	—	2	4	no	—
153	Q₈×C₂₃	—	2	2	no	—
154	Q₈×C₂₃	—	4	2	no	—
155	Q₈×C₂₃	—	4	2	no	—
156	Q₈×C₂₃	—	4	4	no	—
157	D₄×C₂₃	—	2	2	no	—
158	D₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	4	no	—
161	C₂×C₄×C₂₃	—	2	2	no	—
162	C₂×C₄×C₂₃	—	4	4	no	—
163	C₂×C₄×C₂₃	—	4	1	no	—
164	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
165	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
167	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
168	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
169	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
170	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
171	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
173	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
177	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	#3555	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂²×C₂₃	—	2	1	no	—
181	C₂²×C₂₃	—	2	1	no	—
182	C₂²×C₂₃	—	2	2	no	—
183	C₂²×C₂₃	—	2	2	no	—
184	C₂²×C₂₃	—	2	2	no	—
185	C₂²×C₂₃	—	2	1	no	—
186	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
187	C₄×C₂₃	—	2	1	no	—
188	C₄×C₂₃	—	2	2	no	—
189	C₄×C₂₃	—	2	1	no	—
190	C₄×C₂₃	—	2	2	no	—
191	C₄×C₂₃	—	2	2	no	—
192	C₄×C₂₃	—	2	4	no	—
193	C₄×C₂₃	—	2	2	no	—
194	C₄×C₂₃	—	2	2	no	—
195	C₄×C₂₃	—	2	4	no	—
196	C₄×C₂₃	—	2	1	no	—
197	C₄×C₂₃	—	2	1	no	—
198	C₄×C₂₃	—	2	2	no	—
199	C₄×C₂₃	—	2	2	no	—
200	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
201	C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
202	#563	C₂×C₂₃	1	4	yes	complemented
203	Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#557	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#557	C₂×C₂₃	1	2	yes	complemented
206	#556	C₂×C₂₃	1	2	yes	complemented
207	#556	C₂×C₂₃	1	2	yes	complemented
208	#556	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#554	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#552	C₂×C₂₃	1	2	yes	complemented
211	#544	C₂×C₂₃	1	2	yes	complemented
212	#542	C₂×C₂₃	1	2	yes	—
213	#541	C₂×C₂₃	1	2	yes	complemented
214	#541	C₂×C₂₃	1	2	yes	complemented
215	#541	C₂×C₂₃	1	4	yes	complemented
216	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#529	C₂×C₂₃	1	1	yes	characteristic
219	C₂×C₂₃	—	4	1	no	—
220	C₂×C₂₃	—	4	2	no	—
221	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
222	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₂₃	#585	1	1	yes	characteristic
224	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
225	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
226	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
227	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
229	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
230	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
231	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
232	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
233	#126	C₂²×C₂₃	1	2	yes	—
234	#126	C₂²×C₂₃	1	4	yes	—
235	#126	C₂²×C₂₃	1	4	yes	—
236	#126	C₂²×C₂₃	1	4	yes	—
237	#126	C₂²×C₂₃	1	4	yes	—
238	#126	C₂²×C₂₃	1	4	yes	—
239	#126	C₂²×C₂₃	1	4	yes	—
240	#125	C₂²×C₂₃	1	1	yes	characteristic
241	#125	C₂²×C₂₃	1	2	yes	—
242	#125	C₂²×C₂₃	1	2	yes	—
243	#125	C₂²×C₂₃	1	4	yes	—
244	#125	C₂²×C₂₃	1	4	yes	—
245	#125	C₂²×C₂₃	1	4	yes	—
246	#125	C₂²×C₂₃	1	2	yes	—
247	#125	C₂²×C₂₃	1	4	yes	—
248	#125	C₂²×C₂₃	1	4	yes	—
249	#125	C₂²×C₂₃	1	2	yes	—
250	#124	C₂²×C₂₃	1	1	yes	characteristic
251	#124	C₂²×C₂₃	1	2	yes	—
252	#124	C₂²×C₂₃	1	4	yes	—
253	#123	C₂²×C₂₃	1	2	yes	—
254	#123	C₂²×C₂₃	1	4	yes	—
255	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
257	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
260	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
261	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
262	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
263	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
264	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
265	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
266	#121	C₂²×C₂₃	1	1	yes	characteristic
267	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
268	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
269	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
271	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
272	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
273	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
274	C₄×Q₈	C₂²×C₂₃	1	4	yes	—
275	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
276	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
277	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
278	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
279	C₄×D₄	C₂²×C₂₃	1	2	yes	—
280	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
281	C₄×D₄	C₂²×C₂₃	1	2	yes	—
282	#117	C₂²×C₂₃	1	2	yes	—
283	#117	C₂²×C₂₃	1	2	yes	—
284	#117	C₂²×C₂₃	1	2	yes	—
285	#117	C₂²×C₂₃	1	4	yes	—
286	#117	C₂²×C₂₃	1	2	yes	—
287	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
289	C₂₃	#3555	1	1	yes	characteristic, direct product factor, Sylow
290	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
291	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
292	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
294	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
295	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
296	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
297	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
298	C₂×D₄	C₂³×C₂₃	1	2	yes	—
299	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
300	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
301	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
302	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
304	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
306	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
307	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
308	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
309	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
310	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
311	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
312	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
313	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
314	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
315	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
317	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
318	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
320	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
322	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
323	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
324	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
325	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
326	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
327	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
328	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
329	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
330	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
331	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
332	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
333	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
336	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
337	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
338	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
339	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
340	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
341	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
342	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
343	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	C₄²	C₂³×C₂₃	1	1	yes	characteristic
348	C₄²	C₂³×C₂₃	1	4	yes	—
349	C₄²	C₂³×C₂₃	1	2	yes	—
350	C₄²	C₂³×C₂₃	1	2	yes	—
351	C₄²	C₂³×C₂₃	1	2	yes	—
352	C₄²	C₂³×C₂₃	1	4	yes	—
353	C₄²	C₂³×C₂₃	1	4	yes	—
354	C₄²	C₂³×C₂₃	1	4	yes	—
355	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
356	C₂³	C₂⁴×C₂₃	1	2	yes	—
357	Q₈	—	2	2	no	—
358	Q₈	—	2	2	no	—
359	Q₈	—	2	2	no	—
360	Q₈	—	2	4	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	4	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
372	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
373	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
374	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
375	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
376	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
377	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
378	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
379	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
380	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
383	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
384	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	1	no	—
392	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	4	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
407	C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	4	2	no	—
410	C₂	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
411	C₂	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
412	C₂	C₂₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
prime radical	C₂₃×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#563	C₂	semidirect
2. Q₈²×C₂₃	C₂	semidirect
3. C₂₃×#557	C₂	semidirect
4. C₂₃×#557	C₂	semidirect
5. C₂₃×#556	C₂	semidirect
6. C₂₃×#556	C₂	semidirect
7. C₂₃×#556	C₂	semidirect
8. C₂₃×#554	C₂	semidirect
9. C₂₃×#552	C₂	semidirect
10. C₂₃×#544	C₂	semidirect
12. C₂₃×#541	C₂	semidirect
13. C₂₃×#541	C₂	semidirect
14. C₂₃×#541	C₂	semidirect
15. C₂₃×#533	C₂	semidirect
16. C₂₃×#533	C₂	semidirect
179. #3555	C₂₃	direct (N × H)
202. #563	C₂×C₂₃	semidirect
203. Q₈²	C₂×C₂₃	semidirect
204. #557	C₂×C₂₃	semidirect
205. #557	C₂×C₂₃	semidirect
206. #556	C₂×C₂₃	semidirect
207. #556	C₂×C₂₃	semidirect
208. #556	C₂×C₂₃	semidirect
209. #554	C₂×C₂₃	semidirect
210. #552	C₂×C₂₃	semidirect
211. #544	C₂×C₂₃	semidirect
213. #541	C₂×C₂₃	semidirect
214. #541	C₂×C₂₃	semidirect
215. #541	C₂×C₂₃	semidirect
216. #533	C₂×C₂₃	semidirect
217. #533	C₂×C₂₃	semidirect
289. C₂₃	#3555	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#563	2
Q₈²×C₂₃	2
C₂₃×#557	2
C₂₃×#556	2
C₂₃×#554	2
C₂₃×#552	2
C₂₃×#544	2
C₂₃×#542	2
C₂₃×#541	2
C₂₃×#533	2
C₂₃×#529	2
#3555	23

Subgroups — C23×#3555 #15671174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3555 #15671174