Subgroups — C₂₃×#61404 #19209030

Order 5888.

1900 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3400	C₂	1	6	yes	maximal, complemented
4	C₂₃×#3398	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3395	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3392	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3152	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3039	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3037	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#137	C₂	1	3	yes	maximal, complemented
12	C₂×C₂₃×#137	C₂²	1	3	yes	—
13	C₂×C₂₃×#137	C₂²	1	6	yes	—
14	C₂₃×#559	C₂²	1	3	yes	—
15	C₂₃×#557	—	2	1	no	—
16	C₂₃×#557	—	2	3	no	—
17	C₂₃×#557	C₂²	1	2	yes	—
18	C₂₃×#557	C₂²	1	6	yes	—
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#556	C₂²	1	3	yes	—
21	C₂₃×#553	—	2	3	no	—
22	C₂₃×#550	C₂²	1	1	yes	characteristic
23	C₂₃×#548	—	2	1	no	—
24	C₂₃×#532	C₂²	1	3	yes	—
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
26	C₂₃×#529	—	2	3	no	—
27	C₂×C₂₃×#125	—	2	3	no	—
28	C₄×#41×C₂₃	—	2	1	no	—
29	C₄×#41×C₂₃	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
31	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
32	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#497	C₂²	1	12	yes	—
34	C₂₃×#494	C₂²	1	6	yes	—
35	C₂₃×#492	C₂²	1	6	yes	—
36	C₂₃×#464	C₂²	1	12	yes	—
37	C₂₃×#462	C₂²	1	6	yes	—
38	C₂₃×#462	C₂²	1	6	yes	—
39	C₂₃×#462	C₂²	1	6	yes	—
40	C₂₃×#461	C₂²	1	6	yes	—
41	C₂₃×#461	C₂²	1	6	yes	—
42	C₂₃×#460	C₂²	1	6	yes	—
43	C₂₃×#446	C₂²	1	6	yes	—
44	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
45	C₂₃×#441	C₂²	1	6	yes	—
46	C₂₃×#440	C₂²	1	6	yes	—
47	C₄×Q₁₆×C₂₃	C₂²	1	6	yes	—
48	C₄×SD₁₆×C₂₃	C₂²	1	6	yes	—
49	C₂₃×#428	C₂²	1	1	yes	characteristic
50	C₂₃×#423	C₂²	1	3	yes	—
51	C₂₃×#419	C₂²	1	3	yes	—
52	C₂₃×#417	C₂²	1	3	yes	—
53	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
54	C₂×#41×C₂₃	—	2	1	no	—
55	C₂×#41×C₂₃	C₂³	1	3	yes	—
56	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
58	C₂₃×#137	—	2	12	no	—
59	C₂₃×#137	—	4	12	no	—
60	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
61	C₂×Q₁₆×C₂₃	C₂³	1	12	yes	—
62	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
63	C₂×SD₁₆×C₂₃	C₂³	1	12	yes	—
64	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
65	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	—	2	3	no	—
67	C₂₃×C₄⋊Q₈	—	2	3	no	—
68	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
69	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
70	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
71	C₂₃×#127	C₂³	1	3	yes	—
72	C₂₃×#126	—	2	6	no	—
73	C₂₃×#125	—	2	3	no	—
74	C₂₃×#125	—	2	6	no	—
75	C₂₃×#125	—	2	6	no	—
76	C₂₃×#125	—	2	12	no	—
77	C₂₃×#124	—	2	3	no	—
78	C₂₃×#124	C₂³	1	6	yes	—
79	C₂₃×#123	—	2	6	no	—
80	C₂₃×C₂²⋊Q₈	—	2	3	no	—
81	C₂₃×#121	—	2	3	no	—
82	C₂₃×#121	—	2	3	no	—
83	C₂₃×#121	—	2	3	no	—
84	C₄×Q₈×C₂₃	—	2	1	no	—
85	C₄×Q₈×C₂₃	—	2	6	no	—
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	—	2	3	no	—
88	C₄×Q₈×C₂₃	—	2	3	no	—
89	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₂₃	D₄	1	3	yes	—
91	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
92	C₄×Q₈×C₂₃	C₂³	1	6	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	6	yes	—
94	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
97	C₄×D₄×C₂₃	—	2	3	no	—
98	C₄×D₄×C₂₃	—	2	3	no	—
99	C₄×D₄×C₂₃	—	2	3	no	—
100	C₄×D₄×C₂₃	—	2	3	no	—
101	C₄×D₄×C₂₃	—	2	3	no	—
102	C₄×D₄×C₂₃	C₂³	1	3	yes	—
103	C₄×D₄×C₂₃	C₂³	1	3	yes	—
104	C₂₃×#117	—	2	3	no	—
105	C₂₃×#117	—	2	3	no	—
106	C₂₃×#117	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
110	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	—	2	3	no	—
113	C₂×C₄²×C₂₃	—	2	3	no	—
114	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
115	C₂₃×#107	C₂³	1	2	yes	—
116	C₂₃×#106	C₂³	1	2	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
119	C₂₃×#103	C₂³	1	6	yes	—
120	C₂₃×#103	C₂³	1	6	yes	—
121	C₂₃×#103	C₂³	1	6	yes	—
122	C₂₃×#102	C₂³	1	6	yes	—
123	C₂₃×#97	C₂³	1	6	yes	—
124	C₄×C₈×C₂₃	C₂³	1	6	yes	—
125	#41×C₂₃	—	2	6	no	—
126	#41×C₂₃	—	2	6	no	—
127	#41×C₂₃	—	4	4	no	—
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	—	2	3	no	—
130	C₂×Q₈×C₂₃	—	2	6	no	—
131	C₂×Q₈×C₂₃	—	2	6	no	—
132	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
133	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
135	C₂×Q₈×C₂₃	C₂⁴	1	6	yes	—
136	C₂×D₄×C₂₃	—	2	3	no	—
137	C₂×D₄×C₂₃	—	2	3	no	—
138	C₂×D₄×C₂₃	—	2	3	no	—
139	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	3	no	—
142	C₂²×C₄×C₂₃	—	2	3	no	—
143	C₂²×C₄×C₂₃	—	2	3	no	—
144	C₂²×C₄×C₂₃	—	2	3	no	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₂₃	—	2	12	no	—
149	Q₁₆×C₂₃	—	4	12	no	—
150	SD₁₆×C₂₃	—	2	12	no	—
151	SD₁₆×C₂₃	—	4	12	no	—
152	M₁₆×C₂₃	—	2	6	no	—
153	M₁₆×C₂₃	#41	1	4	yes	complemented
154	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
155	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
156	C₄⋊C₄×C₂₃	—	2	6	no	—
157	C₄⋊C₄×C₂₃	—	2	6	no	—
158	C₄⋊C₄×C₂₃	—	2	6	no	—
159	C₄⋊C₄×C₂₃	—	2	6	no	—
160	C₄⋊C₄×C₂₃	—	2	6	no	—
161	C₄⋊C₄×C₂₃	—	2	6	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	3	no	—
164	C₄⋊C₄×C₂₃	—	2	3	no	—
165	C₄⋊C₄×C₂₃	—	2	3	no	—
166	C₄⋊C₄×C₂₃	—	2	3	no	—
167	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
170	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
171	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
173	C₂²⋊C₄×C₂₃	—	2	1	no	—
174	C₂²⋊C₄×C₂₃	—	2	6	no	—
175	C₂²⋊C₄×C₂₃	—	2	3	no	—
176	C₂²⋊C₄×C₂₃	—	2	3	no	—
177	C₂²⋊C₄×C₂₃	—	2	6	no	—
178	C₄²×C₂₃	—	2	1	no	—
179	C₄²×C₂₃	—	2	3	no	—
180	C₄²×C₂₃	—	2	3	no	—
181	C₄²×C₂₃	—	2	6	no	—
182	C₄²×C₂₃	—	2	6	no	—
183	C₄²×C₂₃	C₂×D₄	1	3	yes	—
184	C₄²×C₂₃	C₂×D₄	1	3	yes	—
185	C₄²×C₂₃	C₂⁴	1	3	yes	—
186	C₄²×C₂₃	C₂⁴	1	3	yes	—
187	#61404	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	3	no	—
189	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₂₃	—	2	6	no	—
191	Q₈×C₂₃	—	2	6	no	—
192	Q₈×C₂₃	—	2	3	no	—
193	Q₈×C₂₃	—	2	6	no	—
194	Q₈×C₂₃	—	2	3	no	—
195	Q₈×C₂₃	—	4	6	no	—
196	Q₈×C₂₃	#137	1	4	yes	complemented
197	D₄×C₂₃	—	2	3	no	—
198	D₄×C₂₃	—	2	3	no	—
199	D₄×C₂₃	—	4	6	no	—
200	D₄×C₂₃	—	4	6	no	—
201	D₄×C₂₃	—	8	3	no	—
202	C₂×C₄×C₂₃	—	2	6	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	1	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	3	no	—
207	C₂×C₄×C₂₃	—	2	6	no	—
208	C₂×C₄×C₂₃	—	4	3	no	—
209	C₂×C₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	4	6	no	—
211	C₂×C₄×C₂₃	—	4	6	no	—
212	C₂×C₄×C₂₃	—	4	6	no	—
213	C₂×C₄×C₂₃	—	8	3	no	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
220	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₂₃	—	2	6	no	—
223	C₈×C₂₃	C₂×#41	1	4	yes	—
224	#3480	C₂×C₂₃	1	1	yes	characteristic
225	C₂×#557	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#3400	C₂×C₂₃	1	6	yes	complemented
227	#3398	C₂×C₂₃	1	6	yes	complemented
228	#3395	C₂×C₂₃	1	2	yes	complemented
229	#3392	C₂×C₂₃	1	2	yes	complemented
230	#3152	C₂×C₂₃	1	3	yes	complemented
231	#3039	C₂×C₂₃	1	3	yes	complemented
232	#3037	C₂×C₂₃	1	3	yes	complemented
233	Q₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
234	C₄×#137	C₂×C₂₃	1	3	yes	complemented
235	C₂²×C₂₃	—	2	1	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	4	3	no	—
238	C₂²×C₂₃	—	4	3	no	—
239	C₂²×C₂₃	—	4	3	no	—
240	C₂²×C₂₃	#550	1	2	yes	—
241	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₂₃	—	2	3	no	—
243	C₄×C₂₃	—	2	3	no	—
244	C₄×C₂₃	—	2	3	no	—
245	C₄×C₂₃	—	4	2	no	—
246	C₄×C₂₃	—	4	1	no	—
247	C₄×C₂₃	—	4	3	no	—
248	C₄×C₂₃	—	4	6	no	—
249	C₄×C₂₃	—	4	1	no	—
250	C₄×C₂₃	#550	1	2	yes	—
251	C₄×C₂₃	C₂×#137	1	6	yes	—
252	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
253	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
254	C₂×#137	C₂²×C₂₃	1	3	yes	—
255	C₂×#137	C₂²×C₂₃	1	6	yes	—
256	#559	C₂²×C₂₃	1	3	yes	—
257	#557	—	2	1	no	—
258	#557	—	2	3	no	—
259	#557	C₂²×C₂₃	1	2	yes	—
260	#557	C₂²×C₂₃	1	1	yes	characteristic
261	#557	C₂²×C₂₃	1	6	yes	—
262	#556	C₂²×C₂₃	1	3	yes	—
263	#553	—	2	3	no	—
264	#550	C₂²×C₂₃	1	1	yes	characteristic
265	#548	—	2	1	no	—
266	#532	C₂²×C₂₃	1	3	yes	—
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
268	#529	—	2	3	no	—
269	C₂×#125	—	2	3	no	—
270	C₄×#41	—	2	1	no	—
271	C₄×#41	C₂²×C₂₃	1	3	yes	—
272	C₂×C₄×Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
274	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
275	#497	C₂²×C₂₃	1	12	yes	—
276	#494	C₂²×C₂₃	1	6	yes	—
277	#492	C₂²×C₂₃	1	6	yes	—
278	#464	C₂²×C₂₃	1	12	yes	—
279	#462	C₂²×C₂₃	1	6	yes	—
280	#462	C₂²×C₂₃	1	6	yes	—
281	#462	C₂²×C₂₃	1	6	yes	—
282	#461	C₂²×C₂₃	1	6	yes	—
283	#461	C₂²×C₂₃	1	6	yes	—
284	#460	C₂²×C₂₃	1	6	yes	—
285	#446	C₂²×C₂₃	1	6	yes	—
286	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
287	#441	C₂²×C₂₃	1	6	yes	—
288	#440	C₂²×C₂₃	1	6	yes	—
289	C₄×Q₁₆	C₂²×C₂₃	1	6	yes	—
290	C₄×SD₁₆	C₂²×C₂₃	1	6	yes	—
291	#428	C₂²×C₂₃	1	1	yes	characteristic
292	#423	C₂²×C₂₃	1	3	yes	—
293	#419	C₂²×C₂₃	1	3	yes	—
294	#417	C₂²×C₂₃	1	3	yes	—
295	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
296	C₂×C₂₃	—	2	2	no	—
297	C₂×C₂₃	—	8	3	no	—
298	C₂×C₂₃	C₂×#550	1	1	yes	characteristic
299	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
300	C₂×C₂₃	#3586	1	1	yes	characteristic
301	C₂×#41	—	2	1	no	—
302	C₂×#41	C₂³×C₂₃	1	3	yes	—
303	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
304	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
305	#137	—	2	12	no	—
306	#137	—	4	12	no	—
307	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
308	C₂×Q₁₆	C₂³×C₂₃	1	12	yes	—
309	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
310	C₂×SD₁₆	C₂³×C₂₃	1	12	yes	—
311	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
312	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
313	C₄⋊Q₈	—	2	3	no	—
314	C₄⋊Q₈	—	2	3	no	—
315	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
316	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
317	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
318	#127	C₂³×C₂₃	1	3	yes	—
319	#126	—	2	6	no	—
320	#125	—	2	6	no	—
321	#125	—	2	6	no	—
322	#125	—	2	12	no	—
323	#125	—	2	3	no	—
324	#124	—	2	3	no	—
325	#124	C₂³×C₂₃	1	6	yes	—
326	#123	—	2	6	no	—
327	C₂²⋊Q₈	—	2	3	no	—
328	#121	—	2	3	no	—
329	#121	—	2	3	no	—
330	#121	—	2	3	no	—
331	C₄×Q₈	—	2	6	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	1	no	—
334	C₄×Q₈	—	2	3	no	—
335	C₄×Q₈	—	2	3	no	—
336	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
337	C₄×Q₈	D₄×C₂₃	1	3	yes	—
338	C₄×Q₈	C₂³×C₂₃	1	6	yes	—
339	C₄×Q₈	C₂³×C₂₃	1	6	yes	—
340	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
341	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
342	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
343	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	—	2	3	no	—
348	C₄×D₄	—	2	3	no	—
349	C₄×D₄	C₂³×C₂₃	1	3	yes	—
350	C₄×D₄	C₂³×C₂₃	1	3	yes	—
351	#117	—	2	3	no	—
352	#117	—	2	3	no	—
353	#117	C₂³×C₂₃	1	1	yes	characteristic
354	C₂×C₄⋊C₄	—	2	3	no	—
355	C₂×C₄⋊C₄	—	2	3	no	—
356	C₂×C₄⋊C₄	—	2	3	no	—
357	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
358	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₂×C₄²	—	2	3	no	—
360	C₂×C₄²	—	2	3	no	—
361	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
362	#107	C₂³×C₂₃	1	2	yes	—
363	#106	C₂³×C₂₃	1	2	yes	—
364	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
365	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
366	#103	C₂³×C₂₃	1	6	yes	—
367	#103	C₂³×C₂₃	1	6	yes	—
368	#103	C₂³×C₂₃	1	6	yes	—
369	#102	C₂³×C₂₃	1	6	yes	—
370	#97	C₂³×C₂₃	1	6	yes	—
371	C₄×C₈	C₂³×C₂₃	1	6	yes	—
372	C₂₃	#61404	1	1	yes	characteristic, direct product factor, Sylow
373	#41	—	2	6	no	—
374	#41	—	2	6	no	—
375	#41	—	4	4	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	2	3	no	—
378	C₂×Q₈	—	2	6	no	—
379	C₂×Q₈	—	2	6	no	—
380	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
381	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
382	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
383	C₂×Q₈	C₂⁴×C₂₃	1	6	yes	—
384	C₂×D₄	—	2	3	no	—
385	C₂×D₄	—	2	3	no	—
386	C₂×D₄	—	2	3	no	—
387	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	3	no	—
390	C₂²×C₄	—	2	3	no	—
391	C₂²×C₄	—	2	3	no	—
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
395	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
396	Q₁₆	—	2	12	no	—
397	Q₁₆	—	4	12	no	—
398	SD₁₆	—	2	12	no	—
399	SD₁₆	—	4	12	no	—
400	M₁₆	—	2	6	no	—
401	M₁₆	#41×C₂₃	1	4	yes	complemented
402	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
403	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
404	C₄⋊C₄	—	2	6	no	—
405	C₄⋊C₄	—	2	6	no	—
406	C₄⋊C₄	—	2	6	no	—
407	C₄⋊C₄	—	2	6	no	—
408	C₄⋊C₄	—	2	6	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	3	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
416	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
417	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
418	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
419	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
420	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₂²⋊C₄	—	2	3	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	—	2	6	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	3	no	—
428	C₄²	—	2	3	no	—
429	C₄²	—	2	6	no	—
430	C₄²	—	2	6	no	—
431	C₄²	C₂×D₄×C₂₃	1	3	yes	—
432	C₄²	C₂×D₄×C₂₃	1	3	yes	—
433	C₄²	C₂⁴×C₂₃	1	3	yes	—
434	C₄²	C₂⁴×C₂₃	1	3	yes	—
435	C₂³	—	2	3	no	—
436	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
437	Q₈	—	2	6	no	—
438	Q₈	—	2	6	no	—
439	Q₈	—	2	3	no	—
440	Q₈	—	2	6	no	—
441	Q₈	—	2	3	no	—
442	Q₈	—	4	6	no	—
443	Q₈	C₂₃×#137	1	4	yes	complemented
444	D₄	—	2	3	no	—
445	D₄	—	2	3	no	—
446	D₄	—	4	6	no	—
447	D₄	—	4	6	no	—
448	D₄	—	8	3	no	—
449	C₂×C₄	—	2	6	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	6	no	—
455	C₂×C₄	—	4	3	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	6	no	—
458	C₂×C₄	—	4	6	no	—
459	C₂×C₄	—	4	6	no	—
460	C₂×C₄	—	8	3	no	—
461	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
462	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
464	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
466	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
467	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
468	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
469	C₈	—	2	6	no	—
470	C₈	C₂×#41×C₂₃	1	4	yes	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	4	3	no	—
474	C₂²	—	4	3	no	—
475	C₂²	—	4	3	no	—
476	C₂²	C₂₃×#550	1	2	yes	—
477	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
478	C₄	—	2	3	no	—
479	C₄	—	2	3	no	—
480	C₄	—	2	3	no	—
481	C₄	—	4	3	no	—
482	C₄	—	4	6	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	1	no	—
486	C₄	C₂₃×#550	1	2	yes	—
487	C₄	C₂×C₂₃×#137	1	6	yes	—
488	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
489	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
490	C₂	—	2	2	no	—
491	C₂	—	8	3	no	—
492	C₂	C₂×C₂₃×#550	1	1	yes	characteristic
493	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
494	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3480	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#557	C₂	semidirect
3. C₂₃×#3400	C₂	semidirect
4. C₂₃×#3398	C₂	semidirect
5. C₂₃×#3395	C₂	semidirect
6. C₂₃×#3392	C₂	semidirect
7. C₂₃×#3152	C₂	semidirect
8. C₂₃×#3039	C₂	semidirect
9. C₂₃×#3037	C₂	semidirect
10. Q₈×M₁₆×C₂₃	C₂	semidirect
11. C₄×C₂₃×#137	C₂	semidirect
153. M₁₆×C₂₃	#41	semidirect
187. #61404	C₂₃	direct (N × H)
196. Q₈×C₂₃	#137	semidirect
225. C₂×#557	C₂×C₂₃	semidirect
226. #3400	C₂×C₂₃	semidirect
227. #3398	C₂×C₂₃	semidirect
228. #3395	C₂×C₂₃	semidirect
229. #3392	C₂×C₂₃	semidirect
230. #3152	C₂×C₂₃	semidirect
231. #3039	C₂×C₂₃	semidirect
232. #3037	C₂×C₂₃	semidirect
233. Q₈×M₁₆	C₂×C₂₃	semidirect
234. C₄×#137	C₂×C₂₃	semidirect
372. C₂₃	#61404	direct (N × H)
401. M₁₆	#41×C₂₃	semidirect
443. Q₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3480	2
C₂×C₂₃×#557	2
C₂₃×#3400	2
C₂₃×#3398	2
C₂₃×#3395	2
C₂₃×#3392	2
C₂₃×#3152	2
C₂₃×#3039	2
C₂₃×#3037	2
Q₈×M₁₆×C₂₃	2
C₄×C₂₃×#137	2
#61404	23

Subgroups — C23×#61404 #19209030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61404 #19209030