Subgroups — C₂₃×#60809 #19208435

Order 5888.

2034 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3223	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3222	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3152	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3142	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3037	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3007	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
12	C₂₃×#541	—	2	4	no	—
13	C₂₃×#541	C₂²	1	8	yes	complemented
14	C₂₃×#537	—	2	2	no	—
15	C₂₃×#536	C₂²	1	4	yes	—
16	C₂₃×#535	C₂²	1	4	yes	—
17	C₂₃×#534	—	2	2	no	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#532	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₂₃×#126	—	2	2	no	—
22	C₂×C₂₃×#125	—	2	2	no	—
23	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#525	—	2	4	no	—
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂₃×#497	C₂²	1	8	yes	complemented
29	C₂₃×#494	C₂²	1	4	yes	complemented
30	C₂₃×#492	C₂²	1	4	yes	complemented
31	C₂₃×#491	C₂²	1	4	yes	complemented
32	C₂₃×#490	C₂²	1	4	yes	complemented
33	C₂₃×#488	C₂²	1	8	yes	complemented
34	C₂₃×#464	C₂²	1	16	yes	complemented
35	C₂₃×#462	C₂²	1	8	yes	complemented
36	C₂₃×#460	C₂²	1	8	yes	complemented
37	C₂₃×#452	C₂²	1	16	yes	—
38	C₂₃×#451	C₂²	1	8	yes	complemented
39	C₂₃×#450	C₂²	1	8	yes	complemented
40	C₂₃×#433	C₂²	1	8	yes	complemented
41	C₂₃×#423	C₂²	1	2	yes	complemented
42	C₂₃×#417	C₂²	1	8	yes	complemented
43	C₂₃×#407	C₂²	1	2	yes	—
44	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
45	C₂×#41×C₂₃	—	2	2	no	—
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	4	8	no	—
48	C₂×#41×C₂₃	C₂³	1	4	yes	—
49	C₂×#41×C₂₃	C₂³	1	4	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₂₃	—	2	2	no	—
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#137	—	4	16	no	—
57	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
58	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
60	C₂₃×C₄⋊Q₈	—	2	2	no	—
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
64	C₂₃×#127	C₂³	1	2	yes	—
65	C₂₃×#127	C₂³	1	2	yes	—
66	C₂₃×#126	—	2	16	no	—
67	C₂₃×#125	—	2	16	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	C₂³	1	4	yes	—
70	C₂₃×#124	C₂³	1	4	yes	—
71	C₂₃×#124	C₂³	1	4	yes	—
72	C₂₃×#123	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	8	no	—
75	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
76	C₂₃×#121	—	2	4	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	8	no	—
79	C₂₃×#120	—	2	8	no	—
80	C₄×Q₈×C₂₃	—	2	4	no	—
81	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
82	C₄×D₄×C₂₃	—	2	4	no	—
83	C₄×D₄×C₂₃	—	2	4	no	—
84	C₂₃×#117	D₄	1	4	yes	—
85	C₂₃×#117	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
94	C₂×C₄²×C₂₃	—	2	2	no	—
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
98	C₂₃×#103	C₂³	1	16	yes	—
99	C₂₃×#102	C₂³	1	16	yes	—
100	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
101	C₂₃×#97	C₂³	1	4	yes	—
102	C₄×C₈×C₂₃	C₂³	1	4	yes	—
103	C₂⁴×C₂₃	—	2	2	no	—
104	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
105	#41×C₂₃	—	2	16	no	—
106	#41×C₂₃	—	4	8	no	—
107	#41×C₂₃	—	4	8	no	—
108	#41×C₂₃	—	4	8	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	8	no	—
112	C₂×Q₈×C₂₃	—	4	4	no	—
113	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
114	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
115	C₂×D₄×C₂₃	—	2	4	no	—
116	C₂×D₄×C₂₃	—	2	4	no	—
117	C₂×D₄×C₂₃	—	2	2	no	—
118	C₂×D₄×C₂₃	—	2	4	no	—
119	C₂×D₄×C₂₃	—	2	4	no	—
120	C₂×D₄×C₂₃	—	4	4	no	—
121	C₂×D₄×C₂₃	—	4	8	no	—
122	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	4	no	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	4	8	no	—
130	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
136	Q₁₆×C₂₃	—	4	16	no	—
137	SD₁₆×C₂₃	—	4	16	no	—
138	M₁₆×C₂₃	—	2	8	no	—
139	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	8	no	—
142	C₄⋊C₄×C₂₃	—	2	8	no	—
143	C₄⋊C₄×C₂₃	—	2	8	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	4	no	—
147	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
148	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
149	C₂²⋊C₄×C₂₃	—	2	8	no	—
150	C₂²⋊C₄×C₂₃	—	2	8	no	—
151	C₂²⋊C₄×C₂₃	—	2	8	no	—
152	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
153	C₄²×C₂₃	—	2	8	no	—
154	C₄²×C₂₃	C₂×D₄	1	2	yes	—
155	C₄²×C₂₃	C₂×D₄	1	2	yes	—
156	C₄²×C₂₃	C₂⁴	1	2	yes	—
157	C₄²×C₂₃	C₂⁴	1	2	yes	—
158	#60809	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₂₃	—	2	2	no	—
160	C₂³×C₂₃	—	2	4	no	—
161	C₂³×C₂₃	—	4	2	no	—
162	C₂³×C₂₃	—	4	4	no	—
163	C₂³×C₂₃	—	8	4	no	—
164	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
165	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₂₃	—	2	8	no	—
167	Q₈×C₂₃	—	2	8	no	—
168	Q₈×C₂₃	—	4	4	no	—
169	Q₈×C₂₃	—	4	4	no	—
170	D₄×C₂₃	—	2	8	no	—
171	D₄×C₂₃	—	4	4	no	—
172	D₄×C₂₃	—	4	8	no	—
173	D₄×C₂₃	—	4	8	no	—
174	D₄×C₂₃	—	8	4	no	—
175	D₄×C₂₃	—	8	4	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	4	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	4	no	—
180	C₂×C₄×C₂₃	—	2	4	no	—
181	C₂×C₄×C₂₃	—	2	4	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	8	no	—
185	C₂×C₄×C₂₃	—	4	4	no	—
186	C₂×C₄×C₂₃	—	4	4	no	—
187	C₂×C₄×C₂₃	—	4	4	no	—
188	C₂×C₄×C₂₃	—	4	8	no	—
189	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
190	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
192	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
194	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
195	C₈×C₂₃	—	2	8	no	—
196	C₂×#541	C₂×C₂₃	1	1	yes	characteristic, complemented
197	#3449	C₂×C₂₃	1	1	yes	characteristic
198	#3223	C₂×C₂₃	1	8	yes	complemented
199	#3222	C₂×C₂₃	1	8	yes	complemented
200	#3152	C₂×C₂₃	1	2	yes	complemented
201	#3142	C₂×C₂₃	1	2	yes	complemented
202	#3037	C₂×C₂₃	1	4	yes	complemented
203	#3007	C₂×C₂₃	1	4	yes	complemented
204	#2921	C₂×C₂₃	1	1	yes	characteristic, complemented
205	C₂²×C₂₃	—	2	2	no	—
206	C₂²×C₂₃	—	2	4	no	—
207	C₂²×C₂₃	—	2	2	no	—
208	C₂²×C₂₃	—	4	8	no	—
209	C₂²×C₂₃	—	4	4	no	—
210	C₂²×C₂₃	—	8	2	no	—
211	C₂²×C₂₃	—	8	8	no	—
212	C₂²×C₂₃	#535	1	2	yes	—
213	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
214	C₄×C₂₃	—	2	4	no	—
215	C₄×C₂₃	—	2	4	no	—
216	C₄×C₂₃	—	2	2	no	—
217	C₄×C₂₃	—	4	4	no	—
218	C₄×C₂₃	—	4	4	no	—
219	C₄×C₂₃	—	4	4	no	—
220	C₄×C₂₃	#535	1	2	yes	—
221	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
222	C₂²×#41	C₂²×C₂₃	1	2	yes	—
223	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
224	#541	—	2	4	no	—
225	#541	C₂²×C₂₃	1	8	yes	complemented
226	#537	—	2	2	no	—
227	#536	C₂²×C₂₃	1	4	yes	—
228	#535	C₂²×C₂₃	1	4	yes	—
229	#534	—	2	2	no	—
230	#532	C₂²×C₂₃	1	2	yes	—
231	#532	C₂²×C₂₃	1	2	yes	—
232	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
233	C₂×#126	—	2	2	no	—
234	C₂×#125	—	2	2	no	—
235	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
236	#525	—	2	4	no	—
237	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
238	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
239	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
240	#497	C₂²×C₂₃	1	8	yes	complemented
241	#494	C₂²×C₂₃	1	4	yes	complemented
242	#492	C₂²×C₂₃	1	4	yes	complemented
243	#491	C₂²×C₂₃	1	4	yes	complemented
244	#490	C₂²×C₂₃	1	4	yes	complemented
245	#488	C₂²×C₂₃	1	8	yes	complemented
246	#464	C₂²×C₂₃	1	16	yes	complemented
247	#462	C₂²×C₂₃	1	8	yes	complemented
248	#460	C₂²×C₂₃	1	8	yes	complemented
249	#452	C₂²×C₂₃	1	16	yes	—
250	#451	C₂²×C₂₃	1	8	yes	complemented
251	#450	C₂²×C₂₃	1	8	yes	complemented
252	#433	C₂²×C₂₃	1	8	yes	complemented
253	#423	C₂²×C₂₃	1	2	yes	complemented
254	#417	C₂²×C₂₃	1	8	yes	complemented
255	#407	C₂²×C₂₃	1	2	yes	—
256	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
257	C₂×C₂₃	—	2	2	no	—
258	C₂×C₂₃	—	4	2	no	—
259	C₂×C₂₃	—	8	4	no	—
260	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
261	C₂×C₂₃	#3586	1	2	yes	—
262	C₂×#41	—	2	2	no	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	4	8	no	—
265	C₂×#41	C₂³×C₂₃	1	4	yes	—
266	C₂×#41	C₂³×C₂₃	1	4	yes	—
267	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
268	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
271	C₂³×C₄	—	2	2	no	—
272	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
273	#137	—	4	16	no	—
274	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
275	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
276	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
279	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
280	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
281	#127	C₂³×C₂₃	1	2	yes	—
282	#127	C₂³×C₂₃	1	2	yes	—
283	#126	—	2	16	no	—
284	#125	—	2	16	no	—
285	#125	—	2	2	no	—
286	#125	C₂³×C₂₃	1	4	yes	—
287	#124	C₂³×C₂₃	1	4	yes	—
288	#124	C₂³×C₂₃	1	4	yes	—
289	#123	—	2	8	no	—
290	C₂²⋊Q₈	—	2	4	no	—
291	C₂²⋊Q₈	—	2	8	no	—
292	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
293	#121	—	2	4	no	—
294	#121	—	2	4	no	—
295	#121	—	2	8	no	—
296	#120	—	2	8	no	—
297	C₄×Q₈	—	2	4	no	—
298	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	4	no	—
301	#117	D₄×C₂₃	1	4	yes	—
302	#117	C₂³×C₂₃	1	4	yes	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
314	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
315	#103	C₂³×C₂₃	1	16	yes	—
316	#102	C₂³×C₂₃	1	16	yes	—
317	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
318	#97	C₂³×C₂₃	1	4	yes	—
319	C₄×C₈	C₂³×C₂₃	1	4	yes	—
320	C₂₃	#60809	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	2	no	—
322	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
323	#41	—	2	16	no	—
324	#41	—	4	8	no	—
325	#41	—	4	8	no	—
326	#41	—	4	8	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	8	no	—
330	C₂×Q₈	—	4	4	no	—
331	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
332	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	—	4	8	no	—
340	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
354	Q₁₆	—	4	16	no	—
355	SD₁₆	—	4	16	no	—
356	M₁₆	—	2	8	no	—
357	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
366	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
371	C₄²	—	2	8	no	—
372	C₄²	C₂×D₄×C₂₃	1	2	yes	—
373	C₄²	C₂×D₄×C₂₃	1	2	yes	—
374	C₄²	C₂⁴×C₂₃	1	2	yes	—
375	C₄²	C₂⁴×C₂₃	1	2	yes	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	4	no	—
380	C₂³	—	8	4	no	—
381	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
382	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
383	Q₈	—	2	8	no	—
384	Q₈	—	2	8	no	—
385	Q₈	—	4	4	no	—
386	Q₈	—	4	4	no	—
387	D₄	—	2	8	no	—
388	D₄	—	4	4	no	—
389	D₄	—	4	8	no	—
390	D₄	—	4	8	no	—
391	D₄	—	8	4	no	—
392	D₄	—	8	4	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	8	no	—
402	C₂×C₄	—	4	4	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
407	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
408	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
409	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
410	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
411	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
412	C₈	—	2	8	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	2	4	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	8	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	8	8	no	—
420	C₂²	C₂₃×#535	1	2	yes	—
421	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	4	no	—
425	C₄	—	4	4	no	—
426	C₄	—	4	4	no	—
427	C₄	—	4	4	no	—
428	C₄	C₂₃×#535	1	2	yes	—
429	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
430	C₂	—	2	2	no	—
431	C₂	—	4	2	no	—
432	C₂	—	8	4	no	—
433	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
434	C₂	C₂₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#541	C₂	semidirect
3. C₂₃×#3223	C₂	semidirect
4. C₂₃×#3222	C₂	semidirect
5. C₂₃×#3152	C₂	semidirect
6. C₂₃×#3142	C₂	semidirect
7. C₂₃×#3037	C₂	semidirect
8. C₂₃×#3007	C₂	semidirect
9. C₂₃×#2921	C₂	semidirect
11. C₂×C₂₃×#137	C₂²	semidirect
13. C₂₃×#541	C₂²	semidirect
20. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₂₃×#125	C₂²	semidirect
26. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
28. C₂₃×#497	C₂²	semidirect
29. C₂₃×#494	C₂²	semidirect
30. C₂₃×#492	C₂²	semidirect
31. C₂₃×#491	C₂²	semidirect
32. C₂₃×#490	C₂²	semidirect
33. C₂₃×#488	C₂²	semidirect
34. C₂₃×#464	C₂²	semidirect
35. C₂₃×#462	C₂²	semidirect
36. C₂₃×#460	C₂²	semidirect
38. C₂₃×#451	C₂²	semidirect
39. C₂₃×#450	C₂²	semidirect
40. C₂₃×#433	C₂²	semidirect
41. C₂₃×#423	C₂²	semidirect
42. C₂₃×#417	C₂²	semidirect
44. C₄×M₁₆×C₂₃	C₂²	semidirect
158. #60809	C₂₃	direct (N × H)
196. C₂×#541	C₂×C₂₃	semidirect
198. #3223	C₂×C₂₃	semidirect
199. #3222	C₂×C₂₃	semidirect
200. #3152	C₂×C₂₃	semidirect
201. #3142	C₂×C₂₃	semidirect
202. #3037	C₂×C₂₃	semidirect
203. #3007	C₂×C₂₃	semidirect
204. #2921	C₂×C₂₃	semidirect
223. C₂×#137	C₂²×C₂₃	semidirect
225. #541	C₂²×C₂₃	semidirect
232. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
235. C₂×#125	C₂²×C₂₃	semidirect
238. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
240. #497	C₂²×C₂₃	semidirect
241. #494	C₂²×C₂₃	semidirect
242. #492	C₂²×C₂₃	semidirect
243. #491	C₂²×C₂₃	semidirect
244. #490	C₂²×C₂₃	semidirect
245. #488	C₂²×C₂₃	semidirect
246. #464	C₂²×C₂₃	semidirect
247. #462	C₂²×C₂₃	semidirect
248. #460	C₂²×C₂₃	semidirect
250. #451	C₂²×C₂₃	semidirect
251. #450	C₂²×C₂₃	semidirect
252. #433	C₂²×C₂₃	semidirect
253. #423	C₂²×C₂₃	semidirect
254. #417	C₂²×C₂₃	semidirect
256. C₄×M₁₆	C₂²×C₂₃	semidirect
320. C₂₃	#60809	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×C₂₃×#541	2
C₂₃×#3449	2
C₂₃×#3223	2
C₂₃×#3222	2
C₂₃×#3152	2
C₂₃×#3142	2
C₂₃×#3037	2
C₂₃×#3007	2
C₂₃×#2921	2
#60809	23

Subgroups — C23×#60809 #19208435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60809 #19208435