Subgroups — C₂₃×#61720 #19209346

Order 5888.

2078 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3354	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3325	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3227	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3192	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3100	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3094	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
11	C₂₃×#555	—	2	4	no	—
12	C₂₃×#555	C₂²	1	8	yes	complemented
13	C₂₃×#547	C₂²	1	4	yes	complemented
14	C₂₃×#546	—	2	4	no	—
15	D₄²×C₂₃	C₂²	1	4	yes	complemented
16	C₂₃×#537	—	2	1	no	—
17	C₂₃×#534	—	2	2	no	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂₃×#124	—	2	2	no	—
21	C₂×C₂₃×#121	—	2	1	no	—
22	C₂×C₂₃×#121	C₂²	1	4	yes	—
23	C₂×C₂₃×#121	C₂²	1	2	yes	complemented
24	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂₃×#117	—	2	1	no	—
27	C₂×C₂₃×#117	C₂²	1	2	yes	complemented
28	C₂₃×#498	C₂²	1	4	yes	complemented
29	C₂₃×#486	C₂²	1	4	yes	complemented
30	C₂₃×#482	C₂²	1	16	yes	complemented
31	C₂₃×#481	C₂²	1	8	yes	complemented
32	C₂₃×#481	C₂²	1	8	yes	complemented
33	C₂₃×#480	C₂²	1	4	yes	complemented
34	C₂₃×#476	C₂²	1	8	yes	complemented
35	C₂₃×#474	C₂²	1	4	yes	complemented
36	C₂₃×#469	C₂²	1	16	yes	complemented
37	C₂₃×#459	C₂²	1	4	yes	complemented
38	C₂₃×#447	C₂²	1	4	yes	complemented
39	C₂₃×#447	C₂²	1	8	yes	complemented
40	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
41	C₂₃×#436	C₂²	1	8	yes	complemented
42	C₂₃×#431	C₂²	1	2	yes	complemented
43	C₂₃×#428	C₂²	1	8	yes	complemented
44	C₂₃×#418	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	complemented
46	C₂₃×H(5,𝔽₂)	—	4	8	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	2	no	—
49	C₂²×Q₈×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	2	4	no	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	—	2	4	no	—
53	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
55	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
58	C₂×D₈×C₂₃	C₂³	1	8	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
60	C₂₃×C₄⋊Q₈	—	2	2	no	—
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
63	C₂₃×#127	C₂³	1	2	yes	—
64	C₂₃×#124	—	2	16	no	—
65	C₂₃×#124	—	2	2	no	—
66	C₂₃×#123	—	2	8	no	—
67	C₂₃×C₂²⋊Q₈	—	2	4	no	—
68	C₂₃×#121	—	2	8	no	—
69	C₂₃×#121	—	2	4	no	—
70	C₂₃×#121	—	2	4	no	—
71	C₂₃×#121	—	2	4	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#121	—	2	4	no	—
74	C₂₃×#121	C₂³	1	8	yes	—
75	C₂₃×#121	C₂³	1	8	yes	—
76	C₂₃×#121	C₂³	1	4	yes	—
77	C₂₃×#120	—	2	8	no	—
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	D₄	1	4	yes	—
81	C₄×D₄×C₂₃	C₂³	1	4	yes	—
82	C₄×D₄×C₂₃	C₂³	1	4	yes	—
83	C₄×D₄×C₂₃	C₂³	1	4	yes	—
84	C₂₃×#117	—	2	4	no	—
85	C₂₃×#117	—	2	8	no	—
86	C₂₃×#117	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
97	C₂×C₄²×C₂₃	—	2	2	no	—
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	C₂³	1	8	yes	—
100	C₂₃×#106	C₂³	1	16	yes	—
101	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
102	C₂₃×#102	C₂³	1	8	yes	—
103	C₂₃×#102	C₂³	1	16	yes	—
104	C₂₃×#102	C₂³	1	8	yes	—
105	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
106	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
107	C₄×C₈×C₂₃	C₂³	1	4	yes	—
108	C₂⁴×C₂₃	—	2	1	no	—
109	C₂⁴×C₂₃	—	2	4	no	—
110	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
111	#41×C₂₃	—	4	16	no	—
112	#41×C₂₃	—	4	8	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	8	2	no	—
116	C₂×D₄×C₂₃	—	2	8	no	—
117	C₂×D₄×C₂₃	—	2	4	no	—
118	C₂×D₄×C₂₃	—	2	8	no	—
119	C₂×D₄×C₂₃	—	2	4	no	—
120	C₂×D₄×C₂₃	—	2	1	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	4	2	no	—
125	C₂×D₄×C₂₃	—	4	8	no	—
126	C₂×D₄×C₂₃	—	4	16	no	—
127	C₂×D₄×C₂₃	—	4	16	no	—
128	C₂×D₄×C₂₃	—	8	4	no	—
129	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
130	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
131	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	4	4	no	—
138	C₂²×C₄×C₂₃	—	4	4	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
145	D₈×C₂₃	—	4	8	no	—
146	M₁₆×C₂₃	—	4	8	no	—
147	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
154	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
155	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
156	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	4	no	—
159	C₂²⋊C₄×C₂₃	—	2	8	no	—
160	C₂²⋊C₄×C₂₃	—	2	4	no	—
161	C₂²⋊C₄×C₂₃	—	2	8	no	—
162	C₂²⋊C₄×C₂₃	—	2	4	no	—
163	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
164	C₄²×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	C₂×D₄	1	2	yes	—
166	C₄²×C₂₃	C₂⁴	1	2	yes	—
167	#61720	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₂₃	—	2	4	no	—
169	C₂³×C₂₃	—	2	2	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	4	2	no	—
172	C₂³×C₂₃	—	4	2	no	—
173	C₂³×C₂₃	—	4	2	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	—	4	8	no	—
176	C₂³×C₂₃	—	4	2	no	—
177	C₂³×C₂₃	—	8	4	no	—
178	C₂³×C₂₃	—	8	4	no	—
179	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
180	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₂₃	—	4	4	no	—
182	Q₈×C₂₃	—	8	4	no	—
183	D₄×C₂₃	—	2	8	no	—
184	D₄×C₂₃	—	2	4	no	—
185	D₄×C₂₃	—	2	8	no	—
186	D₄×C₂₃	—	4	2	no	—
187	D₄×C₂₃	—	4	16	no	—
188	D₄×C₂₃	—	4	8	no	—
189	D₄×C₂₃	—	4	4	no	—
190	D₄×C₂₃	—	8	2	no	—
191	D₄×C₂₃	—	8	4	no	—
192	D₄×C₂₃	—	8	8	no	—
193	C₂×C₄×C₂₃	—	2	8	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	4	4	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	8	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	8	no	—
205	C₂×C₄×C₂₃	—	4	8	no	—
206	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
212	C₈×C₂₃	—	2	8	no	—
213	#3496	C₂×C₂₃	1	1	yes	characteristic, complemented
214	C₂×#555	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#3354	C₂×C₂₃	1	8	yes	complemented
216	#3325	C₂×C₂₃	1	8	yes	complemented
217	#3227	C₂×C₂₃	1	2	yes	complemented
218	#3192	C₂×C₂₃	1	4	yes	complemented
219	#3100	C₂×C₂₃	1	2	yes	complemented
220	#3094	C₂×C₂₃	1	4	yes	complemented
221	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	1	no	—
223	C₂²×C₂₃	—	2	1	no	—
224	C₂²×C₂₃	—	2	4	no	—
225	C₂²×C₂₃	—	2	4	no	—
226	C₂²×C₂₃	—	2	4	no	—
227	C₂²×C₂₃	—	4	2	no	—
228	C₂²×C₂₃	—	4	4	no	—
229	C₂²×C₂₃	—	4	2	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	4	2	no	—
232	C₂²×C₂₃	—	4	4	no	—
233	C₂²×C₂₃	—	8	4	no	—
234	C₂²×C₂₃	—	8	4	no	—
235	C₂²×C₂₃	—	16	8	no	—
236	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
237	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₂₃	—	2	2	no	—
239	C₄×C₂₃	—	2	1	no	—
240	C₄×C₂₃	—	2	4	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	—	4	2	no	—
243	C₄×C₂₃	—	4	4	no	—
244	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
245	C₄×C₂₃	#585	1	2	yes	—
246	C₂×H(5,𝔽₂)	—	2	1	no	—
247	#555	—	2	4	no	—
248	#555	C₂²×C₂₃	1	8	yes	complemented
249	#547	C₂²×C₂₃	1	4	yes	complemented
250	#546	—	2	4	no	—
251	D₄²	C₂²×C₂₃	1	4	yes	complemented
252	#537	—	2	1	no	—
253	#534	—	2	2	no	—
254	#532	C₂²×C₂₃	1	2	yes	—
255	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
256	C₂×#124	—	2	2	no	—
257	C₂×#121	—	2	1	no	—
258	C₂×#121	C₂²×C₂₃	1	4	yes	—
259	C₂×#121	C₂²×C₂₃	1	2	yes	complemented
260	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
262	C₂×#117	—	2	1	no	—
263	C₂×#117	C₂²×C₂₃	1	2	yes	complemented
264	#498	C₂²×C₂₃	1	4	yes	complemented
265	#486	C₂²×C₂₃	1	4	yes	complemented
266	#482	C₂²×C₂₃	1	16	yes	complemented
267	#481	C₂²×C₂₃	1	8	yes	complemented
268	#481	C₂²×C₂₃	1	8	yes	complemented
269	#480	C₂²×C₂₃	1	4	yes	complemented
270	#476	C₂²×C₂₃	1	8	yes	complemented
271	#474	C₂²×C₂₃	1	4	yes	complemented
272	#469	C₂²×C₂₃	1	16	yes	complemented
273	#459	C₂²×C₂₃	1	4	yes	complemented
274	#447	C₂²×C₂₃	1	4	yes	complemented
275	#447	C₂²×C₂₃	1	8	yes	complemented
276	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
277	#436	C₂²×C₂₃	1	8	yes	complemented
278	#431	C₂²×C₂₃	1	2	yes	complemented
279	#428	C₂²×C₂₃	1	8	yes	complemented
280	#418	C₂²×C₂₃	1	8	yes	complemented
281	#407	C₂²×C₂₃	1	4	yes	complemented
282	C₂×C₂₃	—	2	2	no	—
283	C₂×C₂₃	—	4	4	no	—
284	C₂×C₂₃	—	8	2	no	—
285	C₂×C₂₃	—	8	4	no	—
286	C₂×C₂₃	#3485	1	1	yes	characteristic
287	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
288	C₂×C₂₃	#3585	1	1	yes	characteristic
289	H(5,𝔽₂)	—	4	8	no	—
290	C₂×#41	—	2	4	no	—
291	C₂×#41	—	2	2	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×D₄	—	2	4	no	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	—	2	4	no	—
296	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
297	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
298	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
299	C₂³×C₄	—	2	1	no	—
300	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
301	C₂×D₈	C₂³×C₂₃	1	8	yes	—
302	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
305	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
306	#127	C₂³×C₂₃	1	2	yes	—
307	#124	—	2	16	no	—
308	#124	—	2	2	no	—
309	#123	—	2	8	no	—
310	C₂²⋊Q₈	—	2	4	no	—
311	#121	—	2	8	no	—
312	#121	—	2	4	no	—
313	#121	—	2	4	no	—
314	#121	—	2	4	no	—
315	#121	—	2	2	no	—
316	#121	—	2	4	no	—
317	#121	C₂³×C₂₃	1	4	yes	—
318	#121	C₂³×C₂₃	1	8	yes	—
319	#121	C₂³×C₂₃	1	8	yes	—
320	#120	—	2	8	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	D₄×C₂₃	1	4	yes	—
324	C₄×D₄	C₂³×C₂₃	1	4	yes	—
325	C₄×D₄	C₂³×C₂₃	1	4	yes	—
326	C₄×D₄	C₂³×C₂₃	1	4	yes	—
327	#117	—	2	4	no	—
328	#117	—	2	8	no	—
329	#117	C₂³×C₂₃	1	8	yes	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₂²⋊C₄	—	2	4	no	—
339	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
342	#107	C₂³×C₂₃	1	8	yes	—
343	#106	C₂³×C₂₃	1	16	yes	—
344	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
345	#102	C₂³×C₂₃	1	8	yes	—
346	#102	C₂³×C₂₃	1	16	yes	—
347	#102	C₂³×C₂₃	1	8	yes	—
348	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
349	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
350	C₄×C₈	C₂³×C₂₃	1	4	yes	—
351	C₂₃	#61720	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	—	2	1	no	—
353	C₂⁴	—	2	4	no	—
354	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
355	#41	—	4	16	no	—
356	#41	—	4	8	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	8	2	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	4	2	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	16	no	—
371	C₂×D₄	—	4	16	no	—
372	C₂×D₄	—	8	4	no	—
373	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
374	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
375	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
389	D₈	—	4	8	no	—
390	M₁₆	—	4	8	no	—
391	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
398	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
399	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
400	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
408	C₄²	—	2	8	no	—
409	C₄²	C₂×D₄×C₂₃	1	2	yes	—
410	C₄²	C₂⁴×C₂₃	1	2	yes	—
411	C₂³	—	2	4	no	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	8	4	no	—
421	C₂³	—	8	4	no	—
422	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
423	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
424	Q₈	—	4	4	no	—
425	Q₈	—	8	4	no	—
426	D₄	—	2	8	no	—
427	D₄	—	2	4	no	—
428	D₄	—	2	8	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	16	no	—
431	D₄	—	4	8	no	—
432	D₄	—	4	4	no	—
433	D₄	—	8	4	no	—
434	D₄	—	8	2	no	—
435	D₄	—	8	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	8	no	—
448	C₂×C₄	—	4	8	no	—
449	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
450	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
451	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
455	C₈	—	2	8	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	4	no	—
459	C₂²	—	2	4	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	4	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	8	4	no	—
468	C₂²	—	8	4	no	—
469	C₂²	—	16	8	no	—
470	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
471	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
472	C₄	—	2	1	no	—
473	C₄	—	2	4	no	—
474	C₄	—	2	2	no	—
475	C₄	—	4	4	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	4	no	—
478	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
479	C₄	C₂₃×#585	1	2	yes	—
480	C₂	—	2	2	no	—
481	C₂	—	4	4	no	—
482	C₂	—	8	2	no	—
483	C₂	—	8	4	no	—
484	C₂	C₂₃×#3485	1	1	yes	characteristic
485	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
486	C₂	C₂₃×#3585	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3496	C₂	semidirect
2. C₂×C₂₃×#555	C₂	semidirect
3. C₂₃×#3354	C₂	semidirect
4. C₂₃×#3325	C₂	semidirect
5. C₂₃×#3227	C₂	semidirect
6. C₂₃×#3192	C₂	semidirect
7. C₂₃×#3100	C₂	semidirect
8. C₂₃×#3094	C₂	semidirect
9. C₂₃×#2975	C₂	semidirect
12. C₂₃×#555	C₂²	semidirect
13. C₂₃×#547	C₂²	semidirect
15. D₄²×C₂₃	C₂²	semidirect
19. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₂₃×#121	C₂²	semidirect
25. C₂×C₄×D₄×C₂₃	C₂²	semidirect
27. C₂×C₂₃×#117	C₂²	semidirect
28. C₂₃×#498	C₂²	semidirect
29. C₂₃×#486	C₂²	semidirect
30. C₂₃×#482	C₂²	semidirect
31. C₂₃×#481	C₂²	semidirect
32. C₂₃×#481	C₂²	semidirect
33. C₂₃×#480	C₂²	semidirect
34. C₂₃×#476	C₂²	semidirect
35. C₂₃×#474	C₂²	semidirect
36. C₂₃×#469	C₂²	semidirect
37. C₂₃×#459	C₂²	semidirect
38. C₂₃×#447	C₂²	semidirect
39. C₂₃×#447	C₂²	semidirect
40. C₄×D₈×C₂₃	C₂²	semidirect
41. C₂₃×#436	C₂²	semidirect
42. C₂₃×#431	C₂²	semidirect
43. C₂₃×#428	C₂²	semidirect
44. C₂₃×#418	C₂²	semidirect
45. C₂₃×#407	C₂²	semidirect
167. #61720	C₂₃	direct (N × H)
213. #3496	C₂×C₂₃	semidirect
214. C₂×#555	C₂×C₂₃	semidirect
215. #3354	C₂×C₂₃	semidirect
216. #3325	C₂×C₂₃	semidirect
217. #3227	C₂×C₂₃	semidirect
218. #3192	C₂×C₂₃	semidirect
219. #3100	C₂×C₂₃	semidirect
220. #3094	C₂×C₂₃	semidirect
221. #2975	C₂×C₂₃	semidirect
248. #555	C₂²×C₂₃	semidirect
249. #547	C₂²×C₂₃	semidirect
251. D₄²	C₂²×C₂₃	semidirect
255. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
259. C₂×#121	C₂²×C₂₃	semidirect
261. C₂×C₄×D₄	C₂²×C₂₃	semidirect
263. C₂×#117	C₂²×C₂₃	semidirect
264. #498	C₂²×C₂₃	semidirect
265. #486	C₂²×C₂₃	semidirect
266. #482	C₂²×C₂₃	semidirect
267. #481	C₂²×C₂₃	semidirect
268. #481	C₂²×C₂₃	semidirect
269. #480	C₂²×C₂₃	semidirect
270. #476	C₂²×C₂₃	semidirect
271. #474	C₂²×C₂₃	semidirect
272. #469	C₂²×C₂₃	semidirect
273. #459	C₂²×C₂₃	semidirect
274. #447	C₂²×C₂₃	semidirect
275. #447	C₂²×C₂₃	semidirect
276. C₄×D₈	C₂²×C₂₃	semidirect
277. #436	C₂²×C₂₃	semidirect
278. #431	C₂²×C₂₃	semidirect
279. #428	C₂²×C₂₃	semidirect
280. #418	C₂²×C₂₃	semidirect
281. #407	C₂²×C₂₃	semidirect
351. C₂₃	#61720	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×#3496	2
C₂×C₂₃×#555	2
C₂₃×#3354	2
C₂₃×#3325	2
C₂₃×#3227	2
C₂₃×#3192	2
C₂₃×#3100	2
C₂₃×#3094	2
C₂₃×#2975	2
#61720	23

Subgroups — C23×#61720 #19209346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61720 #19209346