Subgroups — C₂₃×#3055 #15670674

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#469	C₂	1	2	yes	maximal, complemented
6	C₂₃×#468	C₂	1	2	yes	maximal, complemented
7	C₂₃×#466	C₂	1	2	yes	maximal, complemented
8	C₂₃×#465	C₂	1	2	yes	maximal, complemented
9	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	—	2	1	no	—
14	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
18	C₂×D₈×C₂₃	C₂²	1	2	yes	complemented
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
23	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
24	C₂₃×#123	—	2	1	no	—
25	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#121	—	2	1	no	—
28	C₂₃×#121	—	2	1	no	—
29	C₂₃×#121	C₂²	1	2	yes	—
30	C₂₃×#121	C₂²	1	2	yes	—
31	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#120	—	2	1	no	—
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₂₃×#117	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
39	C₂₃×#107	C₂²	1	2	yes	complemented
40	C₂₃×#106	C₂²	1	2	yes	complemented
41	C₂₃×#103	C₂²	1	2	yes	complemented
42	C₂₃×#102	C₂²	1	2	yes	complemented
43	C₂₃×#102	C₂²	1	2	yes	complemented
44	C₂₃×#102	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	—	2	1	no	—
46	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
47	#41×C₂₃	—	2	2	no	—
48	#41×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	2	1	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	4	4	no	—
59	C₂×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	SD₁₆×C₂₃	—	2	4	no	—
72	D₈×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	D₄	1	4	yes	complemented
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂×C₈×C₂₃	C₂³	1	2	yes	—
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₄²×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	—	2	2	no	—
97	C₂³×C₂₃	—	4	1	no	—
98	C₂³×C₂₃	—	4	2	no	—
99	C₂³×C₂₃	—	4	2	no	—
100	C₂³×C₂₃	—	4	2	no	—
101	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₂₃	—	2	1	no	—
105	Q₈×C₂₃	—	2	1	no	—
106	D₄×C₂₃	—	2	1	no	—
107	D₄×C₂₃	—	2	2	no	—
108	D₄×C₂₃	—	2	2	no	—
109	D₄×C₂₃	—	2	1	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	4	4	no	—
112	D₄×C₂₃	—	4	2	no	—
113	D₄×C₂₃	—	4	2	no	—
114	D₄×C₂₃	—	4	4	no	—
115	D₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	1	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	1	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₂₃	#41	1	2	yes	—
134	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	C₂×D₄	1	4	yes	complemented
137	#3055	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	2	no	—
148	C₂²×C₂₃	—	4	2	no	—
149	C₂²×C₂₃	—	4	1	no	—
150	C₂²×C₂₃	—	4	1	no	—
151	C₂²×C₂₃	—	4	2	no	—
152	C₂²×C₂₃	—	8	4	no	—
153	C₂²×C₂₃	#121	1	2	yes	—
154	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
155	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	1	no	—
162	C₄×C₂₃	#121	1	2	yes	—
163	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
165	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
166	C₂²×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
167	#525	C₂×C₂₃	1	1	yes	characteristic, complemented
168	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
169	#469	C₂×C₂₃	1	2	yes	complemented
170	#468	C₂×C₂₃	1	2	yes	complemented
171	#466	C₂×C₂₃	1	2	yes	complemented
172	#465	C₂×C₂₃	1	2	yes	complemented
173	#428	C₂×C₂₃	1	1	yes	characteristic, complemented
174	#418	C₂×C₂₃	1	1	yes	characteristic, complemented
175	#417	C₂×C₂₃	1	1	yes	characteristic, complemented
176	C₂×C₂₃	—	2	2	no	—
177	C₂×C₂₃	—	2	2	no	—
178	C₂×C₂₃	—	4	1	no	—
179	C₂×C₂₃	—	8	1	no	—
180	C₂×C₂₃	—	8	2	no	—
181	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
182	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×C₂₃	#575	1	1	yes	characteristic
184	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
188	AGL(1,ℤ/8)	—	2	4	no	—
189	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
190	C₂×D₈	C₂²×C₂₃	1	2	yes	complemented
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
193	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
194	C₂×M₁₆	C₂²×C₂₃	1	2	yes	complemented
195	C₂²×C₈	C₂²×C₂₃	1	2	yes	complemented
196	#123	—	2	1	no	—
197	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
198	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
199	#121	—	2	1	no	—
200	#121	—	2	1	no	—
201	#121	C₂²×C₂₃	1	2	yes	—
202	#121	C₂²×C₂₃	1	2	yes	—
203	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
204	#121	C₂²×C₂₃	1	1	yes	characteristic, complemented
205	#120	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	#117	C₂²×C₂₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	#107	C₂²×C₂₃	1	2	yes	complemented
212	#106	C₂²×C₂₃	1	2	yes	complemented
213	#103	C₂²×C₂₃	1	2	yes	complemented
214	#102	C₂²×C₂₃	1	2	yes	complemented
215	#102	C₂²×C₂₃	1	2	yes	complemented
216	#102	C₂²×C₂₃	1	2	yes	complemented
217	C₂₃	#3055	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	—	2	1	no	—
219	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	C₂³×C₂₃	1	2	yes	—
233	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
244	SD₁₆	—	2	4	no	—
245	D₈	—	2	4	no	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	D₄×C₂₃	1	4	yes	complemented
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂³×C₂₃	1	2	yes	—
251	C₂×C₈	C₂³×C₂₃	1	2	yes	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
276	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	2	no	—
281	D₄	—	2	2	no	—
282	D₄	—	2	1	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
306	C₂×C₄	#41×C₂₃	1	2	yes	—
307	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	C₂×D₄×C₂₃	1	4	yes	complemented
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	8	4	no	—
325	C₂²	C₂₃×#121	1	2	yes	—
326	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
327	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂₃×#121	1	2	yes	—
335	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
336	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	2	no	—
341	C₂	—	8	1	no	—
342	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
343	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
344	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×M₁₆×C₂₃	C₂	semidirect
3. C₂₃×#525	C₂	semidirect
4. C₂×C₂₃×#121	C₂	semidirect
5. C₂₃×#469	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#466	C₂	semidirect
8. C₂₃×#465	C₂	semidirect
9. C₂₃×#428	C₂	semidirect
10. C₂₃×#418	C₂	semidirect
11. C₂₃×#417	C₂	semidirect
17. C₂×SD₁₆×C₂₃	C₂²	semidirect
18. C₂×D₈×C₂₃	C₂²	semidirect
22. C₂×M₁₆×C₂₃	C₂²	semidirect
23. C₂²×C₈×C₂₃	C₂²	semidirect
25. C₂₃×C₂²⋊Q₈	C₂²	semidirect
26. C₂₃×C₂²⋊Q₈	C₂²	semidirect
31. C₂₃×#121	C₂²	semidirect
32. C₂₃×#121	C₂²	semidirect
39. C₂₃×#107	C₂²	semidirect
40. C₂₃×#106	C₂²	semidirect
41. C₂₃×#103	C₂²	semidirect
42. C₂₃×#102	C₂²	semidirect
43. C₂₃×#102	C₂²	semidirect
44. C₂₃×#102	C₂²	semidirect
75. M₁₆×C₂₃	D₄	semidirect
136. C₈×C₂₃	C₂×D₄	semidirect
137. #3055	C₂₃	direct (N × H)
154. C₂²×C₂₃	AGL(1,ℤ/8)	semidirect
165. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
166. C₂²×M₁₆	C₂×C₂₃	semidirect
167. #525	C₂×C₂₃	semidirect
168. C₂×#121	C₂×C₂₃	semidirect
169. #469	C₂×C₂₃	semidirect
170. #468	C₂×C₂₃	semidirect
171. #466	C₂×C₂₃	semidirect
172. #465	C₂×C₂₃	semidirect
173. #428	C₂×C₂₃	semidirect
174. #418	C₂×C₂₃	semidirect
175. #417	C₂×C₂₃	semidirect
189. C₂×SD₁₆	C₂²×C₂₃	semidirect
190. C₂×D₈	C₂²×C₂₃	semidirect
194. C₂×M₁₆	C₂²×C₂₃	semidirect
195. C₂²×C₈	C₂²×C₂₃	semidirect
197. C₂²⋊Q₈	C₂²×C₂₃	semidirect
198. C₂²⋊Q₈	C₂²×C₂₃	semidirect
203. #121	C₂²×C₂₃	semidirect
204. #121	C₂²×C₂₃	semidirect
211. #107	C₂²×C₂₃	semidirect
212. #106	C₂²×C₂₃	semidirect
213. #103	C₂²×C₂₃	semidirect
214. #102	C₂²×C₂₃	semidirect
215. #102	C₂²×C₂₃	semidirect
216. #102	C₂²×C₂₃	semidirect
217. C₂₃	#3055	direct (N × H)
248. M₁₆	D₄×C₂₃	semidirect
309. C₈	C₂×D₄×C₂₃	semidirect
326. C₂²	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂²×M₁₆×C₂₃	2
C₂₃×#525	2
C₂×C₂₃×#121	2
C₂₃×#469	2
C₂₃×#468	2
C₂₃×#466	2
C₂₃×#465	2
C₂₃×#428	2
C₂₃×#418	2
C₂₃×#417	2
#3055	23

Subgroups — C23×#3055 #15670674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3055 #15670674