Subgroups — C₂²×D₄²×C₂₃ #19210321

Order 5888.

20038 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₂₃	C₂	1	4	yes	maximal, complemented
2	C₂×D₄²×C₂₃	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂₃×#121	C₂	1	4	yes	maximal, complemented
5	C₂²×C₂₃×#120	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₂₃	C₂²	1	4	yes	complemented
8	C₂³×D₄×C₂₃	—	2	64	no	—
9	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₂₃	C₂²	1	8	yes	complemented
11	C₂³×D₄×C₂₃	C₂²	1	4	yes	complemented
12	C₂³×D₄×C₂₃	C₂²	1	2	yes	complemented
13	C₂³×D₄×C₂₃	C₂²	1	96	yes	complemented
14	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	complemented
15	D₄²×C₂₃	C₂²	1	256	yes	direct product factor
16	C₂×C₂₃×#127	C₂²	1	24	yes	complemented
17	C₂×C₂₃×#121	C₂²	1	96	yes	complemented
18	C₂×C₂₃×#120	C₂²	1	96	yes	complemented
19	C₂×C₄×D₄×C₂₃	C₂²	1	48	yes	complemented
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
21	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
22	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
23	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂⁵×C₂₃	—	2	64	no	—
25	C₂⁵×C₂₃	—	4	16	no	—
26	C₂⁵×C₂₃	C₂³	1	4	yes	—
27	C₂⁵×C₂₃	C₂³	1	4	yes	—
28	C₂⁵×C₂₃	C₂³	1	48	yes	complemented
29	C₂²×D₄×C₂₃	—	2	64	no	—
30	C₂²×D₄×C₂₃	—	2	768	no	—
31	C₂²×D₄×C₂₃	—	2	32	no	—
32	C₂²×D₄×C₂₃	—	4	8	no	—
33	C₂²×D₄×C₂₃	D₄	1	32	yes	direct product factor
34	C₂²×D₄×C₂₃	C₂³	1	24	yes	—
35	C₂²×D₄×C₂₃	C₂³	1	96	yes	complemented
36	C₂²×D₄×C₂₃	C₂³	1	48	yes	complemented
37	C₂²×D₄×C₂₃	C₂³	1	24	yes	complemented
38	C₂²×D₄×C₂₃	C₂³	1	256	yes	complemented
39	C₂³×C₄×C₂₃	—	2	32	no	—
40	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₂₃	C₂³	1	48	yes	complemented
44	C₂₃×#127	C₂³	1	64	yes	complemented
45	C₂₃×#121	C₂³	1	256	yes	complemented
46	C₂₃×#120	C₂³	1	256	yes	complemented
47	C₄×D₄×C₂₃	C₂³	1	128	yes	complemented
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	complemented
49	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	complemented
50	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	complemented
51	C₂×C₄²×C₂₃	C₂³	1	12	yes	complemented
52	C₂⁴×C₂₃	—	2	16	no	—
53	C₂⁴×C₂₃	—	2	16	no	—
54	C₂⁴×C₂₃	—	2	32	no	—
55	C₂⁴×C₂₃	—	2	384	no	—
56	C₂⁴×C₂₃	—	2	16	no	—
57	C₂⁴×C₂₃	—	4	96	no	—
58	C₂⁴×C₂₃	—	4	256	no	—
59	C₂⁴×C₂₃	C₂×D₄	1	32	yes	complemented
60	C₂⁴×C₂₃	C₂⁴	1	24	yes	—
61	C₂⁴×C₂₃	C₂⁴	1	24	yes	—
62	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₂⁴×C₂₃	C₂⁴	1	64	yes	complemented
64	C₂×D₄×C₂₃	—	2	384	no	—
65	C₂×D₄×C₂₃	—	2	1024	no	—
66	C₂×D₄×C₂₃	—	2	192	no	—
67	C₂×D₄×C₂₃	—	4	48	no	—
68	C₂×D₄×C₂₃	C₂×D₄	1	192	yes	direct product factor
69	C₂×D₄×C₂₃	C₂⁴	1	32	yes	—
70	C₂×D₄×C₂₃	C₂⁴	1	128	yes	complemented
71	C₂×D₄×C₂₃	C₂⁴	1	64	yes	complemented
72	C₂×D₄×C₂₃	C₂⁴	1	32	yes	complemented
73	C₂²×C₄×C₂₃	—	2	16	no	—
74	C₂²×C₄×C₂₃	—	2	4	no	—
75	C₂²×C₄×C₂₃	—	2	192	no	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	16	yes	complemented
77	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	64	yes	complemented
81	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	complemented
82	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	complemented
83	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	complemented
84	C₄²×C₂₃	C₂⁴	1	16	yes	complemented
85	C₂²×D₄²	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	128	no	—
87	C₂³×C₂₃	—	2	48	no	—
88	C₂³×C₂₃	—	2	48	no	—
89	C₂³×C₂₃	—	2	96	no	—
90	C₂³×C₂₃	—	2	256	no	—
91	C₂³×C₂₃	—	2	48	no	—
92	C₂³×C₂₃	—	4	64	no	—
93	C₂³×C₂₃	—	4	768	no	—
94	C₂³×C₂₃	—	4	64	no	—
95	C₂³×C₂₃	C₂²×D₄	1	8	yes	—
96	C₂³×C₂₃	C₂²×D₄	1	96	yes	complemented
97	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
98	C₂³×C₂₃	C₂⁵	1	16	yes	—
99	C₂³×C₂₃	C₂⁵	1	16	yes	—
100	C₂³×C₂₃	C₂⁵	1	3	yes	—
101	D₄×C₂₃	—	2	256	no	—
102	D₄×C₂₃	—	2	128	no	—
103	D₄×C₂₃	—	4	32	no	—
104	D₄×C₂₃	C₂²×D₄	1	128	yes	direct product factor
105	C₂×C₄×C₂₃	—	2	48	no	—
106	C₂×C₄×C₂₃	—	2	12	no	—
107	C₂×C₄×C₂₃	—	2	128	no	—
108	C₂×C₄×C₂₃	C₂²×D₄	1	48	yes	complemented
109	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
110	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
111	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
112	C₂⁴×D₄	C₂×C₂₃	1	4	yes	complemented
113	C₂×D₄²	C₂×C₂₃	1	48	yes	direct product factor
114	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic, complemented
115	C₂²×#121	C₂×C₂₃	1	4	yes	complemented
116	C₂²×#120	C₂×C₂₃	1	4	yes	complemented
117	C₂²×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
118	C₂²×C₂₃	—	2	16	no	—
119	C₂²×C₂₃	—	2	16	no	—
120	C₂²×C₂₃	—	2	192	no	—
121	C₂²×C₂₃	—	2	32	no	—
122	C₂²×C₂₃	—	2	16	no	—
123	C₂²×C₂₃	—	4	256	no	—
124	C₂²×C₂₃	—	4	96	no	—
125	C₂²×C₂₃	D₄²	1	16	yes	direct product factor
126	C₂²×C₂₃	C₂³×D₄	1	12	yes	—
127	C₂²×C₂₃	C₂³×D₄	1	32	yes	complemented
128	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	6	yes	—
129	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
130	C₄×C₂₃	—	2	16	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	C₂³×D₄	1	16	yes	complemented
133	C₂⁶	C₂²×C₂₃	1	4	yes	complemented
134	C₂³×D₄	—	2	64	no	—
135	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
136	C₂³×D₄	C₂²×C₂₃	1	8	yes	complemented
137	C₂³×D₄	C₂²×C₂₃	1	4	yes	complemented
138	C₂³×D₄	C₂²×C₂₃	1	2	yes	complemented
139	C₂³×D₄	C₂²×C₂₃	1	96	yes	complemented
140	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	complemented
141	D₄²	C₂²×C₂₃	1	256	yes	direct product factor
142	C₂×#127	C₂²×C₂₃	1	24	yes	complemented
143	C₂×#121	C₂²×C₂₃	1	96	yes	complemented
144	C₂×#120	C₂²×C₂₃	1	96	yes	complemented
145	C₂×C₄×D₄	C₂²×C₂₃	1	48	yes	complemented
146	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
147	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
148	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
149	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
150	C₂×C₂₃	—	2	32	no	—
151	C₂×C₂₃	—	4	16	no	—
152	C₂×C₂₃	C₂×D₄²	1	12	yes	direct product factor
153	C₂×C₂₃	C₂⁴×D₄	1	2	yes	—
154	C₂×C₂₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
155	C₂⁵	—	2	64	no	—
156	C₂⁵	—	4	16	no	—
157	C₂⁵	C₂³×C₂₃	1	4	yes	—
158	C₂⁵	C₂³×C₂₃	1	4	yes	—
159	C₂⁵	C₂³×C₂₃	1	48	yes	complemented
160	C₂²×D₄	—	2	64	no	—
161	C₂²×D₄	—	2	768	no	—
162	C₂²×D₄	—	2	32	no	—
163	C₂²×D₄	—	4	8	no	—
164	C₂²×D₄	D₄×C₂₃	1	32	yes	direct product factor
165	C₂²×D₄	C₂³×C₂₃	1	24	yes	—
166	C₂²×D₄	C₂³×C₂₃	1	96	yes	complemented
167	C₂²×D₄	C₂³×C₂₃	1	48	yes	complemented
168	C₂²×D₄	C₂³×C₂₃	1	24	yes	complemented
169	C₂²×D₄	C₂³×C₂₃	1	256	yes	complemented
170	C₂³×C₄	—	2	32	no	—
171	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
173	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
174	C₂³×C₄	C₂³×C₂₃	1	48	yes	complemented
175	#127	C₂³×C₂₃	1	64	yes	complemented
176	#121	C₂³×C₂₃	1	256	yes	complemented
177	#120	C₂³×C₂₃	1	256	yes	complemented
178	C₄×D₄	C₂³×C₂₃	1	128	yes	complemented
179	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	complemented
180	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	complemented
181	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	complemented
182	C₂×C₄²	C₂³×C₂₃	1	12	yes	complemented
183	C₂₃	C₂²×D₄²	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	16	no	—
185	C₂⁴	—	2	16	no	—
186	C₂⁴	—	2	32	no	—
187	C₂⁴	—	2	384	no	—
188	C₂⁴	—	2	16	no	—
189	C₂⁴	—	4	96	no	—
190	C₂⁴	—	4	256	no	—
191	C₂⁴	C₂×D₄×C₂₃	1	32	yes	complemented
192	C₂⁴	C₂⁴×C₂₃	1	24	yes	—
193	C₂⁴	C₂⁴×C₂₃	1	24	yes	—
194	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
195	C₂⁴	C₂⁴×C₂₃	1	64	yes	complemented
196	C₂×D₄	—	2	384	no	—
197	C₂×D₄	—	2	1024	no	—
198	C₂×D₄	—	2	192	no	—
199	C₂×D₄	—	4	48	no	—
200	C₂×D₄	C₂×D₄×C₂₃	1	192	yes	direct product factor
201	C₂×D₄	C₂⁴×C₂₃	1	32	yes	—
202	C₂×D₄	C₂⁴×C₂₃	1	128	yes	complemented
203	C₂×D₄	C₂⁴×C₂₃	1	64	yes	complemented
204	C₂×D₄	C₂⁴×C₂₃	1	32	yes	complemented
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	192	no	—
208	C₂²×C₄	C₂×D₄×C₂₃	1	16	yes	complemented
209	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
210	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
211	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
212	C₂²×C₄	C₂⁴×C₂₃	1	64	yes	complemented
213	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	complemented
214	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	complemented
215	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	complemented
216	C₄²	C₂⁴×C₂₃	1	16	yes	complemented
217	C₂³	—	2	48	no	—
218	C₂³	—	2	48	no	—
219	C₂³	—	2	128	no	—
220	C₂³	—	2	96	no	—
221	C₂³	—	2	256	no	—
222	C₂³	—	2	48	no	—
223	C₂³	—	4	64	no	—
224	C₂³	—	4	768	no	—
225	C₂³	—	4	64	no	—
226	C₂³	C₂²×D₄×C₂₃	1	8	yes	—
227	C₂³	C₂²×D₄×C₂₃	1	96	yes	complemented
228	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
229	C₂³	C₂⁵×C₂₃	1	16	yes	—
230	C₂³	C₂⁵×C₂₃	1	16	yes	—
231	C₂³	C₂⁵×C₂₃	1	3	yes	—
232	D₄	—	2	256	no	—
233	D₄	—	2	128	no	—
234	D₄	—	4	32	no	—
235	D₄	C₂²×D₄×C₂₃	1	128	yes	direct product factor
236	C₂×C₄	—	2	48	no	—
237	C₂×C₄	—	2	12	no	—
238	C₂×C₄	—	2	128	no	—
239	C₂×C₄	C₂²×D₄×C₂₃	1	48	yes	complemented
240	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
241	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
242	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
243	C₂²	—	2	192	no	—
244	C₂²	—	2	16	no	—
245	C₂²	—	2	16	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	16	no	—
248	C₂²	—	4	256	no	—
249	C₂²	—	4	96	no	—
250	C₂²	D₄²×C₂₃	1	16	yes	direct product factor
251	C₂²	C₂³×D₄×C₂₃	1	12	yes	—
252	C₂²	C₂³×D₄×C₂₃	1	32	yes	complemented
253	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	6	yes	—
254	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
255	C₄	—	2	16	no	—
256	C₄	—	2	4	no	—
257	C₄	C₂³×D₄×C₂₃	1	16	yes	complemented
258	C₂	—	2	32	no	—
259	C₂	—	4	16	no	—
260	C₂	C₂×D₄²×C₂₃	1	12	yes	direct product factor
261	C₂	C₂⁴×D₄×C₂₃	1	2	yes	—
262	C₂	C₂²×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₂₃	C₂	semidirect
2. C₂×D₄²×C₂₃	C₂	direct (N × H)
3. C₂²×C₂₃×#127	C₂	semidirect
4. C₂²×C₂₃×#121	C₂	semidirect
5. C₂²×C₂₃×#120	C₂	semidirect
6. C₂²×C₄×D₄×C₂₃	C₂	semidirect
7. C₂⁶×C₂₃	C₂²	semidirect
10. C₂³×D₄×C₂₃	C₂²	semidirect
11. C₂³×D₄×C₂₃	C₂²	semidirect
12. C₂³×D₄×C₂₃	C₂²	semidirect
13. C₂³×D₄×C₂₃	C₂²	semidirect
14. C₂⁴×C₄×C₂₃	C₂²	semidirect
15. D₄²×C₂₃	C₂²	direct (N × H)
16. C₂×C₂₃×#127	C₂²	semidirect
17. C₂×C₂₃×#121	C₂²	semidirect
18. C₂×C₂₃×#120	C₂²	semidirect
19. C₂×C₄×D₄×C₂₃	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
22. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
23. C₂²×C₄²×C₂₃	C₂²	semidirect
28. C₂⁵×C₂₃	C₂³	semidirect
33. C₂²×D₄×C₂₃	D₄	direct (N × H)
35. C₂²×D₄×C₂₃	C₂³	semidirect
36. C₂²×D₄×C₂₃	C₂³	semidirect
37. C₂²×D₄×C₂₃	C₂³	semidirect
38. C₂²×D₄×C₂₃	C₂³	semidirect
43. C₂³×C₄×C₂₃	C₂³	semidirect
44. C₂₃×#127	C₂³	semidirect
45. C₂₃×#121	C₂³	semidirect
46. C₂₃×#120	C₂³	semidirect
47. C₄×D₄×C₂₃	C₂³	semidirect
48. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
49. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
50. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
51. C₂×C₄²×C₂₃	C₂³	semidirect
59. C₂⁴×C₂₃	C₂×D₄	semidirect
63. C₂⁴×C₂₃	C₂⁴	semidirect
68. C₂×D₄×C₂₃	C₂×D₄	direct (N × H)
70. C₂×D₄×C₂₃	C₂⁴	semidirect
71. C₂×D₄×C₂₃	C₂⁴	semidirect
72. C₂×D₄×C₂₃	C₂⁴	semidirect
76. C₂²×C₄×C₂₃	C₂×D₄	semidirect
80. C₂²×C₄×C₂₃	C₂⁴	semidirect
81. C₄⋊C₄×C₂₃	C₂⁴	semidirect
82. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
83. C₂²⋊C₄×C₂₃	C₂⁴	semidirect
84. C₄²×C₂₃	C₂⁴	semidirect
85. C₂²×D₄²	C₂₃	direct (N × H)
96. C₂³×C₂₃	C₂²×D₄	semidirect
104. D₄×C₂₃	C₂²×D₄	direct (N × H)
108. C₂×C₄×C₂₃	C₂²×D₄	semidirect
112. C₂⁴×D₄	C₂×C₂₃	semidirect
113. C₂×D₄²	C₂×C₂₃	direct (N × H)
114. C₂²×#127	C₂×C₂₃	semidirect
115. C₂²×#121	C₂×C₂₃	semidirect
116. C₂²×#120	C₂×C₂₃	semidirect
117. C₂²×C₄×D₄	C₂×C₂₃	semidirect
125. C₂²×C₂₃	D₄²	direct (N × H)
127. C₂²×C₂₃	C₂³×D₄	semidirect
132. C₄×C₂₃	C₂³×D₄	semidirect
133. C₂⁶	C₂²×C₂₃	semidirect
136. C₂³×D₄	C₂²×C₂₃	semidirect
137. C₂³×D₄	C₂²×C₂₃	semidirect
138. C₂³×D₄	C₂²×C₂₃	semidirect
139. C₂³×D₄	C₂²×C₂₃	semidirect
140. C₂⁴×C₄	C₂²×C₂₃	semidirect
141. D₄²	C₂²×C₂₃	direct (N × H)
142. C₂×#127	C₂²×C₂₃	semidirect
143. C₂×#121	C₂²×C₂₃	semidirect
144. C₂×#120	C₂²×C₂₃	semidirect
145. C₂×C₄×D₄	C₂²×C₂₃	semidirect
146. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
147. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
148. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
149. C₂²×C₄²	C₂²×C₂₃	semidirect
152. C₂×C₂₃	C₂×D₄²	direct (N × H)
159. C₂⁵	C₂³×C₂₃	semidirect
164. C₂²×D₄	D₄×C₂₃	direct (N × H)
166. C₂²×D₄	C₂³×C₂₃	semidirect
167. C₂²×D₄	C₂³×C₂₃	semidirect
168. C₂²×D₄	C₂³×C₂₃	semidirect
169. C₂²×D₄	C₂³×C₂₃	semidirect
174. C₂³×C₄	C₂³×C₂₃	semidirect
175. #127	C₂³×C₂₃	semidirect
176. #121	C₂³×C₂₃	semidirect
177. #120	C₂³×C₂₃	semidirect
178. C₄×D₄	C₂³×C₂₃	semidirect
179. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
180. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
181. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
182. C₂×C₄²	C₂³×C₂₃	semidirect
183. C₂₃	C₂²×D₄²	direct (N × H)
191. C₂⁴	C₂×D₄×C₂₃	semidirect
195. C₂⁴	C₂⁴×C₂₃	semidirect
200. C₂×D₄	C₂×D₄×C₂₃	direct (N × H)
202. C₂×D₄	C₂⁴×C₂₃	semidirect
203. C₂×D₄	C₂⁴×C₂₃	semidirect
204. C₂×D₄	C₂⁴×C₂₃	semidirect
208. C₂²×C₄	C₂×D₄×C₂₃	semidirect
212. C₂²×C₄	C₂⁴×C₂₃	semidirect
213. C₄⋊C₄	C₂⁴×C₂₃	semidirect
214. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
215. C₂²⋊C₄	C₂⁴×C₂₃	semidirect
216. C₄²	C₂⁴×C₂₃	semidirect
227. C₂³	C₂²×D₄×C₂₃	semidirect
235. D₄	C₂²×D₄×C₂₃	direct (N × H)
239. C₂×C₄	C₂²×D₄×C₂₃	semidirect
250. C₂²	D₄²×C₂₃	direct (N × H)
252. C₂²	C₂³×D₄×C₂₃	semidirect
257. C₄	C₂³×D₄×C₂₃	semidirect
260. C₂	C₂×D₄²×C₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₂₃	2
C₂×D₄²×C₂₃	2
C₂²×C₂₃×#127	2
C₂²×C₂₃×#121	2
C₂²×C₂₃×#120	2
C₂²×C₄×D₄×C₂₃	2
C₂²×D₄²	23

Subgroups — C22×D42×C23 #19210321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄²×C₂₃ #19210321