Subgroups — C₂²×C₂₃×#556 #19210332

Order 5888.

7846 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#556	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#126	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂₃×#124	C₂	1	2	yes	maximal
5	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Q₈×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₂₃×#117	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#556	C₂²	1	256	yes	direct product factor
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
16	C₂×C₂₃×#126	C₂²	1	96	yes	complemented
17	C₂×C₂₃×#124	C₂²	1	48	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	48	yes	complemented
19	C₂×C₄×Q₈×C₂₃	C₂²	1	48	yes	complemented
20	C₂×C₄×Q₈×C₂₃	C₂²	1	24	yes	complemented
21	C₂×C₄×D₄×C₂₃	C₂²	1	24	yes	—
22	C₂×C₂₃×#117	C₂²	1	48	yes	complemented
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
27	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
30	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
32	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
33	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
35	C₂⁵×C₂₃	C₂³	1	2	yes	—
36	C₂²×Q₈×C₂₃	—	2	8	no	—
37	C₂²×Q₈×C₂₃	—	2	16	no	—
38	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
39	C₂²×Q₈×C₂₃	C₂³	1	24	yes	—
40	C₂²×D₄×C₂₃	—	2	8	no	—
41	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
42	C₂³×C₄×C₂₃	—	2	16	no	—
43	C₂³×C₄×C₂₃	C₂³	1	24	yes	—
44	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
47	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
52	C₂₃×#126	C₂³	1	256	yes	complemented
53	C₂₃×#124	C₂³	1	128	yes	—
54	C₂₃×C₂²⋊Q₈	C₂³	1	128	yes	complemented
55	C₄×Q₈×C₂₃	C₂³	1	128	yes	complemented
56	C₄×Q₈×C₂₃	C₂³	1	64	yes	complemented
57	C₄×D₄×C₂₃	C₂³	1	64	yes	—
58	C₂₃×#117	C₂³	1	128	yes	complemented
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
61	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
65	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
66	C₂×C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
67	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
68	C₂×C₄²×C₂₃	C₂³	1	48	yes	—
69	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
70	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
71	C₂⁴×C₂₃	—	2	8	no	—
72	C₂⁴×C₂₃	—	2	8	no	—
73	C₂⁴×C₂₃	—	2	8	no	—
74	C₂⁴×C₂₃	C₂⁴	1	12	yes	—
75	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₂₃	—	2	48	no	—
77	C₂×Q₈×C₂₃	—	2	96	no	—
78	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
79	C₂×Q₈×C₂₃	C₂⁴	1	32	yes	—
80	C₂×D₄×C₂₃	—	2	48	no	—
81	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₂₃	—	2	8	no	—
83	C₂²×C₄×C₂₃	—	2	8	no	—
84	C₂²×C₄×C₂₃	—	2	16	no	—
85	C₂²×C₄×C₂₃	—	2	8	no	—
86	C₂²×C₄×C₂₃	—	2	4	no	—
87	C₂²×C₄×C₂₃	—	2	96	no	—
88	C₂²×C₄×C₂₃	#41	1	8	yes	—
89	C₂²×C₄×C₂₃	#41	1	8	yes	—
90	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
92	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
94	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₂₃	C₂⁴	1	32	yes	—
96	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
97	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
98	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
99	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
100	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
101	C₄⋊C₄×C₂₃	C₂⁴	1	64	yes	—
102	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
103	C₄⋊C₄×C₂₃	C₂⁴	1	32	yes	—
104	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
105	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
106	C₂²⋊C₄×C₂₃	C₂⁴	1	64	yes	—
107	C₄²×C₂₃	C₂⁴	1	64	yes	—
108	C₄²×C₂₃	C₂⁴	1	16	yes	—
109	C₄²×C₂₃	C₂⁴	1	32	yes	—
110	C₂²×#556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₂₃	—	2	24	no	—
112	C₂³×C₂₃	—	2	24	no	—
113	C₂³×C₂₃	—	2	24	no	—
114	C₂³×C₂₃	—	4	32	no	—
115	C₂³×C₂₃	C₂×#41	1	4	yes	—
116	C₂³×C₂₃	C₂×#41	1	4	yes	—
117	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
118	C₂³×C₂₃	C₂⁵	1	8	yes	—
119	C₂³×C₂₃	C₂⁵	1	3	yes	—
120	Q₈×C₂₃	—	2	32	no	—
121	Q₈×C₂₃	—	2	64	no	—
122	D₄×C₂₃	—	2	32	no	—
123	C₂×C₄×C₂₃	—	2	24	no	—
124	C₂×C₄×C₂₃	—	2	24	no	—
125	C₂×C₄×C₂₃	—	2	48	no	—
126	C₂×C₄×C₂₃	—	2	24	no	—
127	C₂×C₄×C₂₃	—	2	12	no	—
128	C₂×C₄×C₂₃	—	2	64	no	—
129	C₂×C₄×C₂₃	C₂×#41	1	24	yes	—
130	C₂×C₄×C₂₃	C₂×#41	1	24	yes	—
131	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
132	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
133	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
134	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
135	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
136	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
137	C₂×C₄×C₂₃	C₂⁵	1	4	yes	—
138	C₂×#556	C₂×C₂₃	1	48	yes	direct product factor
139	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂²×#126	C₂×C₂₃	1	4	yes	complemented
141	C₂²×#124	C₂×C₂₃	1	2	yes	—
142	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	complemented
143	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₄×Q₈	C₂×C₂₃	1	2	yes	complemented
145	C₂²×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
146	C₂²×#117	C₂×C₂₃	1	2	yes	complemented
147	C₂²×C₂₃	—	2	8	no	—
148	C₂²×C₂₃	—	2	8	no	—
149	C₂²×C₂₃	—	2	8	no	—
150	C₂²×C₂₃	—	4	48	no	—
151	C₂²×C₂₃	#556	1	16	yes	direct product factor
152	C₂²×C₂₃	C₂²×#41	1	6	yes	—
153	C₂²×C₂₃	C₂²×#41	1	6	yes	—
154	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	6	yes	—
155	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
156	C₄×C₂₃	—	2	8	no	—
157	C₄×C₂₃	—	2	8	no	—
158	C₄×C₂₃	—	2	16	no	—
159	C₄×C₂₃	—	2	8	no	—
160	C₄×C₂₃	—	2	4	no	—
161	C₄×C₂₃	C₂²×#41	1	8	yes	—
162	C₄×C₂₃	C₂²×#41	1	8	yes	—
163	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
165	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
166	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
167	#556	C₂²×C₂₃	1	256	yes	direct product factor
168	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	complemented
169	C₂×#126	C₂²×C₂₃	1	96	yes	complemented
170	C₂×#124	C₂²×C₂₃	1	48	yes	—
171	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	48	yes	complemented
172	C₂×C₄×Q₈	C₂²×C₂₃	1	48	yes	complemented
173	C₂×C₄×Q₈	C₂²×C₂₃	1	24	yes	complemented
174	C₂×C₄×D₄	C₂²×C₂₃	1	24	yes	—
175	C₂×#117	C₂²×C₂₃	1	48	yes	complemented
176	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
178	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
182	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
183	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
184	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
185	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
186	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
187	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
188	C₂×C₂₃	—	4	8	no	—
189	C₂×C₂₃	C₂×#556	1	12	yes	direct product factor
190	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
191	C₂×C₂₃	C₂³×#41	1	1	yes	characteristic
192	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
193	C₂⁵	C₂³×C₂₃	1	2	yes	—
194	C₂²×Q₈	—	2	8	no	—
195	C₂²×Q₈	—	2	16	no	—
196	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
197	C₂²×Q₈	C₂³×C₂₃	1	24	yes	—
198	C₂²×D₄	—	2	8	no	—
199	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
200	C₂³×C₄	—	2	16	no	—
201	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
203	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
204	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
205	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
206	C₂³×C₄	C₂³×C₂₃	1	24	yes	—
207	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
208	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
209	C₄⋊Q₈	C₂³×C₂₃	1	64	yes	complemented
210	#126	C₂³×C₂₃	1	256	yes	complemented
211	#124	C₂³×C₂₃	1	128	yes	—
212	C₂²⋊Q₈	C₂³×C₂₃	1	128	yes	complemented
213	C₄×Q₈	C₂³×C₂₃	1	128	yes	complemented
214	C₄×Q₈	C₂³×C₂₃	1	64	yes	complemented
215	C₄×D₄	C₂³×C₂₃	1	64	yes	—
216	#117	C₂³×C₂₃	1	128	yes	complemented
217	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
218	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
219	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
220	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
221	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
222	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
223	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
224	C₂×C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
225	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
226	C₂×C₄²	C₂³×C₂₃	1	48	yes	—
227	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
228	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
229	C₂₃	C₂²×#556	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	2	8	no	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	8	no	—
233	C₂⁴	C₂⁴×C₂₃	1	12	yes	—
234	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
235	C₂×Q₈	—	2	48	no	—
236	C₂×Q₈	—	2	96	no	—
237	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
238	C₂×Q₈	C₂⁴×C₂₃	1	32	yes	—
239	C₂×D₄	—	2	48	no	—
240	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	16	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	96	no	—
247	C₂²×C₄	#41×C₂₃	1	8	yes	—
248	C₂²×C₄	#41×C₂₃	1	8	yes	—
249	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
250	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
251	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
252	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
253	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
254	C₂²×C₄	C₂⁴×C₂₃	1	32	yes	—
255	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
256	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
257	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
258	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
259	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
260	C₄⋊C₄	C₂⁴×C₂₃	1	64	yes	—
261	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
262	C₄⋊C₄	C₂⁴×C₂₃	1	32	yes	—
263	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
264	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
265	C₂²⋊C₄	C₂⁴×C₂₃	1	64	yes	—
266	C₄²	C₂⁴×C₂₃	1	64	yes	—
267	C₄²	C₂⁴×C₂₃	1	16	yes	—
268	C₄²	C₂⁴×C₂₃	1	32	yes	—
269	C₂³	—	2	24	no	—
270	C₂³	—	2	24	no	—
271	C₂³	—	2	24	no	—
272	C₂³	—	4	32	no	—
273	C₂³	C₂×#41×C₂₃	1	4	yes	—
274	C₂³	C₂×#41×C₂₃	1	4	yes	—
275	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
276	C₂³	C₂⁵×C₂₃	1	8	yes	—
277	C₂³	C₂⁵×C₂₃	1	3	yes	—
278	Q₈	—	2	32	no	—
279	Q₈	—	2	64	no	—
280	D₄	—	2	32	no	—
281	C₂×C₄	—	2	24	no	—
282	C₂×C₄	—	2	24	no	—
283	C₂×C₄	—	2	48	no	—
284	C₂×C₄	—	2	24	no	—
285	C₂×C₄	—	2	12	no	—
286	C₂×C₄	—	2	64	no	—
287	C₂×C₄	C₂×#41×C₂₃	1	24	yes	—
288	C₂×C₄	C₂×#41×C₂₃	1	24	yes	—
289	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
290	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
291	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
292	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
293	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
294	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
295	C₂×C₄	C₂⁵×C₂₃	1	4	yes	—
296	C₂²	—	2	8	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	48	no	—
300	C₂²	C₂₃×#556	1	16	yes	direct product factor
301	C₂²	C₂²×#41×C₂₃	1	6	yes	—
302	C₂²	C₂²×#41×C₂₃	1	6	yes	—
303	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	6	yes	—
304	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
305	C₄	—	2	8	no	—
306	C₄	—	2	8	no	—
307	C₄	—	2	16	no	—
308	C₄	—	2	8	no	—
309	C₄	—	2	4	no	—
310	C₄	C₂²×#41×C₂₃	1	8	yes	—
311	C₄	C₂²×#41×C₂₃	1	8	yes	—
312	C₂	—	4	8	no	—
313	C₂	C₂×C₂₃×#556	1	12	yes	direct product factor
314	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic
315	C₂	C₂³×#41×C₂₃	1	1	yes	characteristic
316	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
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. C₂×C₂₃×#556	C₂	direct (N × H)
2. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₂₃×#126	C₂	semidirect
5. C₂²×C₂₃×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
7. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
9. C₂²×C₂₃×#117	C₂	semidirect
14. C₂₃×#556	C₂²	direct (N × H)
15. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₂₃×#126	C₂²	semidirect
18. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
20. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
22. C₂×C₂₃×#117	C₂²	semidirect
51. C₂₃×C₄⋊Q₈	C₂³	semidirect
52. C₂₃×#126	C₂³	semidirect
54. C₂₃×C₂²⋊Q₈	C₂³	semidirect
55. C₄×Q₈×C₂₃	C₂³	semidirect
56. C₄×Q₈×C₂₃	C₂³	semidirect
58. C₂₃×#117	C₂³	semidirect
110. C₂²×#556	C₂₃	direct (N × H)
138. C₂×#556	C₂×C₂₃	direct (N × H)
139. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
140. C₂²×#126	C₂×C₂₃	semidirect
142. C₂²×C₂²⋊Q₈	C₂×C₂₃	semidirect
143. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
144. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
146. C₂²×#117	C₂×C₂₃	semidirect
151. C₂²×C₂₃	#556	direct (N × H)
167. #556	C₂²×C₂₃	direct (N × H)
168. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
169. C₂×#126	C₂²×C₂₃	semidirect
171. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
172. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
173. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
175. C₂×#117	C₂²×C₂₃	semidirect
189. C₂×C₂₃	C₂×#556	direct (N × H)
209. C₄⋊Q₈	C₂³×C₂₃	semidirect
210. #126	C₂³×C₂₃	semidirect
212. C₂²⋊Q₈	C₂³×C₂₃	semidirect
213. C₄×Q₈	C₂³×C₂₃	semidirect
214. C₄×Q₈	C₂³×C₂₃	semidirect
216. #117	C₂³×C₂₃	semidirect
229. C₂₃	C₂²×#556	direct (N × H)
300. C₂²	C₂₃×#556	direct (N × H)
313. C₂	C₂×C₂₃×#556	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×#556	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂²×C₂₃×#126	2
C₂²×C₂₃×#124	2
C₂²×C₂₃×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₂₃	2
C₂²×C₄×D₄×C₂₃	2
C₂²×C₂₃×#117	2
C₂²×#556	23

Subgroups — C22×C23×#556 #19210332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#556 #19210332