Subgroups — C₂×C₂₃×#3115 #19208164

Order 5888.

2430 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3115	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#482	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#449	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#442	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#532	—	2	4	no	—
18	C₂₃×#532	C₂²	1	8	yes	complemented
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#124	—	2	1	no	—
22	C₂×C₂₃×#121	—	2	1	no	—
23	C₂×C₂₃×#121	C₂²	1	2	yes	—
24	C₄×#41×C₂₃	—	2	4	no	—
25	C₄×#41×C₂₃	C₂²	1	8	yes	complemented
26	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₂₃	—	2	1	no	—
29	C₂×C₄×D₄×C₂₃	—	2	1	no	—
30	C₂×C₄×D₄×C₂₃	—	2	1	no	—
31	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
34	C₂×C₂₃×#117	—	2	1	no	—
35	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₂₃	—	2	1	no	—
37	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#482	C₂²	1	16	yes	complemented
39	C₂₃×#475	C₂²	1	8	yes	complemented
40	C₂₃×#474	C₂²	1	8	yes	complemented
41	C₂₃×#460	C₂²	1	8	yes	complemented
42	C₂₃×#459	C₂²	1	8	yes	complemented
43	C₂₃×#449	C₂²	1	16	yes	complemented
44	C₂₃×#442	C₂²	1	16	yes	complemented
45	C₂₃×#440	C₂²	1	16	yes	complemented
46	C₂₃×#432	C₂²	1	8	yes	complemented
47	C₂×C₂₃×#107	C₂²	1	2	yes	—
48	C₂×C₂₃×#106	C₂²	1	2	yes	—
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#103	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×#102	C₂²	1	2	yes	—
54	C₂×C₂₃×#102	C₂²	1	2	yes	—
55	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
56	C₂×C₂₃×#97	C₂²	1	2	yes	—
57	C₂×#41×C₂₃	—	2	4	no	—
58	C₂×#41×C₂₃	—	2	4	no	—
59	C₂×#41×C₂₃	—	2	2	no	—
60	C₂×#41×C₂₃	—	2	8	no	—
61	C₂×#41×C₂₃	—	4	8	no	—
62	C₂×#41×C₂₃	C₂³	1	4	yes	—
63	C₂²×Q₈×C₂₃	—	2	1	no	—
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	—	2	1	no	—
66	C₂²×D₄×C₂₃	—	2	2	no	—
67	C₂²×D₄×C₂₃	—	2	1	no	—
68	C₂²×D₄×C₂₃	—	2	1	no	—
69	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	—	2	2	no	—
73	C₂³×C₄×C₂₃	—	2	1	no	—
74	C₂³×C₄×C₂₃	—	2	1	no	—
75	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
77	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₂₃	—	2	8	no	—
79	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
80	C₂×D₈×C₂₃	—	2	8	no	—
81	C₂×D₈×C₂₃	C₂³	1	8	yes	—
82	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
83	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
84	C₂₃×C₄⋊Q₈	—	2	2	no	—
85	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
86	C₂₃×#127	—	2	2	no	—
87	C₂₃×#127	C₂³	1	4	yes	—
88	C₂₃×#124	—	2	8	no	—
89	C₂₃×#121	—	2	8	no	—
90	C₂₃×#121	—	2	4	no	—
91	C₂₃×#121	C₂³	1	8	yes	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
96	C₄×D₄×C₂₃	—	2	8	no	—
97	C₄×D₄×C₂₃	—	2	8	no	—
98	C₄×D₄×C₂₃	—	2	8	no	—
99	C₄×D₄×C₂₃	—	2	2	no	—
100	C₄×D₄×C₂₃	—	2	2	no	—
101	C₄×D₄×C₂₃	—	2	4	no	—
102	C₄×D₄×C₂₃	C₂³	1	8	yes	—
103	C₄×D₄×C₂₃	C₂³	1	4	yes	—
104	C₄×D₄×C₂₃	C₂³	1	4	yes	—
105	C₂₃×#117	—	2	2	no	—
106	C₂₃×#117	—	2	8	no	—
107	C₂₃×#117	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
119	C₂×C₄²×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	—	2	1	no	—
121	C₂×C₄²×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	—	2	8	no	—
123	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₂₃	D₄	1	4	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
129	C₂₃×#107	C₂³	1	8	yes	—
130	C₂₃×#106	C₂³	1	8	yes	—
131	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
132	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
133	C₂₃×#103	C₂³	1	8	yes	—
134	C₂₃×#102	C₂³	1	8	yes	—
135	C₂₃×#102	C₂³	1	8	yes	—
136	C₂₃×#102	C₂³	1	8	yes	—
137	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
138	C₂₃×#97	C₂³	1	8	yes	—
139	C₂⁴×C₂₃	—	2	1	no	—
140	C₂⁴×C₂₃	—	2	2	no	—
141	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
142	#41×C₂₃	—	2	16	no	—
143	#41×C₂₃	—	2	8	no	—
144	#41×C₂₃	—	2	8	no	—
145	#41×C₂₃	—	4	32	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	4	no	—
148	C₂×Q₈×C₂₃	—	2	2	no	—
149	C₂×Q₈×C₂₃	—	4	4	no	—
150	C₂×Q₈×C₂₃	#41	1	4	yes	complemented
151	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₂₃	—	2	8	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	8	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	2	no	—
157	C₂×D₄×C₂₃	—	2	4	no	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	2	no	—
160	C₂×D₄×C₂₃	—	2	4	no	—
161	C₂×D₄×C₂₃	—	2	4	no	—
162	C₂×D₄×C₂₃	—	4	4	no	—
163	C₂×D₄×C₂₃	—	4	8	no	—
164	C₂×D₄×C₂₃	—	4	4	no	—
165	C₂×D₄×C₂₃	#41	1	4	yes	complemented
166	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₂₃	—	2	8	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	4	no	—
171	C₂²×C₄×C₂₃	—	2	4	no	—
172	C₂²×C₄×C₂₃	—	2	2	no	—
173	C₂²×C₄×C₂₃	—	2	2	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	2	no	—
177	C₂²×C₄×C₂₃	—	2	4	no	—
178	C₂²×C₄×C₂₃	—	2	4	no	—
179	C₂²×C₄×C₂₃	—	2	8	no	—
180	C₂²×C₄×C₂₃	—	4	2	no	—
181	C₂²×C₄×C₂₃	—	4	4	no	—
182	C₂²×C₄×C₂₃	—	4	8	no	—
183	C₂²×C₄×C₂₃	—	4	4	no	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
194	SD₁₆×C₂₃	—	2	16	no	—
195	D₈×C₂₃	—	2	16	no	—
196	C₂×C₈×C₂₃	—	2	4	no	—
197	C₂×C₈×C₂₃	—	2	4	no	—
198	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
199	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	2	4	no	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	2	no	—
206	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
208	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
209	C₂²⋊C₄×C₂₃	—	2	4	no	—
210	C₂²⋊C₄×C₂₃	—	2	4	no	—
211	C₂²⋊C₄×C₂₃	—	2	4	no	—
212	C₂²⋊C₄×C₂₃	—	2	8	no	—
213	C₂²⋊C₄×C₂₃	—	2	8	no	—
214	C₄²×C₂₃	—	2	4	no	—
215	C₄²×C₂₃	—	2	4	no	—
216	C₄²×C₂₃	—	2	4	no	—
217	C₄²×C₂₃	C₂×D₄	1	4	yes	—
218	C₄²×C₂₃	C₂×D₄	1	4	yes	—
219	C₄²×C₂₃	C₂×D₄	1	2	yes	—
220	C₄²×C₂₃	C₂×D₄	1	2	yes	—
221	C₄²×C₂₃	C₂⁴	1	2	yes	—
222	C₄²×C₂₃	C₂⁴	1	2	yes	—
223	C₂×#3115	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	2	4	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	2	no	—
229	C₂³×C₂₃	—	2	4	no	—
230	C₂³×C₂₃	—	4	2	no	—
231	C₂³×C₂₃	—	4	2	no	—
232	C₂³×C₂₃	—	4	4	no	—
233	C₂³×C₂₃	—	4	4	no	—
234	C₂³×C₂₃	—	4	4	no	—
235	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
236	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
237	Q₈×C₂₃	—	2	4	no	—
238	Q₈×C₂₃	—	2	2	no	—
239	Q₈×C₂₃	—	4	8	no	—
240	Q₈×C₂₃	C₂×#41	1	4	yes	complemented
241	D₄×C₂₃	—	2	16	no	—
242	D₄×C₂₃	—	2	4	no	—
243	D₄×C₂₃	—	2	2	no	—
244	D₄×C₂₃	—	2	4	no	—
245	D₄×C₂₃	—	2	4	no	—
246	D₄×C₂₃	—	4	8	no	—
247	D₄×C₂₃	—	4	16	no	—
248	D₄×C₂₃	—	4	4	no	—
249	D₄×C₂₃	—	4	8	no	—
250	D₄×C₂₃	C₂×#41	1	4	yes	complemented
251	C₂×C₄×C₂₃	—	2	16	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	4	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	4	no	—
259	C₂×C₄×C₂₃	—	2	8	no	—
260	C₂×C₄×C₂₃	—	2	2	no	—
261	C₂×C₄×C₂₃	—	2	2	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	2	no	—
264	C₂×C₄×C₂₃	—	2	4	no	—
265	C₂×C₄×C₂₃	—	4	4	no	—
266	C₂×C₄×C₂₃	—	4	8	no	—
267	C₂×C₄×C₂₃	—	4	16	no	—
268	C₂×C₄×C₂₃	—	4	2	no	—
269	C₂×C₄×C₂₃	—	4	4	no	—
270	C₂×C₄×C₂₃	—	4	2	no	—
271	C₂×C₄×C₂₃	—	4	8	no	—
272	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
273	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
280	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
281	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
282	C₈×C₂₃	—	2	4	no	—
283	C₈×C₂₃	—	2	4	no	—
284	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
285	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#3115	C₂×C₂₃	1	16	yes	direct product factor
287	C₂×#482	C₂×C₂₃	1	2	yes	complemented
288	C₂×#475	C₂×C₂₃	1	1	yes	characteristic, complemented
289	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
290	C₂×#460	C₂×C₂₃	1	1	yes	characteristic, complemented
291	C₂×#459	C₂×C₂₃	1	1	yes	characteristic, complemented
292	C₂×#449	C₂×C₂₃	1	2	yes	complemented
293	C₂×#442	C₂×C₂₃	1	2	yes	complemented
294	C₂×#440	C₂×C₂₃	1	2	yes	complemented
295	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	4	no	—
298	C₂²×C₂₃	—	2	2	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	4	4	no	—
301	C₂²×C₂₃	—	4	8	no	—
302	C₂²×C₂₃	—	4	2	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	—	4	4	no	—
305	C₂²×C₂₃	—	4	4	no	—
306	C₂²×C₂₃	—	4	4	no	—
307	C₂²×C₂₃	—	8	8	no	—
308	C₂²×C₂₃	#525	1	2	yes	—
309	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
310	C₂²×C₂₃	#575	1	2	yes	—
311	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
312	C₄×C₂₃	—	2	2	no	—
313	C₄×C₂₃	—	2	4	no	—
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	2	no	—
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	4	4	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	#525	1	2	yes	—
321	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
322	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
323	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
324	C₂²×#41	—	2	1	no	—
325	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
326	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
327	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
328	#532	—	2	4	no	—
329	#532	C₂²×C₂₃	1	8	yes	complemented
330	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#124	—	2	1	no	—
333	C₂×#121	—	2	1	no	—
334	C₂×#121	C₂²×C₂₃	1	2	yes	—
335	C₄×#41	—	2	4	no	—
336	C₄×#41	C₂²×C₂₃	1	8	yes	complemented
337	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×C₄×D₄	—	2	1	no	—
340	C₂×C₄×D₄	—	2	1	no	—
341	C₂×C₄×D₄	—	2	1	no	—
342	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
345	C₂×#117	—	2	1	no	—
346	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
347	C₂²×C₄²	—	2	1	no	—
348	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
349	#482	C₂²×C₂₃	1	16	yes	complemented
350	#475	C₂²×C₂₃	1	8	yes	complemented
351	#474	C₂²×C₂₃	1	8	yes	complemented
352	#460	C₂²×C₂₃	1	8	yes	complemented
353	#459	C₂²×C₂₃	1	8	yes	complemented
354	#449	C₂²×C₂₃	1	16	yes	complemented
355	#442	C₂²×C₂₃	1	16	yes	complemented
356	#440	C₂²×C₂₃	1	16	yes	complemented
357	#432	C₂²×C₂₃	1	8	yes	complemented
358	C₂×#107	C₂²×C₂₃	1	2	yes	—
359	C₂×#106	C₂²×C₂₃	1	2	yes	—
360	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×#103	C₂²×C₂₃	1	2	yes	—
363	C₂×#102	C₂²×C₂₃	1	2	yes	—
364	C₂×#102	C₂²×C₂₃	1	2	yes	—
365	C₂×#102	C₂²×C₂₃	1	2	yes	—
366	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
367	C₂×#97	C₂²×C₂₃	1	2	yes	—
368	C₂×C₂₃	—	4	2	no	—
369	C₂×C₂₃	—	4	4	no	—
370	C₂×C₂₃	—	8	4	no	—
371	C₂×C₂₃	#3115	1	4	yes	direct product factor
372	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
373	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
374	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
375	C₂×#41	—	2	4	no	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	—	2	2	no	—
378	C₂×#41	—	2	8	no	—
379	C₂×#41	—	4	8	no	—
380	C₂×#41	C₂³×C₂₃	1	4	yes	—
381	C₂²×Q₈	—	2	1	no	—
382	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₂²×D₄	—	2	1	no	—
384	C₂²×D₄	—	2	2	no	—
385	C₂²×D₄	—	2	1	no	—
386	C₂²×D₄	—	2	1	no	—
387	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
388	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂³×C₄	—	2	1	no	—
390	C₂³×C₄	—	2	2	no	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
394	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
396	C₂×SD₁₆	—	2	8	no	—
397	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
398	C₂×D₈	—	2	8	no	—
399	C₂×D₈	C₂³×C₂₃	1	8	yes	—
400	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
401	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
402	C₄⋊Q₈	—	2	2	no	—
403	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
404	#127	—	2	2	no	—
405	#127	C₂³×C₂₃	1	4	yes	—
406	#124	—	2	8	no	—
407	#121	—	2	8	no	—
408	#121	—	2	4	no	—
409	#121	C₂³×C₂₃	1	8	yes	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
413	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
414	C₄×D₄	—	2	8	no	—
415	C₄×D₄	—	2	8	no	—
416	C₄×D₄	—	2	8	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	4	no	—
420	C₄×D₄	C₂³×C₂₃	1	4	yes	—
421	C₄×D₄	C₂³×C₂₃	1	4	yes	—
422	C₄×D₄	C₂³×C₂₃	1	8	yes	—
423	#117	—	2	2	no	—
424	#117	—	2	8	no	—
425	#117	C₂³×C₂₃	1	4	yes	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	2	no	—
436	C₂×C₂²⋊C₄	—	2	2	no	—
437	C₂×C₄²	—	2	1	no	—
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	—	2	1	no	—
440	C₂×C₄²	—	2	8	no	—
441	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
442	C₂×C₄²	D₄×C₂₃	1	4	yes	—
443	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
447	#107	C₂³×C₂₃	1	8	yes	—
448	#106	C₂³×C₂₃	1	8	yes	—
449	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
450	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
451	#103	C₂³×C₂₃	1	8	yes	—
452	#102	C₂³×C₂₃	1	8	yes	—
453	#102	C₂³×C₂₃	1	8	yes	—
454	#102	C₂³×C₂₃	1	8	yes	—
455	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
456	#97	C₂³×C₂₃	1	8	yes	—
457	C₂₃	C₂×#3115	1	1	yes	characteristic, direct product factor, Sylow
458	C₂⁴	—	2	1	no	—
459	C₂⁴	—	2	2	no	—
460	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
461	#41	—	2	16	no	—
462	#41	—	2	8	no	—
463	#41	—	2	8	no	—
464	#41	—	4	32	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	4	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	4	4	no	—
469	C₂×Q₈	#41×C₂₃	1	4	yes	complemented
470	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
471	C₂×D₄	—	2	8	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	8	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	4	4	no	—
482	C₂×D₄	—	4	8	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	#41×C₂₃	1	4	yes	complemented
485	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
486	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	8	no	—
499	C₂²×C₄	—	4	2	no	—
500	C₂²×C₄	—	4	8	no	—
501	C₂²×C₄	—	4	4	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
506	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
510	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
513	SD₁₆	—	2	16	no	—
514	D₈	—	2	16	no	—
515	C₂×C₈	—	2	4	no	—
516	C₂×C₈	—	2	4	no	—
517	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
518	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
526	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
527	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	2	4	no	—
536	C₄²	C₂×D₄×C₂₃	1	4	yes	—
537	C₄²	C₂×D₄×C₂₃	1	4	yes	—
538	C₄²	C₂×D₄×C₂₃	1	2	yes	—
539	C₄²	C₂×D₄×C₂₃	1	2	yes	—
540	C₄²	C₂⁴×C₂₃	1	2	yes	—
541	C₄²	C₂⁴×C₂₃	1	2	yes	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	2	4	no	—
548	C₂³	—	4	2	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	4	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	4	4	no	—
553	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
554	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
555	Q₈	—	2	2	no	—
556	Q₈	—	2	4	no	—
557	Q₈	—	4	8	no	—
558	Q₈	C₂×#41×C₂₃	1	4	yes	complemented
559	D₄	—	2	16	no	—
560	D₄	—	2	2	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	4	no	—
564	D₄	—	4	8	no	—
565	D₄	—	4	16	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	4	no	—
568	D₄	C₂×#41×C₂₃	1	4	yes	complemented
569	C₂×C₄	—	2	16	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	8	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	16	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	8	no	—
590	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
591	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
593	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
598	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
599	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
600	C₈	—	2	4	no	—
601	C₈	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	8	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	8	8	no	—
614	C₂²	C₂₃×#525	1	2	yes	—
615	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
616	C₂²	C₂₃×#575	1	2	yes	—
617	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	4	4	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	C₂₃×#525	1	2	yes	—
627	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
628	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
629	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
630	C₂	—	4	2	no	—
631	C₂	—	4	4	no	—
632	C₂	—	8	4	no	—
633	C₂	C₂₃×#3115	1	4	yes	direct product factor
634	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
635	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
636	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#532	C₂	semidirect
2. C₂×C₄×#41×C₂₃	C₂	semidirect
3. C₂₃×#3115	C₂	direct (N × H)
4. C₂×C₂₃×#482	C₂	semidirect
5. C₂×C₂₃×#475	C₂	semidirect
6. C₂×C₂₃×#474	C₂	semidirect
7. C₂×C₂₃×#460	C₂	semidirect
8. C₂×C₂₃×#459	C₂	semidirect
9. C₂×C₂₃×#449	C₂	semidirect
10. C₂×C₂₃×#442	C₂	semidirect
11. C₂×C₂₃×#440	C₂	semidirect
12. C₂×C₂₃×#432	C₂	semidirect
18. C₂₃×#532	C₂²	semidirect
25. C₄×#41×C₂₃	C₂²	semidirect
38. C₂₃×#482	C₂²	semidirect
39. C₂₃×#475	C₂²	semidirect
40. C₂₃×#474	C₂²	semidirect
41. C₂₃×#460	C₂²	semidirect
42. C₂₃×#459	C₂²	semidirect
43. C₂₃×#449	C₂²	semidirect
44. C₂₃×#442	C₂²	semidirect
45. C₂₃×#440	C₂²	semidirect
46. C₂₃×#432	C₂²	semidirect
150. C₂×Q₈×C₂₃	#41	semidirect
165. C₂×D₄×C₂₃	#41	semidirect
223. C₂×#3115	C₂₃	direct (N × H)
240. Q₈×C₂₃	C₂×#41	semidirect
250. D₄×C₂₃	C₂×#41	semidirect
284. C₂×#532	C₂×C₂₃	semidirect
285. C₂×C₄×#41	C₂×C₂₃	semidirect
286. #3115	C₂×C₂₃	direct (N × H)
287. C₂×#482	C₂×C₂₃	semidirect
288. C₂×#475	C₂×C₂₃	semidirect
289. C₂×#474	C₂×C₂₃	semidirect
290. C₂×#460	C₂×C₂₃	semidirect
291. C₂×#459	C₂×C₂₃	semidirect
292. C₂×#449	C₂×C₂₃	semidirect
293. C₂×#442	C₂×C₂₃	semidirect
294. C₂×#440	C₂×C₂₃	semidirect
295. C₂×#432	C₂×C₂₃	semidirect
329. #532	C₂²×C₂₃	semidirect
336. C₄×#41	C₂²×C₂₃	semidirect
349. #482	C₂²×C₂₃	semidirect
350. #475	C₂²×C₂₃	semidirect
351. #474	C₂²×C₂₃	semidirect
352. #460	C₂²×C₂₃	semidirect
353. #459	C₂²×C₂₃	semidirect
354. #449	C₂²×C₂₃	semidirect
355. #442	C₂²×C₂₃	semidirect
356. #440	C₂²×C₂₃	semidirect
357. #432	C₂²×C₂₃	semidirect
371. C₂×C₂₃	#3115	direct (N × H)
457. C₂₃	C₂×#3115	direct (N × H)
469. C₂×Q₈	#41×C₂₃	semidirect
484. C₂×D₄	#41×C₂₃	semidirect
558. Q₈	C₂×#41×C₂₃	semidirect
568. D₄	C₂×#41×C₂₃	semidirect
633. C₂	C₂₃×#3115	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×#532	2
C₂×C₄×#41×C₂₃	2
C₂₃×#3115	2
C₂×C₂₃×#482	2
C₂×C₂₃×#475	2
C₂×C₂₃×#474	2
C₂×C₂₃×#460	2
C₂×C₂₃×#459	2
C₂×C₂₃×#449	2
C₂×C₂₃×#442	2
C₂×C₂₃×#440	2
C₂×C₂₃×#432	2
C₂×#3115	23

Subgroups — C2×C23×#3115 #19208164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3115 #19208164