Subgroups — C₂×C₂₃×#3289 #19208725

Order 5888.

2620 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#577	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3289	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#473	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#471	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#456	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#454	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#444	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂₃×#578	—	2	8	no	—
15	C₂₃×#578	C₂²	1	16	yes	complemented
16	C₂₃×#577	—	2	8	no	—
17	C₂₃×#577	C₂²	1	16	yes	complemented
18	C₂×C₂₃×#137	—	2	2	no	—
19	C₂×C₂₃×#137	—	2	2	no	—
20	C₂×C₂₃×#137	—	2	2	no	—
21	C₂×C₂₃×#137	C₂²	1	2	yes	—
22	C₂×C₂₃×#137	C₂²	1	2	yes	—
23	C₂×C₂₃×#137	C₂²	1	2	yes	complemented
24	C₂×C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
25	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
26	C₂×C₂₃×#135	—	2	2	no	—
27	C₂×C₂₃×#135	—	2	2	no	—
28	C₂×C₂₃×#135	C₂²	1	2	yes	—
29	C₂²×Q₁₆×C₂₃	—	2	2	no	—
30	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
31	C₂²×SD₁₆×C₂₃	—	2	2	no	—
32	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
33	C₂×C₂₃×#131	C₂²	1	2	yes	—
34	C₂×C₂₃×#131	C₂²	1	2	yes	complemented
35	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
37	C₂₃×#497	C₂²	1	8	yes	complemented
38	C₂₃×#473	C₂²	1	16	yes	complemented
39	C₂₃×#471	C₂²	1	16	yes	complemented
40	C₂₃×#456	C₂²	1	16	yes	complemented
41	C₂₃×#455	C₂²	1	8	yes	complemented
42	C₂₃×#454	C₂²	1	8	yes	complemented
43	C₂₃×#444	C₂²	1	16	yes	complemented
44	C₂×C₂₃×#108	C₂²	1	2	yes	complemented
45	C₂×C₂₃×#104	C₂²	1	2	yes	—
46	C₂×C₂₃×#104	C₂²	1	2	yes	complemented
47	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic, complemented
48	C₂×C₂₃×#101	C₂²	1	2	yes	complemented
49	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic, complemented
51	C₂₃×2₋¹⁺⁴	—	2	4	no	—
52	C₂₃×2₋¹⁺⁴	—	2	4	no	—
53	C₂₃×2₋¹⁺⁴	—	2	4	no	—
54	C₂₃×2₋¹⁺⁴	—	4	2	no	—
55	C₂₃×2₋¹⁺⁴	—	4	1	no	—
56	C₂₃×2₋¹⁺⁴	C₂³	1	8	yes	—
57	C₂₃×2₋¹⁺⁴	C₂³	1	4	yes	complemented
58	C₂₃×H(5,𝔽₂)	—	2	4	no	—
59	C₂₃×H(5,𝔽₂)	—	4	1	no	—
60	C₂₃×H(5,𝔽₂)	C₂³	1	4	yes	—
61	C₂×#41×C₂₃	—	2	2	no	—
62	C₂×#41×C₂₃	—	2	2	no	—
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂×#41×C₂₃	—	2	2	no	—
65	C₂×#41×C₂₃	—	4	2	no	—
66	C₂×#41×C₂₃	—	4	1	no	—
67	C₂×#41×C₂₃	C₂³	1	2	yes	—
68	C₂×#41×C₂₃	C₂³	1	2	yes	—
69	C₂²×Q₈×C₂₃	—	2	2	no	—
70	C₂²×Q₈×C₂₃	—	2	2	no	—
71	C₂²×Q₈×C₂₃	—	2	2	no	—
72	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
73	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₂₃	—	2	2	no	—
75	C₂²×D₄×C₂₃	—	4	1	no	—
76	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×#137	—	2	4	no	—
78	C₂₃×#137	—	2	4	no	—
79	C₂₃×#137	—	2	4	no	—
80	C₂₃×#137	—	2	16	no	—
81	C₂₃×#137	—	2	16	no	—
82	C₂₃×#137	—	2	16	no	—
83	C₂₃×#137	C₂³	1	8	yes	—
84	C₂₃×#137	C₂³	1	8	yes	—
85	C₂₃×#137	C₂³	1	8	yes	complemented
86	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
87	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
88	C₂₃×AGL(1,ℤ/8)	C₂³	1	8	yes	—
89	C₂₃×#135	—	2	16	no	—
90	C₂₃×#135	—	2	16	no	—
91	C₂₃×#135	D₄	1	8	yes	—
92	C₂₃×#135	C₂³	1	8	yes	—
93	C₂×Q₁₆×C₂₃	—	2	2	no	—
94	C₂×Q₁₆×C₂₃	—	2	2	no	—
95	C₂×Q₁₆×C₂₃	—	2	2	no	—
96	C₂×Q₁₆×C₂₃	—	2	16	no	—
97	C₂×Q₁₆×C₂₃	—	2	2	no	—
98	C₂×Q₁₆×C₂₃	—	2	2	no	—
99	C₂×Q₁₆×C₂₃	—	2	4	no	—
100	C₂×Q₁₆×C₂₃	D₄	1	2	yes	—
101	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	complemented
102	C₂×SD₁₆×C₂₃	—	2	2	no	—
103	C₂×SD₁₆×C₂₃	—	2	2	no	—
104	C₂×SD₁₆×C₂₃	—	2	2	no	—
105	C₂×SD₁₆×C₂₃	—	2	2	no	—
106	C₂×SD₁₆×C₂₃	—	2	16	no	—
107	C₂×SD₁₆×C₂₃	—	2	2	no	—
108	C₂×SD₁₆×C₂₃	—	2	2	no	—
109	C₂×SD₁₆×C₂₃	—	2	4	no	—
110	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
111	C₂×SD₁₆×C₂₃	D₄	1	2	yes	—
112	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
113	C₂×D₈×C₂₃	—	2	2	no	—
114	C₂×D₈×C₂₃	D₄	1	2	yes	—
115	C₂₃×#131	—	2	4	no	—
116	C₂₃×#131	—	2	4	no	—
117	C₂₃×#131	C₂³	1	8	yes	—
118	C₂₃×#131	C₂³	1	8	yes	complemented
119	C₂×M₁₆×C₂₃	—	2	2	no	—
120	C₂×M₁₆×C₂₃	—	2	2	no	—
121	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
122	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
123	C₂²×C₈×C₂₃	—	2	2	no	—
124	C₂²×C₈×C₂₃	—	2	2	no	—
125	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
126	C₂₃×C₄⋊Q₈	—	4	1	no	—
127	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
128	C₂₃×#124	—	4	1	no	—
129	C₂₃×#124	C₂³	1	4	yes	—
130	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
131	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
132	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
133	C₂₃×#108	C₂³	1	8	yes	complemented
134	C₂₃×#104	C₂³	1	8	yes	—
135	C₂₃×#104	C₂³	1	8	yes	complemented
136	C₂₃×#101	C₂³	1	4	yes	complemented
137	C₂₃×#101	C₂³	1	8	yes	complemented
138	C₂₃×#100	C₂³	1	4	yes	—
139	C₂₃×#97	C₂³	1	4	yes	complemented
140	C₂⁴×C₂₃	—	2	1	no	—
141	C₂⁴×C₂₃	—	4	1	no	—
142	#41×C₂₃	—	2	2	no	—
143	#41×C₂₃	—	2	2	no	—
144	#41×C₂₃	—	2	2	no	—
145	#41×C₂₃	—	2	2	no	—
146	#41×C₂₃	—	2	8	no	—
147	#41×C₂₃	—	2	8	no	—
148	#41×C₂₃	—	2	8	no	—
149	#41×C₂₃	—	2	8	no	—
150	#41×C₂₃	—	4	8	no	—
151	#41×C₂₃	—	4	4	no	—
152	#41×C₂₃	—	4	4	no	—
153	#41×C₂₃	—	4	4	no	—
154	#41×C₂₃	—	4	4	no	—
155	#41×C₂₃	—	4	4	no	—
156	#41×C₂₃	—	4	8	no	—
157	#41×C₂₃	—	4	4	no	—
158	#41×C₂₃	C₂×D₄	1	4	yes	—
159	#41×C₂₃	C₂×D₄	1	4	yes	—
160	#41×C₂₃	C₂⁴	1	4	yes	—
161	#41×C₂₃	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₂₃	—	2	1	no	—
163	C₂×Q₈×C₂₃	—	2	2	no	—
164	C₂×Q₈×C₂₃	—	2	8	no	—
165	C₂×Q₈×C₂₃	—	2	8	no	—
166	C₂×Q₈×C₂₃	—	2	8	no	—
167	C₂×Q₈×C₂₃	—	2	2	no	—
168	C₂×Q₈×C₂₃	—	2	2	no	—
169	C₂×Q₈×C₂₃	—	2	2	no	—
170	C₂×Q₈×C₂₃	—	2	2	no	—
171	C₂×Q₈×C₂₃	—	2	2	no	—
172	C₂×Q₈×C₂₃	—	2	2	no	—
173	C₂×Q₈×C₂₃	—	4	2	no	—
174	C₂×Q₈×C₂₃	—	4	1	no	—
175	C₂×Q₈×C₂₃	—	4	4	no	—
176	C₂×Q₈×C₂₃	—	4	4	no	—
177	C₂×Q₈×C₂₃	—	4	4	no	—
178	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
179	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
180	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
181	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
182	C₂×D₄×C₂₃	—	2	2	no	—
183	C₂×D₄×C₂₃	—	2	2	no	—
184	C₂×D₄×C₂₃	—	2	8	no	—
185	C₂×D₄×C₂₃	—	2	2	no	—
186	C₂×D₄×C₂₃	—	2	2	no	—
187	C₂×D₄×C₂₃	—	2	1	no	—
188	C₂×D₄×C₂₃	—	4	4	no	—
189	C₂×D₄×C₂₃	—	4	2	no	—
190	C₂×D₄×C₂₃	—	4	4	no	—
191	C₂×D₄×C₂₃	—	4	4	no	—
192	C₂×D₄×C₂₃	—	4	1	no	—
193	C₂×D₄×C₂₃	—	4	2	no	—
194	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
195	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	2	no	—
199	C₂²×C₄×C₂₃	—	2	1	no	—
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	—	2	2	no	—
202	C₂²×C₄×C₂₃	—	4	1	no	—
203	C₂²×C₄×C₂₃	—	4	2	no	—
204	C₂²×C₄×C₂₃	—	4	2	no	—
205	C₂²×C₄×C₂₃	—	4	2	no	—
206	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₂₃	—	2	8	no	—
208	Q₁₆×C₂₃	—	2	8	no	—
209	Q₁₆×C₂₃	—	2	8	no	—
210	Q₁₆×C₂₃	—	2	8	no	—
211	Q₁₆×C₂₃	—	2	8	no	—
212	Q₁₆×C₂₃	C₂×D₄	1	8	yes	—
213	SD₁₆×C₂₃	—	2	8	no	—
214	SD₁₆×C₂₃	—	2	8	no	—
215	SD₁₆×C₂₃	—	2	8	no	—
216	SD₁₆×C₂₃	—	2	8	no	—
217	SD₁₆×C₂₃	—	2	8	no	—
218	SD₁₆×C₂₃	—	2	8	no	—
219	SD₁₆×C₂₃	C₂×D₄	1	8	yes	—
220	SD₁₆×C₂₃	C₂×D₄	1	8	yes	—
221	D₈×C₂₃	—	2	8	no	—
222	D₈×C₂₃	C₂×D₄	1	8	yes	—
223	M₁₆×C₂₃	—	2	2	no	—
224	M₁₆×C₂₃	—	2	2	no	—
225	M₁₆×C₂₃	—	2	8	no	—
226	M₁₆×C₂₃	—	2	8	no	—
227	M₁₆×C₂₃	C₂⁴	1	4	yes	—
228	M₁₆×C₂₃	C₂⁴	1	4	yes	—
229	C₂×C₈×C₂₃	—	2	2	no	—
230	C₂×C₈×C₂₃	—	2	2	no	—
231	C₂×C₈×C₂₃	—	2	8	no	—
232	C₂×C₈×C₂₃	—	2	8	no	—
233	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
234	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
235	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
236	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₂₃	—	4	4	no	—
238	C₂²⋊C₄×C₂₃	—	4	4	no	—
239	C₄²×C₂₃	—	2	1	no	—
240	C₄²×C₂₃	C₂⁴	1	2	yes	—
241	C₂×#3289	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂³×C₂₃	—	2	2	no	—
243	C₂³×C₂₃	—	2	2	no	—
244	C₂³×C₂₃	—	2	2	no	—
245	C₂³×C₂₃	—	4	4	no	—
246	C₂³×C₂₃	—	4	1	no	—
247	C₂³×C₂₃	—	4	1	no	—
248	C₂³×C₂₃	—	8	2	no	—
249	C₂³×C₂₃	—	16	2	no	—
250	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
251	Q₈×C₂₃	—	2	2	no	—
252	Q₈×C₂₃	—	2	2	no	—
253	Q₈×C₂₃	—	2	4	no	—
254	Q₈×C₂₃	—	2	4	no	—
255	Q₈×C₂₃	—	2	4	no	—
256	Q₈×C₂₃	—	4	8	no	—
257	Q₈×C₂₃	—	4	4	no	—
258	Q₈×C₂₃	—	4	2	no	—
259	Q₈×C₂₃	—	4	2	no	—
260	Q₈×C₂₃	—	4	2	no	—
261	Q₈×C₂₃	C₂²×D₄	1	4	yes	—
262	Q₈×C₂₃	C₂²×D₄	1	4	yes	—
263	D₄×C₂₃	—	2	4	no	—
264	D₄×C₂₃	—	2	4	no	—
265	D₄×C₂₃	—	2	4	no	—
266	D₄×C₂₃	—	2	4	no	—
267	D₄×C₂₃	—	2	2	no	—
268	D₄×C₂₃	—	2	2	no	—
269	D₄×C₂₃	—	2	4	no	—
270	D₄×C₂₃	—	4	8	no	—
271	D₄×C₂₃	—	4	4	no	—
272	D₄×C₂₃	—	4	8	no	—
273	D₄×C₂₃	—	4	2	no	—
274	D₄×C₂₃	C₂²×D₄	1	4	yes	—
275	D₄×C₂₃	C₂²×D₄	1	4	yes	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	4	no	—
279	C₂×C₄×C₂₃	—	2	4	no	—
280	C₂×C₄×C₂₃	—	2	4	no	—
281	C₂×C₄×C₂₃	—	2	4	no	—
282	C₂×C₄×C₂₃	—	2	2	no	—
283	C₂×C₄×C₂₃	—	2	2	no	—
284	C₂×C₄×C₂₃	—	2	4	no	—
285	C₂×C₄×C₂₃	—	4	4	no	—
286	C₂×C₄×C₂₃	—	4	8	no	—
287	C₂×C₄×C₂₃	—	4	8	no	—
288	C₂×C₄×C₂₃	—	4	8	no	—
289	C₂×C₄×C₂₃	—	4	2	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	2	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	2	no	—
294	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
296	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
298	C₈×C₂₃	—	2	4	no	—
299	C₈×C₂₃	—	2	4	no	—
300	C₈×C₂₃	C₂²×D₄	1	4	yes	—
301	C₈×C₂₃	C₂²×D₄	1	4	yes	—
302	C₂×#578	C₂×C₂₃	1	2	yes	complemented
303	C₂×#577	C₂×C₂₃	1	2	yes	complemented
304	#3289	C₂×C₂₃	1	16	yes	direct product factor
305	C₂×#497	C₂×C₂₃	1	1	yes	characteristic, complemented
306	C₂×#473	C₂×C₂₃	1	2	yes	complemented
307	C₂×#471	C₂×C₂₃	1	2	yes	complemented
308	C₂×#456	C₂×C₂₃	1	2	yes	complemented
309	C₂×#454	C₂×C₂₃	1	1	yes	characteristic, complemented
310	C₂×#455	C₂×C₂₃	1	1	yes	characteristic, complemented
311	C₂×#444	C₂×C₂₃	1	2	yes	complemented
312	C₂²×C₂₃	—	2	2	no	—
313	C₂²×C₂₃	—	2	4	no	—
314	C₂²×C₂₃	—	2	4	no	—
315	C₂²×C₂₃	—	4	4	no	—
316	C₂²×C₂₃	—	4	4	no	—
317	C₂²×C₂₃	—	4	2	no	—
318	C₂²×C₂₃	—	8	2	no	—
319	C₂²×C₂₃	—	8	2	no	—
320	C₂²×C₂₃	—	16	1	no	—
321	C₂²×C₂₃	—	16	4	no	—
322	C₂²×C₂₃	D₄²	1	1	yes	characteristic
323	C₂²×C₂₃	D₄²	1	1	yes	characteristic
324	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
325	C₄×C₂₃	—	2	4	no	—
326	C₄×C₂₃	—	2	4	no	—
327	C₄×C₂₃	—	4	2	no	—
328	C₄×C₂₃	—	4	2	no	—
329	C₄×C₂₃	—	4	4	no	—
330	C₄×C₂₃	D₄²	1	2	yes	—
331	C₄×C₂₃	C₂³×D₄	1	2	yes	—
332	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
333	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic, complemented
334	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	1	yes	characteristic
335	#578	—	2	8	no	—
336	#578	C₂²×C₂₃	1	16	yes	complemented
337	#577	—	2	8	no	—
338	#577	C₂²×C₂₃	1	16	yes	complemented
339	C₂×#137	—	2	2	no	—
340	C₂×#137	—	2	2	no	—
341	C₂×#137	—	2	2	no	—
342	C₂×#137	C₂²×C₂₃	1	2	yes	—
343	C₂×#137	C₂²×C₂₃	1	2	yes	—
344	C₂×#137	C₂²×C₂₃	1	2	yes	complemented
345	C₂×AGL(1,ℤ/8)	—	2	2	no	—
346	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
347	C₂×#135	—	2	2	no	—
348	C₂×#135	—	2	2	no	—
349	C₂×#135	C₂²×C₂₃	1	2	yes	—
350	C₂²×Q₁₆	—	2	2	no	—
351	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
352	C₂²×SD₁₆	—	2	2	no	—
353	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
354	C₂×#131	C₂²×C₂₃	1	2	yes	—
355	C₂×#131	C₂²×C₂₃	1	2	yes	complemented
356	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
357	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
358	#497	C₂²×C₂₃	1	8	yes	complemented
359	#473	C₂²×C₂₃	1	16	yes	complemented
360	#471	C₂²×C₂₃	1	16	yes	complemented
361	#456	C₂²×C₂₃	1	16	yes	complemented
362	#455	C₂²×C₂₃	1	8	yes	complemented
363	#454	C₂²×C₂₃	1	8	yes	complemented
364	#444	C₂²×C₂₃	1	16	yes	complemented
365	C₂×#108	C₂²×C₂₃	1	2	yes	complemented
366	C₂×#104	C₂²×C₂₃	1	2	yes	—
367	C₂×#104	C₂²×C₂₃	1	2	yes	complemented
368	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic, complemented
369	C₂×#101	C₂²×C₂₃	1	2	yes	complemented
370	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic, complemented
372	C₂×C₂₃	—	2	1	no	—
373	C₂×C₂₃	—	2	1	no	—
374	C₂×C₂₃	—	4	4	no	—
375	C₂×C₂₃	—	4	4	no	—
376	C₂×C₂₃	—	8	2	no	—
377	C₂×C₂₃	#3289	1	2	yes	direct product factor
378	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
379	2₋¹⁺⁴	—	2	4	no	—
380	2₋¹⁺⁴	—	2	4	no	—
381	2₋¹⁺⁴	—	2	4	no	—
382	2₋¹⁺⁴	—	4	2	no	—
383	2₋¹⁺⁴	—	4	1	no	—
384	2₋¹⁺⁴	C₂³×C₂₃	1	8	yes	—
385	2₋¹⁺⁴	C₂³×C₂₃	1	4	yes	complemented
386	H(5,𝔽₂)	—	2	4	no	—
387	H(5,𝔽₂)	—	4	1	no	—
388	H(5,𝔽₂)	C₂³×C₂₃	1	4	yes	—
389	C₂×#41	—	2	2	no	—
390	C₂×#41	—	2	2	no	—
391	C₂×#41	—	2	2	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	4	2	no	—
394	C₂×#41	—	4	1	no	—
395	C₂×#41	C₂³×C₂₃	1	2	yes	—
396	C₂×#41	C₂³×C₂₃	1	2	yes	—
397	C₂²×Q₈	—	2	2	no	—
398	C₂²×Q₈	—	2	2	no	—
399	C₂²×Q₈	—	2	2	no	—
400	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
402	C₂²×D₄	—	2	2	no	—
403	C₂²×D₄	—	4	1	no	—
404	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
405	#137	—	2	4	no	—
406	#137	—	2	4	no	—
407	#137	—	2	4	no	—
408	#137	—	2	16	no	—
409	#137	—	2	16	no	—
410	#137	—	2	16	no	—
411	#137	C₂³×C₂₃	1	8	yes	—
412	#137	C₂³×C₂₃	1	8	yes	—
413	#137	C₂³×C₂₃	1	8	yes	complemented
414	AGL(1,ℤ/8)	—	2	4	no	—
415	AGL(1,ℤ/8)	—	2	16	no	—
416	AGL(1,ℤ/8)	C₂³×C₂₃	1	8	yes	—
417	#135	—	2	16	no	—
418	#135	—	2	16	no	—
419	#135	D₄×C₂₃	1	8	yes	—
420	#135	C₂³×C₂₃	1	8	yes	—
421	C₂×Q₁₆	—	2	2	no	—
422	C₂×Q₁₆	—	2	2	no	—
423	C₂×Q₁₆	—	2	2	no	—
424	C₂×Q₁₆	—	2	16	no	—
425	C₂×Q₁₆	—	2	2	no	—
426	C₂×Q₁₆	—	2	2	no	—
427	C₂×Q₁₆	—	2	4	no	—
428	C₂×Q₁₆	D₄×C₂₃	1	2	yes	—
429	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	complemented
430	C₂×SD₁₆	—	2	2	no	—
431	C₂×SD₁₆	—	2	2	no	—
432	C₂×SD₁₆	—	2	2	no	—
433	C₂×SD₁₆	—	2	2	no	—
434	C₂×SD₁₆	—	2	16	no	—
435	C₂×SD₁₆	—	2	2	no	—
436	C₂×SD₁₆	—	2	2	no	—
437	C₂×SD₁₆	—	2	4	no	—
438	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
439	C₂×SD₁₆	D₄×C₂₃	1	2	yes	—
440	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
441	C₂×D₈	—	2	2	no	—
442	C₂×D₈	D₄×C₂₃	1	2	yes	—
443	#131	—	2	4	no	—
444	#131	—	2	4	no	—
445	#131	C₂³×C₂₃	1	8	yes	—
446	#131	C₂³×C₂₃	1	8	yes	complemented
447	C₂×M₁₆	—	2	2	no	—
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
450	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
451	C₂²×C₈	—	2	2	no	—
452	C₂²×C₈	—	2	2	no	—
453	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
454	C₄⋊Q₈	—	4	1	no	—
455	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
456	#124	—	4	1	no	—
457	#124	C₂³×C₂₃	1	4	yes	—
458	C₂×C₄⋊C₄	—	4	1	no	—
459	C₂×C₂²⋊C₄	—	4	1	no	—
460	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
461	#108	C₂³×C₂₃	1	8	yes	complemented
462	#104	C₂³×C₂₃	1	8	yes	—
463	#104	C₂³×C₂₃	1	8	yes	complemented
464	#101	C₂³×C₂₃	1	4	yes	complemented
465	#101	C₂³×C₂₃	1	8	yes	complemented
466	#100	C₂³×C₂₃	1	4	yes	—
467	#97	C₂³×C₂₃	1	4	yes	complemented
468	C₂₃	C₂×#3289	1	1	yes	characteristic, direct product factor, Sylow
469	C₂⁴	—	2	1	no	—
470	C₂⁴	—	4	1	no	—
471	#41	—	2	2	no	—
472	#41	—	2	2	no	—
473	#41	—	2	2	no	—
474	#41	—	2	2	no	—
475	#41	—	2	8	no	—
476	#41	—	2	8	no	—
477	#41	—	2	8	no	—
478	#41	—	2	8	no	—
479	#41	—	4	8	no	—
480	#41	—	4	4	no	—
481	#41	—	4	4	no	—
482	#41	—	4	4	no	—
483	#41	—	4	4	no	—
484	#41	—	4	4	no	—
485	#41	—	4	8	no	—
486	#41	—	4	4	no	—
487	#41	C₂×D₄×C₂₃	1	4	yes	—
488	#41	C₂×D₄×C₂₃	1	4	yes	—
489	#41	C₂⁴×C₂₃	1	4	yes	—
490	#41	C₂⁴×C₂₃	1	4	yes	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	—	2	8	no	—
494	C₂×Q₈	—	2	8	no	—
495	C₂×Q₈	—	2	8	no	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	4	2	no	—
503	C₂×Q₈	—	4	1	no	—
504	C₂×Q₈	—	4	4	no	—
505	C₂×Q₈	—	4	4	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
508	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
509	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
510	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	8	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	4	4	no	—
518	C₂×D₄	—	4	2	no	—
519	C₂×D₄	—	4	4	no	—
520	C₂×D₄	—	4	4	no	—
521	C₂×D₄	—	4	1	no	—
522	C₂×D₄	—	4	2	no	—
523	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
524	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
525	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	4	1	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	—	4	2	no	—
535	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
536	Q₁₆	—	2	8	no	—
537	Q₁₆	—	2	8	no	—
538	Q₁₆	—	2	8	no	—
539	Q₁₆	—	2	8	no	—
540	Q₁₆	—	2	8	no	—
541	Q₁₆	C₂×D₄×C₂₃	1	8	yes	—
542	SD₁₆	—	2	8	no	—
543	SD₁₆	—	2	8	no	—
544	SD₁₆	—	2	8	no	—
545	SD₁₆	—	2	8	no	—
546	SD₁₆	—	2	8	no	—
547	SD₁₆	—	2	8	no	—
548	SD₁₆	C₂×D₄×C₂₃	1	8	yes	—
549	SD₁₆	C₂×D₄×C₂₃	1	8	yes	—
550	D₈	—	2	8	no	—
551	D₈	C₂×D₄×C₂₃	1	8	yes	—
552	M₁₆	—	2	2	no	—
553	M₁₆	—	2	2	no	—
554	M₁₆	—	2	8	no	—
555	M₁₆	—	2	8	no	—
556	M₁₆	C₂⁴×C₂₃	1	4	yes	—
557	M₁₆	C₂⁴×C₂₃	1	4	yes	—
558	C₂×C₈	—	2	2	no	—
559	C₂×C₈	—	2	2	no	—
560	C₂×C₈	—	2	8	no	—
561	C₂×C₈	—	2	8	no	—
562	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
563	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
564	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
565	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
566	C₄⋊C₄	—	4	4	no	—
567	C₂²⋊C₄	—	4	4	no	—
568	C₄²	—	2	1	no	—
569	C₄²	C₂⁴×C₂₃	1	2	yes	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	4	4	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	8	2	no	—
577	C₂³	—	16	2	no	—
578	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
579	Q₈	—	2	2	no	—
580	Q₈	—	2	2	no	—
581	Q₈	—	2	4	no	—
582	Q₈	—	2	4	no	—
583	Q₈	—	2	4	no	—
584	Q₈	—	4	8	no	—
585	Q₈	—	4	4	no	—
586	Q₈	—	4	2	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	4	2	no	—
589	Q₈	C₂²×D₄×C₂₃	1	4	yes	—
590	Q₈	C₂²×D₄×C₂₃	1	4	yes	—
591	D₄	—	2	4	no	—
592	D₄	—	2	4	no	—
593	D₄	—	2	4	no	—
594	D₄	—	2	4	no	—
595	D₄	—	2	4	no	—
596	D₄	—	2	2	no	—
597	D₄	—	2	2	no	—
598	D₄	—	4	8	no	—
599	D₄	—	4	4	no	—
600	D₄	—	4	8	no	—
601	D₄	—	4	2	no	—
602	D₄	C₂²×D₄×C₂₃	1	4	yes	—
603	D₄	C₂²×D₄×C₂₃	1	4	yes	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	8	no	—
616	C₂×C₄	—	4	8	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
624	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
625	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
626	C₈	—	2	4	no	—
627	C₈	—	2	4	no	—
628	C₈	C₂²×D₄×C₂₃	1	4	yes	—
629	C₈	C₂²×D₄×C₂₃	1	4	yes	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	8	2	no	—
637	C₂²	—	8	2	no	—
638	C₂²	—	16	1	no	—
639	C₂²	—	16	4	no	—
640	C₂²	D₄²×C₂₃	1	1	yes	characteristic
641	C₂²	D₄²×C₂₃	1	1	yes	characteristic
642	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
643	C₄	—	2	4	no	—
644	C₄	—	2	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	4	no	—
648	C₄	D₄²×C₂₃	1	2	yes	—
649	C₄	C₂³×D₄×C₂₃	1	2	yes	—
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	4	4	no	—
653	C₂	—	4	4	no	—
654	C₂	—	8	2	no	—
655	C₂	C₂₃×#3289	1	2	yes	direct product factor
656	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#578	C₂	semidirect
2. C₂×C₂₃×#577	C₂	semidirect
3. C₂₃×#3289	C₂	direct (N × H)
4. C₂×C₂₃×#497	C₂	semidirect
5. C₂×C₂₃×#473	C₂	semidirect
6. C₂×C₂₃×#471	C₂	semidirect
7. C₂×C₂₃×#456	C₂	semidirect
8. C₂×C₂₃×#454	C₂	semidirect
9. C₂×C₂₃×#455	C₂	semidirect
10. C₂×C₂₃×#444	C₂	semidirect
12. C₂×C₂₃×2₋¹⁺⁴	C₂²	semidirect
15. C₂₃×#578	C₂²	semidirect
17. C₂₃×#577	C₂²	semidirect
23. C₂×C₂₃×#137	C₂²	semidirect
30. C₂²×Q₁₆×C₂₃	C₂²	semidirect
34. C₂×C₂₃×#131	C₂²	semidirect
35. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
37. C₂₃×#497	C₂²	semidirect
38. C₂₃×#473	C₂²	semidirect
39. C₂₃×#471	C₂²	semidirect
40. C₂₃×#456	C₂²	semidirect
41. C₂₃×#455	C₂²	semidirect
42. C₂₃×#454	C₂²	semidirect
43. C₂₃×#444	C₂²	semidirect
44. C₂×C₂₃×#108	C₂²	semidirect
46. C₂×C₂₃×#104	C₂²	semidirect
47. C₂×C₂₃×#101	C₂²	semidirect
48. C₂×C₂₃×#101	C₂²	semidirect
50. C₂×C₂₃×#97	C₂²	semidirect
57. C₂₃×2₋¹⁺⁴	C₂³	semidirect
85. C₂₃×#137	C₂³	semidirect
101. C₂×Q₁₆×C₂₃	C₂³	semidirect
118. C₂₃×#131	C₂³	semidirect
127. C₂₃×C₄⋊Q₈	C₂³	semidirect
133. C₂₃×#108	C₂³	semidirect
135. C₂₃×#104	C₂³	semidirect
136. C₂₃×#101	C₂³	semidirect
137. C₂₃×#101	C₂³	semidirect
139. C₂₃×#97	C₂³	semidirect
241. C₂×#3289	C₂₃	direct (N × H)
302. C₂×#578	C₂×C₂₃	semidirect
303. C₂×#577	C₂×C₂₃	semidirect
304. #3289	C₂×C₂₃	direct (N × H)
305. C₂×#497	C₂×C₂₃	semidirect
306. C₂×#473	C₂×C₂₃	semidirect
307. C₂×#471	C₂×C₂₃	semidirect
308. C₂×#456	C₂×C₂₃	semidirect
309. C₂×#454	C₂×C₂₃	semidirect
310. C₂×#455	C₂×C₂₃	semidirect
311. C₂×#444	C₂×C₂₃	semidirect
333. C₂×2₋¹⁺⁴	C₂²×C₂₃	semidirect
336. #578	C₂²×C₂₃	semidirect
338. #577	C₂²×C₂₃	semidirect
344. C₂×#137	C₂²×C₂₃	semidirect
351. C₂²×Q₁₆	C₂²×C₂₃	semidirect
355. C₂×#131	C₂²×C₂₃	semidirect
356. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
358. #497	C₂²×C₂₃	semidirect
359. #473	C₂²×C₂₃	semidirect
360. #471	C₂²×C₂₃	semidirect
361. #456	C₂²×C₂₃	semidirect
362. #455	C₂²×C₂₃	semidirect
363. #454	C₂²×C₂₃	semidirect
364. #444	C₂²×C₂₃	semidirect
365. C₂×#108	C₂²×C₂₃	semidirect
367. C₂×#104	C₂²×C₂₃	semidirect
368. C₂×#101	C₂²×C₂₃	semidirect
369. C₂×#101	C₂²×C₂₃	semidirect
371. C₂×#97	C₂²×C₂₃	semidirect
377. C₂×C₂₃	#3289	direct (N × H)
385. 2₋¹⁺⁴	C₂³×C₂₃	semidirect
413. #137	C₂³×C₂₃	semidirect
429. C₂×Q₁₆	C₂³×C₂₃	semidirect
446. #131	C₂³×C₂₃	semidirect
455. C₄⋊Q₈	C₂³×C₂₃	semidirect
461. #108	C₂³×C₂₃	semidirect
463. #104	C₂³×C₂₃	semidirect
464. #101	C₂³×C₂₃	semidirect
465. #101	C₂³×C₂₃	semidirect
467. #97	C₂³×C₂₃	semidirect
468. C₂₃	C₂×#3289	direct (N × H)
655. C₂	C₂₃×#3289	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×#578	2
C₂×C₂₃×#577	2
C₂₃×#3289	2
C₂×C₂₃×#497	2
C₂×C₂₃×#473	2
C₂×C₂₃×#471	2
C₂×C₂₃×#456	2
C₂×C₂₃×#454	2
C₂×C₂₃×#455	2
C₂×C₂₃×#444	2
C₂×#3289	23

Subgroups — C2×C23×#3289 #19208725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3289 #19208725