Subgroups — C₂×C₂₃×#3213 #19208409

Order 5888.

2522 conjugacy classes of subgroups.

The subgroup lattice has 536 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	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3213	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#496	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#463	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#448	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#447	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	—	2	1	no	—
11	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
14	C₂²×D₈×C₂₃	C₂²	1	4	yes	—
15	C₂₃×#532	—	2	4	no	—
16	C₂₃×#532	C₂²	1	8	yes	complemented
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₂₃×#124	—	2	1	no	—
20	C₂×C₂₃×#124	—	2	1	no	—
21	C₂×C₂₃×#124	—	2	4	no	—
22	C₂×C₂₃×#124	—	2	1	no	—
23	C₂×C₂₃×#124	C₂²	1	2	yes	—
24	C₂×C₂₃×#124	C₂²	1	2	yes	—
25	C₂×C₂₃×#124	C₂²	1	8	yes	complemented
26	C₂×C₂₃×#121	—	2	1	no	—
27	C₂×C₂₃×#121	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₂₃	—	2	1	no	—
29	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
32	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#496	C₂²	1	16	yes	complemented
34	C₂₃×#489	C₂²	1	16	yes	complemented
35	C₂₃×#463	C₂²	1	32	yes	complemented
36	C₂₃×#448	C₂²	1	16	yes	complemented
37	C₂₃×#447	C₂²	1	16	yes	complemented
38	C₂₃×#432	C₂²	1	8	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
41	C₂×C₂₃×#102	C₂²	1	4	yes	—
42	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
44	C₂⁵×C₂₃	—	2	1	no	—
45	C₂×#41×C₂₃	—	2	8	no	—
46	C₂×#41×C₂₃	—	4	8	no	—
47	C₂²×Q₈×C₂₃	—	2	2	no	—
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂²×Q₈×C₂₃	—	2	1	no	—
50	C₂²×Q₈×C₂₃	—	4	4	no	—
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₂₃	—	2	2	no	—
54	C₂²×D₄×C₂₃	—	2	2	no	—
55	C₂²×D₄×C₂₃	—	2	1	no	—
56	C₂²×D₄×C₂₃	—	2	1	no	—
57	C₂²×D₄×C₂₃	—	2	1	no	—
58	C₂²×D₄×C₂₃	—	4	4	no	—
59	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	—	2	2	no	—
64	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×SD₁₆×C₂₃	—	4	8	no	—
68	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	complemented
69	C₂×D₈×C₂₃	—	4	8	no	—
70	C₂×D₈×C₂₃	C₂³	1	16	yes	—
71	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
74	C₂₃×#127	—	2	2	no	—
75	C₂₃×#127	C₂³	1	4	yes	complemented
76	C₂₃×#124	—	2	4	no	—
77	C₂₃×#124	—	2	4	no	—
78	C₂₃×#124	—	2	8	no	—
79	C₂₃×#124	—	2	8	no	—
80	C₂₃×#124	—	2	8	no	—
81	C₂₃×#124	C₂³	1	8	yes	—
82	C₂₃×#124	C₂³	1	8	yes	—
83	C₂₃×#121	—	2	8	no	—
84	C₂₃×#121	—	2	4	no	—
85	C₂₃×#121	C₂³	1	8	yes	—
86	C₄×D₄×C₂₃	—	2	8	no	—
87	C₄×D₄×C₂₃	—	2	4	no	—
88	C₄×D₄×C₂₃	C₂³	1	8	yes	complemented
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	D₄	1	4	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
105	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
106	C₂₃×#103	C₂³	1	16	yes	complemented
107	C₂₃×#102	C₂³	1	16	yes	—
108	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₂₃×#97	C₂³	1	8	yes	complemented
110	C₂⁴×C₂₃	—	2	4	no	—
111	C₂⁴×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	2	2	no	—
113	C₂⁴×C₂₃	—	4	4	no	—
114	C₂⁴×C₂₃	—	4	4	no	—
115	C₂⁴×C₂₃	—	4	4	no	—
116	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
117	#41×C₂₃	—	4	32	no	—
118	C₂×Q₈×C₂₃	—	2	4	no	—
119	C₂×Q₈×C₂₃	—	2	2	no	—
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	—	2	8	no	—
122	C₂×Q₈×C₂₃	—	4	8	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×Q₈×C₂₃	—	4	8	no	—
125	C₂×Q₈×C₂₃	—	4	4	no	—
126	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
127	C₂×D₄×C₂₃	—	2	8	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	—	2	2	no	—
132	C₂×D₄×C₂₃	—	2	4	no	—
133	C₂×D₄×C₂₃	—	2	8	no	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	—	4	8	no	—
136	C₂×D₄×C₂₃	—	4	4	no	—
137	C₂×D₄×C₂₃	—	4	4	no	—
138	C₂×D₄×C₂₃	—	4	8	no	—
139	C₂×D₄×C₂₃	—	4	4	no	—
140	C₂×D₄×C₂₃	—	4	8	no	—
141	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
142	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	4	no	—
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	2	no	—
148	C₂²×C₄×C₂₃	—	2	8	no	—
149	C₂²×C₄×C₂₃	—	4	2	no	—
150	C₂²×C₄×C₂₃	—	4	8	no	—
151	C₂²×C₄×C₂₃	—	4	4	no	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₂₃	—	4	16	no	—
163	D₈×C₂₃	—	4	16	no	—
164	C₂×C₈×C₂₃	—	2	8	no	—
165	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	—	2	2	no	—
168	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	—	2	8	no	—
171	C₂²⋊C₄×C₂₃	—	2	8	no	—
172	C₂²⋊C₄×C₂₃	—	2	8	no	—
173	C₂²⋊C₄×C₂₃	—	2	8	no	—
174	C₂²⋊C₄×C₂₃	—	2	8	no	—
175	C₄²×C₂₃	C₂×D₄	1	4	yes	—
176	C₄²×C₂₃	C₂×D₄	1	2	yes	—
177	C₄²×C₂₃	C₂×D₄	1	4	yes	—
178	C₄²×C₂₃	C₂×D₄	1	2	yes	—
179	C₄²×C₂₃	C₂⁴	1	2	yes	—
180	C₄²×C₂₃	C₂⁴	1	2	yes	—
181	C₂×#3213	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₂₃	—	2	2	no	—
183	C₂³×C₂₃	—	2	2	no	—
184	C₂³×C₂₃	—	2	4	no	—
185	C₂³×C₂₃	—	2	4	no	—
186	C₂³×C₂₃	—	4	8	no	—
187	C₂³×C₂₃	—	4	8	no	—
188	C₂³×C₂₃	—	4	4	no	—
189	C₂³×C₂₃	—	4	4	no	—
190	C₂³×C₂₃	—	4	4	no	—
191	C₂³×C₂₃	—	4	8	no	—
192	C₂³×C₂₃	—	4	4	no	—
193	C₂³×C₂₃	—	4	4	no	—
194	C₂³×C₂₃	—	4	4	no	—
195	C₂³×C₂₃	—	8	16	no	—
196	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
197	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
198	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₂₃	—	2	8	no	—
200	Q₈×C₂₃	—	4	8	no	—
201	Q₈×C₂₃	—	4	8	no	—
202	Q₈×C₂₃	—	4	4	no	—
203	D₄×C₂₃	—	2	8	no	—
204	D₄×C₂₃	—	2	8	no	—
205	D₄×C₂₃	—	4	8	no	—
206	D₄×C₂₃	—	4	8	no	—
207	D₄×C₂₃	—	4	8	no	—
208	D₄×C₂₃	—	4	16	no	—
209	D₄×C₂₃	—	4	4	no	—
210	D₄×C₂₃	—	4	4	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	8	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	2	4	no	—
221	C₂×C₄×C₂₃	—	4	4	no	—
222	C₂×C₄×C₂₃	—	4	16	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	—	4	4	no	—
225	C₂×C₄×C₂₃	—	4	4	no	—
226	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
233	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
234	C₈×C₂₃	—	2	8	no	—
235	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
237	#3213	C₂×C₂₃	1	16	yes	direct product factor
238	C₂×#496	C₂×C₂₃	1	2	yes	complemented
239	C₂×#489	C₂×C₂₃	1	2	yes	complemented
240	C₂×#463	C₂×C₂₃	1	4	yes	complemented
241	C₂×#448	C₂×C₂₃	1	2	yes	complemented
242	C₂×#447	C₂×C₂₃	1	2	yes	complemented
243	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	8	no	—
246	C₂²×C₂₃	—	2	2	no	—
247	C₂²×C₂₃	—	4	4	no	—
248	C₂²×C₂₃	—	4	4	no	—
249	C₂²×C₂₃	—	4	4	no	—
250	C₂²×C₂₃	—	4	4	no	—
251	C₂²×C₂₃	—	4	4	no	—
252	C₂²×C₂₃	—	4	4	no	—
253	C₂²×C₂₃	—	8	16	no	—
254	C₂²×C₂₃	—	8	8	no	—
255	C₂²×C₂₃	—	8	8	no	—
256	C₂²×C₂₃	#535	1	2	yes	—
257	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
258	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
259	C₂²×C₂₃	#575	1	2	yes	—
260	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
261	C₄×C₂₃	—	2	2	no	—
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	2	2	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	2	no	—
266	C₄×C₂₃	—	4	4	no	—
267	C₄×C₂₃	—	4	4	no	—
268	C₄×C₂₃	#535	1	2	yes	—
269	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
270	C₂²×#41	—	2	1	no	—
271	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
273	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
274	C₂²×D₈	C₂²×C₂₃	1	4	yes	—
275	#532	—	2	4	no	—
276	#532	C₂²×C₂₃	1	8	yes	complemented
277	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
279	C₂×#124	—	2	1	no	—
280	C₂×#124	—	2	1	no	—
281	C₂×#124	—	2	4	no	—
282	C₂×#124	—	2	1	no	—
283	C₂×#124	C₂²×C₂₃	1	2	yes	—
284	C₂×#124	C₂²×C₂₃	1	2	yes	—
285	C₂×#124	C₂²×C₂₃	1	8	yes	complemented
286	C₂×#121	—	2	1	no	—
287	C₂×#121	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
290	C₂²×C₂²⋊C₄	—	2	1	no	—
291	C₂²×C₂²⋊C₄	—	2	1	no	—
292	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
293	#496	C₂²×C₂₃	1	16	yes	complemented
294	#489	C₂²×C₂₃	1	16	yes	complemented
295	#463	C₂²×C₂₃	1	32	yes	complemented
296	#448	C₂²×C₂₃	1	16	yes	complemented
297	#447	C₂²×C₂₃	1	16	yes	complemented
298	#432	C₂²×C₂₃	1	8	yes	complemented
299	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
300	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
301	C₂×#102	C₂²×C₂₃	1	4	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
303	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
304	C₂×C₂₃	—	2	4	no	—
305	C₂×C₂₃	—	8	4	no	—
306	C₂×C₂₃	—	8	4	no	—
307	C₂×C₂₃	#3213	1	4	yes	direct product factor
308	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
309	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
311	C₂⁵	—	2	1	no	—
312	C₂×#41	—	2	8	no	—
313	C₂×#41	—	4	8	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	—	2	2	no	—
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×Q₈	—	4	4	no	—
318	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
319	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	2	no	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	—	4	4	no	—
326	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
327	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
328	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	—	2	2	no	—
331	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
332	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
334	C₂×SD₁₆	—	4	8	no	—
335	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	complemented
336	C₂×D₈	—	4	8	no	—
337	C₂×D₈	C₂³×C₂₃	1	16	yes	—
338	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
341	#127	—	2	2	no	—
342	#127	C₂³×C₂₃	1	4	yes	complemented
343	#124	—	2	4	no	—
344	#124	—	2	4	no	—
345	#124	—	2	8	no	—
346	#124	—	2	8	no	—
347	#124	—	2	8	no	—
348	#124	C₂³×C₂₃	1	8	yes	—
349	#124	C₂³×C₂₃	1	8	yes	—
350	#121	—	2	8	no	—
351	#121	—	2	4	no	—
352	#121	C₂³×C₂₃	1	8	yes	—
353	C₄×D₄	—	2	8	no	—
354	C₄×D₄	—	2	4	no	—
355	C₄×D₄	C₂³×C₂₃	1	8	yes	complemented
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	8	no	—
363	C₂×C₂²⋊C₄	—	2	8	no	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₂₃	1	4	yes	—
369	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
372	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
373	#103	C₂³×C₂₃	1	16	yes	complemented
374	#102	C₂³×C₂₃	1	16	yes	—
375	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
376	#97	C₂³×C₂₃	1	8	yes	complemented
377	C₂₃	C₂×#3213	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	—	2	4	no	—
379	C₂⁴	—	2	2	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	4	4	no	—
382	C₂⁴	—	4	4	no	—
383	C₂⁴	—	4	4	no	—
384	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
385	#41	—	4	32	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	8	no	—
390	C₂×Q₈	—	4	8	no	—
391	C₂×Q₈	—	4	4	no	—
392	C₂×Q₈	—	4	8	no	—
393	C₂×Q₈	—	4	4	no	—
394	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	4	8	no	—
404	C₂×D₄	—	4	4	no	—
405	C₂×D₄	—	4	8	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	4	8	no	—
409	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
410	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	8	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
422	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
427	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
430	SD₁₆	—	4	16	no	—
431	D₈	—	4	16	no	—
432	C₂×C₈	—	2	8	no	—
433	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₄²	C₂×D₄×C₂₃	1	4	yes	—
444	C₄²	C₂×D₄×C₂₃	1	2	yes	—
445	C₄²	C₂×D₄×C₂₃	1	4	yes	—
446	C₄²	C₂×D₄×C₂₃	1	2	yes	—
447	C₄²	C₂⁴×C₂₃	1	2	yes	—
448	C₄²	C₂⁴×C₂₃	1	2	yes	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	4	8	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	8	no	—
457	C₂³	—	4	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	—	4	4	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	4	4	no	—
462	C₂³	—	8	16	no	—
463	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
464	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
465	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
466	Q₈	—	2	8	no	—
467	Q₈	—	4	8	no	—
468	Q₈	—	4	8	no	—
469	Q₈	—	4	4	no	—
470	D₄	—	2	8	no	—
471	D₄	—	2	8	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	8	no	—
474	D₄	—	4	8	no	—
475	D₄	—	4	16	no	—
476	D₄	—	4	4	no	—
477	D₄	—	4	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	16	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	4	4	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
498	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
500	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
501	C₈	—	2	8	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	8	16	no	—
512	C₂²	—	8	8	no	—
513	C₂²	—	8	8	no	—
514	C₂²	C₂₃×#535	1	2	yes	—
515	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
516	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
517	C₂²	C₂₃×#575	1	2	yes	—
518	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
519	C₄	—	2	2	no	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	4	no	—
526	C₄	C₂₃×#535	1	2	yes	—
527	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
528	C₂	—	2	4	no	—
529	C₂	—	8	4	no	—
530	C₂	—	8	4	no	—
531	C₂	C₂₃×#3213	1	4	yes	direct product factor
532	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
533	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
534	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. C₂²×C₂₃×#124	C₂	semidirect
3. C₂₃×#3213	C₂	direct (N × H)
4. C₂×C₂₃×#496	C₂	semidirect
5. C₂×C₂₃×#489	C₂	semidirect
6. C₂×C₂₃×#463	C₂	semidirect
7. C₂×C₂₃×#448	C₂	semidirect
8. C₂×C₂₃×#447	C₂	semidirect
9. C₂×C₂₃×#432	C₂	semidirect
13. C₂²×SD₁₆×C₂₃	C₂²	semidirect
16. C₂₃×#532	C₂²	semidirect
17. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂×C₂₃×#127	C₂²	semidirect
25. C₂×C₂₃×#124	C₂²	semidirect
29. C₂×C₄×D₄×C₂₃	C₂²	semidirect
33. C₂₃×#496	C₂²	semidirect
34. C₂₃×#489	C₂²	semidirect
35. C₂₃×#463	C₂²	semidirect
36. C₂₃×#448	C₂²	semidirect
37. C₂₃×#447	C₂²	semidirect
38. C₂₃×#432	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂×C₂₃×#103	C₂²	semidirect
43. C₂×C₂₃×#97	C₂²	semidirect
68. C₂×SD₁₆×C₂₃	C₂³	semidirect
73. C₂₃×C₄⋊Q₈	C₂³	semidirect
75. C₂₃×#127	C₂³	semidirect
88. C₄×D₄×C₂₃	C₂³	semidirect
105. C₂₃×C₄⋊C₈	C₂³	semidirect
106. C₂₃×#103	C₂³	semidirect
109. C₂₃×#97	C₂³	semidirect
181. C₂×#3213	C₂₃	direct (N × H)
235. C₂×#532	C₂×C₂₃	semidirect
236. C₂²×#124	C₂×C₂₃	semidirect
237. #3213	C₂×C₂₃	direct (N × H)
238. C₂×#496	C₂×C₂₃	semidirect
239. C₂×#489	C₂×C₂₃	semidirect
240. C₂×#463	C₂×C₂₃	semidirect
241. C₂×#448	C₂×C₂₃	semidirect
242. C₂×#447	C₂×C₂₃	semidirect
243. C₂×#432	C₂×C₂₃	semidirect
273. C₂²×SD₁₆	C₂²×C₂₃	semidirect
276. #532	C₂²×C₂₃	semidirect
277. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
278. C₂×#127	C₂²×C₂₃	semidirect
285. C₂×#124	C₂²×C₂₃	semidirect
289. C₂×C₄×D₄	C₂²×C₂₃	semidirect
293. #496	C₂²×C₂₃	semidirect
294. #489	C₂²×C₂₃	semidirect
295. #463	C₂²×C₂₃	semidirect
296. #448	C₂²×C₂₃	semidirect
297. #447	C₂²×C₂₃	semidirect
298. #432	C₂²×C₂₃	semidirect
299. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
300. C₂×#103	C₂²×C₂₃	semidirect
303. C₂×#97	C₂²×C₂₃	semidirect
307. C₂×C₂₃	#3213	direct (N × H)
335. C₂×SD₁₆	C₂³×C₂₃	semidirect
340. C₄⋊Q₈	C₂³×C₂₃	semidirect
342. #127	C₂³×C₂₃	semidirect
355. C₄×D₄	C₂³×C₂₃	semidirect
372. C₄⋊C₈	C₂³×C₂₃	semidirect
373. #103	C₂³×C₂₃	semidirect
376. #97	C₂³×C₂₃	semidirect
377. C₂₃	C₂×#3213	direct (N × H)
531. C₂	C₂₃×#3213	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
C₂²×C₂₃×#124	2
C₂₃×#3213	2
C₂×C₂₃×#496	2
C₂×C₂₃×#489	2
C₂×C₂₃×#463	2
C₂×C₂₃×#448	2
C₂×C₂₃×#447	2
C₂×C₂₃×#432	2
C₂×#3213	23

Subgroups — C2×C23×#3213 #19208409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3213 #19208409