Subgroups — C₂×C₂₃×#3112 #19208170

Order 5888.

2398 conjugacy classes of subgroups.

The subgroup lattice has 496 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
2	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3112	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#480	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#477	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#460	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#448	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#440	C₂	1	4	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	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂₃×#532	—	2	4	no	—
14	C₂₃×#532	C₂²	1	8	yes	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#124	—	2	1	no	—
18	C₂×C₂₃×#121	—	2	1	no	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	—	2	4	no	—
23	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
26	C₂×C₄×D₄×C₂₃	—	2	1	no	—
27	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₂₃	—	2	1	no	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#480	C₂²	1	16	yes	complemented
33	C₂₃×#477	C₂²	1	16	yes	complemented
34	C₂₃×#460	C₂²	1	16	yes	complemented
35	C₂₃×#448	C₂²	1	16	yes	complemented
36	C₂₃×#440	C₂²	1	32	yes	complemented
37	C₂₃×#432	C₂²	1	8	yes	complemented
38	C₂×C₂₃×#107	C₂²	1	4	yes	—
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂×C₂₃×#103	C₂²	1	4	yes	—
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	—
44	C₂×#41×C₂₃	—	2	8	no	—
45	C₂×#41×C₂₃	—	4	8	no	—
46	C₂²×Q₈×C₂₃	—	2	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	C₂³	1	4	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂³×C₄×C₂₃	—	2	1	no	—
58	C₂³×C₄×C₂₃	—	2	2	no	—
59	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₂₃	—	2	16	no	—
63	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₂₃×#127	—	2	2	no	—
68	C₂₃×#127	C₂³	1	4	yes	—
69	C₂₃×#124	—	2	8	no	—
70	C₂₃×#121	—	2	8	no	—
71	C₂₃×#121	—	2	4	no	—
72	C₂₃×#121	C₂³	1	8	yes	—
73	C₄×Q₈×C₂₃	—	2	4	no	—
74	C₄×Q₈×C₂₃	—	2	4	no	—
75	C₄×Q₈×C₂₃	—	2	8	no	—
76	C₄×Q₈×C₂₃	—	2	8	no	—
77	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
78	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
79	C₄×D₄×C₂₃	—	2	8	no	—
80	C₄×D₄×C₂₃	—	2	4	no	—
81	C₄×D₄×C₂₃	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₄²×C₂₃	—	2	2	no	—
94	C₂×C₄²×C₂₃	—	2	2	no	—
95	C₂×C₄²×C₂₃	—	2	8	no	—
96	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	D₄	1	4	yes	—
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
102	C₂₃×#107	C₂³	1	16	yes	—
103	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
104	C₂₃×#103	C₂³	1	16	yes	—
105	C₂₃×#102	C₂³	1	16	yes	—
106	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
107	C₂₃×#97	C₂³	1	8	yes	—
108	C₂⁴×C₂₃	—	2	2	no	—
109	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
110	#41×C₂₃	—	4	32	no	—
111	C₂×Q₈×C₂₃	—	2	16	no	—
112	C₂×Q₈×C₂₃	—	2	8	no	—
113	C₂×Q₈×C₂₃	—	2	16	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	—	2	4	no	—
118	C₂×Q₈×C₂₃	—	4	4	no	—
119	C₂×Q₈×C₂₃	#41	1	8	yes	complemented
120	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
121	C₂×D₄×C₂₃	—	2	8	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	8	no	—
126	C₂×D₄×C₂₃	—	4	4	no	—
127	C₂×D₄×C₂₃	—	4	8	no	—
128	C₂×D₄×C₂₃	—	4	4	no	—
129	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₂₃	—	2	8	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	8	no	—
140	C₂²×C₄×C₂₃	—	4	8	no	—
141	C₂²×C₄×C₂₃	—	4	4	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	SD₁₆×C₂₃	—	2	32	no	—
153	C₂×C₈×C₂₃	—	2	8	no	—
154	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
155	C₄⋊C₄×C₂₃	—	2	8	no	—
156	C₄⋊C₄×C₂₃	—	2	8	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	2	no	—
161	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	—	2	8	no	—
166	C₄²×C₂₃	—	2	8	no	—
167	C₄²×C₂₃	C₂×D₄	1	2	yes	—
168	C₄²×C₂₃	C₂×D₄	1	4	yes	—
169	C₄²×C₂₃	C₂×D₄	1	4	yes	—
170	C₄²×C₂₃	C₂×D₄	1	2	yes	—
171	C₄²×C₂₃	C₂⁴	1	2	yes	—
172	C₄²×C₂₃	C₂⁴	1	2	yes	—
173	C₂×#3112	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₂₃	—	2	2	no	—
175	C₂³×C₂₃	—	2	2	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	4	4	no	—
178	C₂³×C₂₃	—	4	4	no	—
179	C₂³×C₂₃	—	4	4	no	—
180	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
181	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
182	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₂₃	—	2	16	no	—
184	Q₈×C₂₃	—	2	8	no	—
185	Q₈×C₂₃	—	2	4	no	—
186	Q₈×C₂₃	—	4	8	no	—
187	Q₈×C₂₃	C₂×#41	1	8	yes	complemented
188	D₄×C₂₃	—	2	8	no	—
189	D₄×C₂₃	—	4	8	no	—
190	D₄×C₂₃	—	4	16	no	—
191	D₄×C₂₃	—	4	4	no	—
192	D₄×C₂₃	—	4	8	no	—
193	C₂×C₄×C₂₃	—	2	16	no	—
194	C₂×C₄×C₂₃	—	2	8	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	8	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	4	16	no	—
207	C₂×C₄×C₂₃	—	4	8	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	#137	1	4	yes	—
211	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
218	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
219	C₈×C₂₃	—	2	8	no	—
220	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
221	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#3112	C₂×C₂₃	1	16	yes	direct product factor
223	C₂×#480	C₂×C₂₃	1	2	yes	complemented
224	C₂×#477	C₂×C₂₃	1	2	yes	complemented
225	C₂×#460	C₂×C₂₃	1	2	yes	complemented
226	C₂×#448	C₂×C₂₃	1	2	yes	complemented
227	C₂×#440	C₂×C₂₃	1	4	yes	complemented
228	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂²×C₂₃	—	2	8	no	—
230	C₂²×C₂₃	—	2	2	no	—
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	4	4	no	—
233	C₂²×C₂₃	—	4	4	no	—
234	C₂²×C₂₃	—	4	4	no	—
235	C₂²×C₂₃	—	8	8	no	—
236	C₂²×C₂₃	#525	1	2	yes	—
237	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
238	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
239	C₂²×C₂₃	C₂×#137	1	2	yes	—
240	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₂₃	—	2	8	no	—
242	C₄×C₂₃	—	2	2	no	—
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	2	2	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	—	4	4	no	—
248	C₄×C₂₃	#525	1	2	yes	—
249	C₄×C₂₃	C₂×#137	1	4	yes	—
250	C₄×C₂₃	C₂²×#41	1	2	yes	—
251	C₂²×#41	—	2	1	no	—
252	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
254	#532	—	2	4	no	—
255	#532	C₂²×C₂₃	1	8	yes	—
256	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×#124	—	2	1	no	—
259	C₂×#121	—	2	1	no	—
260	C₂×#121	C₂²×C₂₃	1	2	yes	—
261	C₂×C₄×Q₈	—	2	1	no	—
262	C₂×C₄×Q₈	—	2	1	no	—
263	C₂×C₄×Q₈	—	2	4	no	—
264	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
265	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
266	C₂×C₄×Q₈	C₂²×C₂₃	1	8	yes	complemented
267	C₂×C₄×D₄	—	2	1	no	—
268	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
269	C₂²×C₄⋊C₄	—	2	1	no	—
270	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
271	C₂²×C₄²	—	2	1	no	—
272	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
273	#480	C₂²×C₂₃	1	16	yes	complemented
274	#477	C₂²×C₂₃	1	16	yes	complemented
275	#460	C₂²×C₂₃	1	16	yes	complemented
276	#448	C₂²×C₂₃	1	16	yes	complemented
277	#440	C₂²×C₂₃	1	32	yes	complemented
278	#432	C₂²×C₂₃	1	8	yes	complemented
279	C₂×#107	C₂²×C₂₃	1	4	yes	—
280	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
281	C₂×#103	C₂²×C₂₃	1	4	yes	—
282	C₂×#102	C₂²×C₂₃	1	4	yes	—
283	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
284	C₂×#97	C₂²×C₂₃	1	2	yes	—
285	C₂×C₂₃	—	2	4	no	—
286	C₂×C₂₃	—	8	4	no	—
287	C₂×C₂₃	#3112	1	4	yes	direct product factor
288	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
289	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
291	C₂×#41	—	2	8	no	—
292	C₂×#41	—	4	8	no	—
293	C₂²×Q₈	—	2	8	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
298	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
307	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
309	C₂×SD₁₆	—	2	16	no	—
310	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
311	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
314	#127	—	2	2	no	—
315	#127	C₂³×C₂₃	1	4	yes	—
316	#124	—	2	8	no	—
317	#121	—	2	8	no	—
318	#121	—	2	4	no	—
319	#121	C₂³×C₂₃	1	8	yes	—
320	C₄×Q₈	—	2	4	no	—
321	C₄×Q₈	—	2	4	no	—
322	C₄×Q₈	—	2	8	no	—
323	C₄×Q₈	—	2	8	no	—
324	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
325	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
326	C₄×D₄	—	2	8	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	C₂³×C₂₃	1	8	yes	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	—	2	8	no	—
343	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
344	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
345	C₂×C₄²	D₄×C₂₃	1	4	yes	—
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
349	#107	C₂³×C₂₃	1	16	yes	—
350	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
351	#103	C₂³×C₂₃	1	16	yes	—
352	#102	C₂³×C₂₃	1	16	yes	—
353	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
354	#97	C₂³×C₂₃	1	8	yes	—
355	C₂₃	C₂×#3112	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	2	no	—
357	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
358	#41	—	4	32	no	—
359	C₂×Q₈	—	2	16	no	—
360	C₂×Q₈	—	2	8	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×Q₈	#41×C₂₃	1	8	yes	complemented
368	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	4	8	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
400	SD₁₆	—	2	32	no	—
401	C₂×C₈	—	2	8	no	—
402	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
410	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₄²	—	2	8	no	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×D₄×C₂₃	1	2	yes	—
416	C₄²	C₂×D₄×C₂₃	1	4	yes	—
417	C₄²	C₂×D₄×C₂₃	1	4	yes	—
418	C₄²	C₂×D₄×C₂₃	1	2	yes	—
419	C₄²	C₂⁴×C₂₃	1	2	yes	—
420	C₄²	C₂⁴×C₂₃	1	2	yes	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	4	4	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
428	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
430	Q₈	—	2	16	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	2	8	no	—
433	Q₈	—	4	8	no	—
434	Q₈	C₂×#41×C₂₃	1	8	yes	complemented
435	D₄	—	2	8	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	16	no	—
438	D₄	—	4	8	no	—
439	D₄	—	4	4	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	C₂₃×#137	1	4	yes	—
458	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
465	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
466	C₈	—	2	8	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	8	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	8	no	—
474	C₂²	C₂₃×#525	1	2	yes	—
475	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
476	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
477	C₂²	C₂×C₂₃×#137	1	2	yes	—
478	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
479	C₄	—	2	8	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	C₂₃×#525	1	2	yes	—
487	C₄	C₂×C₂₃×#137	1	4	yes	—
488	C₄	C₂²×#41×C₂₃	1	2	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	8	4	no	—
491	C₂	C₂₃×#3112	1	4	yes	direct product factor
492	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
493	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
494	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#532	C₂
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
2. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#3112	C₂	direct (N × H)
4. C₂×C₂₃×#480	C₂	semidirect
5. C₂×C₂₃×#477	C₂	semidirect
6. C₂×C₂₃×#460	C₂	semidirect
7. C₂×C₂₃×#448	C₂	semidirect
8. C₂×C₂₃×#440	C₂	semidirect
9. C₂×C₂₃×#432	C₂	semidirect
25. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
32. C₂₃×#480	C₂²	semidirect
33. C₂₃×#477	C₂²	semidirect
34. C₂₃×#460	C₂²	semidirect
35. C₂₃×#448	C₂²	semidirect
36. C₂₃×#440	C₂²	semidirect
37. C₂₃×#432	C₂²	semidirect
119. C₂×Q₈×C₂₃	#41	semidirect
173. C₂×#3112	C₂₃	direct (N × H)
187. Q₈×C₂₃	C₂×#41	semidirect
221. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
222. #3112	C₂×C₂₃	direct (N × H)
223. C₂×#480	C₂×C₂₃	semidirect
224. C₂×#477	C₂×C₂₃	semidirect
225. C₂×#460	C₂×C₂₃	semidirect
226. C₂×#448	C₂×C₂₃	semidirect
227. C₂×#440	C₂×C₂₃	semidirect
228. C₂×#432	C₂×C₂₃	semidirect
266. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
273. #480	C₂²×C₂₃	semidirect
274. #477	C₂²×C₂₃	semidirect
275. #460	C₂²×C₂₃	semidirect
276. #448	C₂²×C₂₃	semidirect
277. #440	C₂²×C₂₃	semidirect
278. #432	C₂²×C₂₃	semidirect
287. C₂×C₂₃	#3112	direct (N × H)
355. C₂₃	C₂×#3112	direct (N × H)
367. C₂×Q₈	#41×C₂₃	semidirect
434. Q₈	C₂×#41×C₂₃	semidirect
491. C₂	C₂₃×#3112	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×C₄×Q₈×C₂₃	2
C₂₃×#3112	2
C₂×C₂₃×#480	2
C₂×C₂₃×#477	2
C₂×C₂₃×#460	2
C₂×C₂₃×#448	2
C₂×C₂₃×#440	2
C₂×C₂₃×#432	2
C₂×#3112	23

Subgroups — C2×C23×#3112 #19208170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3112 #19208170