Subgroups — C₂×C₂₃×#3077 #19208106

Order 5888.

2530 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3077	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#478	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#476	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#474	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#418	C₂	1	2	yes	maximal
10	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	—	2	1	no	—
12	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#533	—	2	4	no	—
14	C₂₃×#533	C₂²	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₂₃×#125	C₂²	1	2	yes	complemented
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	—	2	1	no	—
23	C₂×C₄×D₄×C₂₃	—	2	1	no	—
24	C₂×C₄×D₄×C₂₃	—	2	4	no	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
28	C₂×C₂₃×#117	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#478	C₂²	1	32	yes	complemented
33	C₂₃×#476	C₂²	1	16	yes	complemented
34	C₂₃×#474	C₂²	1	16	yes	complemented
35	C₂₃×#428	C₂²	1	16	yes	complemented
36	C₂×C₂₃×#107	C₂²	1	4	yes	complemented
37	C₂×C₂₃×#106	C₂²	1	4	yes	complemented
38	C₂₃×#423	C₂²	1	8	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
41	C₂₃×#418	C₂²	1	16	yes	—
42	C₂×C₂₃×#102	C₂²	1	4	yes	—
43	C₂×C₂₃×#102	C₂²	1	4	yes	—
44	C₂⁵×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	—	2	1	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	4	no	—
50	C₂²×D₄×C₂₃	Q₈	1	8	yes	complemented
51	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂³×C₄×C₂₃	—	2	1	no	—
54	C₂³×C₄×C₂₃	—	2	8	no	—
55	C₂³×C₄×C₂₃	—	2	2	no	—
56	C₂³×C₄×C₂₃	—	2	2	no	—
57	C₂³×C₄×C₂₃	—	4	8	no	—
58	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	—	2	8	no	—
62	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
67	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
68	C₂₃×#125	—	2	4	no	—
69	C₂₃×#125	C₂³	1	8	yes	complemented
70	C₂₃×C₂²⋊Q₈	—	2	8	no	—
71	C₂₃×C₂²⋊Q₈	—	2	8	no	—
72	C₄×Q₈×C₂₃	—	2	8	no	—
73	C₄×Q₈×C₂₃	—	2	8	no	—
74	C₄×D₄×C₂₃	—	2	8	no	—
75	C₄×D₄×C₂₃	—	2	8	no	—
76	C₄×D₄×C₂₃	—	2	4	no	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	C₂³	1	8	yes	—
79	C₄×D₄×C₂₃	C₂³	1	8	yes	—
80	C₂₃×#117	—	2	4	no	—
81	C₂₃×#117	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	2	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	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₄²×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	D₄	1	4	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
103	C₂₃×#107	C₂³	1	16	yes	complemented
104	C₂₃×#106	C₂³	1	16	yes	complemented
105	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
106	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
107	C₂₃×#102	C₂³	1	16	yes	—
108	C₂₃×#102	C₂³	1	16	yes	—
109	C₂⁴×C₂₃	—	2	8	no	—
110	C₂⁴×C₂₃	—	2	4	no	—
111	C₂⁴×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	4	4	no	—
113	C₂⁴×C₂₃	—	4	4	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₂₃	—	2	4	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	—	4	4	no	—
118	C₂×Q₈×C₂₃	—	4	4	no	—
119	C₂×D₄×C₂₃	—	2	16	no	—
120	C₂×D₄×C₂₃	—	2	8	no	—
121	C₂×D₄×C₂₃	—	2	4	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	C₂×Q₈	1	8	yes	—
126	C₂×D₄×C₂₃	C₂×Q₈	1	16	yes	complemented
127	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	2	no	—
132	C₂²×C₄×C₂₃	—	2	2	no	—
133	C₂²×C₄×C₂₃	—	2	8	no	—
134	C₂²×C₄×C₂₃	—	2	8	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	8	no	—
137	C₂²×C₄×C₂₃	—	4	32	no	—
138	C₂²×C₄×C₂₃	—	4	4	no	—
139	C₂²×C₄×C₂₃	—	4	8	no	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
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×Q₈	1	4	yes	—
149	C₂²×C₄×C₂₃	#41	1	4	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₂₃	—	2	16	no	—
153	C₂×C₈×C₂₃	—	2	8	no	—
154	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	8	no	—
158	C₄⋊C₄×C₂₃	—	2	8	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	2	8	no	—
168	C₂²⋊C₄×C₂₃	—	2	8	no	—
169	C₄²×C₂₃	—	2	8	no	—
170	C₄²×C₂₃	C₂×D₄	1	4	yes	—
171	C₄²×C₂₃	C₂×D₄	1	4	yes	—
172	C₄²×C₂₃	C₂×D₄	1	2	yes	—
173	C₄²×C₂₃	C₂×D₄	1	2	yes	—
174	C₄²×C₂₃	C₂⁴	1	2	yes	—
175	C₄²×C₂₃	C₂⁴	1	2	yes	—
176	C₂×#3077	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	8	no	—
180	C₂³×C₂₃	—	2	16	no	—
181	C₂³×C₂₃	—	2	4	no	—
182	C₂³×C₂₃	—	4	8	no	—
183	C₂³×C₂₃	—	4	32	no	—
184	C₂³×C₂₃	—	4	8	no	—
185	C₂³×C₂₃	—	4	4	no	—
186	C₂³×C₂₃	—	4	4	no	—
187	C₂³×C₂₃	C₂²⋊Q₈	1	4	yes	—
188	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
189	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₂₃	—	4	8	no	—
191	Q₈×C₂₃	—	4	8	no	—
192	D₄×C₂₃	—	2	8	no	—
193	D₄×C₂₃	—	2	16	no	—
194	D₄×C₂₃	—	2	4	no	—
195	D₄×C₂₃	C₂²×Q₈	1	8	yes	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	4	4	no	—
204	C₂×C₄×C₂₃	—	4	16	no	—
205	C₂×C₄×C₂₃	—	4	8	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	16	no	—
210	C₂×C₄×C₂₃	C₂²⋊Q₈	1	8	yes	—
211	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
212	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	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	2	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
220	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
221	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
222	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
223	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
224	C₈×C₂₃	—	2	8	no	—
225	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂²×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
227	#3077	C₂×C₂₃	1	16	yes	direct product factor
228	C₂×#478	C₂×C₂₃	1	4	yes	complemented
229	C₂×#476	C₂×C₂₃	1	2	yes	complemented
230	C₂×#474	C₂×C₂₃	1	2	yes	complemented
231	C₂×#428	C₂×C₂₃	1	2	yes	complemented
232	C₂×#423	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂×#418	C₂×C₂₃	1	2	yes	—
234	C₂²×C₂₃	—	2	8	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	2	no	—
237	C₂²×C₂₃	—	2	8	no	—
238	C₂²×C₂₃	—	4	32	no	—
239	C₂²×C₂₃	—	4	16	no	—
240	C₂²×C₂₃	—	4	4	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	4	yes	—
243	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
244	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
245	C₂²×C₂₃	#575	1	2	yes	—
246	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	2	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	—	4	4	no	—
253	C₄×C₂₃	C₂×C₂²⋊Q₈	1	4	yes	—
254	C₄×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
255	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
256	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
257	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
258	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
259	C₂⁴×C₄	—	2	1	no	—
260	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
261	#533	—	2	4	no	—
262	#533	C₂²×C₂₃	1	8	yes	complemented
263	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
264	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
265	C₂×#125	C₂²×C₂₃	1	2	yes	complemented
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×C₂²⋊Q₈	—	2	1	no	—
268	C₂×C₄×Q₈	—	2	1	no	—
269	C₂×C₄×Q₈	—	2	1	no	—
270	C₂×C₄×D₄	—	2	1	no	—
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	—	2	4	no	—
273	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
275	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
276	C₂×#117	C₂²×C₂₃	1	2	yes	—
277	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
278	C₂²×C₂²⋊C₄	—	2	1	no	—
279	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
280	#478	C₂²×C₂₃	1	32	yes	complemented
281	#476	C₂²×C₂₃	1	16	yes	complemented
282	#474	C₂²×C₂₃	1	16	yes	complemented
283	#428	C₂²×C₂₃	1	16	yes	complemented
284	C₂×#107	C₂²×C₂₃	1	4	yes	complemented
285	C₂×#106	C₂²×C₂₃	1	4	yes	complemented
286	#423	C₂²×C₂₃	1	8	yes	complemented
287	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
288	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
289	#418	C₂²×C₂₃	1	16	yes	—
290	C₂×#102	C₂²×C₂₃	1	4	yes	—
291	C₂×#102	C₂²×C₂₃	1	4	yes	—
292	C₂×C₂₃	—	2	4	no	—
293	C₂×C₂₃	—	4	8	no	—
294	C₂×C₂₃	#3077	1	4	yes	direct product factor
295	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
296	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
298	C₂⁵	—	2	1	no	—
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	—	2	4	no	—
304	C₂²×D₄	Q₈×C₂₃	1	8	yes	complemented
305	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
306	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	—	2	2	no	—
309	C₂³×C₄	—	2	8	no	—
310	C₂³×C₄	—	2	2	no	—
311	C₂³×C₄	—	4	8	no	—
312	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
313	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂×M₁₆	—	2	8	no	—
316	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
317	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
321	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
322	#125	—	2	4	no	—
323	#125	C₂³×C₂₃	1	8	yes	complemented
324	C₂²⋊Q₈	—	2	8	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₄×Q₈	—	2	8	no	—
327	C₄×Q₈	—	2	8	no	—
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	—	2	8	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	C₂³×C₂₃	1	8	yes	—
333	C₄×D₄	C₂³×C₂₃	1	8	yes	—
334	#117	—	2	4	no	—
335	#117	C₂³×C₂₃	1	8	yes	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	8	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₂₃	1	4	yes	—
354	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
357	#107	C₂³×C₂₃	1	16	yes	complemented
358	#106	C₂³×C₂₃	1	16	yes	complemented
359	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
360	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
361	#102	C₂³×C₂₃	1	16	yes	—
362	#102	C₂³×C₂₃	1	16	yes	—
363	C₂₃	C₂×#3077	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	2	4	no	—
367	C₂⁴	—	4	4	no	—
368	C₂⁴	—	4	4	no	—
369	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	4	4	no	—
373	C₂×Q₈	—	4	4	no	—
374	C₂×D₄	—	2	16	no	—
375	C₂×D₄	—	2	8	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	C₂×Q₈×C₂₃	1	8	yes	—
381	C₂×D₄	C₂×Q₈×C₂₃	1	16	yes	complemented
382	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	8	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	8	no	—
392	C₂²×C₄	—	4	32	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	8	no	—
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
400	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
403	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
404	C₂²×C₄	#41×C₂₃	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
407	M₁₆	—	2	16	no	—
408	C₂×C₈	—	2	8	no	—
409	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
419	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
420	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₄²	—	2	8	no	—
425	C₄²	C₂×D₄×C₂₃	1	4	yes	—
426	C₄²	C₂×D₄×C₂₃	1	4	yes	—
427	C₄²	C₂×D₄×C₂₃	1	2	yes	—
428	C₄²	C₂×D₄×C₂₃	1	2	yes	—
429	C₄²	C₂⁴×C₂₃	1	2	yes	—
430	C₄²	C₂⁴×C₂₃	1	2	yes	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	8	no	—
435	C₂³	—	2	16	no	—
436	C₂³	—	4	8	no	—
437	C₂³	—	4	8	no	—
438	C₂³	—	4	32	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	4	no	—
441	C₂³	C₂₃×C₂²⋊Q₈	1	4	yes	—
442	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
443	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
444	Q₈	—	4	8	no	—
445	Q₈	—	4	8	no	—
446	D₄	—	2	4	no	—
447	D₄	—	2	8	no	—
448	D₄	—	2	16	no	—
449	D₄	C₂²×Q₈×C₂₃	1	8	yes	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	8	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	—	4	8	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	16	no	—
464	C₂×C₄	C₂₃×C₂²⋊Q₈	1	8	yes	—
465	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
466	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
471	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
472	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
473	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
474	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
475	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
476	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
477	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
478	C₈	—	2	8	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	8	no	—
482	C₂²	—	2	8	no	—
483	C₂²	—	4	32	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	16	no	—
487	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	4	yes	—
488	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	2	yes	—
489	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
490	C₂²	C₂₃×#575	1	2	yes	—
491	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	4	no	—
498	C₄	C₂×C₂₃×C₂²⋊Q₈	1	4	yes	—
499	C₄	C₂×C₂₃×C₂²⋊Q₈	1	2	yes	—
500	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
501	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
502	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
503	C₂	—	2	4	no	—
504	C₂	—	4	8	no	—
505	C₂	C₂₃×#3077	1	4	yes	direct product factor
506	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
507	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
508	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
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×#533	C₂	semidirect
3. C₂₃×#3077	C₂	direct (N × H)
4. C₂×C₂₃×#478	C₂	semidirect
5. C₂×C₂₃×#476	C₂	semidirect
6. C₂×C₂₃×#474	C₂	semidirect
7. C₂×C₂₃×#428	C₂	semidirect
8. C₂×C₂₃×#423	C₂	semidirect
14. C₂₃×#533	C₂²	semidirect
15. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₂₃×#125	C₂²	semidirect
32. C₂₃×#478	C₂²	semidirect
33. C₂₃×#476	C₂²	semidirect
34. C₂₃×#474	C₂²	semidirect
35. C₂₃×#428	C₂²	semidirect
36. C₂×C₂₃×#107	C₂²	semidirect
37. C₂×C₂₃×#106	C₂²	semidirect
38. C₂₃×#423	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
50. C₂²×D₄×C₂₃	Q₈	semidirect
66. C₂₃×C₄⋊Q₈	C₂³	semidirect
67. C₂₃×C₄⋊Q₈	C₂³	semidirect
69. C₂₃×#125	C₂³	semidirect
103. C₂₃×#107	C₂³	semidirect
104. C₂₃×#106	C₂³	semidirect
105. C₂₃×C₄⋊C₈	C₂³	semidirect
106. C₂₃×C₄⋊C₈	C₂³	semidirect
126. C₂×D₄×C₂₃	C₂×Q₈	semidirect
176. C₂×#3077	C₂₃	direct (N × H)
225. C₂×#533	C₂×C₂₃	semidirect
227. #3077	C₂×C₂₃	direct (N × H)
228. C₂×#478	C₂×C₂₃	semidirect
229. C₂×#476	C₂×C₂₃	semidirect
230. C₂×#474	C₂×C₂₃	semidirect
231. C₂×#428	C₂×C₂₃	semidirect
232. C₂×#423	C₂×C₂₃	semidirect
262. #533	C₂²×C₂₃	semidirect
263. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
264. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
265. C₂×#125	C₂²×C₂₃	semidirect
280. #478	C₂²×C₂₃	semidirect
281. #476	C₂²×C₂₃	semidirect
282. #474	C₂²×C₂₃	semidirect
283. #428	C₂²×C₂₃	semidirect
284. C₂×#107	C₂²×C₂₃	semidirect
285. C₂×#106	C₂²×C₂₃	semidirect
286. #423	C₂²×C₂₃	semidirect
287. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
288. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
294. C₂×C₂₃	#3077	direct (N × H)
304. C₂²×D₄	Q₈×C₂₃	semidirect
320. C₄⋊Q₈	C₂³×C₂₃	semidirect
321. C₄⋊Q₈	C₂³×C₂₃	semidirect
323. #125	C₂³×C₂₃	semidirect
357. #107	C₂³×C₂₃	semidirect
358. #106	C₂³×C₂₃	semidirect
359. C₄⋊C₈	C₂³×C₂₃	semidirect
360. C₄⋊C₈	C₂³×C₂₃	semidirect
363. C₂₃	C₂×#3077	direct (N × H)
381. C₂×D₄	C₂×Q₈×C₂₃	semidirect
505. C₂	C₂₃×#3077	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×#533	2
C₂²×C₄×D₄×C₂₃	2
C₂₃×#3077	2
C₂×C₂₃×#478	2
C₂×C₂₃×#476	2
C₂×C₂₃×#474	2
C₂×C₂₃×#428	2
C₂×C₂₃×#423	2
C₂×C₂₃×#418	2
C₂×#3077	23

Subgroups — C2×C23×#3077 #19208106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3077 #19208106