Subgroups — C₂×C₂₃×#3218 #19208412

Order 5888.

2146 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3218	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#497	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#491	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#462	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#461	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#451	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#448	C₂	1	2	yes	maximal
10	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₂₃×C₄⋊Q₈	—	2	4	no	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₂₃×#529	—	2	4	no	—
21	C₂₃×#529	C₂²	1	8	yes	—
22	C₂×C₂₃×#126	—	2	1	no	—
23	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#123	—	2	1	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₂₃	—	2	1	no	—
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	—	2	1	no	—
32	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#497	C₂²	1	16	yes	complemented
36	C₂₃×#491	C₂²	1	16	yes	complemented
37	C₂₃×#462	C₂²	1	16	yes	complemented
38	C₂₃×#461	C₂²	1	16	yes	complemented
39	C₂₃×#451	C₂²	1	16	yes	complemented
40	C₂₃×#448	C₂²	1	16	yes	—
41	C₂₃×#432	C₂²	1	8	yes	complemented
42	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
43	C₂×C₂₃×#103	C₂²	1	4	yes	—
44	C₂×C₂₃×#103	C₂²	1	4	yes	—
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
46	C₂×C₂₃×#97	C₂²	1	2	yes	—
47	C₂²×Q₈×C₂₃	—	2	4	no	—
48	C₂²×Q₈×C₂₃	—	2	2	no	—
49	C₂²×Q₈×C₂₃	—	4	8	no	—
50	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	—	2	2	no	—
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₂₃	—	4	8	no	—
61	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
62	C₂×SD₁₆×C₂₃	—	4	8	no	—
63	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
65	C₂₃×C₄⋊Q₈	—	2	4	no	—
66	C₂₃×C₄⋊Q₈	—	2	4	no	—
67	C₂₃×C₄⋊Q₈	—	2	8	no	—
68	C₂₃×C₄⋊Q₈	—	2	8	no	—
69	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
70	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
71	C₂₃×#126	—	2	8	no	—
72	C₂₃×#125	—	2	2	no	—
73	C₂₃×#125	C₂³	1	4	yes	—
74	C₂₃×#124	—	2	2	no	—
75	C₂₃×#124	C₂³	1	4	yes	—
76	C₂₃×#123	—	2	8	no	—
77	C₂₃×C₂²⋊Q₈	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
79	C₂₃×#121	—	2	2	no	—
80	C₂₃×#121	C₂³	1	4	yes	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
83	C₄×D₄×C₂₃	—	2	8	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	C₂³	1	4	yes	—
86	C₂₃×#117	—	2	8	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄²×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	D₄	1	4	yes	—
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
111	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
112	C₂₃×#103	C₂³	1	16	yes	—
113	C₂₃×#103	C₂³	1	16	yes	—
114	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
115	C₂₃×#97	C₂³	1	8	yes	—
116	C₂⁴×C₂₃	—	2	1	no	—
117	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	—	2	8	no	—
122	C₂×Q₈×C₂₃	—	2	4	no	—
123	C₂×Q₈×C₂₃	—	2	8	no	—
124	C₂×Q₈×C₂₃	—	4	16	no	—
125	C₂×Q₈×C₂₃	—	4	8	no	—
126	C₂×Q₈×C₂₃	—	4	16	no	—
127	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
128	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	2	1	no	—
131	C₂×D₄×C₂₃	—	2	4	no	—
132	C₂×D₄×C₂₃	—	4	4	no	—
133	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
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	8	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	4	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	4	8	no	—
144	C₂²×C₄×C₂₃	—	4	4	no	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
155	Q₁₆×C₂₃	—	4	16	no	—
156	SD₁₆×C₂₃	—	4	16	no	—
157	C₂×C₈×C₂₃	—	2	8	no	—
158	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
159	C₄⋊C₄×C₂₃	—	2	8	no	—
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	8	no	—
162	C₄⋊C₄×C₂₃	—	2	8	no	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	2	4	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	—	2	4	no	—
173	C₂²⋊C₄×C₂₃	—	2	4	no	—
174	C₂²⋊C₄×C₂₃	—	2	4	no	—
175	C₂²⋊C₄×C₂₃	—	2	4	no	—
176	C₄²×C₂₃	—	2	4	no	—
177	C₄²×C₂₃	C₂×D₄	1	4	yes	—
178	C₄²×C₂₃	C₂×D₄	1	2	yes	—
179	C₄²×C₂₃	C₂×D₄	1	2	yes	—
180	C₄²×C₂₃	C₂×D₄	1	4	yes	—
181	C₄²×C₂₃	C₂⁴	1	2	yes	—
182	C₄²×C₂₃	C₂⁴	1	2	yes	—
183	C₂×#3218	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₂₃	—	2	2	no	—
185	C₂³×C₂₃	—	2	2	no	—
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	4	2	no	—
188	C₂³×C₂₃	—	4	2	no	—
189	C₂³×C₂₃	—	4	2	no	—
190	C₂³×C₂₃	#137	1	4	yes	—
191	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
192	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
193	Q₈×C₂₃	—	2	8	no	—
194	Q₈×C₂₃	—	2	4	no	—
195	Q₈×C₂₃	—	2	8	no	—
196	Q₈×C₂₃	—	4	16	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	8	no	—
199	D₄×C₂₃	—	2	4	no	—
200	D₄×C₂₃	—	4	8	no	—
201	D₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	8	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	8	no	—
209	C₂×C₄×C₂₃	—	2	4	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	8	no	—
214	C₂×C₄×C₂₃	—	4	16	no	—
215	C₂×C₄×C₂₃	—	4	8	no	—
216	C₂×C₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	4	2	no	—
218	C₂×C₄×C₂₃	—	4	2	no	—
219	C₂×C₄×C₂₃	—	4	8	no	—
220	C₂×C₄×C₂₃	#137	1	4	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
228	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
229	C₈×C₂₃	—	2	8	no	—
230	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
232	#3218	C₂×C₂₃	1	16	yes	direct product factor
233	C₂×#497	C₂×C₂₃	1	2	yes	complemented
234	C₂×#491	C₂×C₂₃	1	2	yes	complemented
235	C₂×#462	C₂×C₂₃	1	2	yes	complemented
236	C₂×#461	C₂×C₂₃	1	2	yes	complemented
237	C₂×#451	C₂×C₂₃	1	2	yes	complemented
238	C₂×#448	C₂×C₂₃	1	2	yes	—
239	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂²×C₂₃	—	2	2	no	—
241	C₂²×C₂₃	—	2	8	no	—
242	C₂²×C₂₃	—	2	2	no	—
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	—	4	2	no	—
246	C₂²×C₂₃	—	8	4	no	—
247	C₂²×C₂₃	#535	1	2	yes	—
248	C₂²×C₂₃	C₂×#137	1	4	yes	—
249	C₂²×C₂₃	C₂×#137	1	2	yes	—
250	C₂²×C₂₃	C₂×#137	1	2	yes	—
251	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₂₃	—	2	2	no	—
253	C₄×C₂₃	—	2	2	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	4	8	no	—
257	C₄×C₂₃	—	4	2	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	#535	1	2	yes	—
261	C₄×C₂₃	C₂×#137	1	4	yes	—
262	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
263	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
264	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
265	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
266	C₂×C₄⋊Q₈	—	2	1	no	—
267	C₂×C₄⋊Q₈	—	2	1	no	—
268	C₂×C₄⋊Q₈	—	2	4	no	—
269	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
271	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	complemented
272	#529	—	2	4	no	—
273	#529	C₂²×C₂₃	1	8	yes	—
274	C₂×#126	—	2	1	no	—
275	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×#123	—	2	1	no	—
278	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄×D₄	—	2	1	no	—
282	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×#117	—	2	1	no	—
284	C₂²×C₄⋊C₄	—	2	1	no	—
285	C₂²×C₄⋊C₄	—	2	1	no	—
286	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
287	#497	C₂²×C₂₃	1	16	yes	complemented
288	#491	C₂²×C₂₃	1	16	yes	complemented
289	#462	C₂²×C₂₃	1	16	yes	complemented
290	#461	C₂²×C₂₃	1	16	yes	complemented
291	#451	C₂²×C₂₃	1	16	yes	complemented
292	#448	C₂²×C₂₃	1	16	yes	—
293	#432	C₂²×C₂₃	1	8	yes	complemented
294	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
295	C₂×#103	C₂²×C₂₃	1	4	yes	—
296	C₂×#103	C₂²×C₂₃	1	4	yes	—
297	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
298	C₂×#97	C₂²×C₂₃	1	2	yes	—
299	C₂×C₂₃	—	2	4	no	—
300	C₂×C₂₃	—	8	2	no	—
301	C₂×C₂₃	#3218	1	4	yes	direct product factor
302	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
303	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
304	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
305	C₂²×Q₈	—	2	2	no	—
306	C₂²×Q₈	—	2	4	no	—
307	C₂²×Q₈	—	4	8	no	—
308	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
310	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂³×C₄	—	2	2	no	—
314	C₂³×C₄	—	2	1	no	—
315	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
316	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₂×Q₁₆	—	4	8	no	—
319	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
320	C₂×SD₁₆	—	4	8	no	—
321	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
322	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
323	C₄⋊Q₈	—	2	4	no	—
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	—	2	8	no	—
326	C₄⋊Q₈	—	2	8	no	—
327	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
328	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
329	#126	—	2	8	no	—
330	#125	—	2	2	no	—
331	#125	C₂³×C₂₃	1	4	yes	—
332	#124	—	2	2	no	—
333	#124	C₂³×C₂₃	1	4	yes	—
334	#123	—	2	8	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
337	#121	—	2	2	no	—
338	#121	C₂³×C₂₃	1	4	yes	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
341	C₄×D₄	—	2	8	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	C₂³×C₂₃	1	4	yes	—
344	#117	—	2	8	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	8	no	—
350	C₂×C₄⋊C₄	—	2	8	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₄²	—	2	1	no	—
363	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
364	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₂₃	1	4	yes	—
366	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
369	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
370	#103	C₂³×C₂₃	1	16	yes	—
371	#103	C₂³×C₂₃	1	16	yes	—
372	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
373	#97	C₂³×C₂₃	1	8	yes	—
374	C₂₃	C₂×#3218	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁴	—	2	1	no	—
376	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	8	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	8	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	8	no	—
383	C₂×Q₈	—	4	16	no	—
384	C₂×Q₈	—	4	8	no	—
385	C₂×Q₈	—	4	16	no	—
386	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
387	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
410	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
411	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
414	Q₁₆	—	4	16	no	—
415	SD₁₆	—	4	16	no	—
416	C₂×C₈	—	2	8	no	—
417	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
429	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₄²	—	2	4	no	—
436	C₄²	C₂×D₄×C₂₃	1	4	yes	—
437	C₄²	C₂×D₄×C₂₃	1	2	yes	—
438	C₄²	C₂×D₄×C₂₃	1	2	yes	—
439	C₄²	C₂×D₄×C₂₃	1	4	yes	—
440	C₄²	C₂⁴×C₂₃	1	2	yes	—
441	C₄²	C₂⁴×C₂₃	1	2	yes	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	C₂₃×#137	1	4	yes	—
449	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
450	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	2	8	no	—
454	Q₈	—	4	16	no	—
455	Q₈	—	4	8	no	—
456	Q₈	—	4	2	no	—
457	D₄	—	2	4	no	—
458	D₄	—	4	8	no	—
459	D₄	—	4	2	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	4	16	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	C₂₃×#137	1	4	yes	—
479	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
483	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
486	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
487	C₈	—	2	8	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	8	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	8	4	no	—
495	C₂²	C₂₃×#535	1	2	yes	—
496	C₂²	C₂×C₂₃×#137	1	4	yes	—
497	C₂²	C₂×C₂₃×#137	1	2	yes	—
498	C₂²	C₂×C₂₃×#137	1	2	yes	—
499	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	8	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	C₂₃×#535	1	2	yes	—
509	C₄	C₂×C₂₃×#137	1	4	yes	—
510	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
511	C₂	—	2	4	no	—
512	C₂	—	8	2	no	—
513	C₂	C₂₃×#3218	1	4	yes	direct product factor
514	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
515	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
516	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×#121	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. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#3218	C₂	direct (N × H)
4. C₂×C₂₃×#497	C₂	semidirect
5. C₂×C₂₃×#491	C₂	semidirect
6. C₂×C₂₃×#462	C₂	semidirect
7. C₂×C₂₃×#461	C₂	semidirect
8. C₂×C₂₃×#451	C₂	semidirect
10. C₂×C₂₃×#432	C₂	semidirect
19. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
35. C₂₃×#497	C₂²	semidirect
36. C₂₃×#491	C₂²	semidirect
37. C₂₃×#462	C₂²	semidirect
38. C₂₃×#461	C₂²	semidirect
39. C₂₃×#451	C₂²	semidirect
41. C₂₃×#432	C₂²	semidirect
183. C₂×#3218	C₂₃	direct (N × H)
230. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
232. #3218	C₂×C₂₃	direct (N × H)
233. C₂×#497	C₂×C₂₃	semidirect
234. C₂×#491	C₂×C₂₃	semidirect
235. C₂×#462	C₂×C₂₃	semidirect
236. C₂×#461	C₂×C₂₃	semidirect
237. C₂×#451	C₂×C₂₃	semidirect
239. C₂×#432	C₂×C₂₃	semidirect
271. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
287. #497	C₂²×C₂₃	semidirect
288. #491	C₂²×C₂₃	semidirect
289. #462	C₂²×C₂₃	semidirect
290. #461	C₂²×C₂₃	semidirect
291. #451	C₂²×C₂₃	semidirect
293. #432	C₂²×C₂₃	semidirect
301. C₂×C₂₃	#3218	direct (N × H)
374. C₂₃	C₂×#3218	direct (N × H)
513. C₂	C₂₃×#3218	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#529	2
C₂₃×#3218	2
C₂×C₂₃×#497	2
C₂×C₂₃×#491	2
C₂×C₂₃×#462	2
C₂×C₂₃×#461	2
C₂×C₂₃×#451	2
C₂×C₂₃×#448	2
C₂×C₂₃×#432	2
C₂×#3218	23

Subgroups — C2×C23×#3218 #19208412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3218 #19208412