Subgroups — C₂×C₂₃×#3206 #19208403

Order 5888.

2414 conjugacy classes of subgroups.

The subgroup lattice has 448 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×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3206	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#492	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#487	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#461	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#448	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₂₃	—	2	1	no	—
10	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₂₃	C₂²	1	8	yes	—
12	C₂₃×#532	—	2	4	no	—
13	C₂₃×#532	C₂²	1	8	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	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
21	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#124	—	2	1	no	—
23	C₂×C₂₃×#121	—	2	1	no	—
24	C₂×C₂₃×#121	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₂₃	—	2	1	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#492	C₂²	1	16	yes	complemented
31	C₂₃×#487	C₂²	1	16	yes	complemented
32	C₂₃×#461	C₂²	1	32	yes	complemented
33	C₂₃×#448	C₂²	1	32	yes	complemented
34	C₂₃×#431	C₂²	1	8	yes	complemented
35	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂×C₂₃×#103	C₂²	1	8	yes	—
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
39	C₂×#41×C₂₃	—	2	8	no	—
40	C₂×#41×C₂₃	—	4	8	no	—
41	C₂²×Q₈×C₂₃	—	2	4	no	—
42	C₂²×Q₈×C₂₃	—	2	2	no	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	—	4	8	no	—
45	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
46	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	—	2	2	no	—
53	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×SD₁₆×C₂₃	—	4	16	no	—
57	C₂×SD₁₆×C₂₃	C₂³	1	32	yes	—
58	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
59	C₂₃×C₄⋊Q₈	—	2	4	no	—
60	C₂₃×C₄⋊Q₈	—	2	4	no	—
61	C₂₃×C₄⋊Q₈	—	2	8	no	—
62	C₂₃×C₄⋊Q₈	—	2	8	no	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
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×D₄×C₂₃	—	2	8	no	—
74	C₄×D₄×C₂₃	—	2	4	no	—
75	C₄×D₄×C₂₃	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	D₄	1	4	yes	—
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
92	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
93	C₂₃×#103	C₂³	1	32	yes	—
94	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
95	C₄×C₈×C₂₃	C₂³	1	8	yes	—
96	C₂⁴×C₂₃	—	2	2	no	—
97	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
98	#41×C₂₃	—	4	32	no	—
99	C₂×Q₈×C₂₃	—	2	4	no	—
100	C₂×Q₈×C₂₃	—	2	8	no	—
101	C₂×Q₈×C₂₃	—	2	4	no	—
102	C₂×Q₈×C₂₃	—	2	16	no	—
103	C₂×Q₈×C₂₃	—	4	16	no	—
104	C₂×Q₈×C₂₃	—	4	8	no	—
105	C₂×Q₈×C₂₃	—	4	16	no	—
106	C₂×Q₈×C₂₃	—	4	4	no	—
107	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
108	C₂×D₄×C₂₃	—	2	8	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	—	2	4	no	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	8	no	—
113	C₂×D₄×C₂₃	—	4	8	no	—
114	C₂×D₄×C₂₃	—	4	4	no	—
115	C₂×D₄×C₂₃	—	4	4	no	—
116	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₂₃	—	2	4	no	—
118	C₂²×C₄×C₂₃	—	2	4	no	—
119	C₂²×C₄×C₂₃	—	2	8	no	—
120	C₂²×C₄×C₂₃	—	2	2	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	2	8	no	—
123	C₂²×C₄×C₂₃	—	4	8	no	—
124	C₂²×C₄×C₂₃	—	4	8	no	—
125	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	complemented
126	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₂₃	—	4	32	no	—
137	C₂×C₈×C₂₃	—	2	8	no	—
138	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
139	C₄⋊C₄×C₂₃	—	2	8	no	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	8	no	—
142	C₄⋊C₄×C₂₃	—	2	8	no	—
143	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
146	C₂²⋊C₄×C₂₃	—	2	8	no	—
147	C₂²⋊C₄×C₂₃	—	2	8	no	—
148	C₄²×C₂₃	C₂×D₄	1	4	yes	—
149	C₄²×C₂₃	C₂×D₄	1	2	yes	—
150	C₄²×C₂₃	C₂×D₄	1	4	yes	—
151	C₄²×C₂₃	C₂×D₄	1	2	yes	—
152	C₄²×C₂₃	C₂⁴	1	2	yes	—
153	C₄²×C₂₃	C₂⁴	1	2	yes	—
154	C₂×#3206	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₂₃	—	2	2	no	—
156	C₂³×C₂₃	—	2	2	no	—
157	C₂³×C₂₃	—	2	4	no	—
158	C₂³×C₂₃	—	4	4	no	—
159	C₂³×C₂₃	—	4	4	no	—
160	C₂³×C₂₃	—	4	4	no	—
161	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
162	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
163	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₂₃	—	2	16	no	—
165	Q₈×C₂₃	—	4	16	no	—
166	Q₈×C₂₃	—	4	8	no	—
167	Q₈×C₂₃	—	4	8	no	—
168	D₄×C₂₃	—	2	8	no	—
169	D₄×C₂₃	—	4	16	no	—
170	D₄×C₂₃	—	4	4	no	—
171	D₄×C₂₃	—	4	8	no	—
172	D₄×C₂₃	—	4	8	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	4	no	—
176	C₂×C₄×C₂₃	—	2	8	no	—
177	C₂×C₄×C₂₃	—	2	4	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	8	no	—
180	C₂×C₄×C₂₃	—	4	16	no	—
181	C₂×C₄×C₂₃	—	4	8	no	—
182	C₂×C₄×C₂₃	—	4	4	no	—
183	C₂×C₄×C₂₃	—	4	16	no	—
184	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
185	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
186	C₂×C₄×C₂₃	C₂×SD₁₆	1	16	yes	complemented
187	C₂×C₄×C₂₃	#137	1	4	yes	—
188	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
192	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
193	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
195	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
196	C₈×C₂₃	—	2	8	no	—
197	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
198	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#3206	C₂×C₂₃	1	16	yes	direct product factor
200	C₂×#492	C₂×C₂₃	1	2	yes	complemented
201	C₂×#487	C₂×C₂₃	1	2	yes	complemented
202	C₂×#461	C₂×C₂₃	1	4	yes	complemented
203	C₂×#448	C₂×C₂₃	1	4	yes	complemented
204	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
205	C₂²×C₂₃	—	2	2	no	—
206	C₂²×C₂₃	—	2	8	no	—
207	C₂²×C₂₃	—	2	2	no	—
208	C₂²×C₂₃	—	4	4	no	—
209	C₂²×C₂₃	—	4	4	no	—
210	C₂²×C₂₃	—	4	4	no	—
211	C₂²×C₂₃	—	8	8	no	—
212	C₂²×C₂₃	#535	1	2	yes	—
213	C₂²×C₂₃	C₂²×SD₁₆	1	4	yes	—
214	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
215	C₂²×C₂₃	C₂×#137	1	2	yes	—
216	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
217	C₄×C₂₃	—	2	2	no	—
218	C₄×C₂₃	—	2	2	no	—
219	C₄×C₂₃	—	4	8	no	—
220	C₄×C₂₃	—	4	4	no	—
221	C₄×C₂₃	#535	1	2	yes	—
222	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
223	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
224	C₄×C₂₃	C₂×#137	1	4	yes	—
225	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
226	C₂²×#41	—	2	1	no	—
227	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
228	C₂²×SD₁₆	C₂²×C₂₃	1	8	yes	—
229	#532	—	2	4	no	—
230	#532	C₂²×C₂₃	1	8	yes	—
231	C₂×C₄⋊Q₈	—	2	1	no	—
232	C₂×C₄⋊Q₈	—	2	1	no	—
233	C₂×C₄⋊Q₈	—	2	4	no	—
234	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
235	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
236	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
237	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	complemented
238	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
239	C₂×#124	—	2	1	no	—
240	C₂×#121	—	2	1	no	—
241	C₂×#121	C₂²×C₂₃	1	2	yes	—
242	C₂×C₄×D₄	—	2	1	no	—
243	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
244	C₂²×C₄⋊C₄	—	2	1	no	—
245	C₂²×C₄⋊C₄	—	2	1	no	—
246	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
247	#492	C₂²×C₂₃	1	16	yes	complemented
248	#487	C₂²×C₂₃	1	16	yes	complemented
249	#461	C₂²×C₂₃	1	32	yes	complemented
250	#448	C₂²×C₂₃	1	32	yes	complemented
251	#431	C₂²×C₂₃	1	8	yes	complemented
252	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
253	C₂×#103	C₂²×C₂₃	1	8	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
256	C₂×C₂₃	—	2	4	no	—
257	C₂×C₂₃	—	8	4	no	—
258	C₂×C₂₃	#3206	1	4	yes	direct product factor
259	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
260	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
261	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
262	C₂×#41	—	2	8	no	—
263	C₂×#41	—	4	8	no	—
264	C₂²×Q₈	—	2	1	no	—
265	C₂²×Q₈	—	2	2	no	—
266	C₂²×Q₈	—	2	4	no	—
267	C₂²×Q₈	—	4	8	no	—
268	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
269	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
270	C₂²×D₄	—	2	2	no	—
271	C₂²×D₄	—	2	1	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
274	C₂³×C₄	—	2	2	no	—
275	C₂³×C₄	—	2	2	no	—
276	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
277	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
279	C₂×SD₁₆	—	4	16	no	—
280	C₂×SD₁₆	C₂³×C₂₃	1	32	yes	—
281	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
282	C₄⋊Q₈	—	2	4	no	—
283	C₄⋊Q₈	—	2	4	no	—
284	C₄⋊Q₈	—	2	8	no	—
285	C₄⋊Q₈	—	2	8	no	—
286	C₄⋊Q₈	—	2	2	no	—
287	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
288	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
289	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
290	#127	—	2	2	no	—
291	#127	C₂³×C₂₃	1	4	yes	—
292	#124	—	2	8	no	—
293	#121	—	2	8	no	—
294	#121	—	2	4	no	—
295	#121	C₂³×C₂₃	1	8	yes	—
296	C₄×D₄	—	2	8	no	—
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	C₂³×C₂₃	1	8	yes	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	8	no	—
304	C₂×C₄⋊C₄	—	2	8	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
310	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
311	C₂×C₄²	D₄×C₂₃	1	4	yes	—
312	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
315	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
316	#103	C₂³×C₂₃	1	32	yes	—
317	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
318	C₄×C₈	C₂³×C₂₃	1	8	yes	—
319	C₂₃	C₂×#3206	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁴	—	2	2	no	—
321	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
322	#41	—	4	32	no	—
323	C₂×Q₈	—	2	4	no	—
324	C₂×Q₈	—	2	8	no	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	2	16	no	—
327	C₂×Q₈	—	4	16	no	—
328	C₂×Q₈	—	4	8	no	—
329	C₂×Q₈	—	4	16	no	—
330	C₂×Q₈	—	4	4	no	—
331	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	4	8	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	—	4	8	no	—
349	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	complemented
350	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
354	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
357	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
359	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
360	SD₁₆	—	4	32	no	—
361	C₂×C₈	—	2	8	no	—
362	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
363	C₄⋊C₄	—	2	8	no	—
364	C₄⋊C₄	—	2	8	no	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₂²⋊C₄	—	2	8	no	—
372	C₄²	C₂×D₄×C₂₃	1	4	yes	—
373	C₄²	C₂×D₄×C₂₃	1	2	yes	—
374	C₄²	C₂×D₄×C₂₃	1	4	yes	—
375	C₄²	C₂×D₄×C₂₃	1	2	yes	—
376	C₄²	C₂⁴×C₂₃	1	2	yes	—
377	C₄²	C₂⁴×C₂₃	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	4	4	no	—
382	C₂³	—	4	4	no	—
383	C₂³	—	4	4	no	—
384	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
385	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
386	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
387	Q₈	—	2	16	no	—
388	Q₈	—	4	16	no	—
389	Q₈	—	4	8	no	—
390	Q₈	—	4	8	no	—
391	D₄	—	2	8	no	—
392	D₄	—	4	16	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	8	no	—
395	D₄	—	4	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	8	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	4	16	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
408	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
409	C₂×C₄	C₂×SD₁₆×C₂₃	1	16	yes	complemented
410	C₂×C₄	C₂₃×#137	1	4	yes	—
411	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
416	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
418	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
419	C₈	—	2	8	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	8	8	no	—
427	C₂²	C₂₃×#535	1	2	yes	—
428	C₂²	C₂²×SD₁₆×C₂₃	1	4	yes	—
429	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
430	C₂²	C₂×C₂₃×#137	1	2	yes	—
431	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	4	8	no	—
435	C₄	—	4	4	no	—
436	C₄	C₂₃×#535	1	2	yes	—
437	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
438	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
439	C₄	C₂×C₂₃×#137	1	4	yes	—
440	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
441	C₂	—	2	4	no	—
442	C₂	—	8	4	no	—
443	C₂	C₂₃×#3206	1	4	yes	direct product factor
444	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
445	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
446	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×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#3206	C₂	direct (N × H)
4. C₂×C₂₃×#492	C₂	semidirect
5. C₂×C₂₃×#487	C₂	semidirect
6. C₂×C₂₃×#461	C₂	semidirect
7. C₂×C₂₃×#448	C₂	semidirect
8. C₂×C₂₃×#431	C₂	semidirect
20. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
30. C₂₃×#492	C₂²	semidirect
31. C₂₃×#487	C₂²	semidirect
32. C₂₃×#461	C₂²	semidirect
33. C₂₃×#448	C₂²	semidirect
34. C₂₃×#431	C₂²	semidirect
125. C₂²×C₄×C₂₃	SD₁₆	semidirect
154. C₂×#3206	C₂₃	direct (N × H)
186. C₂×C₄×C₂₃	C₂×SD₁₆	semidirect
198. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
199. #3206	C₂×C₂₃	direct (N × H)
200. C₂×#492	C₂×C₂₃	semidirect
201. C₂×#487	C₂×C₂₃	semidirect
202. C₂×#461	C₂×C₂₃	semidirect
203. C₂×#448	C₂×C₂₃	semidirect
204. C₂×#431	C₂×C₂₃	semidirect
237. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
247. #492	C₂²×C₂₃	semidirect
248. #487	C₂²×C₂₃	semidirect
249. #461	C₂²×C₂₃	semidirect
250. #448	C₂²×C₂₃	semidirect
251. #431	C₂²×C₂₃	semidirect
258. C₂×C₂₃	#3206	direct (N × H)
319. C₂₃	C₂×#3206	direct (N × H)
349. C₂²×C₄	SD₁₆×C₂₃	semidirect
409. C₂×C₄	C₂×SD₁₆×C₂₃	semidirect
443. C₂	C₂₃×#3206	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×C₄⋊Q₈	2
C₂₃×#3206	2
C₂×C₂₃×#492	2
C₂×C₂₃×#487	2
C₂×C₂₃×#461	2
C₂×C₂₃×#448	2
C₂×C₂₃×#431	2
C₂×#3206	23

Subgroups — C2×C23×#3206 #19208403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3206 #19208403