Subgroups — C₂×C₂₃×#3287 #19208673

Order 5888.

2910 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#550	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3287	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#466	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#460	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#449	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	4	yes	—
11	C₂×C₂₃×#135	—	2	16	no	—
12	C₂×C₂₃×#135	C₂²	1	16	yes	complemented
13	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
15	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#550	—	2	8	no	—
19	C₂₃×#550	C₂²	1	16	yes	complemented
20	C₂×C₂₃×#127	—	2	2	no	—
21	C₂×C₂₃×#127	C₂²	1	2	yes	—
22	C₂×C₂₃×#121	—	2	4	no	—
23	C₂×C₂₃×#121	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₂₃	—	2	2	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#493	C₂²	1	8	yes	complemented
28	C₂₃×#466	C₂²	1	16	yes	complemented
29	C₂₃×#460	C₂²	1	16	yes	complemented
30	C₂₃×#449	C₂²	1	32	yes	complemented
31	C₄×Q₁₆×C₂₃	C₂²	1	8	yes	complemented
32	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
33	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#103	C₂²	1	2	yes	—
36	C₂×C₂₃×#102	C₂²	1	4	yes	—
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂×#41×C₂₃	—	2	8	no	—
40	C₂×#41×C₂₃	—	2	16	no	—
41	C₂×#41×C₂₃	—	4	8	no	—
42	C₂×#41×C₂₃	C₂³	1	16	yes	—
43	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
44	C₂²×D₄×C₂₃	—	2	4	no	—
45	C₂²×D₄×C₂₃	—	2	4	no	—
46	C₂²×D₄×C₂₃	—	2	2	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₂₃	—	2	4	no	—
51	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂₃×#135	—	2	64	no	—
53	C₂×Q₁₆×C₂₃	D₄	1	8	yes	complemented
54	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
55	C₂×SD₁₆×C₂₃	—	2	16	no	—
56	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
57	C₂×D₈×C₂₃	—	2	8	no	—
58	C₂×D₈×C₂₃	—	4	4	no	—
59	C₂×D₈×C₂₃	C₂³	1	8	yes	—
60	C₂×D₈×C₂₃	C₂³	1	8	yes	—
61	C₂²×C₈×C₂₃	—	2	8	no	—
62	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×#127	—	2	16	no	—
67	C₂₃×#127	—	2	4	no	—
68	C₂₃×#127	C₂³	1	8	yes	—
69	C₂₃×#121	—	2	8	no	—
70	C₂₃×#121	—	2	32	no	—
71	C₂₃×#121	C₂³	1	16	yes	—
72	C₄×Q₈×C₂₃	—	2	4	no	—
73	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
74	C₄×D₄×C₂₃	—	2	16	no	—
75	C₄×D₄×C₂₃	D₄	1	4	yes	—
76	C₄×D₄×C₂₃	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
82	C₂×C₄²×C₂₃	—	2	2	no	—
83	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
84	C₂₃×#107	C₂³	1	4	yes	—
85	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
86	C₂₃×#103	C₂³	1	8	yes	—
87	C₂₃×#102	C₂³	1	16	yes	—
88	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
89	C₄×C₈×C₂₃	C₂³	1	4	yes	—
90	C₂⁴×C₂₃	—	2	4	no	—
91	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
92	#41×C₂₃	—	2	32	no	—
93	#41×C₂₃	—	4	16	no	—
94	#41×C₂₃	—	4	32	no	—
95	C₂×Q₈×C₂₃	—	2	2	no	—
96	C₂×Q₈×C₂₃	—	2	4	no	—
97	C₂×Q₈×C₂₃	C₂×D₄	1	8	yes	—
98	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
99	C₂×D₄×C₂₃	—	2	16	no	—
100	C₂×D₄×C₂₃	—	2	16	no	—
101	C₂×D₄×C₂₃	—	2	4	no	—
102	C₂×D₄×C₂₃	—	2	8	no	—
103	C₂×D₄×C₂₃	—	2	8	no	—
104	C₂×D₄×C₂₃	—	2	4	no	—
105	C₂×D₄×C₂₃	—	2	8	no	—
106	C₂×D₄×C₂₃	—	2	8	no	—
107	C₂×D₄×C₂₃	—	4	16	no	—
108	C₂×D₄×C₂₃	—	4	8	no	—
109	C₂×D₄×C₂₃	—	8	4	no	—
110	C₂×D₄×C₂₃	—	8	8	no	—
111	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
112	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
113	C₂²×C₄×C₂₃	—	2	2	no	—
114	C₂²×C₄×C₂₃	—	2	4	no	—
115	C₂²×C₄×C₂₃	—	2	4	no	—
116	C₂²×C₄×C₂₃	—	2	2	no	—
117	C₂²×C₄×C₂₃	—	2	16	no	—
118	C₂²×C₄×C₂₃	—	4	16	no	—
119	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₂₃	C₂×D₄	1	16	yes	complemented
126	SD₁₆×C₂₃	—	2	32	no	—
127	D₈×C₂₃	—	2	16	no	—
128	D₈×C₂₃	—	4	8	no	—
129	C₂×C₈×C₂₃	—	2	4	no	—
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₂×C₈×C₂₃	—	2	16	no	—
132	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
133	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
134	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
135	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
136	C₄⋊C₄×C₂₃	—	2	8	no	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
139	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
140	C₂²⋊C₄×C₂₃	—	2	16	no	—
141	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
142	C₄²×C₂₃	—	2	8	no	—
143	C₄²×C₂₃	C₂×D₄	1	2	yes	—
144	C₄²×C₂₃	C₂⁴	1	2	yes	—
145	C₂×#3287	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₂₃	—	2	4	no	—
147	C₂³×C₂₃	—	2	4	no	—
148	C₂³×C₂₃	—	2	4	no	—
149	C₂³×C₂₃	—	2	8	no	—
150	C₂³×C₂₃	—	4	8	no	—
151	C₂³×C₂₃	—	4	8	no	—
152	C₂³×C₂₃	—	4	8	no	—
153	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
154	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₂₃	—	2	8	no	—
156	Q₈×C₂₃	—	2	4	no	—
157	Q₈×C₂₃	C₂²×D₄	1	8	yes	—
158	D₄×C₂₃	—	2	8	no	—
159	D₄×C₂₃	—	2	8	no	—
160	D₄×C₂₃	—	2	8	no	—
161	D₄×C₂₃	—	4	32	no	—
162	D₄×C₂₃	—	4	8	no	—
163	D₄×C₂₃	—	4	16	no	—
164	D₄×C₂₃	—	8	8	no	—
165	D₄×C₂₃	—	8	16	no	—
166	C₂×C₄×C₂₃	—	2	4	no	—
167	C₂×C₄×C₂₃	—	2	8	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	4	no	—
170	C₂×C₄×C₂₃	—	2	8	no	—
171	C₂×C₄×C₂₃	—	2	4	no	—
172	C₂×C₄×C₂₃	—	2	4	no	—
173	C₂×C₄×C₂₃	—	2	4	no	—
174	C₂×C₄×C₂₃	—	4	32	no	—
175	C₂×C₄×C₂₃	—	4	4	no	—
176	C₂×C₄×C₂₃	#135	1	4	yes	complemented
177	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
183	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
184	C₈×C₂₃	—	2	4	no	—
185	C₈×C₂₃	—	2	2	no	—
186	C₈×C₂₃	C₂²×D₄	1	4	yes	—
187	C₂²×#135	C₂×C₂₃	1	2	yes	complemented
188	C₂×#550	C₂×C₂₃	1	2	yes	complemented
189	#3287	C₂×C₂₃	1	16	yes	direct product factor
190	C₂×#493	C₂×C₂₃	1	1	yes	characteristic, complemented
191	C₂×#466	C₂×C₂₃	1	2	yes	complemented
192	C₂×#460	C₂×C₂₃	1	2	yes	complemented
193	C₂×#449	C₂×C₂₃	1	4	yes	complemented
194	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
195	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
196	C₂²×C₂₃	—	2	4	no	—
197	C₂²×C₂₃	—	2	4	no	—
198	C₂²×C₂₃	—	2	4	no	—
199	C₂²×C₂₃	—	4	8	no	—
200	C₂²×C₂₃	—	4	8	no	—
201	C₂²×C₂₃	—	4	8	no	—
202	C₂²×C₂₃	—	4	8	no	—
203	C₂²×C₂₃	—	8	16	no	—
204	C₂²×C₂₃	D₄²	1	2	yes	—
205	C₂²×C₂₃	C₂×#135	1	2	yes	—
206	C₂²×C₂₃	#576	1	2	yes	—
207	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
208	C₄×C₂₃	—	2	4	no	—
209	C₄×C₂₃	—	2	8	no	—
210	C₄×C₂₃	—	2	2	no	—
211	C₄×C₂₃	—	4	4	no	—
212	C₄×C₂₃	D₄²	1	2	yes	—
213	C₄×C₂₃	C₂×#135	1	4	yes	complemented
214	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²×#41	C₂²×C₂₃	1	4	yes	—
217	C₂×#135	—	2	16	no	—
218	C₂×#135	C₂²×C₂₃	1	16	yes	complemented
219	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
220	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
221	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
222	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
223	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
224	#550	—	2	8	no	—
225	#550	C₂²×C₂₃	1	16	yes	complemented
226	C₂×#127	—	2	2	no	—
227	C₂×#127	C₂²×C₂₃	1	2	yes	—
228	C₂×#121	—	2	4	no	—
229	C₂×#121	C₂²×C₂₃	1	4	yes	—
230	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
231	C₂×C₄×D₄	—	2	2	no	—
232	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
233	#493	C₂²×C₂₃	1	8	yes	complemented
234	#466	C₂²×C₂₃	1	16	yes	complemented
235	#460	C₂²×C₂₃	1	16	yes	complemented
236	#449	C₂²×C₂₃	1	32	yes	complemented
237	C₄×Q₁₆	C₂²×C₂₃	1	8	yes	complemented
238	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
239	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
241	C₂×#103	C₂²×C₂₃	1	2	yes	—
242	C₂×#102	C₂²×C₂₃	1	4	yes	—
243	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
244	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₂₃	—	4	4	no	—
246	C₂×C₂₃	—	8	8	no	—
247	C₂×C₂₃	#3287	1	4	yes	direct product factor
248	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
249	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
250	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
251	C₂×#41	—	2	8	no	—
252	C₂×#41	—	2	16	no	—
253	C₂×#41	—	4	8	no	—
254	C₂×#41	C₂³×C₂₃	1	16	yes	—
255	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
256	C₂²×D₄	—	2	4	no	—
257	C₂²×D₄	—	2	4	no	—
258	C₂²×D₄	—	2	2	no	—
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
261	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
262	C₂³×C₄	—	2	4	no	—
263	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
264	#135	—	2	64	no	—
265	C₂×Q₁₆	D₄×C₂₃	1	8	yes	complemented
266	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
267	C₂×SD₁₆	—	2	16	no	—
268	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
269	C₂×D₈	—	2	8	no	—
270	C₂×D₈	—	4	4	no	—
271	C₂×D₈	C₂³×C₂₃	1	8	yes	—
272	C₂×D₈	C₂³×C₂₃	1	8	yes	—
273	C₂²×C₈	—	2	8	no	—
274	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
275	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
276	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
278	#127	—	2	16	no	—
279	#127	—	2	4	no	—
280	#127	C₂³×C₂₃	1	8	yes	—
281	#121	—	2	8	no	—
282	#121	—	2	32	no	—
283	#121	C₂³×C₂₃	1	16	yes	—
284	C₄×Q₈	—	2	4	no	—
285	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
286	C₄×D₄	—	2	16	no	—
287	C₄×D₄	D₄×C₂₃	1	4	yes	—
288	C₄×D₄	C₂³×C₂₃	1	4	yes	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
292	C₂×C₂²⋊C₄	—	2	4	no	—
293	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
294	C₂×C₄²	—	2	2	no	—
295	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
296	#107	C₂³×C₂₃	1	4	yes	—
297	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
298	#103	C₂³×C₂₃	1	8	yes	—
299	#102	C₂³×C₂₃	1	16	yes	—
300	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
301	C₄×C₈	C₂³×C₂₃	1	4	yes	—
302	C₂₃	C₂×#3287	1	1	yes	characteristic, direct product factor, Sylow
303	C₂⁴	—	2	4	no	—
304	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
305	#41	—	2	32	no	—
306	#41	—	4	16	no	—
307	#41	—	4	32	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×Q₈	C₂×D₄×C₂₃	1	8	yes	—
311	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
312	C₂×D₄	—	2	16	no	—
313	C₂×D₄	—	2	16	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	8	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	8	no	—
319	C₂×D₄	—	2	8	no	—
320	C₂×D₄	—	4	16	no	—
321	C₂×D₄	—	4	8	no	—
322	C₂×D₄	—	8	4	no	—
323	C₂×D₄	—	8	8	no	—
324	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
325	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	16	no	—
331	C₂²×C₄	—	4	16	no	—
332	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
335	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
337	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
338	Q₁₆	C₂×D₄×C₂₃	1	16	yes	complemented
339	SD₁₆	—	2	32	no	—
340	D₈	—	2	16	no	—
341	D₈	—	4	8	no	—
342	C₂×C₈	—	2	4	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	16	no	—
345	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
346	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
347	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
348	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
349	C₄⋊C₄	—	2	8	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
352	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
353	C₂²⋊C₄	—	2	16	no	—
354	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
355	C₄²	—	2	8	no	—
356	C₄²	C₂×D₄×C₂₃	1	2	yes	—
357	C₄²	C₂⁴×C₂₃	1	2	yes	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	2	8	no	—
362	C₂³	—	4	8	no	—
363	C₂³	—	4	8	no	—
364	C₂³	—	4	8	no	—
365	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
366	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
367	Q₈	—	2	4	no	—
368	Q₈	—	2	8	no	—
369	Q₈	C₂²×D₄×C₂₃	1	8	yes	—
370	D₄	—	2	8	no	—
371	D₄	—	2	8	no	—
372	D₄	—	2	8	no	—
373	D₄	—	4	32	no	—
374	D₄	—	4	16	no	—
375	D₄	—	4	8	no	—
376	D₄	—	8	8	no	—
377	D₄	—	8	16	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	4	32	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	C₂₃×#135	1	4	yes	complemented
389	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
390	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
391	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
393	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
395	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
396	C₈	—	2	4	no	—
397	C₈	—	2	2	no	—
398	C₈	C₂²×D₄×C₂₃	1	4	yes	—
399	C₂²	—	2	4	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	4	8	no	—
403	C₂²	—	4	8	no	—
404	C₂²	—	4	8	no	—
405	C₂²	—	4	8	no	—
406	C₂²	—	8	16	no	—
407	C₂²	D₄²×C₂₃	1	2	yes	—
408	C₂²	C₂×C₂₃×#135	1	2	yes	—
409	C₂²	C₂₃×#576	1	2	yes	—
410	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
411	C₄	—	2	4	no	—
412	C₄	—	2	8	no	—
413	C₄	—	2	2	no	—
414	C₄	—	4	4	no	—
415	C₄	D₄²×C₂₃	1	2	yes	—
416	C₄	C₂×C₂₃×#135	1	4	yes	complemented
417	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
418	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
419	C₂	—	4	4	no	—
420	C₂	—	8	8	no	—
421	C₂	C₂₃×#3287	1	4	yes	direct product factor
422	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic
423	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
424	C₂	C₂×C₂₃×#576	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 ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₂₃×#135	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂₃×#3287	C₂	direct (N × H)
4. C₂×C₂₃×#493	C₂	semidirect
5. C₂×C₂₃×#466	C₂	semidirect
6. C₂×C₂₃×#460	C₂	semidirect
7. C₂×C₂₃×#449	C₂	semidirect
8. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
9. C₂×C₈×D₄×C₂₃	C₂	semidirect
12. C₂×C₂₃×#135	C₂²	semidirect
19. C₂₃×#550	C₂²	semidirect
27. C₂₃×#493	C₂²	semidirect
28. C₂₃×#466	C₂²	semidirect
29. C₂₃×#460	C₂²	semidirect
30. C₂₃×#449	C₂²	semidirect
31. C₄×Q₁₆×C₂₃	C₂²	semidirect
32. C₈×D₄×C₂₃	C₂²	semidirect
53. C₂×Q₁₆×C₂₃	D₄	semidirect
125. Q₁₆×C₂₃	C₂×D₄	semidirect
145. C₂×#3287	C₂₃	direct (N × H)
176. C₂×C₄×C₂₃	#135	semidirect
187. C₂²×#135	C₂×C₂₃	semidirect
188. C₂×#550	C₂×C₂₃	semidirect
189. #3287	C₂×C₂₃	direct (N × H)
190. C₂×#493	C₂×C₂₃	semidirect
191. C₂×#466	C₂×C₂₃	semidirect
192. C₂×#460	C₂×C₂₃	semidirect
193. C₂×#449	C₂×C₂₃	semidirect
194. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
195. C₂×C₈×D₄	C₂×C₂₃	semidirect
213. C₄×C₂₃	C₂×#135	semidirect
218. C₂×#135	C₂²×C₂₃	semidirect
225. #550	C₂²×C₂₃	semidirect
233. #493	C₂²×C₂₃	semidirect
234. #466	C₂²×C₂₃	semidirect
235. #460	C₂²×C₂₃	semidirect
236. #449	C₂²×C₂₃	semidirect
237. C₄×Q₁₆	C₂²×C₂₃	semidirect
238. C₈×D₄	C₂²×C₂₃	semidirect
247. C₂×C₂₃	#3287	direct (N × H)
265. C₂×Q₁₆	D₄×C₂₃	semidirect
302. C₂₃	C₂×#3287	direct (N × H)
338. Q₁₆	C₂×D₄×C₂₃	semidirect
388. C₂×C₄	C₂₃×#135	semidirect
416. C₄	C₂×C₂₃×#135	semidirect
421. C₂	C₂₃×#3287	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×#135	2
C₂×C₂₃×#550	2
C₂₃×#3287	2
C₂×C₂₃×#493	2
C₂×C₂₃×#466	2
C₂×C₂₃×#460	2
C₂×C₂₃×#449	2
C₂×C₄×Q₁₆×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂×#3287	23

Subgroups — C2×C23×#3287 #19208673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3287 #19208673