Subgroups — C₂₃×#22803 #19170429

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2925	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1975	C₂	1	2	yes	maximal
4	C₂₃×#1929	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1806	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1773	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
10	C₄×#41×C₂₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
12	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
13	C₄×SD₁₆×C₂₃	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₂₃	C₄	1	2	yes	complemented
15	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
16	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
17	C₂₃×#433	—	2	4	no	—
18	C₂₃×#432	—	2	2	no	—
19	C₂₃×#431	—	2	2	no	—
20	C₂×C₂₃×#108	C₂²	1	2	yes	—
21	C₂×C₂₃×#108	C₂²	1	2	yes	—
22	C₂₃×#427	C₂²	1	1	yes	characteristic
23	C₂₃×#423	C₂²	1	2	yes	—
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#104	C₂²	1	4	yes	—
26	C₂₃×#416	C₂²	1	2	yes	—
27	C₂₃×#408	C₂²	1	4	yes	—
28	C₂₃×#405	C₂²	1	4	yes	—
29	C₂×C₂₃×#97	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#339	C₂²	1	4	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂×#41×C₂₃	C₂³	1	2	yes	—
35	C₂₃×#135	—	2	2	no	—
36	C₂₃×#135	—	2	2	no	—
37	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	2	2	no	—
48	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂₃×#117	—	2	2	no	—
51	C₂₃×#117	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₂×C₄²×C₂₃	—	2	2	no	—
54	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#108	—	2	4	no	—
57	C₂₃×#108	—	4	2	no	—
58	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
59	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
60	C₂₃×#106	C₂×C₄	1	2	yes	—
61	C₂₃×C₄⋊C₈	—	2	2	no	—
62	C₂₃×C₄⋊C₈	—	2	2	no	—
63	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
64	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
65	C₂₃×#104	—	4	4	no	—
66	C₂₃×#103	C₂×C₄	1	2	yes	—
67	C₂₃×#103	C₂×C₄	1	2	yes	—
68	C₂₃×#102	C₂×C₄	1	2	yes	—
69	C₂₃×#102	C₂×C₄	1	2	yes	—
70	C₂₃×C₂²⋊C₈	—	2	4	no	—
71	C₂₃×C₂²⋊C₈	—	2	4	no	—
72	C₂₃×#97	—	2	4	no	—
73	C₂₃×#97	—	2	4	no	—
74	C₄×C₈×C₂₃	—	2	4	no	—
75	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
79	C₂₃×#95	—	2	1	no	—
80	#41×C₂₃	—	2	2	no	—
81	#41×C₂₃	—	2	2	no	—
82	#41×C₂₃	—	2	2	no	—
83	#41×C₂₃	—	2	2	no	—
84	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×D₄×C₂₃	—	2	2	no	—
86	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	2	2	no	—
89	C₂²×C₄×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	—	2	2	no	—
91	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
92	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
93	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	Q₁₆×C₂₃	—	2	2	no	—
95	SD₁₆×C₂₃	—	2	4	no	—
96	D₈×C₂₃	—	2	2	no	—
97	M₁₆×C₂₃	—	2	4	no	—
98	M₁₆×C₂₃	—	2	4	no	—
99	C₂×C₈×C₂₃	—	2	2	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	4	no	—
104	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₂₃	#41	1	2	yes	—
110	C₂×C₈×C₂₃	#41	1	2	yes	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₄⋊C₄×C₂₃	—	4	2	no	—
114	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
115	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
116	C₂²⋊C₄×C₂₃	—	2	2	no	—
117	C₂²⋊C₄×C₂₃	—	2	2	no	—
118	C₂²⋊C₄×C₂₃	—	2	2	no	—
119	C₄²×C₂₃	—	2	2	no	—
120	C₄²×C₂₃	—	2	2	no	—
121	C₄²×C₂₃	—	2	2	no	—
122	C₄²×C₂₃	—	2	2	no	—
123	C₄²×C₂₃	—	2	2	no	—
124	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
128	#22803	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	2	2	no	—
130	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
131	Q₈×C₂₃	—	2	2	no	—
132	Q₈×C₂₃	—	2	2	no	—
133	D₄×C₂₃	—	2	2	no	—
134	D₄×C₂₃	—	2	2	no	—
135	D₄×C₂₃	—	4	2	no	—
136	D₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	2	2	no	—
138	C₂×C₄×C₂₃	—	2	2	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	2	no	—
141	C₂×C₄×C₂₃	—	4	1	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	4	4	no	—
145	C₂×C₄×C₂₃	—	4	2	no	—
146	C₂×C₄×C₂₃	—	4	2	no	—
147	C₂×C₄×C₂₃	—	4	2	no	—
148	C₂×C₄×C₂₃	—	8	2	no	—
149	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
159	C₈×C₂₃	—	2	1	no	—
160	C₈×C₂₃	—	2	1	no	—
161	C₈×C₂₃	—	2	4	no	—
162	C₈×C₂₃	—	4	4	no	—
163	C₈×C₂₃	#117	1	2	yes	—
164	C₈×C₂₃	#117	1	2	yes	—
165	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
166	#2925	C₂×C₂₃	1	2	yes	complemented
167	#1975	C₂×C₂₃	1	2	yes	—
168	#1929	C₂×C₂₃	1	4	yes	complemented
169	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
170	#1806	C₂×C₂₃	1	2	yes	complemented
171	#1773	C₂×C₂₃	1	2	yes	complemented
172	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
173	C₂²×C₂₃	—	2	1	no	—
174	C₂²×C₂₃	—	2	1	no	—
175	C₂²×C₂₃	—	4	2	no	—
176	C₂²×C₂₃	—	4	2	no	—
177	C₂²×C₂₃	—	4	2	no	—
178	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂²×C₂₃	#555	1	1	yes	characteristic
180	C₂²×C₂₃	#556	1	1	yes	characteristic
181	C₄×C₂₃	—	2	1	no	—
182	C₄×C₂₃	—	2	1	no	—
183	C₄×C₂₃	—	2	1	no	—
184	C₄×C₂₃	—	4	2	no	—
185	C₄×C₂₃	—	4	2	no	—
186	C₄×C₂₃	—	4	2	no	—
187	C₄×C₂₃	—	8	2	no	—
188	C₄×C₂₃	#444	1	2	yes	—
189	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
190	C₄×C₂₃	#520	1	1	yes	characteristic
191	C₄×C₂₃	#547	1	1	yes	characteristic
192	C₄×C₂₃	#550	1	1	yes	characteristic
193	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
194	C₄×#41	C₂²×C₂₃	1	2	yes	—
195	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
196	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
197	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
198	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
199	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
200	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
201	#433	—	2	4	no	—
202	#432	—	2	2	no	—
203	#431	—	2	2	no	—
204	C₂×#108	C₂²×C₂₃	1	2	yes	—
205	C₂×#108	C₂²×C₂₃	1	2	yes	—
206	#427	C₂²×C₂₃	1	1	yes	characteristic
207	#423	C₂²×C₂₃	1	2	yes	—
208	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
209	C₂×#104	C₂²×C₂₃	1	4	yes	—
210	#416	C₂²×C₂₃	1	2	yes	—
211	#408	C₂²×C₂₃	1	4	yes	—
212	#405	C₂²×C₂₃	1	4	yes	—
213	C₂×#97	C₂²×C₂₃	1	2	yes	—
214	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
216	#339	C₂²×C₂₃	1	4	yes	—
217	#336	C₂²×C₂₃	1	2	yes	—
218	C₂×C₂₃	—	2	1	no	—
219	C₂×C₂₃	—	2	1	no	—
220	C₂×C₂₃	—	8	2	no	—
221	C₂×C₂₃	#2354	1	1	yes	characteristic
222	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
223	C₂×C₂₃	#2957	1	1	yes	characteristic
224	C₂×#41	C₂³×C₂₃	1	2	yes	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
229	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
230	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
231	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
233	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
234	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
235	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
239	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
240	#117	—	2	2	no	—
241	#117	C₂³×C₂₃	1	2	yes	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄²	—	2	2	no	—
244	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
245	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
246	#108	—	2	4	no	—
247	#108	—	4	2	no	—
248	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
249	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
250	#106	C₂×C₄×C₂₃	1	2	yes	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
254	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
255	#104	—	4	4	no	—
256	#103	C₂×C₄×C₂₃	1	2	yes	—
257	#103	C₂×C₄×C₂₃	1	2	yes	—
258	#102	C₂×C₄×C₂₃	1	2	yes	—
259	#102	C₂×C₄×C₂₃	1	2	yes	—
260	C₂²⋊C₈	—	2	4	no	—
261	C₂²⋊C₈	—	2	4	no	—
262	#97	—	2	4	no	—
263	#97	—	2	4	no	—
264	C₄×C₈	—	2	4	no	—
265	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
266	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
267	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
268	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
269	#95	—	2	1	no	—
270	C₂₃	#22803	1	1	yes	characteristic, direct product factor, Sylow
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
283	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
284	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
285	Q₁₆	—	2	2	no	—
286	SD₁₆	—	2	4	no	—
287	D₈	—	2	2	no	—
288	M₁₆	—	2	4	no	—
289	M₁₆	—	2	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
296	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
297	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
298	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
299	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
300	C₂×C₈	#41×C₂₃	1	2	yes	—
301	C₂×C₈	#41×C₂₃	1	2	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
306	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
316	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
317	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
340	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
341	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
342	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
343	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
345	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
347	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
348	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	4	no	—
353	C₈	C₂₃×#117	1	2	yes	—
354	C₈	C₂₃×#117	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
361	C₂²	C₂₃×#555	1	1	yes	characteristic
362	C₂²	C₂₃×#556	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	8	2	no	—
370	C₄	C₂₃×#444	1	2	yes	—
371	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
372	C₄	C₂₃×#520	1	1	yes	characteristic
373	C₄	C₂₃×#547	1	1	yes	characteristic
374	C₄	C₂₃×#550	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	8	2	no	—
378	C₂	C₂₃×#2354	1	1	yes	characteristic
379	C₂	C₂×C₂₃×#444	1	1	yes	characteristic
380	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2925	C₂	semidirect
4. C₂₃×#1929	C₂	semidirect
5. C₂₃×#1848	C₂	semidirect
6. C₂₃×#1806	C₂	semidirect
7. C₂₃×#1773	C₂	semidirect
8. C₂₃×#1770	C₂	semidirect
11. C₄×Q₁₆×C₂₃	C₄	semidirect
12. C₄×Q₁₆×C₂₃	C₄	semidirect
13. C₄×SD₁₆×C₂₃	C₄	semidirect
14. C₄×SD₁₆×C₂₃	C₄	semidirect
15. C₄×D₈×C₂₃	C₄	semidirect
16. C₄×D₈×C₂₃	C₄	semidirect
128. #22803	C₂₃	direct (N × H)
166. #2925	C₂×C₂₃	semidirect
168. #1929	C₂×C₂₃	semidirect
169. #1848	C₂×C₂₃	semidirect
170. #1806	C₂×C₂₃	semidirect
171. #1773	C₂×C₂₃	semidirect
172. #1770	C₂×C₂₃	semidirect
195. C₄×Q₁₆	C₄×C₂₃	semidirect
196. C₄×Q₁₆	C₄×C₂₃	semidirect
197. C₄×SD₁₆	C₄×C₂₃	semidirect
198. C₄×SD₁₆	C₄×C₂₃	semidirect
199. C₄×D₈	C₄×C₂₃	semidirect
200. C₄×D₈	C₄×C₂₃	semidirect
270. C₂₃	#22803	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#135	2
C₂₃×#2925	2
C₂₃×#1975	2
C₂₃×#1929	2
C₂₃×#1848	2
C₂₃×#1806	2
C₂₃×#1773	2
C₂₃×#1770	2
#22803	23

Subgroups — C23×#22803 #19170429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22803 #19170429