Subgroups — C₂×C₂₃×#3025 #19208020

Order 5888.

5086 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#525	C₂	1	3	yes	maximal, complemented
3	C₂₃×#3025	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#458	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×B(4,𝔽₂)	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#410	C₂	1	3	yes	maximal, complemented
7	C₂₃×#585	—	2	12	no	—
8	C₂₃×#585	C₂²	1	8	yes	complemented
9	C₂×C₂₃×2₋¹⁺⁴	—	2	3	no	—
10	C₂×C₂₃×H(5,𝔽₂)	—	2	3	no	—
11	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	4	yes	—
12	C₂²×#41×C₂₃	—	2	6	no	—
13	C₂²×#41×C₂₃	C₂²	1	3	yes	—
14	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#525	—	2	12	no	—
16	C₂₃×#525	C₂²	1	24	yes	complemented
17	C₂×C₂₃×#123	C₂²	1	6	yes	complemented
18	C₂×C₂₃×C₂²⋊Q₈	—	2	3	no	—
19	C₂×C₂₃×#121	—	2	3	no	—
20	C₂×C₂₃×#120	C₂²	1	6	yes	complemented
21	C₂×C₄×D₄×C₂₃	—	2	6	no	—
22	C₂×C₂₃×#117	C₂²	1	3	yes	complemented
23	C₂₃×#458	C₂²	1	32	yes	complemented
24	C₂₃×B(4,𝔽₂)	C₂²	1	32	yes	complemented
25	C₂₃×#410	C₂²	1	24	yes	complemented
26	C₂×C₂₃×#99	C₂²	1	12	yes	complemented
27	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×2₋¹⁺⁴	—	2	24	no	—
29	C₂₃×2₋¹⁺⁴	—	2	24	no	—
30	C₂₃×H(5,𝔽₂)	—	2	24	no	—
31	C₂₃×H(5,𝔽₂)	—	2	24	no	—
32	C₂₃×H(5,𝔽₂)	—	2	24	no	—
33	C₂₃×H(5,𝔽₂)	C₂³	1	16	yes	—
34	C₂×#41×C₂₃	—	2	4	no	—
35	C₂×#41×C₂₃	—	2	12	no	—
36	C₂×#41×C₂₃	—	2	48	no	—
37	C₂×#41×C₂₃	—	2	6	no	—
38	C₂×#41×C₂₃	—	2	12	no	—
39	C₂×#41×C₂₃	—	2	6	no	—
40	C₂×#41×C₂₃	—	2	12	no	—
41	C₂×#41×C₂₃	—	2	48	no	—
42	C₂×#41×C₂₃	D₄	1	12	yes	complemented
43	C₂×#41×C₂₃	C₂³	1	12	yes	—
44	C₂²×Q₈×C₂₃	—	2	6	no	—
45	C₂²×Q₈×C₂₃	D₄	1	3	yes	complemented
46	C₂²×D₄×C₂₃	—	2	6	no	—
47	C₂²×D₄×C₂₃	—	2	12	no	—
48	C₂²×D₄×C₂₃	—	2	6	no	—
49	C₂²×D₄×C₂₃	D₄	1	3	yes	complemented
50	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
51	C₂³×C₄×C₂₃	—	2	4	no	—
52	C₂³×C₄×C₂₃	—	2	3	no	—
53	C₂³×C₄×C₂₃	—	2	3	no	—
54	C₂³×C₄×C₂₃	—	2	6	no	—
55	C₂³×C₄×C₂₃	D₄	1	6	yes	complemented
56	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	C₂³	1	4	yes	complemented
59	C₂₃×#123	—	2	12	no	—
60	C₂₃×#123	C₂³	1	24	yes	complemented
61	C₂₃×C₂²⋊Q₈	—	2	24	no	—
62	C₂₃×#121	—	2	24	no	—
63	C₂₃×#120	—	2	12	no	—
64	C₂₃×#120	C₂³	1	24	yes	complemented
65	C₄×D₄×C₂₃	—	2	48	no	—
66	C₂₃×#117	—	2	6	no	—
67	C₂₃×#117	C₂³	1	12	yes	complemented
68	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	6	no	—
72	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
73	C₂×C₄²×C₂₃	—	2	3	no	—
74	C₂₃×#99	C₂³	1	48	yes	complemented
75	C₂⁴×C₂₃	—	2	4	no	—
76	C₂⁴×C₂₃	—	2	3	no	—
77	C₂⁴×C₂₃	—	2	3	no	—
78	C₂⁴×C₂₃	—	4	4	no	—
79	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
80	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
81	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
82	#41×C₂₃	—	2	32	no	—
83	#41×C₂₃	—	2	48	no	—
84	#41×C₂₃	—	2	96	no	—
85	#41×C₂₃	—	2	24	no	—
86	#41×C₂₃	—	2	24	no	—
87	#41×C₂₃	—	2	24	no	—
88	#41×C₂₃	—	2	48	no	—
89	#41×C₂₃	—	2	24	no	—
90	C₂×Q₈×C₂₃	—	2	4	no	—
91	C₂×Q₈×C₂₃	—	2	48	no	—
92	C₂×Q₈×C₂₃	—	2	6	no	—
93	C₂×Q₈×C₂₃	—	2	6	no	—
94	C₂×Q₈×C₂₃	C₂×D₄	1	12	yes	complemented
95	C₂×D₄×C₂₃	—	2	12	no	—
96	C₂×D₄×C₂₃	—	2	48	no	—
97	C₂×D₄×C₂₃	—	2	48	no	—
98	C₂×D₄×C₂₃	—	2	12	no	—
99	C₂×D₄×C₂₃	—	2	48	no	—
100	C₂×D₄×C₂₃	—	2	12	no	—
101	C₂×D₄×C₂₃	—	2	24	no	—
102	C₂×D₄×C₂₃	—	2	6	no	—
103	C₂×D₄×C₂₃	—	2	6	no	—
104	C₂×D₄×C₂₃	—	4	24	no	—
105	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	—
106	C₂×D₄×C₂₃	C₂×D₄	1	12	yes	complemented
107	C₂×D₄×C₂₃	C₂⁴	1	12	yes	—
108	C₂²×C₄×C₂₃	—	2	12	no	—
109	C₂²×C₄×C₂₃	—	2	32	no	—
110	C₂²×C₄×C₂₃	—	2	6	no	—
111	C₂²×C₄×C₂₃	—	2	12	no	—
112	C₂²×C₄×C₂₃	—	2	6	no	—
113	C₂²×C₄×C₂₃	—	2	6	no	—
114	C₂²×C₄×C₂₃	—	2	3	no	—
115	C₂²×C₄×C₂₃	—	2	3	no	—
116	C₂²×C₄×C₂₃	—	2	12	no	—
117	C₂²×C₄×C₂₃	—	2	12	no	—
118	C₂²×C₄×C₂₃	—	4	3	no	—
119	C₂²×C₄×C₂₃	—	4	6	no	—
120	C₂²×C₄×C₂₃	—	4	6	no	—
121	C₂²×C₄×C₂₃	—	4	1	no	—
122	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
123	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	complemented
127	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
129	C₄⋊C₄×C₂₃	—	2	12	no	—
130	C₄⋊C₄×C₂₃	—	2	24	no	—
131	C₂²⋊C₄×C₂₃	—	2	24	no	—
132	C₂²⋊C₄×C₂₃	—	2	6	no	—
133	C₂²⋊C₄×C₂₃	—	2	24	no	—
134	C₂²⋊C₄×C₂₃	C₂⁴	1	12	yes	—
135	C₄²×C₂₃	—	2	12	no	—
136	C₂×#3025	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₂₃	—	2	12	no	—
138	C₂³×C₂₃	—	2	16	no	—
139	C₂³×C₂₃	—	2	6	no	—
140	C₂³×C₂₃	—	2	6	no	—
141	C₂³×C₂₃	—	2	6	no	—
142	C₂³×C₂₃	—	4	2	no	—
143	C₂³×C₂₃	—	4	4	no	—
144	C₂³×C₂₃	—	4	6	no	—
145	C₂³×C₂₃	—	4	12	no	—
146	C₂³×C₂₃	—	4	12	no	—
147	C₂³×C₂₃	—	4	3	no	—
148	C₂³×C₂₃	—	4	1	no	—
149	C₂³×C₂₃	—	8	8	no	—
150	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
151	C₂³×C₂₃	C₂²×D₄	1	12	yes	—
152	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
153	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
154	Q₈×C₂₃	—	2	16	no	—
155	Q₈×C₂₃	—	2	24	no	—
156	D₄×C₂₃	—	2	48	no	—
157	D₄×C₂₃	—	2	24	no	—
158	D₄×C₂₃	—	2	24	no	—
159	D₄×C₂₃	—	2	24	no	—
160	D₄×C₂₃	—	4	48	no	—
161	C₂×C₄×C₂₃	—	2	48	no	—
162	C₂×C₄×C₂₃	—	2	6	no	—
163	C₂×C₄×C₂₃	—	2	6	no	—
164	C₂×C₄×C₂₃	—	2	24	no	—
165	C₂×C₄×C₂₃	—	2	24	no	—
166	C₂×C₄×C₂₃	—	2	12	no	—
167	C₂×C₄×C₂₃	—	2	6	no	—
168	C₂×C₄×C₂₃	—	4	12	no	—
169	C₂×C₄×C₂₃	—	4	3	no	—
170	C₂×C₄×C₂₃	—	4	1	no	—
171	C₂×C₄×C₂₃	—	4	4	no	—
172	C₂×C₄×C₂₃	—	4	12	no	—
173	C₂×C₄×C₂₃	#120	1	3	yes	—
174	C₂×C₄×C₂₃	#120	1	6	yes	—
175	C₂×C₄×C₂₃	#120	1	6	yes	—
176	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
178	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
179	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
180	C₂×#585	C₂×C₂₃	1	1	yes	characteristic, complemented
181	C₂×#525	C₂×C₂₃	1	3	yes	complemented
182	#3025	C₂×C₂₃	1	16	yes	direct product factor
183	C₂×#458	C₂×C₂₃	1	4	yes	complemented
184	C₂×B(4,𝔽₂)	C₂×C₂₃	1	4	yes	complemented
185	C₂×#410	C₂×C₂₃	1	3	yes	complemented
186	C₂²×C₂₃	—	2	6	no	—
187	C₂²×C₂₃	—	2	12	no	—
188	C₂²×C₂₃	—	2	12	no	—
189	C₂²×C₂₃	—	4	3	no	—
190	C₂²×C₂₃	—	4	1	no	—
191	C₂²×C₂₃	—	4	12	no	—
192	C₂²×C₂₃	—	4	6	no	—
193	C₂²×C₂₃	—	4	12	no	—
194	C₂²×C₂₃	—	4	2	no	—
195	C₂²×C₂₃	—	4	4	no	—
196	C₂²×C₂₃	—	4	12	no	—
197	C₂²×C₂₃	—	8	6	no	—
198	C₂²×C₂₃	—	8	16	no	—
199	C₂²×C₂₃	C₂×#120	1	3	yes	—
200	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
201	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
202	C₄×C₂₃	—	2	12	no	—
203	C₄×C₂₃	—	4	12	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	C₂×#120	1	2	yes	—
206	C₄×C₂₃	C₂×#120	1	6	yes	—
207	#585	—	2	12	no	—
208	#585	C₂²×C₂₃	1	8	yes	complemented
209	C₂×2₋¹⁺⁴	—	2	3	no	—
210	C₂×H(5,𝔽₂)	—	2	3	no	—
211	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	4	yes	—
212	C₂²×#41	—	2	6	no	—
213	C₂²×#41	C₂²×C₂₃	1	3	yes	—
214	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
215	#525	—	2	12	no	—
216	#525	C₂²×C₂₃	1	24	yes	complemented
217	C₂×#123	C₂²×C₂₃	1	6	yes	complemented
218	C₂×C₂²⋊Q₈	—	2	3	no	—
219	C₂×#121	—	2	3	no	—
220	C₂×#120	C₂²×C₂₃	1	6	yes	complemented
221	C₂×C₄×D₄	—	2	6	no	—
222	C₂×#117	C₂²×C₂₃	1	3	yes	complemented
223	#458	C₂²×C₂₃	1	32	yes	complemented
224	B(4,𝔽₂)	C₂²×C₂₃	1	32	yes	complemented
225	#410	C₂²×C₂₃	1	24	yes	complemented
226	C₂×#99	C₂²×C₂₃	1	12	yes	complemented
227	C₂×C₂₃	—	2	3	no	—
228	C₂×C₂₃	—	2	3	no	—
229	C₂×C₂₃	—	4	4	no	—
230	C₂×C₂₃	—	4	12	no	—
231	C₂×C₂₃	#3025	1	2	yes	direct product factor
232	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
233	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
234	2₋¹⁺⁴	—	2	24	no	—
235	2₋¹⁺⁴	—	2	24	no	—
236	H(5,𝔽₂)	—	2	24	no	—
237	H(5,𝔽₂)	—	2	24	no	—
238	H(5,𝔽₂)	—	2	24	no	—
239	H(5,𝔽₂)	C₂³×C₂₃	1	16	yes	—
240	C₂×#41	—	2	4	no	—
241	C₂×#41	—	2	12	no	—
242	C₂×#41	—	2	48	no	—
243	C₂×#41	—	2	6	no	—
244	C₂×#41	—	2	12	no	—
245	C₂×#41	—	2	6	no	—
246	C₂×#41	—	2	12	no	—
247	C₂×#41	—	2	48	no	—
248	C₂×#41	D₄×C₂₃	1	12	yes	complemented
249	C₂×#41	C₂³×C₂₃	1	12	yes	—
250	C₂²×Q₈	—	2	6	no	—
251	C₂²×Q₈	D₄×C₂₃	1	3	yes	complemented
252	C₂²×D₄	—	2	6	no	—
253	C₂²×D₄	—	2	12	no	—
254	C₂²×D₄	—	2	6	no	—
255	C₂²×D₄	D₄×C₂₃	1	3	yes	complemented
256	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
257	C₂³×C₄	—	2	4	no	—
258	C₂³×C₄	—	2	3	no	—
259	C₂³×C₄	—	2	3	no	—
260	C₂³×C₄	—	2	6	no	—
261	C₂³×C₄	D₄×C₂₃	1	6	yes	complemented
262	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
263	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₂₃	1	4	yes	complemented
265	#123	—	2	12	no	—
266	#123	C₂³×C₂₃	1	24	yes	complemented
267	C₂²⋊Q₈	—	2	24	no	—
268	#121	—	2	24	no	—
269	#120	—	2	12	no	—
270	#120	C₂³×C₂₃	1	24	yes	complemented
271	C₄×D₄	—	2	48	no	—
272	#117	—	2	6	no	—
273	#117	C₂³×C₂₃	1	12	yes	complemented
274	C₂×C₄⋊C₄	—	2	3	no	—
275	C₂×C₄⋊C₄	—	2	6	no	—
276	C₂×C₂²⋊C₄	—	2	6	no	—
277	C₂×C₂²⋊C₄	—	2	6	no	—
278	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
279	C₂×C₄²	—	2	3	no	—
280	#99	C₂³×C₂₃	1	48	yes	complemented
281	C₂₃	C₂×#3025	1	1	yes	characteristic, direct product factor, Sylow
282	C₂⁴	—	2	4	no	—
283	C₂⁴	—	2	3	no	—
284	C₂⁴	—	2	3	no	—
285	C₂⁴	—	4	4	no	—
286	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
287	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
288	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
289	#41	—	2	32	no	—
290	#41	—	2	48	no	—
291	#41	—	2	96	no	—
292	#41	—	2	24	no	—
293	#41	—	2	24	no	—
294	#41	—	2	24	no	—
295	#41	—	2	48	no	—
296	#41	—	2	24	no	—
297	C₂×Q₈	—	2	4	no	—
298	C₂×Q₈	—	2	48	no	—
299	C₂×Q₈	—	2	6	no	—
300	C₂×Q₈	—	2	6	no	—
301	C₂×Q₈	C₂×D₄×C₂₃	1	12	yes	complemented
302	C₂×D₄	—	2	12	no	—
303	C₂×D₄	—	2	48	no	—
304	C₂×D₄	—	2	48	no	—
305	C₂×D₄	—	2	12	no	—
306	C₂×D₄	—	2	48	no	—
307	C₂×D₄	—	2	12	no	—
308	C₂×D₄	—	2	24	no	—
309	C₂×D₄	—	2	6	no	—
310	C₂×D₄	—	2	6	no	—
311	C₂×D₄	—	4	24	no	—
312	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	—
313	C₂×D₄	C₂×D₄×C₂₃	1	12	yes	complemented
314	C₂×D₄	C₂⁴×C₂₃	1	12	yes	—
315	C₂²×C₄	—	2	12	no	—
316	C₂²×C₄	—	2	32	no	—
317	C₂²×C₄	—	2	6	no	—
318	C₂²×C₄	—	2	6	no	—
319	C₂²×C₄	—	2	12	no	—
320	C₂²×C₄	—	2	6	no	—
321	C₂²×C₄	—	2	3	no	—
322	C₂²×C₄	—	2	3	no	—
323	C₂²×C₄	—	2	12	no	—
324	C₂²×C₄	—	2	12	no	—
325	C₂²×C₄	—	4	1	no	—
326	C₂²×C₄	—	4	3	no	—
327	C₂²×C₄	—	4	6	no	—
328	C₂²×C₄	—	4	6	no	—
329	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
330	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
331	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
332	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
333	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	complemented
334	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
336	C₄⋊C₄	—	2	12	no	—
337	C₄⋊C₄	—	2	24	no	—
338	C₂²⋊C₄	—	2	24	no	—
339	C₂²⋊C₄	—	2	6	no	—
340	C₂²⋊C₄	—	2	24	no	—
341	C₂²⋊C₄	C₂⁴×C₂₃	1	12	yes	—
342	C₄²	—	2	12	no	—
343	C₂³	—	2	12	no	—
344	C₂³	—	2	16	no	—
345	C₂³	—	2	6	no	—
346	C₂³	—	2	6	no	—
347	C₂³	—	2	6	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	4	no	—
350	C₂³	—	4	6	no	—
351	C₂³	—	4	12	no	—
352	C₂³	—	4	12	no	—
353	C₂³	—	4	3	no	—
354	C₂³	—	4	1	no	—
355	C₂³	—	8	8	no	—
356	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
357	C₂³	C₂²×D₄×C₂₃	1	12	yes	—
358	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
359	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
360	Q₈	—	2	16	no	—
361	Q₈	—	2	24	no	—
362	D₄	—	2	48	no	—
363	D₄	—	2	24	no	—
364	D₄	—	2	24	no	—
365	D₄	—	2	24	no	—
366	D₄	—	4	48	no	—
367	C₂×C₄	—	2	48	no	—
368	C₂×C₄	—	2	6	no	—
369	C₂×C₄	—	2	6	no	—
370	C₂×C₄	—	2	24	no	—
371	C₂×C₄	—	2	24	no	—
372	C₂×C₄	—	2	12	no	—
373	C₂×C₄	—	2	6	no	—
374	C₂×C₄	—	4	12	no	—
375	C₂×C₄	—	4	3	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	—	4	12	no	—
379	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
380	C₂×C₄	C₂₃×#120	1	3	yes	—
381	C₂×C₄	C₂₃×#120	1	6	yes	—
382	C₂×C₄	C₂₃×#120	1	6	yes	—
383	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
384	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
385	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
386	C₂²	—	2	6	no	—
387	C₂²	—	2	12	no	—
388	C₂²	—	2	12	no	—
389	C₂²	—	4	3	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	12	no	—
392	C₂²	—	4	6	no	—
393	C₂²	—	4	12	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	12	no	—
397	C₂²	—	8	6	no	—
398	C₂²	—	8	16	no	—
399	C₂²	C₂×C₂₃×#120	1	3	yes	—
400	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
401	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
402	C₄	—	2	12	no	—
403	C₄	—	4	12	no	—
404	C₄	—	4	2	no	—
405	C₄	C₂×C₂₃×#120	1	2	yes	—
406	C₄	C₂×C₂₃×#120	1	6	yes	—
407	C₂	—	2	3	no	—
408	C₂	—	2	3	no	—
409	C₂	—	4	4	no	—
410	C₂	—	4	12	no	—
411	C₂	C₂₃×#3025	1	2	yes	direct product factor
412	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂×#120
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#585	C₂	semidirect
2. C₂×C₂₃×#525	C₂	semidirect
3. C₂₃×#3025	C₂	direct (N × H)
4. C₂×C₂₃×#458	C₂	semidirect
5. C₂×C₂₃×B(4,𝔽₂)	C₂	semidirect
6. C₂×C₂₃×#410	C₂	semidirect
8. C₂₃×#585	C₂²	semidirect
14. C₂⁴×C₄×C₂₃	C₂²	semidirect
16. C₂₃×#525	C₂²	semidirect
17. C₂×C₂₃×#123	C₂²	semidirect
20. C₂×C₂₃×#120	C₂²	semidirect
22. C₂×C₂₃×#117	C₂²	semidirect
23. C₂₃×#458	C₂²	semidirect
24. C₂₃×B(4,𝔽₂)	C₂²	semidirect
25. C₂₃×#410	C₂²	semidirect
26. C₂×C₂₃×#99	C₂²	semidirect
42. C₂×#41×C₂₃	D₄	semidirect
45. C₂²×Q₈×C₂₃	D₄	semidirect
49. C₂²×D₄×C₂₃	D₄	semidirect
55. C₂³×C₄×C₂₃	D₄	semidirect
58. C₂³×C₄×C₂₃	C₂³	semidirect
60. C₂₃×#123	C₂³	semidirect
64. C₂₃×#120	C₂³	semidirect
67. C₂₃×#117	C₂³	semidirect
74. C₂₃×#99	C₂³	semidirect
94. C₂×Q₈×C₂₃	C₂×D₄	semidirect
106. C₂×D₄×C₂₃	C₂×D₄	semidirect
126. C₂²×C₄×C₂₃	C₂×D₄	semidirect
136. C₂×#3025	C₂₃	direct (N × H)
180. C₂×#585	C₂×C₂₃	semidirect
181. C₂×#525	C₂×C₂₃	semidirect
182. #3025	C₂×C₂₃	direct (N × H)
183. C₂×#458	C₂×C₂₃	semidirect
184. C₂×B(4,𝔽₂)	C₂×C₂₃	semidirect
185. C₂×#410	C₂×C₂₃	semidirect
208. #585	C₂²×C₂₃	semidirect
214. C₂⁴×C₄	C₂²×C₂₃	semidirect
216. #525	C₂²×C₂₃	semidirect
217. C₂×#123	C₂²×C₂₃	semidirect
220. C₂×#120	C₂²×C₂₃	semidirect
222. C₂×#117	C₂²×C₂₃	semidirect
223. #458	C₂²×C₂₃	semidirect
224. B(4,𝔽₂)	C₂²×C₂₃	semidirect
225. #410	C₂²×C₂₃	semidirect
226. C₂×#99	C₂²×C₂₃	semidirect
231. C₂×C₂₃	#3025	direct (N × H)
248. C₂×#41	D₄×C₂₃	semidirect
251. C₂²×Q₈	D₄×C₂₃	semidirect
255. C₂²×D₄	D₄×C₂₃	semidirect
261. C₂³×C₄	D₄×C₂₃	semidirect
264. C₂³×C₄	C₂³×C₂₃	semidirect
266. #123	C₂³×C₂₃	semidirect
270. #120	C₂³×C₂₃	semidirect
273. #117	C₂³×C₂₃	semidirect
280. #99	C₂³×C₂₃	semidirect
281. C₂₃	C₂×#3025	direct (N × H)
301. C₂×Q₈	C₂×D₄×C₂₃	semidirect
313. C₂×D₄	C₂×D₄×C₂₃	semidirect
333. C₂²×C₄	C₂×D₄×C₂₃	semidirect
411. C₂	C₂₃×#3025	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×#585	2
C₂×C₂₃×#525	2
C₂₃×#3025	2
C₂×C₂₃×#458	2
C₂×C₂₃×B(4,𝔽₂)	2
C₂×C₂₃×#410	2
C₂×#3025	23

Subgroups — C2×C23×#3025 #19208020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3025 #19208020