Subgroups — C₂×C₄×C₂₃×#120 #19181653

Order 5888.

8582 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#120	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×D₄×C₂₃	C₂	1	3	yes	maximal, complemented
4	C₄×C₂₃×#120	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#386	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×#385	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	3	yes	maximal, complemented
9	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₂₃	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₂₃	—	2	24	no	—
12	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
13	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
14	C₂⁴×C₄×C₂₃	C₂²	1	8	yes	complemented
15	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₂₃×#120	C₄	1	16	yes	direct product factor
17	C₂×C₂₃×#120	C₂²	1	8	yes	—
18	C₂×C₄×D₄×C₂₃	—	2	24	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	24	yes	complemented
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	complemented
21	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
22	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
24	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	complemented
25	C₂²×C₄²×C₂₃	C₂²	1	3	yes	complemented
26	C₂₃×#386	C₂²	1	24	yes	complemented
27	C₂₃×#385	C₂²	1	24	yes	complemented
28	C₂₃×#379	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊C₄×C₂₃	C₂²	1	24	yes	complemented
30	C₂×C₂₃×#95	C₂²	1	6	yes	complemented
31	C₂⁵×C₂₃	—	2	12	no	—
32	C₂⁵×C₂₃	—	2	12	no	—
33	C₂⁵×C₂₃	C₂×C₄	1	8	yes	complemented
34	C₂⁵×C₂₃	C₂³	1	4	yes	—
35	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
36	C₂⁵×C₂₃	C₂³	1	3	yes	—
37	C₂²×D₄×C₂₃	—	2	12	no	—
38	C₂²×D₄×C₂₃	—	2	12	no	—
39	C₂²×D₄×C₂₃	C₂×C₄	1	24	yes	complemented
40	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
41	C₂³×C₄×C₂₃	—	2	128	no	—
42	C₂³×C₄×C₂₃	—	2	12	no	—
43	C₂³×C₄×C₂₃	—	2	24	no	—
44	C₂³×C₄×C₂₃	—	2	24	no	—
45	C₂³×C₄×C₂₃	—	2	96	no	—
46	C₂³×C₄×C₂₃	—	2	12	no	—
47	C₂³×C₄×C₂₃	D₄	1	24	yes	complemented
48	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
51	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
52	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
53	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
54	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	C₂³	1	4	yes	complemented
56	C₂₃×#120	C₂×C₄	1	64	yes	direct product factor
57	C₄×D₄×C₂₃	—	2	96	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	complemented
60	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
61	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
62	C₂×C₂²⋊C₄×C₂₃	—	2	12	no	—
63	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	24	yes	complemented
64	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
65	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
66	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
67	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	complemented
68	C₂×C₄²×C₂₃	—	2	12	no	—
69	C₂×C₄²×C₂₃	C₂³	1	12	yes	complemented
70	C₂₃×#95	C₂³	1	24	yes	complemented
71	C₂⁴×C₂₃	—	2	32	no	—
72	C₂⁴×C₂₃	—	2	96	no	—
73	C₂⁴×C₂₃	—	2	96	no	—
74	C₂⁴×C₂₃	—	2	24	no	—
75	C₂⁴×C₂₃	—	2	12	no	—
76	C₂⁴×C₂₃	—	2	24	no	—
77	C₂⁴×C₂₃	—	2	12	no	—
78	C₂⁴×C₂₃	—	2	6	no	—
79	C₂⁴×C₂₃	—	2	6	no	—
80	C₂⁴×C₂₃	C₂²×C₄	1	12	yes	—
81	C₂⁴×C₂₃	C₂²×C₄	1	16	yes	complemented
82	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	complemented
83	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
84	C₂⁴×C₂₃	#41	1	12	yes	—
85	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
86	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
87	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂×D₄×C₂₃	—	2	96	no	—
89	C₂×D₄×C₂₃	—	2	24	no	—
90	C₂×D₄×C₂₃	—	2	24	no	—
91	C₂×D₄×C₂₃	C₂²×C₄	1	48	yes	complemented
92	C₂²×C₄×C₂₃	—	2	96	no	—
93	C₂²×C₄×C₂₃	—	2	256	no	—
94	C₂²×C₄×C₂₃	—	2	6	no	—
95	C₂²×C₄×C₂₃	—	2	24	no	—
96	C₂²×C₄×C₂₃	—	2	12	no	—
97	C₂²×C₄×C₂₃	—	2	6	no	—
98	C₂²×C₄×C₂₃	—	2	96	no	—
99	C₂²×C₄×C₂₃	—	2	48	no	—
100	C₂²×C₄×C₂₃	—	2	48	no	—
101	C₂²×C₄×C₂₃	—	2	6	no	—
102	C₂²×C₄×C₂₃	—	2	6	no	—
103	C₂²×C₄×C₂₃	—	2	12	no	—
104	C₂²×C₄×C₂₃	—	2	24	no	—
105	C₂²×C₄×C₂₃	—	4	16	no	—
106	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	complemented
107	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
108	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
109	C₂²×C₄×C₂₃	C₂×D₄	1	48	yes	complemented
110	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
111	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
112	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
113	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
114	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
115	C₄⋊C₄×C₂₃	—	2	24	no	—
116	C₂²⋊C₄×C₂₃	—	2	24	no	—
117	C₂²⋊C₄×C₂₃	—	2	24	no	—
118	C₂²⋊C₄×C₂₃	—	2	24	no	—
119	C₂²⋊C₄×C₂₃	C₂²×C₄	1	48	yes	complemented
120	C₄²×C₂₃	—	2	24	no	—
121	C₂×C₄×#120	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₂₃	—	2	32	no	—
123	C₂³×C₂₃	—	2	96	no	—
124	C₂³×C₂₃	—	2	256	no	—
125	C₂³×C₂₃	—	2	48	no	—
126	C₂³×C₂₃	—	2	96	no	—
127	C₂³×C₂₃	—	2	24	no	—
128	C₂³×C₂₃	—	2	24	no	—
129	C₂³×C₂₃	—	2	48	no	—
130	C₂³×C₂₃	—	2	12	no	—
131	C₂³×C₂₃	—	2	12	no	—
132	C₂³×C₂₃	—	2	24	no	—
133	C₂³×C₂₃	—	2	6	no	—
134	C₂³×C₂₃	—	2	6	no	—
135	C₂³×C₂₃	—	4	4	no	—
136	C₂³×C₂₃	—	4	12	no	—
137	C₂³×C₂₃	C₄×D₄	1	48	yes	complemented
138	C₂³×C₂₃	C₂³×C₄	1	12	yes	—
139	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
140	C₂³×C₂₃	C₂³×C₄	1	4	yes	complemented
141	C₂³×C₂₃	C₂²×D₄	1	12	yes	—
142	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
143	C₂³×C₂₃	C₂×#41	1	12	yes	—
144	C₂³×C₂₃	C₂×#41	1	6	yes	—
145	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
146	D₄×C₂₃	—	2	96	no	—
147	C₂×C₄×C₂₃	—	2	96	no	—
148	C₂×C₄×C₂₃	—	2	24	no	—
149	C₂×C₄×C₂₃	—	2	24	no	—
150	C₂×C₄×C₂₃	—	2	6	no	—
151	C₂×C₄×C₂₃	—	2	12	no	—
152	C₂×C₄×C₂₃	—	2	6	no	—
153	C₂×C₄×C₂₃	—	2	96	no	—
154	C₂×C₄×C₂₃	—	2	6	no	—
155	C₂×C₄×C₂₃	—	2	6	no	—
156	C₂×C₄×C₂₃	—	2	12	no	—
157	C₂×C₄×C₂₃	—	4	4	no	—
158	C₂×C₄×C₂₃	—	4	12	no	—
159	C₂×C₄×C₂₃	—	4	16	no	—
160	C₂×C₄×C₂₃	#120	1	16	yes	direct product factor
161	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	complemented
162	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
163	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
164	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
165	C₂²×#120	C₂×C₂₃	1	1	yes	characteristic
166	C₂²×C₄×D₄	C₂×C₂₃	1	3	yes	complemented
167	C₄×#120	C₂×C₂₃	1	16	yes	direct product factor
168	C₂×#386	C₂×C₂₃	1	3	yes	complemented
169	C₂×#385	C₂×C₂₃	1	3	yes	complemented
170	C₂×#379	C₂×C₂₃	1	1	yes	characteristic, complemented
171	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	3	yes	complemented
172	C₂²×C₂₃	—	2	128	no	—
173	C₂²×C₂₃	—	2	24	no	—
174	C₂²×C₂₃	—	2	12	no	—
175	C₂²×C₂₃	—	2	12	no	—
176	C₂²×C₂₃	—	2	24	no	—
177	C₂²×C₂₃	—	2	96	no	—
178	C₂²×C₂₃	—	2	12	no	—
179	C₂²×C₂₃	—	4	2	no	—
180	C₂²×C₂₃	—	4	6	no	—
181	C₂²×C₂₃	—	4	16	no	—
182	C₂²×C₂₃	C₂×C₄×D₄	1	12	yes	—
183	C₂²×C₂₃	C₂×C₄×D₄	1	24	yes	complemented
184	C₂²×C₂₃	C₂×#120	1	4	yes	—
185	C₂²×C₂₃	#525	1	12	yes	—
186	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
187	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
188	C₂²×C₂₃	C₂²×#41	1	3	yes	—
189	C₄×C₂₃	—	2	12	no	—
190	C₄×C₂₃	—	2	12	no	—
191	C₄×C₂₃	—	4	2	no	—
192	C₄×C₂₃	—	4	6	no	—
193	C₄×C₂₃	C₂×#120	1	8	yes	direct product factor
194	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
195	C₂³×D₄	C₂²×C₂₃	1	3	yes	—
196	C₂⁴×C₄	—	2	24	no	—
197	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
198	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
199	C₂⁴×C₄	C₂²×C₂₃	1	8	yes	complemented
200	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
201	C₂×#120	C₄×C₂₃	1	16	yes	direct product factor
202	C₂×#120	C₂²×C₂₃	1	8	yes	—
203	C₂×C₄×D₄	—	2	24	no	—
204	C₂×C₄×D₄	C₂²×C₂₃	1	24	yes	complemented
205	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	complemented
206	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
207	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
208	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
209	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	complemented
210	C₂²×C₄²	C₂²×C₂₃	1	3	yes	complemented
211	#386	C₂²×C₂₃	1	24	yes	complemented
212	#385	C₂²×C₂₃	1	24	yes	complemented
213	#379	C₂²×C₂₃	1	8	yes	complemented
214	C₄×C₂²⋊C₄	C₂²×C₂₃	1	24	yes	complemented
215	C₂×#95	C₂²×C₂₃	1	6	yes	complemented
216	C₂×C₂₃	—	2	24	no	—
217	C₂×C₂₃	—	4	4	no	—
218	C₂×C₂₃	C₄×#120	1	8	yes	direct product factor
219	C₂×C₂₃	C₂²×C₄×D₄	1	3	yes	—
220	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
221	C₂×C₂₃	C₂×#525	1	3	yes	—
222	C₂⁵	—	2	12	no	—
223	C₂⁵	—	2	12	no	—
224	C₂⁵	C₂×C₄×C₂₃	1	8	yes	complemented
225	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
226	C₂⁵	C₂³×C₂₃	1	4	yes	—
227	C₂⁵	C₂³×C₂₃	1	3	yes	—
228	C₂²×D₄	—	2	12	no	—
229	C₂²×D₄	—	2	12	no	—
230	C₂²×D₄	C₂×C₄×C₂₃	1	24	yes	complemented
231	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
232	C₂³×C₄	—	2	128	no	—
233	C₂³×C₄	—	2	12	no	—
234	C₂³×C₄	—	2	24	no	—
235	C₂³×C₄	—	2	24	no	—
236	C₂³×C₄	—	2	96	no	—
237	C₂³×C₄	—	2	12	no	—
238	C₂³×C₄	D₄×C₂₃	1	24	yes	complemented
239	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
240	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
242	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
243	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
244	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
245	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₂₃	1	4	yes	complemented
247	#120	C₂×C₄×C₂₃	1	64	yes	direct product factor
248	C₄×D₄	—	2	96	no	—
249	C₂×C₄⋊C₄	—	2	12	no	—
250	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	complemented
251	C₂×C₂²⋊C₄	—	2	12	no	—
252	C₂×C₂²⋊C₄	—	2	12	no	—
253	C₂×C₂²⋊C₄	—	2	12	no	—
254	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	24	yes	complemented
255	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
256	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	complemented
259	C₂×C₄²	—	2	12	no	—
260	C₂×C₄²	C₂³×C₂₃	1	12	yes	complemented
261	#95	C₂³×C₂₃	1	24	yes	complemented
262	C₂₃	C₂×C₄×#120	1	1	yes	characteristic, direct product factor, Sylow
263	C₂⁴	—	2	32	no	—
264	C₂⁴	—	2	96	no	—
265	C₂⁴	—	2	96	no	—
266	C₂⁴	—	2	24	no	—
267	C₂⁴	—	2	12	no	—
268	C₂⁴	—	2	24	no	—
269	C₂⁴	—	2	12	no	—
270	C₂⁴	—	2	6	no	—
271	C₂⁴	—	2	6	no	—
272	C₂⁴	C₂²×C₄×C₂₃	1	12	yes	—
273	C₂⁴	C₂²×C₄×C₂₃	1	16	yes	complemented
274	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	complemented
275	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
276	C₂⁴	#41×C₂₃	1	12	yes	—
277	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
278	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
279	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
280	C₂×D₄	—	2	96	no	—
281	C₂×D₄	—	2	24	no	—
282	C₂×D₄	—	2	24	no	—
283	C₂×D₄	C₂²×C₄×C₂₃	1	48	yes	complemented
284	C₂²×C₄	—	2	96	no	—
285	C₂²×C₄	—	2	256	no	—
286	C₂²×C₄	—	2	6	no	—
287	C₂²×C₄	—	2	24	no	—
288	C₂²×C₄	—	2	12	no	—
289	C₂²×C₄	—	2	6	no	—
290	C₂²×C₄	—	2	96	no	—
291	C₂²×C₄	—	2	48	no	—
292	C₂²×C₄	—	2	48	no	—
293	C₂²×C₄	—	2	6	no	—
294	C₂²×C₄	—	2	6	no	—
295	C₂²×C₄	—	2	12	no	—
296	C₂²×C₄	—	2	24	no	—
297	C₂²×C₄	—	4	16	no	—
298	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	complemented
299	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
300	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
301	C₂²×C₄	C₂×D₄×C₂₃	1	48	yes	complemented
302	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
303	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
304	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
305	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
306	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
307	C₄⋊C₄	—	2	24	no	—
308	C₂²⋊C₄	—	2	24	no	—
309	C₂²⋊C₄	—	2	24	no	—
310	C₂²⋊C₄	—	2	24	no	—
311	C₂²⋊C₄	C₂²×C₄×C₂₃	1	48	yes	complemented
312	C₄²	—	2	24	no	—
313	C₂³	—	2	32	no	—
314	C₂³	—	2	96	no	—
315	C₂³	—	2	256	no	—
316	C₂³	—	2	12	no	—
317	C₂³	—	2	12	no	—
318	C₂³	—	2	48	no	—
319	C₂³	—	2	96	no	—
320	C₂³	—	2	24	no	—
321	C₂³	—	2	24	no	—
322	C₂³	—	2	48	no	—
323	C₂³	—	2	6	no	—
324	C₂³	—	2	6	no	—
325	C₂³	—	2	24	no	—
326	C₂³	—	4	4	no	—
327	C₂³	—	4	12	no	—
328	C₂³	C₄×D₄×C₂₃	1	48	yes	complemented
329	C₂³	C₂³×C₄×C₂₃	1	12	yes	—
330	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
331	C₂³	C₂³×C₄×C₂₃	1	4	yes	complemented
332	C₂³	C₂²×D₄×C₂₃	1	12	yes	—
333	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
334	C₂³	C₂×#41×C₂₃	1	12	yes	—
335	C₂³	C₂×#41×C₂₃	1	6	yes	—
336	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
337	D₄	—	2	96	no	—
338	C₂×C₄	—	2	96	no	—
339	C₂×C₄	—	2	6	no	—
340	C₂×C₄	—	2	12	no	—
341	C₂×C₄	—	2	6	no	—
342	C₂×C₄	—	2	24	no	—
343	C₂×C₄	—	2	24	no	—
344	C₂×C₄	—	2	96	no	—
345	C₂×C₄	—	2	6	no	—
346	C₂×C₄	—	2	6	no	—
347	C₂×C₄	—	2	12	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	12	no	—
350	C₂×C₄	—	4	16	no	—
351	C₂×C₄	C₂₃×#120	1	16	yes	direct product factor
352	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	complemented
353	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
354	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
355	C₂²	—	2	128	no	—
356	C₂²	—	2	24	no	—
357	C₂²	—	2	12	no	—
358	C₂²	—	2	12	no	—
359	C₂²	—	2	24	no	—
360	C₂²	—	2	96	no	—
361	C₂²	—	2	12	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	4	6	no	—
364	C₂²	—	4	16	no	—
365	C₂²	C₂×C₄×D₄×C₂₃	1	12	yes	—
366	C₂²	C₂×C₄×D₄×C₂₃	1	24	yes	complemented
367	C₂²	C₂×C₂₃×#120	1	4	yes	—
368	C₂²	C₂₃×#525	1	12	yes	—
369	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
370	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
371	C₂²	C₂²×#41×C₂₃	1	3	yes	—
372	C₄	—	2	12	no	—
373	C₄	—	2	12	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	6	no	—
376	C₄	C₂×C₂₃×#120	1	8	yes	direct product factor
377	C₂	—	2	24	no	—
378	C₂	—	4	4	no	—
379	C₂	C₄×C₂₃×#120	1	8	yes	direct product factor
380	C₂	C₂²×C₄×D₄×C₂₃	1	3	yes	—
381	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic
382	C₂	C₂×C₂₃×#525	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂⁴×C₂₃	C₂⁴
prime radical	C₂²×C₂₃×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₄×C₂₃	C₂	semidirect
3. C₂²×C₄×D₄×C₂₃	C₂	semidirect
4. C₄×C₂₃×#120	C₂	direct (N × H)
5. C₂×C₂₃×#386	C₂	semidirect
6. C₂×C₂₃×#385	C₂	semidirect
7. C₂×C₂₃×#379	C₂	semidirect
8. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
14. C₂⁴×C₄×C₂₃	C₂²	semidirect
15. C₂⁴×C₄×C₂₃	C₂²	semidirect
16. C₂×C₂₃×#120	C₄	direct (N × H)
19. C₂×C₄×D₄×C₂₃	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
25. C₂²×C₄²×C₂₃	C₂²	semidirect
26. C₂₃×#386	C₂²	semidirect
27. C₂₃×#385	C₂²	semidirect
28. C₂₃×#379	C₂²	semidirect
29. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
30. C₂×C₂₃×#95	C₂²	semidirect
33. C₂⁵×C₂₃	C₂×C₄	semidirect
39. C₂²×D₄×C₂₃	C₂×C₄	semidirect
47. C₂³×C₄×C₂₃	D₄	semidirect
55. C₂³×C₄×C₂₃	C₂³	semidirect
56. C₂₃×#120	C₂×C₄	direct (N × H)
59. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
63. C₂×C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
67. C₂×C₂²⋊C₄×C₂₃	C₂³	semidirect
69. C₂×C₄²×C₂₃	C₂³	semidirect
70. C₂₃×#95	C₂³	semidirect
81. C₂⁴×C₂₃	C₂²×C₄	semidirect
82. C₂⁴×C₂₃	C₂²×C₄	semidirect
91. C₂×D₄×C₂₃	C₂²×C₄	semidirect
106. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
109. C₂²×C₄×C₂₃	C₂×D₄	semidirect
119. C₂²⋊C₄×C₂₃	C₂²×C₄	semidirect
121. C₂×C₄×#120	C₂₃	direct (N × H)
137. C₂³×C₂₃	C₄×D₄	semidirect
140. C₂³×C₂₃	C₂³×C₄	semidirect
160. C₂×C₄×C₂₃	#120	direct (N × H)
161. C₂×C₄×C₂₃	C₂³×C₄	semidirect
164. C₂⁵×C₄	C₂×C₂₃	semidirect
166. C₂²×C₄×D₄	C₂×C₂₃	semidirect
167. C₄×#120	C₂×C₂₃	direct (N × H)
168. C₂×#386	C₂×C₂₃	semidirect
169. C₂×#385	C₂×C₂₃	semidirect
170. C₂×#379	C₂×C₂₃	semidirect
171. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
183. C₂²×C₂₃	C₂×C₄×D₄	semidirect
193. C₄×C₂₃	C₂×#120	direct (N × H)
199. C₂⁴×C₄	C₂²×C₂₃	semidirect
200. C₂⁴×C₄	C₂²×C₂₃	semidirect
201. C₂×#120	C₄×C₂₃	direct (N × H)
204. C₂×C₄×D₄	C₂²×C₂₃	semidirect
205. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
209. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
210. C₂²×C₄²	C₂²×C₂₃	semidirect
211. #386	C₂²×C₂₃	semidirect
212. #385	C₂²×C₂₃	semidirect
213. #379	C₂²×C₂₃	semidirect
214. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
215. C₂×#95	C₂²×C₂₃	semidirect
218. C₂×C₂₃	C₄×#120	direct (N × H)
224. C₂⁵	C₂×C₄×C₂₃	semidirect
230. C₂²×D₄	C₂×C₄×C₂₃	semidirect
238. C₂³×C₄	D₄×C₂₃	semidirect
246. C₂³×C₄	C₂³×C₂₃	semidirect
247. #120	C₂×C₄×C₂₃	direct (N × H)
250. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
254. C₂×C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
258. C₂×C₂²⋊C₄	C₂³×C₂₃	semidirect
260. C₂×C₄²	C₂³×C₂₃	semidirect
261. #95	C₂³×C₂₃	semidirect
262. C₂₃	C₂×C₄×#120	direct (N × H)
273. C₂⁴	C₂²×C₄×C₂₃	semidirect
274. C₂⁴	C₂²×C₄×C₂₃	semidirect
283. C₂×D₄	C₂²×C₄×C₂₃	semidirect
298. C₂²×C₄	C₂²×C₄×C₂₃	semidirect
301. C₂²×C₄	C₂×D₄×C₂₃	semidirect
311. C₂²⋊C₄	C₂²×C₄×C₂₃	semidirect
328. C₂³	C₄×D₄×C₂₃	semidirect
331. C₂³	C₂³×C₄×C₂₃	semidirect
351. C₂×C₄	C₂₃×#120	direct (N × H)
352. C₂×C₄	C₂³×C₄×C₂₃	semidirect
366. C₂²	C₂×C₄×D₄×C₂₃	semidirect
376. C₄	C₂×C₂₃×#120	direct (N × H)
379. C₂	C₄×C₂₃×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₂₃	2
C₂²×C₂₃×#120	2
C₂²×C₄×D₄×C₂₃	2
C₄×C₂₃×#120	2
C₂×C₂₃×#386	2
C₂×C₂₃×#385	2
C₂×C₂₃×#379	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂×C₄×#120	23

Subgroups — C2×C4×C23×#120 #19181653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂₃×#120 #19181653