Subgroups — C₂×C₂₃×#2396 #19182169

Order 5888.

2966 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2396	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#399	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#398	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#396	C₂	1	2	yes	maximal
8	C₂×C₂₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#393	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
13	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₂₃	—	2	8	no	—
16	C₂×C₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
17	C₂×C₂₃×#117	—	2	8	no	—
18	C₂×C₂₃×#117	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
32	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
33	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#400	C₂²	1	8	yes	complemented
35	C₂₃×#399	C₂²	1	8	yes	—
36	C₂₃×#398	C₂²	1	16	yes	complemented
37	C₂₃×#396	C₂²	1	16	yes	—
38	C₂₃×#395	C₂²	1	8	yes	complemented
39	C₂₃×#393	C₂²	1	8	yes	—
40	C₄×C₄⋊C₄×C₂₃	C₂²	1	8	yes	complemented
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	16	yes	complemented
42	C₄×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
43	C₂×C₂₃×#95	C₂²	1	2	yes	—
44	C₂×C₂₃×#95	C₂²	1	2	yes	—
45	C₂×C₂₃×#95	C₂²	1	2	yes	—
46	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#95	C₂²	1	2	yes	—
49	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₂₃	—	2	4	no	—
51	C₂²×Q₈×C₂₃	—	2	8	no	—
52	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
53	C₂³×C₄×C₂₃	—	2	4	no	—
54	C₂³×C₄×C₂₃	—	2	4	no	—
55	C₂³×C₄×C₂₃	Q₈	1	8	yes	complemented
56	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
64	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₂₃	—	2	32	no	—
68	C₂₃×#117	—	2	32	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₄²×C₂₃	—	2	4	no	—
92	C₂×C₄²×C₂₃	—	2	8	no	—
93	C₂×C₄²×C₂₃	—	2	8	no	—
94	C₂×C₄²×C₂₃	D₄	1	8	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
98	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
99	C₂₃×#95	C₂³	1	8	yes	—
100	C₂₃×#95	C₂³	1	8	yes	—
101	C₂₃×#95	C₂³	1	8	yes	—
102	C₂₃×#95	C₂³	1	4	yes	—
103	C₂₃×#95	C₂³	1	4	yes	—
104	C₂₃×#95	C₂³	1	8	yes	—
105	C₂⁴×C₂₃	—	2	2	no	—
106	C₂⁴×C₂₃	—	2	4	no	—
107	C₂⁴×C₂₃	—	2	4	no	—
108	C₂⁴×C₂₃	—	2	2	no	—
109	C₂⁴×C₂₃	C₂×Q₈	1	4	yes	—
110	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
111	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₂×Q₈×C₂₃	—	2	32	no	—
113	C₂×Q₈×C₂₃	—	2	8	no	—
114	C₂×Q₈×C₂₃	—	2	16	no	—
115	C₂²×C₄×C₂₃	—	2	4	no	—
116	C₂²×C₄×C₂₃	—	2	2	no	—
117	C₂²×C₄×C₂₃	—	2	4	no	—
118	C₂²×C₄×C₂₃	—	2	4	no	—
119	C₂²×C₄×C₂₃	—	2	4	no	—
120	C₂²×C₄×C₂₃	—	2	2	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	2	4	no	—
123	C₂²×C₄×C₂₃	—	2	32	no	—
124	C₂²×C₄×C₂₃	—	2	8	no	—
125	C₂²×C₄×C₂₃	—	2	8	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂×Q₈	1	16	yes	complemented
142	C₂²×C₄×C₂₃	#41	1	8	yes	—
143	C₂²×C₄×C₂₃	#41	1	8	yes	—
144	C₂²×C₄×C₂₃	#41	1	4	yes	—
145	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
155	C₄⋊C₄×C₂₃	—	2	8	no	—
156	C₄⋊C₄×C₂₃	—	2	8	no	—
157	C₄⋊C₄×C₂₃	—	2	16	no	—
158	C₄⋊C₄×C₂₃	—	2	8	no	—
159	C₄⋊C₄×C₂₃	—	2	16	no	—
160	C₄⋊C₄×C₂₃	—	2	16	no	—
161	C₄⋊C₄×C₂₃	—	2	16	no	—
162	C₂²⋊C₄×C₂₃	—	2	8	no	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	16	no	—
165	C₂²⋊C₄×C₂₃	—	2	16	no	—
166	C₄²×C₂₃	—	2	8	no	—
167	C₄²×C₂₃	—	2	16	no	—
168	C₄²×C₂₃	—	2	16	no	—
169	C₄²×C₂₃	C₂×D₄	1	16	yes	—
170	C₂×#2396	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	2	8	no	—
172	C₂³×C₂₃	—	2	16	no	—
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	4	no	—
175	C₂³×C₂₃	—	2	4	no	—
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	4	4	no	—
178	C₂³×C₂₃	—	4	8	no	—
179	C₂³×C₂₃	—	4	4	no	—
180	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
181	C₂³×C₂₃	C₂²×Q₈	1	4	yes	—
182	C₂³×C₂₃	C₂²×Q₈	1	2	yes	—
183	C₂³×C₂₃	C₂×#41	1	2	yes	—
184	C₂³×C₂₃	C₂×#41	1	4	yes	—
185	C₂³×C₂₃	C₂×#41	1	4	yes	—
186	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
187	Q₈×C₂₃	—	2	32	no	—
188	C₂×C₄×C₂₃	—	2	8	no	—
189	C₂×C₄×C₂₃	—	2	8	no	—
190	C₂×C₄×C₂₃	—	2	16	no	—
191	C₂×C₄×C₂₃	—	2	16	no	—
192	C₂×C₄×C₂₃	—	2	8	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	32	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	4	no	—
210	C₂×C₄×C₂₃	—	2	4	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	4	4	no	—
213	C₂×C₄×C₂₃	—	4	8	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	4	8	no	—
216	C₂×C₄×C₂₃	C₂²⋊Q₈	1	16	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
221	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
222	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
223	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
224	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
225	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic
227	#2396	C₂×C₂₃	1	16	yes	direct product factor
228	C₂×#400	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×#399	C₂×C₂₃	1	1	yes	characteristic
230	C₂×#398	C₂×C₂₃	1	2	yes	complemented
231	C₂×#396	C₂×C₂₃	1	2	yes	—
232	C₂×#395	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂×#393	C₂×C₂₃	1	1	yes	characteristic
234	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
235	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	2	8	no	—
239	C₂²×C₂₃	—	4	16	no	—
240	C₂²×C₂₃	—	4	2	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	4	yes	—
244	C₂²×C₂₃	#525	1	4	yes	—
245	C₂²×C₂₃	#529	1	8	yes	—
246	C₂²×C₂₃	#532	1	4	yes	—
247	C₂²×C₂₃	#533	1	8	yes	—
248	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
249	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
250	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
251	C₂²×C₂₃	C₂²×#41	1	2	yes	—
252	C₂²×C₂₃	C₂²×#41	1	2	yes	—
253	C₄×C₂₃	—	2	4	no	—
254	C₄×C₂₃	—	2	4	no	—
255	C₄×C₂₃	—	2	8	no	—
256	C₄×C₂₃	—	2	8	no	—
257	C₄×C₂₃	—	2	4	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	4	4	no	—
260	C₄×C₂₃	—	4	2	no	—
261	C₄×C₂₃	—	4	4	no	—
262	C₄×C₂₃	C₂×C₂²⋊Q₈	1	8	yes	—
263	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
264	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄×Q₈	—	2	8	no	—
266	C₂×C₄×Q₈	C₂²×C₂₃	1	8	yes	complemented
267	C₂×#117	—	2	8	no	—
268	C₂×#117	C₂²×C₂₃	1	8	yes	—
269	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
270	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
271	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
272	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
273	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
276	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
277	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
278	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
279	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
280	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
281	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
282	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
283	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
284	#400	C₂²×C₂₃	1	8	yes	complemented
285	#399	C₂²×C₂₃	1	8	yes	—
286	#398	C₂²×C₂₃	1	16	yes	complemented
287	#396	C₂²×C₂₃	1	16	yes	—
288	#395	C₂²×C₂₃	1	8	yes	complemented
289	#393	C₂²×C₂₃	1	8	yes	—
290	C₄×C₄⋊C₄	C₂²×C₂₃	1	16	yes	complemented
291	C₄×C₄⋊C₄	C₂²×C₂₃	1	8	yes	complemented
292	C₄×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
293	C₂×#95	C₂²×C₂₃	1	2	yes	—
294	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×#95	C₂²×C₂₃	1	2	yes	—
296	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#95	C₂²×C₂₃	1	2	yes	—
298	C₂×#95	C₂²×C₂₃	1	2	yes	—
299	C₂×C₂₃	—	4	4	no	—
300	C₂×C₂₃	#2396	1	8	yes	direct product factor
301	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
302	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
303	C₂×C₂₃	C₂×#529	1	2	yes	—
304	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
305	C₂×C₂₃	C₂×#533	1	2	yes	—
306	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
307	C₂²×Q₈	—	2	4	no	—
308	C₂²×Q₈	—	2	8	no	—
309	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
310	C₂³×C₄	—	2	4	no	—
311	C₂³×C₄	—	2	4	no	—
312	C₂³×C₄	Q₈×C₂₃	1	8	yes	complemented
313	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
316	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
317	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
321	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
322	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
323	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
324	C₄×Q₈	—	2	32	no	—
325	#117	—	2	32	no	—
326	C₂×C₄⋊C₄	—	2	4	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	—	2	8	no	—
329	C₂×C₄⋊C₄	—	2	4	no	—
330	C₂×C₄⋊C₄	—	2	8	no	—
331	C₂×C₄⋊C₄	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	8	no	—
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
338	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
340	C₂×C₂²⋊C₄	—	2	4	no	—
341	C₂×C₂²⋊C₄	—	2	4	no	—
342	C₂×C₂²⋊C₄	—	2	8	no	—
343	C₂×C₂²⋊C₄	—	2	8	no	—
344	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
348	C₂×C₄²	—	2	4	no	—
349	C₂×C₄²	—	2	8	no	—
350	C₂×C₄²	—	2	8	no	—
351	C₂×C₄²	D₄×C₂₃	1	8	yes	—
352	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
353	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
354	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
355	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
356	#95	C₂³×C₂₃	1	8	yes	—
357	#95	C₂³×C₂₃	1	8	yes	—
358	#95	C₂³×C₂₃	1	8	yes	—
359	#95	C₂³×C₂₃	1	4	yes	—
360	#95	C₂³×C₂₃	1	4	yes	—
361	#95	C₂³×C₂₃	1	8	yes	—
362	C₂₃	C₂×#2396	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	2	no	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	—	2	2	no	—
367	C₂⁴	C₂×Q₈×C₂₃	1	4	yes	—
368	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
369	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
370	C₂×Q₈	—	2	32	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	16	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	32	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
397	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
398	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂×Q₈×C₂₃	1	16	yes	complemented
400	C₂²×C₄	#41×C₂₃	1	8	yes	—
401	C₂²×C₄	#41×C₂₃	1	8	yes	—
402	C₂²×C₄	#41×C₂₃	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	16	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	16	no	—
418	C₄⋊C₄	—	2	16	no	—
419	C₄⋊C₄	—	2	16	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	16	no	—
423	C₂²⋊C₄	—	2	16	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	16	no	—
426	C₄²	—	2	16	no	—
427	C₄²	C₂×D₄×C₂₃	1	16	yes	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	8	no	—
433	C₂³	—	2	16	no	—
434	C₂³	—	4	4	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	4	no	—
437	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
438	C₂³	C₂²×Q₈×C₂₃	1	4	yes	—
439	C₂³	C₂²×Q₈×C₂₃	1	2	yes	—
440	C₂³	C₂×#41×C₂₃	1	2	yes	—
441	C₂³	C₂×#41×C₂₃	1	4	yes	—
442	C₂³	C₂×#41×C₂₃	1	4	yes	—
443	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
444	Q₈	—	2	32	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	8	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	16	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	32	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	8	no	—
473	C₂×C₄	C₂₃×C₂²⋊Q₈	1	16	yes	—
474	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
475	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
476	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
477	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
478	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
479	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
480	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
481	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	8	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	16	no	—
488	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	4	yes	—
489	C₂²	C₂₃×#525	1	4	yes	—
490	C₂²	C₂₃×#529	1	8	yes	—
491	C₂²	C₂₃×#532	1	4	yes	—
492	C₂²	C₂₃×#533	1	8	yes	—
493	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
494	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
495	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
496	C₂²	C₂²×#41×C₂₃	1	2	yes	—
497	C₂²	C₂²×#41×C₂₃	1	2	yes	—
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	8	no	—
501	C₄	—	2	8	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	4	no	—
507	C₄	C₂×C₂₃×C₂²⋊Q₈	1	8	yes	—
508	C₂	—	4	4	no	—
509	C₂	C₂₃×#2396	1	8	yes	direct product factor
510	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
511	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
512	C₂	C₂×C₂₃×#529	1	2	yes	—
513	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
514	C₂	C₂×C₂₃×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#2396	C₂	direct (N × H)
4. C₂×C₂₃×#400	C₂	semidirect
6. C₂×C₂₃×#398	C₂	semidirect
8. C₂×C₂₃×#395	C₂	semidirect
10. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
16. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
34. C₂₃×#400	C₂²	semidirect
36. C₂₃×#398	C₂²	semidirect
38. C₂₃×#395	C₂²	semidirect
40. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
41. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
55. C₂³×C₄×C₂₃	Q₈	semidirect
141. C₂²×C₄×C₂₃	C₂×Q₈	semidirect
170. C₂×#2396	C₂₃	direct (N × H)
225. C₂²×C₄×Q₈	C₂×C₂₃	semidirect
227. #2396	C₂×C₂₃	direct (N × H)
228. C₂×#400	C₂×C₂₃	semidirect
230. C₂×#398	C₂×C₂₃	semidirect
232. C₂×#395	C₂×C₂₃	semidirect
234. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
235. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
266. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
284. #400	C₂²×C₂₃	semidirect
286. #398	C₂²×C₂₃	semidirect
288. #395	C₂²×C₂₃	semidirect
290. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
291. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
300. C₂×C₂₃	#2396	direct (N × H)
312. C₂³×C₄	Q₈×C₂₃	semidirect
362. C₂₃	C₂×#2396	direct (N × H)
399. C₂²×C₄	C₂×Q₈×C₂₃	semidirect
509. C₂	C₂₃×#2396	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×C₄×Q₈×C₂₃	2
C₂²×C₂₃×#117	2
C₂₃×#2396	2
C₂×C₂₃×#400	2
C₂×C₂₃×#399	2
C₂×C₂₃×#398	2
C₂×C₂₃×#396	2
C₂×C₂₃×#395	2
C₂×C₂₃×#393	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂×#2396	23

Subgroups — C2×C23×#2396 #19182169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2396 #19182169