Subgroups — #15681918

Order 2944.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625155	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625072	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092898	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092422	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092420	C₂	1	2	yes	maximal, complemented
6	#13092416	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092415	C₂	1	2	yes	maximal, complemented
8	#13092414	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092410	C₂	1	2	yes	maximal, complemented
10	#102×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₂₃	—	2	2	no	—
14	#10625155	—	2	4	no	—
15	#10625155	C₄	1	8	yes	complemented
16	C₂×#67290	—	2	1	no	—
17	C₂×#67290	C₂²	1	2	yes	—
18	C₂×D₄×D₂₃	—	2	1	no	—
19	C₂×D₄×D₂₃	C₂²	1	2	yes	—
20	C₂×#67288	C₂²	1	1	yes	characteristic
21	C₂×#67288	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₄×C₂²⋊D₂₃	—	2	2	no	—
25	#10625097	C₂²	1	1	yes	characteristic, complemented
26	C₂×#67267	C₂²	1	1	yes	characteristic, complemented
27	#10625072	—	2	4	no	—
28	C₂×#67262	C₂²	1	2	yes	complemented
29	C₂×C₈×D₂₃	C₂²	1	2	yes	complemented
30	#10625064	—	2	1	no	—
31	#10625063	C₂²	1	2	yes	complemented
32	C₄⋊C₄×D₂₃	C₂²	1	2	yes	complemented
33	#10625058	—	2	1	no	—
34	C₂₃×#102	C₂²	1	2	yes	complemented
35	C₂₃×#102	C₂²	1	2	yes	complemented
36	#10625014	C₂²	1	2	yes	complemented
37	#10625014	C₂²	1	2	yes	complemented
38	#10625003	C₂²	1	1	yes	characteristic, complemented
39	#10625003	C₂²	1	1	yes	characteristic, complemented
40	#10624998	C₂²	1	1	yes	characteristic, complemented
41	#10624998	C₂²	1	1	yes	characteristic, complemented
42	#10624992	C₂²	1	2	yes	complemented
43	#10624991	C₂²	1	2	yes	complemented
44	C₂⁴×C₂₃	—	2	1	no	—
45	C₂³×D₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
49	C₂×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₂₃	—	2	1	no	—
53	C₂×C₂²⋊D₂₃	—	2	2	no	—
54	C₂×C₂²⋊D₂₃	—	2	16	no	—
55	C₂×C₂²⋊D₂₃	—	2	2	no	—
56	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
57	C₂²×Dic₂₃	—	2	2	no	—
58	#67290	—	2	4	no	—
59	#67290	—	2	8	no	—
60	#67290	C₂×C₄	1	8	yes	complemented
61	D₄×D₂₃	—	2	8	no	—
62	D₄×D₂₃	—	2	4	no	—
63	D₄×D₂₃	C₂×C₄	1	8	yes	complemented
64	#67288	—	2	2	no	—
65	#67288	C₂×C₄	1	4	yes	—
66	#67288	C₂×C₄	1	4	yes	complemented
67	#67288	D₄	1	4	yes	—
68	C₂×D₉₂	D₄	1	1	yes	characteristic
69	C₂×D₉₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₂₃	—	2	1	no	—
71	C₂×C₄×D₂₃	—	2	2	no	—
72	C₂×C₄×D₂₃	C₂³	1	2	yes	—
73	C₂×Dic₄₆	D₄	1	1	yes	characteristic
74	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₂₃	—	2	1	no	—
81	(C₂×D₂₃)⋊C₄	—	2	2	no	—
82	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
84	Dic₂₃⋊C₄	—	2	2	no	—
85	C₄×Dic₂₃	—	2	1	no	—
86	C₄×Dic₂₃	—	2	2	no	—
87	#67267	—	2	2	no	—
88	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
89	#67262	—	2	4	no	—
90	C₈×D₂₃	—	2	4	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	4	no	—
93	C₂³×C₂₃	—	8	1	no	—
94	C₂³×C₂₃	—	8	1	no	—
95	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×D₂₃	—	2	1	no	—
97	C₂²×D₂₃	—	2	8	no	—
98	C₂²×D₂₃	—	8	2	no	—
99	C₂²×D₂₃	—	8	2	no	—
100	C₂²×D₂₃	C₂×D₄	1	2	yes	—
101	D₄×C₂₃	—	2	4	no	—
102	D₄×C₂₃	—	2	2	no	—
103	D₄×C₂₃	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	8	1	no	—
106	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₂₃	—	2	4	no	—
111	C₂²⋊D₂₃	—	2	8	no	—
112	C₂²⋊D₂₃	—	2	8	no	—
113	C₂²⋊D₂₃	C₂²⋊C₄	1	8	yes	—
114	C₂×Dic₂₃	—	2	1	no	—
115	C₂×Dic₂₃	—	2	8	no	—
116	C₂×Dic₂₃	—	2	1	no	—
117	C₂×Dic₂₃	—	2	1	no	—
118	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
119	D₉₂	—	2	1	no	—
120	D₉₂	C₂²⋊C₄	1	4	yes	—
121	D₉₂	C₂²×C₄	1	2	yes	—
122	C₄×D₂₃	—	2	4	no	—
123	C₄×D₂₃	—	8	2	no	—
124	C₄×D₂₃	C₂²×C₄	1	4	yes	—
125	C₄×D₂₃	C₂×D₄	1	4	yes	—
126	Dic₄₆	—	2	1	no	—
127	Dic₄₆	C₂²⋊C₄	1	4	yes	—
128	Dic₄₆	C₂²×C₄	1	2	yes	—
129	C₈×C₂₃	—	2	1	no	—
130	C₈×C₂₃	—	2	1	no	—
131	C₂₃⋊C₈	—	2	2	no	—
132	#2897	—	23	1	no	maximal, Sylow
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	4	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	—	8	4	no	—
139	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₂₃	—	2	2	no	—
142	C₂×D₂₃	—	2	2	no	—
143	C₂×D₂₃	—	2	2	no	—
144	C₂×D₂₃	—	4	1	no	—
145	C₂×D₂₃	—	4	1	no	—
146	C₂×D₂₃	—	8	8	no	—
147	C₂×D₂₃	C₂×C₂²⋊C₄	1	4	yes	—
148	C₄×C₂₃	—	4	2	no	—
149	C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
151	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
152	Dic₂₃	—	2	2	no	—
153	Dic₂₃	—	4	1	no	—
154	Dic₂₃	—	4	1	no	—
155	Dic₂₃	C₂×C₂²⋊C₄	1	4	yes	—
156	C₂×H(5,𝔽₂)	—	23	1	no	—
157	C₂×#131	—	23	1	no	—
158	#520	—	23	1	no	—
159	#418	—	23	1	no	—
160	#418	—	23	2	no	—
161	#417	—	23	1	no	—
162	#417	—	23	2	no	—
163	#416	—	23	1	no	—
164	#416	—	23	2	no	—
165	C₂×#102	—	23	2	no	—
166	C₂×#102	D₂₃	1	1	yes	characteristic, complemented
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	4	4	no	—
169	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₂₃	#576	1	1	yes	characteristic
171	C₂×C₂₃	#577	1	1	yes	characteristic
172	D₂₃	—	2	4	no	—
173	D₂₃	—	4	2	no	—
174	H(5,𝔽₂)	—	23	8	no	—
175	H(5,𝔽₂)	—	46	4	no	—
176	C₂×#41	—	23	1	no	—
177	C₂×#41	—	23	2	no	—
178	C₂×#41	—	23	1	no	—
179	C₂×#41	—	46	1	no	—
180	C₂²×D₄	—	23	2	no	—
181	C₂²×D₄	—	46	1	no	—
182	C₂²×D₄	—	46	2	no	—
183	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
184	#131	—	46	4	no	—
185	C₂×M₁₆	—	23	1	no	—
186	C₂×M₁₆	—	23	2	no	—
187	C₂²×C₈	—	23	2	no	—
188	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
189	C₄×Q₈	—	46	1	no	—
190	C₄×D₄	—	46	1	no	—
191	C₄×D₄	—	46	2	no	—
192	#117	—	23	1	no	—
193	#117	—	23	2	no	—
194	C₂×C₄⋊C₄	—	23	2	no	—
195	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic, complemented
196	#103	—	23	1	no	—
197	#103	—	23	1	no	—
198	#103	—	23	2	no	—
199	#102	—	23	2	no	—
200	#102	—	23	2	no	—
201	#102	—	23	1	no	—
202	#102	—	23	1	no	—
203	#102	—	23	2	no	—
204	#102	C₂×D₂₃	1	2	yes	complemented
205	#102	C₂×D₂₃	1	2	yes	complemented
206	C₂₃	#2897	1	1	yes	characteristic, complemented, Sylow
207	C₂⁴	—	2	1	no	—
208	C₂⁴	—	46	2	no	—
209	#41	—	23	4	no	—
210	#41	—	23	4	no	—
211	#41	—	23	8	no	—
212	#41	—	23	4	no	—
213	#41	—	46	4	no	—
214	#41	—	46	8	no	—
215	#41	—	46	2	no	—
216	C₂×Q₈	—	23	1	no	—
217	C₂×Q₈	—	23	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	23	2	no	—
221	C₂×D₄	—	23	8	no	—
222	C₂×D₄	—	23	1	no	—
223	C₂×D₄	—	23	1	no	—
224	C₂×D₄	—	46	4	no	—
225	C₂×D₄	—	46	2	no	—
226	C₂×D₄	—	46	8	no	—
227	C₂×D₄	—	46	16	no	—
228	C₂×D₄	—	46	2	no	—
229	C₂×D₄	—	46	1	no	—
230	C₂×D₄	C₄×D₂₃	1	4	yes	complemented
231	C₂×D₄	C₂²×D₂₃	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	23	2	no	—
234	C₂²×C₄	—	46	2	no	—
235	C₂²×C₄	—	46	2	no	—
236	C₂²×C₄	—	46	1	no	—
237	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
238	M₁₆	—	46	4	no	—
239	M₁₆	—	46	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	23	2	no	—
242	C₂×C₈	—	46	4	no	—
243	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	23	1	no	—
247	C₄⋊C₄	—	23	1	no	—
248	C₄⋊C₄	—	46	2	no	—
249	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
250	C₂²⋊C₄	—	46	2	no	—
251	C₂²⋊C₄	—	46	1	no	—
252	C₄²	—	46	2	no	—
253	C₄²	—	46	1	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	8	1	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	23	2	no	—
259	C₂³	—	46	8	no	—
260	C₂³	—	46	1	no	—
261	C₂³	—	184	2	no	—
262	C₂³	—	184	2	no	—
263	C₂³	D₄×D₂₃	1	1	yes	characteristic
264	Q₈	—	23	4	no	—
265	Q₈	—	23	2	no	—
266	Q₈	—	46	1	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	—	23	8	no	—
270	D₄	—	23	4	no	—
271	D₄	—	23	2	no	—
272	D₄	—	46	8	no	—
273	D₄	—	46	8	no	—
274	D₄	—	46	4	no	—
275	D₄	—	46	1	no	—
276	D₄	C₂×C₄×D₂₃	1	4	yes	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	23	2	no	—
280	C₂×C₄	—	23	4	no	—
281	C₂×C₄	—	23	4	no	—
282	C₂×C₄	—	46	8	no	—
283	C₂×C₄	—	46	1	no	—
284	C₂×C₄	—	46	4	no	—
285	C₂×C₄	—	46	1	no	—
286	C₂×C₄	—	46	1	no	—
287	C₂×C₄	—	184	2	no	—
288	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
289	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
290	C₂×C₄	D₄×D₂₃	1	2	yes	—
291	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	46	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	23	4	no	—
302	C₂²	—	46	2	no	—
303	C₂²	—	46	2	no	—
304	C₂²	—	46	2	no	—
305	C₂²	—	92	1	no	—
306	C₂²	—	92	1	no	—
307	C₂²	—	184	8	no	—
308	C₂²	C₂²⋊C₄×D₂₃	1	2	yes	—
309	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	23	4	no	—
312	C₄	—	46	2	no	—
313	C₄	—	92	1	no	—
314	C₄	—	92	1	no	—
315	C₄	C₂²⋊C₄×D₂₃	1	2	yes	—
316	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
317	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	4	no	—
320	C₂	—	46	4	no	—
321	C₂	—	92	2	no	—
322	C₂	C₂×C₂²⋊C₄×D₂₃	1	1	yes	characteristic
323	C₂	#13093218	1	1	yes	characteristic
324	C₂	#13093223	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#102	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625155	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#102	D₂₃
hypocenter	C₂₃	#2897
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#102

Complements

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

N	H ≅ G/N	type
2. C₂×#10625072	C₂	semidirect
3. #13092898	C₂	semidirect
4. #13092422	C₂	semidirect
5. #13092420	C₂	semidirect
6. #13092416	C₂	semidirect
7. #13092415	C₂	semidirect
8. #13092414	C₂	semidirect
9. #13092410	C₂	semidirect
10. #102×D₂₃	C₂	semidirect
11. C₂×C₂₃×#102	C₂	semidirect
15. #10625155	C₄	semidirect
22. C₂²×C₈×C₂₃	C₂²	semidirect
23. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
25. #10625097	C₂²	semidirect
26. C₂×#67267	C₂²	semidirect
28. C₂×#67262	C₂²	semidirect
29. C₂×C₈×D₂₃	C₂²	semidirect
31. #10625063	C₂²	semidirect
32. C₄⋊C₄×D₂₃	C₂²	semidirect
34. C₂₃×#102	C₂²	semidirect
35. C₂₃×#102	C₂²	semidirect
36. #10625014	C₂²	semidirect
37. #10625014	C₂²	semidirect
38. #10625003	C₂²	semidirect
39. #10625003	C₂²	semidirect
40. #10624998	C₂²	semidirect
41. #10624998	C₂²	semidirect
42. #10624992	C₂²	semidirect
43. #10624991	C₂²	semidirect
48. C₂×D₄×C₂₃	C₂×C₄	semidirect
60. #67290	C₂×C₄	semidirect
63. D₄×D₂₃	C₂×C₄	semidirect
66. #67288	C₂×C₄	semidirect
166. C₂×#102	D₂₃	semidirect
188. C₂²×C₈	C₂×D₂₃	semidirect
195. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
204. #102	C₂×D₂₃	semidirect
205. #102	C₂×D₂₃	semidirect
206. C₂₃	#2897	semidirect
230. C₂×D₄	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625155	2
C₂×#10625072	2
#13092898	2
#13092422	2
#13092420	2
#13092416	2
#13092415	2
#13092414	2
#13092410	2
#102×D₂₃	2
C₂×C₂₃×#102	2
#2897	23