Subgroups — #15685184

Order 2944.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093234	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093224	C₂	1	2	yes	maximal, complemented
3	#13093222	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092801	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092692	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092687	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092681	C₂	1	2	yes	maximal, complemented
8	#13092596	C₂	1	2	yes	maximal, complemented
9	#13092591	C₂	1	2	yes	maximal, complemented
10	#13092535	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#10625161	C₂²	1	2	yes	complemented
13	#10625160	C₂²	1	2	yes	—
14	C₂₃×#135	C₂²	1	1	yes	characteristic
15	C₂₃×#131	C₂²	1	1	yes	characteristic
16	C₂₃×#131	C₂²	1	1	yes	characteristic, complemented
17	#10625122	C₂²	1	2	yes	—
18	#10625120	C₂²	1	2	yes	complemented
19	#10625089	—	2	2	no	—
20	#10625087	—	2	2	no	—
21	SD₁₆×D₂₃	—	2	2	no	—
22	#10625082	—	2	2	no	—
23	#10625080	—	2	2	no	—
24	#10625080	C₂²	1	2	yes	complemented
25	#10625079	—	2	2	no	—
26	#10625079	C₂²	1	2	yes	—
27	C₂×Dic₉₂	—	2	1	no	—
28	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
29	#10625075	—	2	1	no	—
30	#10625075	—	2	1	no	—
31	#10625075	C₂²	1	1	yes	characteristic
32	C₂×D₁₈₄	—	2	1	no	—
33	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
34	C₂×#67263	C₂²	1	2	yes	complemented
35	#10625072	C₂²	1	2	yes	complemented
36	#10625046	C₂²	1	2	yes	complemented
37	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#104	C₂²	1	1	yes	characteristic
39	C₂₃×#104	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
41	#10625006	C₂²	1	2	yes	complemented
42	#10625005	C₂²	1	2	yes	complemented
43	#10625001	C₂²	1	1	yes	characteristic, complemented
44	#10624987	C₂²	1	2	yes	complemented
45	#41×C₂₃	C₂³	1	1	yes	characteristic
46	#41×C₂₃	C₂³	1	1	yes	characteristic
47	#67292	—	2	2	no	—
48	Q₈×D₂₃	—	2	2	no	—
49	#67290	—	2	2	no	—
50	#67290	—	4	2	no	—
51	D₄×D₂₃	—	2	2	no	—
52	D₄×D₂₃	—	4	2	no	—
53	#67288	—	2	2	no	—
54	#67288	—	2	2	no	—
55	#67288	C₂³	1	2	yes	—
56	C₂×D₉₂	—	2	2	no	—
57	C₂×D₉₂	C₂³	1	2	yes	—
58	C₂×Dic₄₆	—	2	2	no	—
59	C₂×Dic₄₆	C₂³	1	2	yes	—
60	Q₁₆×C₂₃	D₄	1	1	yes	characteristic
61	SD₁₆×C₂₃	D₄	1	1	yes	characteristic
62	SD₁₆×C₂₃	D₄	1	1	yes	characteristic
63	D₈×C₂₃	D₄	1	1	yes	characteristic
64	M₁₆×C₂₃	—	2	1	no	—
65	M₁₆×C₂₃	—	2	1	no	—
66	M₁₆×C₂₃	C₂³	1	1	yes	characteristic
67	M₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	#67275	—	2	2	no	—
74	#67274	—	2	2	no	—
75	#67273	—	2	2	no	—
76	#67272	—	2	2	no	—
77	(C₂×D₂₃)⋊C₄	—	4	2	no	—
78	#67267	C₂³	1	2	yes	—
79	Dic₉₂	—	2	2	no	—
80	Dic₉₂	—	2	1	no	—
81	Dic₉₂	—	2	1	no	—
82	Dic₉₂	D₄	1	1	yes	characteristic
83	D₁₈₄	—	2	2	no	—
84	D₁₈₄	—	2	1	no	—
85	D₁₈₄	—	2	1	no	—
86	D₁₈₄	D₄	1	1	yes	characteristic
87	#67263	—	2	2	no	—
88	#67263	—	2	2	no	—
89	#67263	—	2	2	no	—
90	#67263	—	2	2	no	—
91	#67263	D₄	1	2	yes	—
92	#67262	—	2	2	no	—
93	C₈×D₂₃	—	2	2	no	—
94	C₂²×D₂₃	—	2	2	no	—
95	C₂²×D₂₃	—	4	2	no	—
96	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	D₄×C₂₃	—	2	1	no	—
99	D₄×C₂₃	—	2	1	no	—
100	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₂₃	—	2	2	no	—
107	C₂²⋊D₂₃	—	4	2	no	—
108	C₂²⋊D₂₃	—	4	2	no	—
109	C₂×Dic₂₃	—	2	2	no	—
110	C₂×Dic₂₃	—	4	2	no	—
111	D₉₂	—	2	2	no	—
112	D₉₂	—	2	2	no	—
113	D₉₂	—	4	2	no	—
114	D₉₂	C₂×D₄	1	2	yes	—
115	C₄×D₂₃	—	2	2	no	—
116	C₄×D₂₃	—	4	2	no	—
117	C₄×D₂₃	—	4	2	no	—
118	Dic₄₆	—	2	2	no	—
119	Dic₄₆	—	2	2	no	—
120	Dic₄₆	—	4	2	no	—
121	Dic₄₆	C₂×D₄	1	2	yes	—
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	1	no	—
124	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂₃⋊C₈	—	2	2	no	—
129	#3290	—	23	1	no	maximal, Sylow
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×D₂₃	—	2	2	no	—
134	C₂×D₂₃	—	4	2	no	—
135	C₂×D₂₃	—	8	2	no	—
136	C₂×D₂₃	—	16	2	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₂₃	—	2	2	no	—
143	Dic₂₃	—	4	2	no	—
144	#578	—	23	1	no	—
145	#577	—	23	2	no	—
146	#576	—	23	1	no	—
147	#495	—	23	1	no	—
148	#473	—	23	1	no	—
149	#472	—	23	1	no	—
150	#471	—	23	2	no	—
151	#455	—	23	2	no	—
152	#454	—	23	2	no	—
153	#443	—	23	1	no	—
154	#443	D₂₃	1	1	yes	characteristic, complemented
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	—	4	1	no	—
158	C₂×C₂₃	D₄²	1	1	yes	characteristic
159	D₂₃	—	4	2	no	—
160	D₂₃	—	8	2	no	—
161	2₋¹⁺⁴	—	23	2	no	—
162	H(5,𝔽₂)	—	23	2	no	—
163	#137	—	23	2	no	—
164	#137	—	23	2	no	—
165	#137	—	46	2	no	—
166	#137	—	46	2	no	—
167	AGL(1,ℤ/8)	—	23	2	no	—
168	AGL(1,ℤ/8)	—	23	2	no	—
169	AGL(1,ℤ/8)	—	46	2	no	—
170	AGL(1,ℤ/8)	—	46	2	no	—
171	#135	—	23	1	no	—
172	#135	—	46	1	no	—
173	#135	—	46	1	no	—
174	#135	—	46	2	no	—
175	#135	C₂×D₂₃	1	1	yes	characteristic
176	C₂×Q₁₆	—	23	1	no	—
177	C₂×Q₁₆	—	46	1	no	—
178	C₂×SD₁₆	—	23	2	no	—
179	C₂×SD₁₆	—	46	2	no	—
180	C₂×D₈	—	23	1	no	—
181	C₂×D₈	—	46	1	no	—
182	#131	—	23	2	no	—
183	#131	C₂×D₂₃	1	1	yes	characteristic
184	#131	C₂×D₂₃	1	1	yes	characteristic, complemented
185	#124	—	23	2	no	—
186	#108	—	23	1	no	—
187	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
188	#104	—	23	2	no	—
189	#104	C₂×D₂₃	1	1	yes	characteristic
190	#104	C₂×D₂₃	1	1	yes	characteristic, complemented
191	#101	—	23	2	no	—
192	#100	—	23	2	no	—
193	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
194	C₂₃	#3290	1	1	yes	characteristic, complemented, Sylow
195	#41	—	23	2	no	—
196	#41	—	46	2	no	—
197	#41	—	46	2	no	—
198	#41	—	46	2	no	—
199	#41	—	46	2	no	—
200	#41	—	92	2	no	—
201	#41	C₂²×D₂₃	1	1	yes	characteristic
202	#41	C₂²×D₂₃	1	1	yes	characteristic
203	C₂×Q₈	—	23	2	no	—
204	C₂×Q₈	—	46	2	no	—
205	C₂×Q₈	—	46	2	no	—
206	C₂×D₄	—	23	2	no	—
207	C₂×D₄	—	46	2	no	—
208	C₂×D₄	—	46	2	no	—
209	C₂×D₄	—	92	2	no	—
210	Q₁₆	—	23	1	no	—
211	Q₁₆	—	46	2	no	—
212	Q₁₆	—	46	2	no	—
213	Q₁₆	—	46	1	no	—
214	Q₁₆	—	46	1	no	—
215	Q₁₆	D₉₂	1	1	yes	characteristic
216	SD₁₆	—	23	2	no	—
217	SD₁₆	—	46	2	no	—
218	SD₁₆	—	46	2	no	—
219	SD₁₆	—	46	2	no	—
220	SD₁₆	—	46	2	no	—
221	SD₁₆	—	46	2	no	—
222	SD₁₆	—	46	2	no	—
223	SD₁₆	D₉₂	1	1	yes	characteristic
224	SD₁₆	D₉₂	1	1	yes	characteristic
225	D₈	—	23	1	no	—
226	D₈	—	46	2	no	—
227	D₈	—	46	2	no	—
228	D₈	—	46	1	no	—
229	D₈	—	46	1	no	—
230	D₈	D₉₂	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	23	2	no	—
234	M₁₆	—	46	2	no	—
235	M₁₆	C₂²×D₂₃	1	1	yes	characteristic
236	M₁₆	C₂²×D₂₃	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	46	2	no	—
240	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
242	C₂²⋊C₄	—	92	2	no	—
243	C₄²	C₂²×D₂₃	1	1	yes	characteristic
244	C₂³	—	46	2	no	—
245	C₂³	—	92	2	no	—
246	Q₈	—	23	2	no	—
247	Q₈	—	46	2	no	—
248	Q₈	—	46	2	no	—
249	Q₈	—	92	2	no	—
250	Q₈	C₂×D₉₂	1	1	yes	characteristic
251	Q₈	C₂×D₉₂	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	23	2	no	—
255	D₄	—	46	2	no	—
256	D₄	—	46	2	no	—
257	D₄	—	46	2	no	—
258	D₄	—	92	2	no	—
259	D₄	—	92	2	no	—
260	D₄	—	92	2	no	—
261	D₄	C₂×D₉₂	1	1	yes	characteristic
262	D₄	C₂×D₉₂	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	46	2	no	—
267	C₂×C₄	—	46	2	no	—
268	C₂×C₄	—	92	2	no	—
269	C₂×C₄	—	92	2	no	—
270	C₂×C₄	—	92	2	no	—
271	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	46	2	no	—
275	C₈	C₂×D₉₂	1	1	yes	characteristic
276	C₈	C₂×D₉₂	1	1	yes	characteristic
277	C₈	D₄×D₂₃	1	1	yes	characteristic
278	C₈	D₄×D₂₃	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	46	2	no	—
282	C₂²	—	92	2	no	—
283	C₂²	—	368	2	no	—
284	C₂²	—	184	2	no	—
285	C₂²	#10625160	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	46	2	no	—
290	C₄	—	92	2	no	—
291	C₄	C₂²×D₉₂	1	1	yes	characteristic
292	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	92	2	no	—
297	C₂	—	184	2	no	—
298	C₂	D₄×D₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#443	C₂
socle	C₂×C₂₃	D₄²
hypercenter	#443	D₂₃
hypocenter	C₂₃	#3290
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #13093234	C₂	semidirect
2. #13093224	C₂	semidirect
3. #13093222	C₂	semidirect
4. #13092801	C₂	semidirect
5. #13092692	C₂	semidirect
6. #13092687	C₂	semidirect
7. #13092681	C₂	semidirect
8. #13092596	C₂	semidirect
9. #13092591	C₂	semidirect
10. #13092535	C₂	semidirect
11. C₂₃×#443	C₂	semidirect
12. #10625161	C₂²	semidirect
16. C₂₃×#131	C₂²	semidirect
18. #10625120	C₂²	semidirect
24. #10625080	C₂²	semidirect
28. C₂×Dic₉₂	C₂²	semidirect
34. C₂×#67263	C₂²	semidirect
35. #10625072	C₂²	semidirect
36. #10625046	C₂²	semidirect
37. C₂₃×#108	C₂²	semidirect
39. C₂₃×#104	C₂²	semidirect
40. C₄×C₈×C₂₃	C₂²	semidirect
41. #10625006	C₂²	semidirect
42. #10625005	C₂²	semidirect
43. #10625001	C₂²	semidirect
44. #10624987	C₂²	semidirect
154. #443	D₂₃	semidirect
184. #131	C₂×D₂₃	semidirect
187. #108	C₂×D₂₃	semidirect
190. #104	C₂×D₂₃	semidirect
193. C₄×C₈	C₂×D₂₃	semidirect
194. C₂₃	#3290	semidirect

Maximal subgroups

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

subgroup	index
#13093234	2
#13093224	2
#13093222	2
#13092801	2
#13092692	2
#13092687	2
#13092681	2
#13092596	2
#13092591	2
#13092535	2
C₂₃×#443	2
#3290	23