Subgroups — #15682204

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625079	C₂	1	1	yes	maximal, characteristic
2	#13092885	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625064	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092528	C₂	1	2	yes	maximal, complemented
5	#13092516	C₂	1	2	yes	maximal, complemented
6	#13092504	C₂	1	2	yes	maximal, complemented
7	#13092499	C₂	1	2	yes	maximal, complemented
8	#13092421	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092416	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₂₃)⋊C₄	—	2	1	no	—
19	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×#67267	C₂²	1	1	yes	characteristic
21	#10625079	C₄	1	8	yes	complemented
22	C₂×D₁₈₄	C₂²	1	2	yes	—
23	C₂×#67263	C₂²	1	2	yes	—
24	#10625064	—	2	1	no	—
25	#10625064	—	2	1	no	—
26	#10625064	C₂²	1	2	yes	—
27	#10625064	C₂²	1	2	yes	—
28	#10625064	C₂²	1	1	yes	characteristic, complemented
29	#10625064	C₂²	1	1	yes	characteristic, complemented
30	#10625058	C₂²	1	1	yes	characteristic, complemented
31	#10625058	C₂²	1	1	yes	characteristic, complemented
32	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
33	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
34	#10625043	—	2	1	no	—
35	C₄²×D₂₃	—	2	1	no	—
36	C₂₃×#107	C₂²	1	2	yes	complemented
37	C₂₃×#106	C₂²	1	2	yes	complemented
38	#10624997	C₂²	1	2	yes	complemented
39	C₈×Dic₂₃	C₂²	1	2	yes	complemented
40	#10624992	C₂²	1	2	yes	complemented
41	#10624991	C₂²	1	2	yes	—
42	#10624991	C₂²	1	2	yes	—
43	#10624991	C₂²	1	2	yes	complemented
44	C₂³×D₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
49	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
50	#67288	—	2	2	no	—
51	#67288	C₂×C₄	1	4	yes	complemented
52	C₂×D₉₂	—	2	1	no	—
53	C₂×D₉₂	—	2	2	no	—
54	C₂×D₉₂	C₂×C₄	1	4	yes	complemented
55	C₂×D₉₂	C₂³	1	2	yes	—
56	C₂×D₉₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₂₃	—	2	1	no	—
58	C₂×C₄×D₂₃	—	2	1	no	—
59	C₂×C₄×D₂₃	—	2	1	no	—
60	C₂×C₄×D₂₃	—	2	2	no	—
61	C₂×C₄×D₂₃	—	2	2	no	—
62	C₂×C₄×D₂₃	—	4	4	no	—
63	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
64	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
65	C₂×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	—	2	1	no	—
72	(C₂×D₂₃)⋊C₄	—	2	1	no	—
73	(C₂×D₂₃)⋊C₄	—	2	1	no	—
74	(C₂×D₂₃)⋊C₄	—	2	2	no	—
75	(C₂×D₂₃)⋊C₄	—	2	2	no	—
76	Dic₂₃⋊C₄	—	2	1	no	—
77	Dic₂₃⋊C₄	—	2	1	no	—
78	C₄×Dic₂₃	—	2	1	no	—
79	C₄×Dic₂₃	—	2	1	no	—
80	C₄×Dic₂₃	D₄	1	1	yes	characteristic
81	C₄×Dic₂₃	D₄	1	1	yes	characteristic
82	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
84	#67267	—	2	2	no	—
85	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
86	D₁₈₄	C₂×C₄	1	8	yes	complemented
87	#67263	C₂×C₄	1	8	yes	complemented
88	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₂₃	—	2	4	no	—
90	C₂²×D₂₃	—	2	1	no	—
91	C₂²×D₂₃	—	2	2	no	—
92	C₂²×D₂₃	—	4	2	no	—
93	C₂²×D₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#41	1	2	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₂₃	—	2	4	no	—
103	C₂×Dic₂₃	—	2	1	no	—
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	—	4	1	no	—
106	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
110	D₉₂	—	2	4	no	—
111	D₉₂	—	2	2	no	—
112	D₉₂	—	2	1	no	—
113	D₉₂	C₂²×C₄	1	4	yes	—
114	D₉₂	C₂²×C₄	1	2	yes	complemented
115	C₄×D₂₃	—	2	4	no	—
116	C₄×D₂₃	—	4	2	no	—
117	C₄×D₂₃	—	4	4	no	—
118	C₄×D₂₃	—	4	2	no	—
119	C₄×D₂₃	—	4	4	no	—
120	Dic₄₆	—	2	1	no	—
121	Dic₄₆	C₂²×C₄	1	2	yes	complemented
122	C₈×C₂₃	C₂²×C₄	1	4	yes	complemented
123	C₂₃⋊C₈	—	2	2	no	—
124	C₄×AGL(1,ℤ/8)	—	23	1	no	maximal, Sylow
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	C₄×D₄	1	2	yes	—
128	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×D₂₃	—	2	2	no	—
130	C₂×D₂₃	—	2	4	no	—
131	C₂×D₂₃	—	4	8	no	—
132	C₂×D₂₃	—	4	2	no	—
133	C₂×D₂₃	—	4	1	no	—
134	C₂×D₂₃	—	4	1	no	—
135	C₂×D₂₃	—	4	2	no	—
136	C₄×C₂₃	—	4	2	no	—
137	C₄×C₂₃	—	4	1	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	C₄×D₄	1	2	yes	—
140	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
142	Dic₂₃	—	2	2	no	—
143	Dic₂₃	—	2	1	no	—
144	Dic₂₃	—	2	1	no	—
145	Dic₂₃	—	2	1	no	—
146	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
147	C₂×AGL(1,ℤ/8)	—	23	1	no	—
148	C₄×#41	—	23	1	no	—
149	C₂×C₄×D₄	—	23	1	no	—
150	#442	—	23	2	no	—
151	#440	—	23	2	no	—
152	C₄×SD₁₆	—	23	2	no	—
153	C₄×D₈	—	23	2	no	—
154	#428	D₂₃	1	1	yes	characteristic, complemented
155	#418	—	23	1	no	—
156	#417	—	23	1	no	—
157	C₄×M₁₆	—	23	1	no	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₂₃	#575	1	1	yes	characteristic
162	D₂₃	—	4	2	no	—
163	D₂₃	—	4	4	no	—
164	C₂×#41	—	23	1	no	—
165	C₂²×D₄	—	23	1	no	—
166	C₂³×C₄	—	46	1	no	—
167	AGL(1,ℤ/8)	—	23	8	no	—
168	C₂×SD₁₆	—	23	2	no	—
169	C₂×D₈	—	23	2	no	—
170	C₂×M₁₆	—	23	1	no	—
171	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
172	C₄×Q₈	—	23	1	no	—
173	C₄×Q₈	—	23	1	no	—
174	C₄×D₄	—	23	2	no	—
175	C₄×D₄	—	23	2	no	—
176	C₄×D₄	—	23	1	no	—
177	C₄×D₄	—	23	1	no	—
178	C₄×D₄	—	46	1	no	—
179	C₄×D₄	—	46	1	no	—
180	C₄×D₄	—	46	1	no	—
181	C₄×D₄	—	46	1	no	—
182	#117	—	46	1	no	—
183	#117	C₂×D₂₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	46	1	no	—
186	C₂×C₄²	—	23	1	no	—
187	C₂×C₄²	—	46	1	no	—
188	#107	C₂×D₂₃	1	2	yes	complemented
189	#106	C₂×D₂₃	1	2	yes	complemented
190	#103	—	23	2	no	—
191	#102	—	23	2	no	—
192	#102	—	23	2	no	—
193	#102	—	23	2	no	—
194	#97	—	23	2	no	—
195	C₄×C₈	—	23	2	no	—
196	C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	—	46	1	no	—
198	#41	—	23	4	no	—
199	#41	—	46	2	no	—
200	C₂×Q₈	—	23	1	no	—
201	C₂×D₄	—	23	4	no	—
202	C₂×D₄	—	23	1	no	—
203	C₂×D₄	—	23	2	no	—
204	C₂×D₄	—	46	1	no	—
205	C₂×D₄	—	46	1	no	—
206	C₂×D₄	—	46	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	23	1	no	—
209	C₂²×C₄	—	23	1	no	—
210	C₂²×C₄	—	46	1	no	—
211	C₂²×C₄	—	46	1	no	—
212	C₂²×C₄	—	46	2	no	—
213	C₂²×C₄	—	46	2	no	—
214	C₂²×C₄	—	46	1	no	—
215	C₂²×C₄	—	92	4	no	—
216	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
217	SD₁₆	—	23	8	no	—
218	D₈	—	23	8	no	—
219	M₁₆	—	46	2	no	—
220	M₁₆	C₄×D₂₃	1	4	yes	complemented
221	C₂×C₈	—	23	2	no	—
222	C₂×C₈	C₂²×D₂₃	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	46	1	no	—
225	C₄⋊C₄	—	46	1	no	—
226	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	46	2	no	—
231	C₂²⋊C₄	—	46	2	no	—
232	C₂²⋊C₄	—	46	1	no	—
233	C₂²⋊C₄	—	46	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	23	1	no	—
236	C₄²	—	23	1	no	—
237	C₄²	—	23	1	no	—
238	C₄²	—	23	1	no	—
239	C₄²	—	46	1	no	—
240	C₄²	—	46	1	no	—
241	C₂³	—	46	4	no	—
242	C₂³	—	46	1	no	—
243	C₂³	—	46	2	no	—
244	C₂³	—	92	2	no	—
245	C₂³	—	92	2	no	—
246	C₂³	D₄×D₂₃	1	1	yes	characteristic
247	Q₈	—	23	2	no	—
248	Q₈	—	46	1	no	—
249	D₄	—	23	2	no	—
250	D₄	—	23	4	no	—
251	D₄	—	46	4	no	—
252	D₄	—	46	4	no	—
253	D₄	—	46	2	no	—
254	D₄	—	46	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	23	1	no	—
260	C₂×C₄	—	23	1	no	—
261	C₂×C₄	—	23	1	no	—
262	C₂×C₄	—	23	1	no	—
263	C₂×C₄	—	46	2	no	—
264	C₂×C₄	—	46	1	no	—
265	C₂×C₄	—	46	4	no	—
266	C₂×C₄	—	92	4	no	—
267	C₂×C₄	—	92	2	no	—
268	C₂×C₄	—	92	4	no	—
269	C₂×C₄	—	92	2	no	—
270	C₂×C₄	—	92	1	no	—
271	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
272	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
273	C₂×C₄	#67292	1	2	yes	—
274	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
275	C₈	—	46	2	no	—
276	C₈	C₂×C₄×D₂₃	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	46	2	no	—
280	C₂²	—	46	4	no	—
281	C₂²	—	92	8	no	—
282	C₂²	—	92	2	no	—
283	C₂²	—	92	1	no	—
284	C₂²	—	92	1	no	—
285	C₂²	—	92	2	no	—
286	C₂²	#10625064	1	2	yes	—
287	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	23	2	no	—
292	C₄	—	46	2	no	—
293	C₄	—	46	1	no	—
294	C₄	—	46	1	no	—
295	C₄	—	46	1	no	—
296	C₄	#10625064	1	2	yes	—
297	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
298	C₄	C₂×#67292	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	92	4	no	—
301	C₂	—	92	2	no	—
302	C₂	C₂×#10625064	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
304	C₂	#13093216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#428	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625079	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #13092885	C₂	semidirect
3. C₂×#10625064	C₂	semidirect
4. #13092528	C₂	semidirect
5. #13092516	C₂	semidirect
6. #13092504	C₂	semidirect
7. #13092499	C₂	semidirect
8. #13092421	C₂	semidirect
9. #13092416	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
21. #10625079	C₄	semidirect
28. #10625064	C₂²	semidirect
29. #10625064	C₂²	semidirect
30. #10625058	C₂²	semidirect
31. #10625058	C₂²	semidirect
36. C₂₃×#107	C₂²	semidirect
37. C₂₃×#106	C₂²	semidirect
38. #10624997	C₂²	semidirect
39. C₈×Dic₂₃	C₂²	semidirect
40. #10624992	C₂²	semidirect
43. #10624991	C₂²	semidirect
51. #67288	C₂×C₄	semidirect
54. C₂×D₉₂	C₂×C₄	semidirect
64. M₁₆×C₂₃	C₂×C₄	semidirect
86. D₁₈₄	C₂×C₄	semidirect
87. #67263	C₂×C₄	semidirect
114. D₉₂	C₂²×C₄	semidirect
121. Dic₄₆	C₂²×C₄	semidirect
122. C₈×C₂₃	C₂²×C₄	semidirect
146. Dic₂₃	AGL(1,ℤ/8)	semidirect
154. #428	D₂₃	semidirect
188. #107	C₂×D₂₃	semidirect
189. #106	C₂×D₂₃	semidirect
196. C₂₃	C₄×AGL(1,ℤ/8)	semidirect
220. M₁₆	C₄×D₂₃	semidirect
276. C₈	C₂×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×#10625079	2
#13092885	2
C₂×#10625064	2
#13092528	2
#13092516	2
#13092504	2
#13092499	2
#13092421	2
#13092416	2
M₁₆×Dic₂₃	2
C₂₃×#428	2
C₄×AGL(1,ℤ/8)	23