Subgroups — #46824493

Order 34432.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771341	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771316	C₂	1	2	yes	maximal, complemented
3	#31771302	C₂	1	4	yes	maximal, complemented
4	#31771285	C₂	1	4	yes	maximal, complemented
5	#31771252	C₂	1	4	yes	maximal, complemented
6	#31771240	C₂	1	1	yes	maximal, characteristic, complemented
7	#31771137	C₂	1	4	yes	maximal, complemented
8	#31771112	C₂	1	4	yes	maximal, complemented
9	C₄×#16864402	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₂₆₉	C₂	1	4	yes	maximal, complemented
11	#532×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864402	C₂²	1	2	yes	—
13	C₂×D₄×D₂₆₉	C₂²	1	4	yes	—
14	C₂²×C₄×D₂₆₉	C₂²	1	4	yes	complemented
15	#22575446	C₂²	1	8	yes	complemented
16	#22575444	C₂²	1	4	yes	complemented
17	#22575440	C₂²	1	4	yes	complemented
18	#22575435	C₂²	1	8	yes	complemented
19	#22575434	C₂²	1	4	yes	complemented
20	#22575434	C₂²	1	4	yes	complemented
21	#22575433	C₂²	1	4	yes	complemented
22	#22575429	C₂²	1	8	yes	complemented
23	#22575424	C₂²	1	4	yes	—
24	#22575424	C₂²	1	8	yes	complemented
25	#22575421	C₂²	1	4	yes	—
26	#22575421	C₂²	1	4	yes	complemented
27	C₄×Dic₅₃₈	C₂²	1	2	yes	complemented
28	C₄×Dic₅₃₈	C₂²	1	2	yes	complemented
29	C₄×D₁₀₇₆	C₂²	1	2	yes	—
30	C₄×D₁₀₇₆	C₂²	1	2	yes	—
31	D₄×Dic₂₆₉	C₂²	1	4	yes	complemented
32	#22575415	C₂²	1	4	yes	—
33	C₄×C₂²⋊D₂₆₉	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊D₂₆₉	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊D₂₆₉	C₂²	1	4	yes	complemented
36	#22575413	C₂²	1	8	yes	complemented
37	#22575412	C₂²	1	4	yes	complemented
38	#22575412	C₂²	1	2	yes	complemented
39	#22575410	C₂²	1	4	yes	complemented
40	C₄⋊C₄×D₂₆₉	C₂²	1	4	yes	complemented
41	C₂²⋊C₄×D₂₆₉	C₂²	1	8	yes	complemented
42	C₄²×D₂₆₉	C₂²	1	4	yes	complemented
43	C₂×#41×C₂₆₉	C₂²	1	2	yes	complemented
44	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
45	#127×C₂₆₉	C₂²	1	1	yes	characteristic
46	#124×C₂₆₉	C₂²	1	2	yes	complemented
47	#121×C₂₆₉	C₂²	1	4	yes	complemented
48	C₄×D₄×C₂₆₉	C₂²	1	4	yes	complemented
49	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₂₆₉	C₂³	1	4	yes	—
51	#16864402	—	2	8	no	—
52	C₂×Dic₅₃₈	C₂³	1	2	yes	—
53	C₂×D₁₀₇₆	C₂³	1	2	yes	—
54	C₂×C₂²⋊D₂₆₉	C₂³	1	8	yes	—
55	C₂×C₂²⋊D₂₆₉	C₂³	1	4	yes	—
56	D₄×D₂₆₉	—	2	16	no	—
57	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
58	C₂×C₄×D₂₆₉	—	4	8	no	—
59	C₂×C₄×D₂₆₉	C₂³	1	4	yes	—
60	C₂×C₄×D₂₆₉	C₂³	1	4	yes	—
61	C₂×C₄×D₂₆₉	C₂³	1	4	yes	—
62	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
63	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
64	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
65	C₄⋊Dic₂₆₉	C₂³	1	8	yes	—
66	#16864381	C₂³	1	8	yes	—
67	#16864381	C₂³	1	4	yes	—
68	#16864381	C₂³	1	4	yes	—
69	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
70	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
71	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
72	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
73	#16864379	C₂³	1	4	yes	—
74	#16864379	C₂³	1	8	yes	—
75	#16864379	C₂³	1	4	yes	—
76	C₄×Dic₂₆₉	C₂³	1	4	yes	—
77	C₄×Dic₂₆₉	C₂³	1	4	yes	—
78	C₄×Dic₂₆₉	C₂³	1	4	yes	—
79	#41×C₂₆₉	—	4	4	no	—
80	C₂×Q₈×C₂₆₉	C₂³	1	2	yes	—
81	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
82	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
83	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
84	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
85	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
86	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
89	C₄²×C₂₆₉	C₂³	1	2	yes	—
90	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
91	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
92	C₂²×D₂₆₉	—	2	16	no	—
93	C₂²×D₂₆₉	—	4	4	no	—
94	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
95	Dic₅₃₈	—	2	4	no	—
96	D₁₀₇₆	#41	1	8	yes	complemented
97	C₂²⋊D₂₆₉	—	2	16	no	—
98	C₂²⋊D₂₆₉	—	2	8	no	—
99	C₂×Dic₂₆₉	—	4	4	no	—
100	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
101	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
102	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
103	C₄×D₂₆₉	—	2	8	no	—
104	C₄×D₂₆₉	—	2	8	no	—
105	C₄×D₂₆₉	—	2	8	no	—
106	C₄×D₂₆₉	—	4	8	no	—
107	C₂³×C₂₆₉	C₂⁴	1	4	yes	—
108	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
109	Q₈×C₂₆₉	—	4	2	no	—
110	D₄×C₂₆₉	—	2	8	no	—
111	D₄×C₂₆₉	—	4	2	no	—
112	D₄×C₂₆₉	—	4	4	no	—
113	C₂×C₄×C₂₆₉	—	2	4	no	—
114	C₂×C₄×C₂₆₉	—	2	2	no	—
115	C₂×C₄×C₂₆₉	—	2	8	no	—
116	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
117	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
119	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
120	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
121	C₂²×C₂₆₉	—	2	2	no	—
122	C₂²×C₂₆₉	—	2	8	no	—
123	C₂²×C₂₆₉	—	2	1	no	—
124	C₂²×C₂₆₉	—	2	4	no	—
125	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
126	C₂×D₂₆₉	—	2	4	no	—
127	C₂×D₂₆₉	—	2	4	no	—
128	C₂×D₂₆₉	—	4	16	no	—
129	C₂×D₂₆₉	C₂×#41	1	8	yes	—
130	C₄×C₂₆₉	—	2	2	no	—
131	C₄×C₂₆₉	—	2	1	no	—
132	C₄×C₂₆₉	—	2	4	no	—
133	C₄×C₂₆₉	C₂×#41	1	4	yes	—
134	C₄×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
135	Dic₂₆₉	—	2	4	no	—
136	Dic₂₆₉	—	2	4	no	—
137	Dic₂₆₉	—	2	4	no	—
138	C₂×C₂₆₉	—	4	4	no	—
139	C₂×C₂₆₉	—	4	1	no	—
140	C₂×C₂₆₉	C₂²×#41	1	2	yes	—
141	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	D₂₆₉	—	2	8	no	—
143	C₂₆₉	#3481	1	1	yes	characteristic, complemented, Sylow
144	#3481	—	269	1	no	maximal, Sylow
145	#559	—	269	1	no	—
146	#554	—	269	2	no	—
147	#553	—	269	4	no	—
148	#551	—	269	4	no	—
149	#546	—	269	4	no	—
150	D₄²	—	269	1	no	—
151	#532	D₂₆₉	1	1	yes	characteristic, complemented
152	#529	—	269	4	no	—
153	#525	—	269	4	no	—
154	C₄×#41	—	269	2	no	—
155	C₂×C₄×D₄	—	269	4	no	—
156	C₂×#41	—	269	2	no	—
157	C₂×#41	C₂×D₂₆₉	1	2	yes	complemented
158	C₂²×D₄	—	269	4	no	—
159	C₂³×C₄	—	269	4	no	—
160	C₄⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
161	#127	C₂×D₂₆₉	1	1	yes	characteristic
162	#126	—	269	8	no	—
163	#125	—	269	4	no	—
164	#124	—	269	4	no	—
165	#124	C₂×D₂₆₉	1	2	yes	complemented
166	#123	—	269	4	no	—
167	#123	—	269	4	no	—
168	#123	—	269	8	no	—
169	C₂²⋊Q₈	—	269	8	no	—
170	C₂²⋊Q₈	—	269	4	no	—
171	#121	—	269	8	no	—
172	#121	—	269	4	no	—
173	#121	C₂×D₂₆₉	1	4	yes	complemented
174	#120	—	269	4	no	—
175	#120	—	269	4	no	—
176	C₄×Q₈	—	269	2	no	—
177	C₄×Q₈	—	269	2	no	—
178	C₄×D₄	—	269	4	no	—
179	C₄×D₄	—	269	8	no	—
180	C₄×D₄	—	269	4	no	—
181	C₄×D₄	—	269	4	no	—
182	C₄×D₄	—	269	4	no	—
183	C₄×D₄	—	269	8	no	—
184	C₄×D₄	—	269	2	no	—
185	C₄×D₄	—	269	2	no	—
186	C₄×D₄	C₂×D₂₆₉	1	4	yes	complemented
187	#117	—	269	2	no	—
188	#117	—	269	4	no	—
189	#117	—	269	4	no	—
190	C₂×C₄⋊C₄	—	269	4	no	—
191	C₂×C₂²⋊C₄	—	269	8	no	—
192	C₂×C₄²	—	269	4	no	—
193	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic, complemented
194	C₂⁴	—	269	4	no	—
195	#41	—	4	4	no	—
196	#41	—	538	8	no	—
197	C₂×Q₈	—	269	2	no	—
198	C₂×Q₈	C₂²×D₂₆₉	1	2	yes	—
199	C₂×D₄	—	269	4	no	—
200	C₂×D₄	—	269	8	no	—
201	C₂×D₄	—	269	2	no	—
202	C₂×D₄	—	538	16	no	—
203	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
204	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
205	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
206	C₂²×C₄	—	269	4	no	—
207	C₂²×C₄	—	269	4	no	—
208	C₂²×C₄	—	269	4	no	—
209	C₂²×C₄	—	269	4	no	—
210	C₂²×C₄	—	269	8	no	—
211	C₂²×C₄	—	1076	8	no	—
212	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
213	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
215	C₄⋊C₄	—	269	2	no	—
216	C₄⋊C₄	—	269	8	no	—
217	C₄⋊C₄	—	269	2	no	—
218	C₄⋊C₄	—	269	4	no	—
219	C₄⋊C₄	—	269	4	no	—
220	C₄⋊C₄	—	269	8	no	—
221	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
222	C₂²⋊C₄	—	269	8	no	—
223	C₂²⋊C₄	—	269	4	no	—
224	C₂²⋊C₄	—	269	4	no	—
225	C₂²⋊C₄	—	269	4	no	—
226	C₂²⋊C₄	—	269	4	no	—
227	C₂²⋊C₄	—	269	8	no	—
228	C₂²⋊C₄	—	269	4	no	—
229	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
230	C₄²	—	269	4	no	—
231	C₄²	—	269	4	no	—
232	C₄²	—	269	4	no	—
233	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
234	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
235	C₄²	C₂²×D₂₆₉	1	2	yes	—
236	C₂³	—	269	4	no	—
237	C₂³	—	538	16	no	—
238	C₂³	—	1076	4	no	—
239	C₂³	C₂³×D₂₆₉	1	4	yes	—
240	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
241	Q₈	—	4	2	no	—
242	Q₈	—	538	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	D₄	—	269	8	no	—
247	D₄	—	538	16	no	—
248	D₄	—	538	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	269	4	no	—
253	C₂×C₄	—	269	4	no	—
254	C₂×C₄	—	269	4	no	—
255	C₂×C₄	—	538	8	no	—
256	C₂×C₄	—	538	8	no	—
257	C₂×C₄	—	538	8	no	—
258	C₂×C₄	—	1076	8	no	—
259	C₂×C₄	—	1076	4	no	—
260	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
261	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
263	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
264	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	269	8	no	—
270	C₂²	—	538	4	no	—
271	C₂²	—	538	4	no	—
272	C₂²	—	1076	16	no	—
273	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	538	4	no	—
278	C₄	—	538	4	no	—
279	C₄	—	538	4	no	—
280	C₄	#41×D₂₆₉	1	4	yes	—
281	C₄	#22575498	1	2	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	538	8	no	—
285	C₂	C₂×#41×D₂₆₉	1	2	yes	—
286	C₂	C₂×#22575498	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	#532×C₂₆₉	C₂
hypercenter	#532	D₂₆₉
hypocenter	C₂₆₉	#3481
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #31771341	C₂	semidirect
2. #31771316	C₂	semidirect
3. #31771302	C₂	semidirect
4. #31771285	C₂	semidirect
5. #31771252	C₂	semidirect
6. #31771240	C₂	semidirect
7. #31771137	C₂	semidirect
8. #31771112	C₂	semidirect
9. C₄×#16864402	C₂	semidirect
10. C₄×D₄×D₂₆₉	C₂	semidirect
11. #532×C₂₆₉	C₂	semidirect
14. C₂²×C₄×D₂₆₉	C₂²	semidirect
15. #22575446	C₂²	semidirect
16. #22575444	C₂²	semidirect
17. #22575440	C₂²	semidirect
18. #22575435	C₂²	semidirect
19. #22575434	C₂²	semidirect
20. #22575434	C₂²	semidirect
21. #22575433	C₂²	semidirect
22. #22575429	C₂²	semidirect
24. #22575424	C₂²	semidirect
26. #22575421	C₂²	semidirect
27. C₄×Dic₅₃₈	C₂²	semidirect
28. C₄×Dic₅₃₈	C₂²	semidirect
31. D₄×Dic₂₆₉	C₂²	semidirect
33. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
34. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
35. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
36. #22575413	C₂²	semidirect
37. #22575412	C₂²	semidirect
38. #22575412	C₂²	semidirect
39. #22575410	C₂²	semidirect
40. C₄⋊C₄×D₂₆₉	C₂²	semidirect
41. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
42. C₄²×D₂₆₉	C₂²	semidirect
43. C₂×#41×C₂₆₉	C₂²	semidirect
44. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
46. #124×C₂₆₉	C₂²	semidirect
47. #121×C₂₆₉	C₂²	semidirect
48. C₄×D₄×C₂₆₉	C₂²	semidirect
49. C₂×C₄²×C₂₆₉	C₂²	semidirect
96. D₁₀₇₆	#41	semidirect
143. C₂₆₉	#3481	semidirect
151. #532	D₂₆₉	semidirect
157. C₂×#41	C₂×D₂₆₉	semidirect
160. C₄⋊Q₈	C₂×D₂₆₉	semidirect
165. #124	C₂×D₂₆₉	semidirect
173. #121	C₂×D₂₆₉	semidirect
186. C₄×D₄	C₂×D₂₆₉	semidirect
193. C₂×C₄²	C₂×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771341	2
#31771316	2
#31771302	2
#31771285	2
#31771252	2
#31771240	2
#31771137	2
#31771112	2
C₄×#16864402	2
C₄×D₄×D₂₆₉	2
#532×C₂₆₉	2
#3481	269