Subgroups — #46825042

Order 34432.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771344	C₂	1	3	yes	maximal, complemented
2	#31771285	C₂	1	3	yes	maximal, complemented
3	#31771248	C₂	1	6	yes	maximal, complemented
4	#31771237	C₂	1	3	yes	maximal, complemented
5	#31771189	C₂	1	6	yes	maximal, complemented
6	#31771173	C₂	1	3	yes	maximal, complemented
7	#31771165	C₂	1	3	yes	maximal, complemented
8	#120×D₂₆₉	C₂	1	3	yes	maximal, complemented
9	H(3,𝔽₄)×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₂₆₉	C₂²	1	1	yes	characteristic, complemented
11	C₂×#16864403	C₂²	1	3	yes	complemented
12	C₂²×C₂²⋊D₂₆₉	C₂²	1	3	yes	complemented
13	C₂×D₄×D₂₆₉	C₂²	1	3	yes	complemented
14	C₂×D₄×D₂₆₉	C₂²	1	6	yes	complemented
15	#22575450	C₂²	1	1	yes	characteristic, complemented
16	#22575449	C₂²	1	3	yes	complemented
17	#22575447	C₂²	1	2	yes	complemented
18	#22575446	C₂²	1	6	yes	complemented
19	#22575442	C₂²	1	3	yes	complemented
20	#22575437	C₂²	1	3	yes	complemented
21	#22575435	C₂²	1	6	yes	complemented
22	#22575435	C₂²	1	6	yes	complemented
23	#22575434	C₂²	1	3	yes	complemented
24	#22575429	C₂²	1	6	yes	complemented
25	#22575426	C₂²	1	6	yes	complemented
26	#22575426	C₂²	1	6	yes	complemented
27	#22575426	C₂²	1	6	yes	complemented
28	#22575425	C₂²	1	6	yes	complemented
29	#22575424	C₂²	1	6	yes	complemented
30	#22575423	C₂²	1	6	yes	complemented
31	#22575423	C₂²	1	3	yes	complemented
32	#22575423	C₂²	1	6	yes	complemented
33	#22575422	C₂²	1	3	yes	complemented
34	#22575422	C₂²	1	1	yes	characteristic, complemented
35	#22575421	C₂²	1	3	yes	—
36	#22575421	C₂²	1	6	yes	complemented
37	C₄×D₁₀₇₆	C₂²	1	6	yes	complemented
38	#22575413	C₂²	1	6	yes	complemented
39	#22575410	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₂₆₉	C₂²	1	3	yes	complemented
41	C₂²⋊C₄×D₂₆₉	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₂₆₉	C₂²	1	3	yes	complemented
43	#124×C₂₆₉	C₂²	1	3	yes	complemented
44	#124×C₂₆₉	C₂²	1	6	yes	complemented
45	#120×C₂₆₉	C₂²	1	3	yes	complemented
46	#120×C₂₆₉	C₂²	1	3	yes	complemented
47	C₂³×D₂₆₉	—	4	6	no	—
48	C₂³×D₂₆₉	C₂³	1	3	yes	—
49	C₂³×D₂₆₉	C₂³	1	3	yes	—
50	C₂³×D₂₆₉	C₂³	1	3	yes	—
51	#16864403	—	4	6	no	—
52	C₂×D₁₀₇₆	C₂³	1	6	yes	—
53	C₂×D₁₀₇₆	C₂³	1	3	yes	—
54	C₂×D₁₀₇₆	C₂³	1	3	yes	complemented
55	C₂×C₂²⋊D₂₆₉	—	4	6	no	—
56	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₂₆₉	C₂³	1	3	yes	—
58	C₂×C₂²⋊D₂₆₉	C₂³	1	3	yes	—
59	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	—
60	C₂×C₂²⋊D₂₆₉	C₂³	1	6	yes	complemented
61	D₄×D₂₆₉	—	4	6	no	—
62	D₄×D₂₆₉	—	4	12	no	—
63	C₂²×Dic₂₆₉	C₂³	1	3	yes	complemented
64	C₂×C₄×D₂₆₉	C₂³	1	3	yes	—
65	C₂×C₄×D₂₆₉	C₂³	1	6	yes	—
66	C₂×C₄×D₂₆₉	C₂³	1	6	yes	complemented
67	C₄⋊Dic₂₆₉	C₂³	1	6	yes	complemented
68	#16864381	C₂³	1	6	yes	complemented
69	#16864381	C₂³	1	6	yes	complemented
70	C₂²⋊Dic₂₆₉	C₂³	1	3	yes	—
71	C₂²⋊Dic₂₆₉	C₂³	1	6	yes	—
72	C₂²⋊Dic₂₆₉	C₂³	1	3	yes	complemented
73	#16864379	C₂³	1	6	yes	—
74	#16864379	C₂³	1	6	yes	—
75	#16864379	C₂³	1	6	yes	—
76	#16864379	C₂³	1	6	yes	complemented
77	C₄×Dic₂₆₉	C₂³	1	3	yes	complemented
78	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
79	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic, complemented
80	C₂×Q₈×C₂₆₉	C₂³	1	3	yes	complemented
81	C₂×D₄×C₂₆₉	C₂³	1	3	yes	—
82	C₂×D₄×C₂₆₉	C₂³	1	6	yes	—
83	C₂²⋊C₄×C₂₆₉	C₂³	1	6	yes	—
84	C₂²⋊C₄×C₂₆₉	C₂³	1	3	yes	—
85	C₂²⋊C₄×C₂₆₉	C₂³	1	3	yes	complemented
86	C₂²⋊C₄×C₂₆₉	C₂³	1	6	yes	complemented
87	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic, complemented
88	C₄²×C₂₆₉	C₂³	1	2	yes	complemented
89	C₂²×D₂₆₉	—	2	12	no	—
90	C₂²×D₂₆₉	—	2	6	no	—
91	C₂²×D₂₆₉	—	2	6	no	—
92	C₂²×D₂₆₉	—	4	3	no	—
93	C₂²×D₂₆₉	—	4	6	no	—
94	C₂²×D₂₆₉	—	4	6	no	—
95	C₂²×D₂₆₉	—	4	16	no	—
96	C₂²×D₂₆₉	C₂⁴	1	1	yes	characteristic
97	C₂²×D₂₆₉	C₂⁴	1	3	yes	—
98	C₂²×D₂₆₉	C₂⁴	1	3	yes	—
99	D₁₀₇₆	—	2	12	no	—
100	D₁₀₇₆	—	4	3	no	—
101	D₁₀₇₆	—	4	3	no	—
102	C₂²⋊D₂₆₉	—	2	12	no	—
103	C₂²⋊D₂₆₉	—	4	3	no	—
104	C₂²⋊D₂₆₉	—	4	3	no	—
105	C₂²⋊D₂₆₉	—	4	6	no	—
106	C₂²⋊D₂₆₉	—	4	6	no	—
107	C₂×Dic₂₆₉	—	4	3	no	—
108	C₂×Dic₂₆₉	C₂⁴	1	6	yes	—
109	C₂×Dic₂₆₉	C₂⁴	1	3	yes	—
110	C₄×D₂₆₉	—	2	6	no	—
111	C₄×D₂₆₉	—	4	6	no	—
112	C₄×D₂₆₉	—	4	6	no	—
113	C₂³×C₂₆₉	—	4	3	no	—
114	C₂³×C₂₆₉	—	4	3	no	—
115	C₂³×C₂₆₉	C₂⁴	1	3	yes	—
116	C₂³×C₂₆₉	C₂⁴	1	3	yes	—
117	Q₈×C₂₆₉	—	4	3	no	—
118	D₄×C₂₆₉	—	4	3	no	—
119	D₄×C₂₆₉	—	4	6	no	—
120	C₂×C₄×C₂₆₉	C₂⁴	1	3	yes	—
121	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
122	C₂²×C₂₆₉	—	2	3	no	—
123	C₂²×C₂₆₉	—	2	6	no	—
124	C₂²×C₂₆₉	—	2	3	no	—
125	C₂²×C₂₆₉	—	2	6	no	—
126	C₂²×C₂₆₉	—	4	4	no	—
127	C₂²×C₂₆₉	—	8	2	no	—
128	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
129	C₂×D₂₆₉	—	2	6	no	—
130	C₂×D₂₆₉	—	2	6	no	—
131	C₂×D₂₆₉	—	2	3	no	—
132	C₂×D₂₆₉	—	2	3	no	—
133	C₂×D₂₆₉	—	4	12	no	—
134	C₂×D₂₆₉	—	4	12	no	—
135	C₂×D₂₆₉	—	8	6	no	—
136	C₂×D₂₆₉	H(5,𝔽₂)	1	6	yes	—
137	C₄×C₂₆₉	—	2	3	no	—
138	C₄×C₂₆₉	—	2	6	no	—
139	Dic₂₆₉	—	2	3	no	—
140	Dic₂₆₉	—	2	6	no	—
141	C₂×C₂₆₉	—	4	3	no	—
142	C₂×C₂₆₉	—	4	3	no	—
143	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	3	yes	—
144	D₂₆₉	—	2	6	no	—
145	D₂₆₉	—	4	1	no	—
146	D₂₆₉	—	4	3	no	—
147	C₂₆₉	#3532	1	1	yes	characteristic, complemented, Sylow
148	#3532	—	269	1	no	maximal, Sylow
149	H(3,𝔽₄)	D₂₆₉	1	1	yes	characteristic, complemented
150	#560	—	269	3	no	—
151	#551	—	269	3	no	—
152	#546	—	269	6	no	—
153	D₄²	—	269	3	no	—
154	#538	—	269	6	no	—
155	#535	—	269	3	no	—
156	#534	—	269	3	no	—
157	C₂×#120	—	269	3	no	—
158	C₂⁵	—	269	1	no	—
159	C₂×#41	—	269	3	no	—
160	C₂²×D₄	—	269	3	no	—
161	C₂²×D₄	—	269	3	no	—
162	C₂²×D₄	—	269	6	no	—
163	#127	—	269	3	no	—
164	#127	—	269	1	no	—
165	#126	—	269	6	no	—
166	#126	—	269	2	no	—
167	#124	—	269	3	no	—
168	#124	C₂×D₂₆₉	1	3	yes	complemented
169	#124	C₂×D₂₆₉	1	6	yes	complemented
170	#123	—	269	3	no	—
171	#123	—	269	6	no	—
172	#123	—	269	3	no	—
173	#123	—	269	6	no	—
174	C₂²⋊Q₈	—	269	6	no	—
175	#121	—	269	6	no	—
176	#121	—	269	6	no	—
177	#121	—	269	6	no	—
178	#121	—	269	6	no	—
179	#121	—	269	6	no	—
180	#120	—	269	3	no	—
181	#120	—	269	1	no	—
182	#120	—	269	6	no	—
183	#120	—	269	3	no	—
184	#120	—	269	6	no	—
185	#120	—	269	3	no	—
186	#120	—	269	6	no	—
187	#120	C₂×D₂₆₉	1	3	yes	complemented
188	#120	C₂×D₂₆₉	1	3	yes	complemented
189	C₄×D₄	—	269	6	no	—
190	C₄×D₄	—	269	6	no	—
191	#117	—	269	3	no	—
192	C₂×C₂²⋊C₄	—	269	3	no	—
193	C₂×C₂²⋊C₄	—	269	6	no	—
194	C₂×C₂²⋊C₄	—	269	3	no	—
195	C₂⁴	—	269	3	no	—
196	C₂⁴	—	269	3	no	—
197	C₂⁴	—	269	3	no	—
198	C₂⁴	—	1076	6	no	—
199	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
200	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic, complemented
201	#41	—	1076	6	no	—
202	C₂×Q₈	C₂²×D₂₆₉	1	3	yes	complemented
203	C₂×D₄	—	269	6	no	—
204	C₂×D₄	—	269	3	no	—
205	C₂×D₄	—	269	3	no	—
206	C₂×D₄	—	269	6	no	—
207	C₂×D₄	—	269	6	no	—
208	C₂×D₄	—	269	6	no	—
209	C₂×D₄	—	269	3	no	—
210	C₂×D₄	—	269	3	no	—
211	C₂×D₄	—	1076	6	no	—
212	C₂×D₄	—	1076	12	no	—
213	C₂×D₄	—	1076	6	no	—
214	C₂×D₄	C₂²×D₂₆₉	1	6	yes	—
215	C₂×D₄	C₂²×D₂₆₉	1	3	yes	—
216	C₂²×C₄	—	269	3	no	—
217	C₂²×C₄	—	269	3	no	—
218	C₂²×C₄	—	269	6	no	—
219	C₂²×C₄	—	269	6	no	—
220	C₄⋊C₄	—	269	6	no	—
221	C₄⋊C₄	—	269	6	no	—
222	C₄⋊C₄	—	269	6	no	—
223	C₂²⋊C₄	—	269	3	no	—
224	C₂²⋊C₄	—	269	6	no	—
225	C₂²⋊C₄	—	269	3	no	—
226	C₂²⋊C₄	—	269	6	no	—
227	C₂²⋊C₄	—	269	6	no	—
228	C₂²⋊C₄	—	269	6	no	—
229	C₂²⋊C₄	—	269	6	no	—
230	C₂²⋊C₄	C₂²×D₂₆₉	1	6	yes	—
231	C₂²⋊C₄	C₂²×D₂₆₉	1	3	yes	—
232	C₂²⋊C₄	C₂²×D₂₆₉	1	3	yes	complemented
233	C₂²⋊C₄	C₂²×D₂₆₉	1	6	yes	complemented
234	C₄²	—	269	3	no	—
235	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic, complemented
236	C₄²	C₂²×D₂₆₉	1	2	yes	complemented
237	C₂³	—	4	3	no	—
238	C₂³	—	4	3	no	—
239	C₂³	—	269	3	no	—
240	C₂³	—	269	1	no	—
241	C₂³	—	269	3	no	—
242	C₂³	—	538	12	no	—
243	C₂³	—	538	6	no	—
244	C₂³	—	538	6	no	—
245	C₂³	—	1076	3	no	—
246	C₂³	—	1076	6	no	—
247	C₂³	—	1076	6	no	—
248	C₂³	—	1076	16	no	—
249	C₂³	C₂³×D₂₆₉	1	3	yes	—
250	C₂³	C₂³×D₂₆₉	1	3	yes	—
251	Q₈	—	4	3	no	—
252	D₄	—	4	3	no	—
253	D₄	—	4	6	no	—
254	D₄	—	538	12	no	—
255	D₄	—	538	12	no	—
256	D₄	—	1076	3	no	—
257	D₄	—	1076	3	no	—
258	D₄	—	1076	3	no	—
259	D₄	—	1076	3	no	—
260	D₄	—	1076	6	no	—
261	D₄	—	1076	6	no	—
262	C₂×C₄	—	269	6	no	—
263	C₂×C₄	—	269	3	no	—
264	C₂×C₄	—	538	6	no	—
265	C₂×C₄	—	1076	6	no	—
266	C₂×C₄	—	1076	6	no	—
267	C₂×C₄	—	1076	3	no	—
268	C₂×C₄	C₂³×D₂₆₉	1	3	yes	—
269	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
270	C₂²	—	2	3	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	3	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	269	6	no	—
277	C₂²	—	538	6	no	—
278	C₂²	—	538	6	no	—
279	C₂²	—	538	3	no	—
280	C₂²	—	538	3	no	—
281	C₂²	—	1076	12	no	—
282	C₂²	—	1076	12	no	—
283	C₂²	—	2152	6	no	—
284	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
285	C₄	—	2	3	no	—
286	C₄	—	2	6	no	—
287	C₄	—	538	3	no	—
288	C₄	—	538	6	no	—
289	C₂	—	4	3	no	—
290	C₂	—	4	3	no	—
291	C₂	—	538	6	no	—
292	C₂	—	1076	1	no	—
293	C₂	—	1076	3	no	—
294	C₂	H(5,𝔽₂)×D₂₆₉	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁵
Fitting	H(3,𝔽₄)×C₂₆₉	C₂
hypercenter	H(3,𝔽₄)	D₂₆₉
hypocenter	C₂₆₉	#3532
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #31771344	C₂	semidirect
2. #31771285	C₂	semidirect
3. #31771248	C₂	semidirect
4. #31771237	C₂	semidirect
5. #31771189	C₂	semidirect
6. #31771173	C₂	semidirect
7. #31771165	C₂	semidirect
8. #120×D₂₆₉	C₂	semidirect
9. H(3,𝔽₄)×C₂₆₉	C₂	semidirect
10. C₂⁴×D₂₆₉	C₂²	semidirect
11. C₂×#16864403	C₂²	semidirect
12. C₂²×C₂²⋊D₂₆₉	C₂²	semidirect
13. C₂×D₄×D₂₆₉	C₂²	semidirect
14. C₂×D₄×D₂₆₉	C₂²	semidirect
15. #22575450	C₂²	semidirect
16. #22575449	C₂²	semidirect
17. #22575447	C₂²	semidirect
18. #22575446	C₂²	semidirect
19. #22575442	C₂²	semidirect
20. #22575437	C₂²	semidirect
21. #22575435	C₂²	semidirect
22. #22575435	C₂²	semidirect
23. #22575434	C₂²	semidirect
24. #22575429	C₂²	semidirect
25. #22575426	C₂²	semidirect
26. #22575426	C₂²	semidirect
27. #22575426	C₂²	semidirect
28. #22575425	C₂²	semidirect
29. #22575424	C₂²	semidirect
30. #22575423	C₂²	semidirect
31. #22575423	C₂²	semidirect
32. #22575423	C₂²	semidirect
33. #22575422	C₂²	semidirect
34. #22575422	C₂²	semidirect
36. #22575421	C₂²	semidirect
37. C₄×D₁₀₇₆	C₂²	semidirect
38. #22575413	C₂²	semidirect
39. #22575410	C₂²	semidirect
40. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
41. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
42. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
43. #124×C₂₆₉	C₂²	semidirect
44. #124×C₂₆₉	C₂²	semidirect
45. #120×C₂₆₉	C₂²	semidirect
46. #120×C₂₆₉	C₂²	semidirect
54. C₂×D₁₀₇₆	C₂³	semidirect
60. C₂×C₂²⋊D₂₆₉	C₂³	semidirect
63. C₂²×Dic₂₆₉	C₂³	semidirect
66. C₂×C₄×D₂₆₉	C₂³	semidirect
67. C₄⋊Dic₂₆₉	C₂³	semidirect
68. #16864381	C₂³	semidirect
69. #16864381	C₂³	semidirect
72. C₂²⋊Dic₂₆₉	C₂³	semidirect
76. #16864379	C₂³	semidirect
77. C₄×Dic₂₆₉	C₂³	semidirect
79. C₂⁴×C₂₆₉	C₂³	semidirect
80. C₂×Q₈×C₂₆₉	C₂³	semidirect
85. C₂²⋊C₄×C₂₆₉	C₂³	semidirect
86. C₂²⋊C₄×C₂₆₉	C₂³	semidirect
87. C₄²×C₂₆₉	C₂³	semidirect
88. C₄²×C₂₆₉	C₂³	semidirect
147. C₂₆₉	#3532	semidirect
149. H(3,𝔽₄)	D₂₆₉	semidirect
168. #124	C₂×D₂₆₉	semidirect
169. #124	C₂×D₂₆₉	semidirect
187. #120	C₂×D₂₆₉	semidirect
188. #120	C₂×D₂₆₉	semidirect
200. C₂⁴	C₂²×D₂₆₉	semidirect
202. C₂×Q₈	C₂²×D₂₆₉	semidirect
232. C₂²⋊C₄	C₂²×D₂₆₉	semidirect
233. C₂²⋊C₄	C₂²×D₂₆₉	semidirect
235. C₄²	C₂²×D₂₆₉	semidirect
236. C₄²	C₂²×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771344	2
#31771285	2
#31771248	2
#31771237	2
#31771189	2
#31771173	2
#31771165	2
#120×D₂₆₉	2
H(3,𝔽₄)×C₂₆₉	2
#3532	269