Subgroups — #46825661

Order 34432.

1464 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771441	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22575498	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771415	C₂	1	4	yes	maximal, complemented
4	#31771411	C₂	1	4	yes	maximal, complemented
5	#31771409	C₂	1	4	yes	maximal, complemented
6	#31771406	C₂	1	4	yes	maximal, complemented
7	#31771405	C₂	1	1	yes	maximal, characteristic, complemented
8	#31771403	C₂	1	1	yes	maximal, characteristic, complemented
9	#31771401	C₂	1	4	yes	maximal, complemented
10	C₂×#22575479	C₂	1	1	yes	maximal, characteristic, complemented
11	AGL(1,ℤ/8)×D₂₆₉	C₂	1	4	yes	maximal, complemented
12	#31771371	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×AGL(1,ℤ/8)×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22575501	C₂²	1	2	yes	complemented
15	#22575500	—	2	1	no	—
16	#22575500	C₂²	1	2	yes	complemented
17	#22575499	—	2	1	no	—
18	#22575499	C₂²	1	2	yes	complemented
19	#22575498	—	2	2	no	—
20	#22575498	—	2	2	no	—
21	#22575498	—	2	2	no	—
22	#22575498	C₂²	1	4	yes	complemented
23	#22575498	C₂²	1	4	yes	complemented
24	#22575498	C₂²	1	2	yes	complemented
25	C₂×#16864402	—	2	2	no	—
26	C₂×#16864402	C₂²	1	1	yes	characteristic
27	C₂×#16864402	C₂²	1	1	yes	characteristic, complemented
28	C₂×#16864402	C₂²	1	1	yes	characteristic, complemented
29	C₂×#16864401	—	2	1	no	—
30	C₂×#16864401	C₂²	1	2	yes	complemented
31	#41×D₂₆₉	—	2	2	no	—
32	#41×D₂₆₉	C₂²	1	4	yes	complemented
33	C₂²×C₂²⋊D₂₆₉	—	2	2	no	—
34	C₂×D₄×D₂₆₉	—	2	1	no	—
35	C₂×D₄×D₂₆₉	C₂²	1	2	yes	complemented
36	#22575486	C₂²	1	2	yes	complemented
37	#22575486	C₂²	1	2	yes	complemented
38	#22575484	C₂²	1	2	yes	complemented
39	#22575483	C₂²	1	2	yes	complemented
40	#22575482	C₂²	1	8	yes	complemented
41	#22575481	C₂²	1	2	yes	complemented
42	#22575481	C₂²	1	2	yes	complemented
43	#22575480	C₂²	1	8	yes	complemented
44	#22575479	C₂²	1	4	yes	complemented
45	#22575479	C₂²	1	4	yes	complemented
46	#22575479	C₂²	1	2	yes	complemented
47	#22575478	C₂²	1	2	yes	complemented
48	#22575477	C₂²	1	8	yes	complemented
49	#22575477	C₂²	1	8	yes	complemented
50	#22575475	C₂²	1	4	yes	complemented
51	#22575474	C₂²	1	8	yes	complemented
52	#22575473	C₂²	1	4	yes	complemented
53	#22575472	C₂²	1	8	yes	complemented
54	C₂×#16864390	C₂²	1	2	yes	complemented
55	SD₁₆×D₂₆₉	C₂²	1	8	yes	complemented
56	C₂×#16864387	C₂²	1	2	yes	complemented
57	D₈×D₂₆₉	C₂²	1	8	yes	complemented
58	#22575461	C₂²	1	4	yes	complemented
59	#22575459	C₂²	1	4	yes	complemented
60	C₂×#16864386	C₂²	1	1	yes	characteristic, complemented
61	M₁₆×D₂₆₉	C₂²	1	4	yes	complemented
62	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
63	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
64	AGL(1,ℤ/8)×C₂₆₉	C₂²	1	4	yes	complemented
65	AGL(1,ℤ/8)×C₂₆₉	C₂²	1	4	yes	complemented
66	C₂×SD₁₆×C₂₆₉	C₂²	1	2	yes	complemented
67	C₂×D₈×C₂₆₉	C₂²	1	2	yes	complemented
68	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
69	C₂³×D₂₆₉	—	2	2	no	—
70	#16864403	—	2	2	no	—
71	#16864403	C₂³	1	4	yes	complemented
72	#16864402	—	2	2	no	—
73	#16864402	—	2	2	no	—
74	#16864402	—	2	4	no	—
75	#16864402	—	2	4	no	—
76	#16864402	—	2	1	no	—
77	#16864402	—	2	1	no	—
78	#16864402	—	2	1	no	—
79	#16864402	—	2	1	no	—
80	#16864402	D₄	1	2	yes	—
81	#16864402	D₄	1	2	yes	—
82	#16864402	C₂³	1	2	yes	—
83	#16864402	C₂³	1	2	yes	—
84	#16864402	C₂³	1	2	yes	—
85	#16864402	C₂³	1	2	yes	—
86	#16864402	C₂³	1	2	yes	—
87	#16864402	C₂³	1	2	yes	complemented
88	#16864401	—	2	2	no	—
89	#16864401	—	2	4	no	—
90	#16864401	—	2	4	no	—
91	#16864401	—	2	4	no	—
92	#16864401	—	2	4	no	—
93	#16864401	—	2	4	no	—
94	#16864401	C₂³	1	8	yes	—
95	#16864401	C₂³	1	4	yes	complemented
96	C₂×Dic₅₃₈	—	2	2	no	—
97	C₂×Dic₅₃₈	D₄	1	1	yes	characteristic
98	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
99	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic, complemented
100	Q₈×D₂₆₉	—	2	2	no	—
101	Q₈×D₂₆₉	C₂³	1	4	yes	complemented
102	C₂×D₁₀₇₆	—	2	2	no	—
103	C₂×D₁₀₇₆	D₄	1	1	yes	characteristic
104	C₂×D₁₀₇₆	C₂³	1	1	yes	characteristic
105	C₂×D₁₀₇₆	C₂³	1	1	yes	characteristic, complemented
106	C₂×C₂²⋊D₂₆₉	—	2	2	no	—
107	C₂×C₂²⋊D₂₆₉	—	2	8	no	—
108	C₂×C₂²⋊D₂₆₉	—	2	8	no	—
109	C₂×C₂²⋊D₂₆₉	—	2	4	no	—
110	C₂×C₂²⋊D₂₆₉	—	2	2	no	—
111	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
112	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
113	C₂×C₂²⋊D₂₆₉	D₄	1	2	yes	—
114	D₄×D₂₆₉	—	2	4	no	—
115	D₄×D₂₆₉	—	2	4	no	—
116	D₄×D₂₆₉	—	2	4	no	—
117	D₄×D₂₆₉	—	2	2	no	—
118	D₄×D₂₆₉	—	2	4	no	—
119	D₄×D₂₆₉	—	2	4	no	—
120	D₄×D₂₆₉	C₂³	1	8	yes	—
121	D₄×D₂₆₉	C₂³	1	4	yes	complemented
122	C₂²×Dic₂₆₉	—	2	2	no	—
123	C₂×C₄×D₂₆₉	—	2	4	no	—
124	C₂×C₄×D₂₆₉	—	2	1	no	—
125	C₂×C₄×D₂₆₉	—	2	1	no	—
126	C₂×C₄×D₂₆₉	C₂³	1	2	yes	—
127	Dic₁₀₇₆	C₂³	1	4	yes	complemented
128	#16864392	C₂³	1	4	yes	complemented
129	#16864391	C₂³	1	4	yes	complemented
130	#16864391	C₂³	1	4	yes	complemented
131	#16864390	C₂³	1	8	yes	complemented
132	#16864390	C₂³	1	4	yes	complemented
133	#16864389	C₂³	1	4	yes	complemented
134	D₂₁₅₂	C₂³	1	4	yes	complemented
135	#16864387	C₂³	1	4	yes	complemented
136	#16864387	C₂³	1	8	yes	complemented
137	#16864386	C₂³	1	2	yes	—
138	#16864386	C₂³	1	2	yes	complemented
139	#16864385	C₂³	1	8	yes	complemented
140	C₂×#14524130	C₂³	1	2	yes	complemented
141	C₈×D₂₆₉	C₂³	1	8	yes	complemented
142	C₂⁴×C₂₆₉	—	2	1	no	—
143	#41×C₂₆₉	—	2	1	no	—
144	#41×C₂₆₉	—	2	1	no	—
145	#41×C₂₆₉	C₂³	1	2	yes	—
146	#41×C₂₆₉	C₂³	1	2	yes	complemented
147	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic, complemented
148	C₂×D₄×C₂₆₉	—	2	1	no	—
149	C₂×D₄×C₂₆₉	—	2	1	no	—
150	C₂×D₄×C₂₆₉	—	2	1	no	—
151	C₂×D₄×C₂₆₉	—	2	1	no	—
152	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
153	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
154	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
155	C₂×D₄×C₂₆₉	C₂³	1	1	yes	characteristic, complemented
156	C₂²×C₄×C₂₆₉	—	2	1	no	—
157	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
158	SD₁₆×C₂₆₉	C₂³	1	8	yes	complemented
159	D₈×C₂₆₉	C₂³	1	8	yes	complemented
160	M₁₆×C₂₆₉	C₂³	1	2	yes	—
161	M₁₆×C₂₆₉	C₂³	1	2	yes	complemented
162	C₂×C₈×C₂₆₉	C₂³	1	2	yes	complemented
163	C₂²×D₂₆₉	—	2	1	no	—
164	C₂²×D₂₆₉	—	2	1	no	—
165	C₂²×D₂₆₉	—	2	4	no	—
166	C₂²×D₂₆₉	—	2	8	no	—
167	C₂²×D₂₆₉	—	8	4	no	—
168	C₂²×D₂₆₉	C₂×D₄	1	2	yes	—
169	Dic₅₃₈	—	2	1	no	—
170	Dic₅₃₈	—	2	1	no	—
171	Dic₅₃₈	—	2	4	no	—
172	Dic₅₃₈	C₂×D₄	1	2	yes	—
173	Dic₅₃₈	C₂×D₄	1	2	yes	—
174	Dic₅₃₈	C₂⁴	1	2	yes	—
175	Dic₅₃₈	C₂⁴	1	2	yes	—
176	D₁₀₇₆	—	2	4	no	—
177	D₁₀₇₆	—	2	1	no	—
178	D₁₀₇₆	—	2	1	no	—
179	D₁₀₇₆	C₂×D₄	1	2	yes	—
180	D₁₀₇₆	C₂×D₄	1	2	yes	—
181	D₁₀₇₆	C₂⁴	1	2	yes	—
182	D₁₀₇₆	C₂⁴	1	2	yes	—
183	C₂²⋊D₂₆₉	—	2	2	no	—
184	C₂²⋊D₂₆₉	—	2	2	no	—
185	C₂²⋊D₂₆₉	—	2	2	no	—
186	C₂²⋊D₂₆₉	—	2	2	no	—
187	C₂²⋊D₂₆₉	—	2	8	no	—
188	C₂²⋊D₂₆₉	—	2	4	no	—
189	C₂²⋊D₂₆₉	—	2	8	no	—
190	C₂²⋊D₂₆₉	—	2	4	no	—
191	C₂²⋊D₂₆₉	C₂×D₄	1	4	yes	—
192	C₂²⋊D₂₆₉	C₂×D₄	1	4	yes	—
193	C₂×Dic₂₆₉	—	2	1	no	—
194	C₂×Dic₂₆₉	—	2	1	no	—
195	C₂×Dic₂₆₉	—	2	4	no	—
196	C₂×Dic₂₆₉	—	2	8	no	—
197	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
198	C₄×D₂₆₉	—	2	4	no	—
199	C₄×D₂₆₉	—	2	2	no	—
200	C₄×D₂₆₉	—	2	2	no	—
201	C₄×D₂₆₉	—	2	2	no	—
202	C₄×D₂₆₉	—	2	2	no	—
203	C₄×D₂₆₉	—	2	4	no	—
204	C₄×D₂₆₉	C₂×D₄	1	4	yes	—
205	C₄×D₂₆₉	C₂⁴	1	4	yes	—
206	#14524130	C₂⁴	1	4	yes	—
207	C₂³×C₂₆₉	—	2	2	no	—
208	C₂³×C₂₆₉	—	2	1	no	—
209	C₂³×C₂₆₉	—	2	2	no	—
210	C₂³×C₂₆₉	—	2	2	no	—
211	C₂³×C₂₆₉	—	8	2	no	—
212	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
213	Q₈×C₂₆₉	—	2	1	no	—
214	Q₈×C₂₆₉	C₂⁴	1	2	yes	—
215	D₄×C₂₆₉	—	2	2	no	—
216	D₄×C₂₆₉	—	2	2	no	—
217	D₄×C₂₆₉	—	2	2	no	—
218	D₄×C₂₆₉	—	2	2	no	—
219	D₄×C₂₆₉	—	2	2	no	—
220	D₄×C₂₆₉	—	2	1	no	—
221	D₄×C₂₆₉	C₂⁴	1	2	yes	—
222	D₄×C₂₆₉	C₂⁴	1	4	yes	—
223	C₂×C₄×C₂₆₉	—	2	1	no	—
224	C₂×C₄×C₂₆₉	—	2	2	no	—
225	C₂×C₄×C₂₆₉	—	2	2	no	—
226	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
228	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
229	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
230	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
231	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
232	C₈×C₂₆₉	C₂⁴	1	4	yes	—
233	C₂²×C₂₆₉	—	2	2	no	—
234	C₂²×C₂₆₉	—	2	4	no	—
235	C₂²×C₂₆₉	—	4	1	no	—
236	C₂²×C₂₆₉	—	4	4	no	—
237	C₂²×C₂₆₉	—	8	1	no	—
238	C₂²×C₂₆₉	—	8	2	no	—
239	C₂²×C₂₆₉	—	8	2	no	—
240	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
241	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
242	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
243	C₂×D₂₆₉	—	2	2	no	—
244	C₂×D₂₆₉	—	2	2	no	—
245	C₂×D₂₆₉	—	4	2	no	—
246	C₂×D₂₆₉	—	4	2	no	—
247	C₂×D₂₆₉	—	8	4	no	—
248	C₂×D₂₆₉	—	8	8	no	—
249	C₂×D₂₆₉	—	8	1	no	—
250	C₂×D₂₆₉	C₂²×D₄	1	4	yes	—
251	C₄×C₂₆₉	—	2	2	no	—
252	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
253	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
254	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
255	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
256	Dic₂₆₉	—	2	2	no	—
257	Dic₂₆₉	—	2	2	no	—
258	Dic₂₆₉	C₂²×D₄	1	4	yes	—
259	C₂×C₂₆₉	—	2	1	no	—
260	C₂×C₂₆₉	—	2	1	no	—
261	C₂×C₂₆₉	—	2	1	no	—
262	C₂×C₂₆₉	—	4	4	no	—
263	C₂×C₂₆₉	—	4	2	no	—
264	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
265	D₂₆₉	—	2	4	no	—
266	D₂₆₉	—	4	2	no	—
267	D₂₆₉	—	4	2	no	—
268	C₂₆₉	#3585	1	1	yes	characteristic, complemented, Sylow
269	#3585	—	269	1	no	maximal, Sylow
270	#585	—	269	1	no	—
271	C₂×H(5,𝔽₂)	—	269	1	no	—
272	#577	—	269	4	no	—
273	#577	—	269	4	no	—
274	#576	—	269	4	no	—
275	#576	—	269	4	no	—
276	#575	—	269	1	no	—
277	#575	—	269	4	no	—
278	#575	—	269	1	no	—
279	C₂×AGL(1,ℤ/8)	—	269	1	no	—
280	C₂×AGL(1,ℤ/8)	—	269	4	no	—
281	C₂×AGL(1,ℤ/8)	D₂₆₉	1	1	yes	characteristic, complemented
282	#568	—	269	1	no	—
283	2₋¹⁺⁴	—	269	2	no	—
284	2₋¹⁺⁴	—	269	2	no	—
285	2₋¹⁺⁴	—	538	1	no	—
286	H(5,𝔽₂)	—	269	2	no	—
287	H(5,𝔽₂)	—	269	4	no	—
288	H(5,𝔽₂)	—	269	4	no	—
289	H(5,𝔽₂)	—	269	2	no	—
290	H(5,𝔽₂)	—	538	2	no	—
291	H(5,𝔽₂)	—	538	2	no	—
292	H(5,𝔽₂)	—	538	2	no	—
293	H(5,𝔽₂)	—	538	1	no	—
294	C₂×#41	—	269	1	no	—
295	C₂×#41	—	269	4	no	—
296	C₂×#41	—	269	2	no	—
297	C₂×#41	—	269	1	no	—
298	C₂×#41	—	269	1	no	—
299	C₂×#41	—	538	2	no	—
300	C₂×#41	—	538	1	no	—
301	C₂×#41	—	538	2	no	—
302	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic, complemented
303	C₂²×D₄	—	269	2	no	—
304	C₂²×D₄	—	538	1	no	—
305	C₂²×D₄	—	538	2	no	—
306	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
307	#137	—	269	2	no	—
308	#137	—	269	2	no	—
309	#137	—	269	8	no	—
310	#137	—	269	2	no	—
311	#137	—	269	2	no	—
312	AGL(1,ℤ/8)	—	269	2	no	—
313	AGL(1,ℤ/8)	—	269	4	no	—
314	AGL(1,ℤ/8)	—	269	4	no	—
315	AGL(1,ℤ/8)	—	269	2	no	—
316	AGL(1,ℤ/8)	—	269	8	no	—
317	AGL(1,ℤ/8)	—	269	8	no	—
318	AGL(1,ℤ/8)	—	269	8	no	—
319	AGL(1,ℤ/8)	—	269	2	no	—
320	AGL(1,ℤ/8)	—	269	2	no	—
321	AGL(1,ℤ/8)	C₂×D₂₆₉	1	4	yes	complemented
322	AGL(1,ℤ/8)	C₂×D₂₆₉	1	4	yes	complemented
323	#135	—	269	4	no	—
324	#135	—	269	8	no	—
325	#135	—	269	8	no	—
326	#135	—	269	4	no	—
327	C₂×SD₁₆	—	269	2	no	—
328	C₂×SD₁₆	—	269	8	no	—
329	C₂×SD₁₆	C₂×D₂₆₉	1	2	yes	complemented
330	C₂×D₈	—	269	2	no	—
331	C₂×D₈	—	269	8	no	—
332	C₂×D₈	C₂×D₂₆₉	1	2	yes	complemented
333	#131	—	269	4	no	—
334	#131	—	269	4	no	—
335	C₂×M₁₆	—	269	1	no	—
336	C₂×M₁₆	—	269	4	no	—
337	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic, complemented
338	C₂⁴	—	2	1	no	—
339	C₂⁴	—	538	2	no	—
340	#41	—	2	1	no	—
341	#41	—	2	1	no	—
342	#41	—	269	2	no	—
343	#41	—	269	2	no	—
344	#41	—	269	2	no	—
345	#41	—	269	2	no	—
346	#41	—	269	4	no	—
347	#41	—	269	4	no	—
348	#41	—	269	8	no	—
349	#41	—	269	2	no	—
350	#41	—	269	2	no	—
351	#41	—	269	2	no	—
352	#41	—	269	2	no	—
353	#41	—	538	4	no	—
354	#41	—	538	4	no	—
355	#41	—	538	2	no	—
356	#41	—	538	2	no	—
357	#41	—	538	4	no	—
358	#41	—	538	4	no	—
359	#41	—	538	4	no	—
360	#41	—	538	4	no	—
361	#41	—	538	4	no	—
362	#41	—	538	2	no	—
363	#41	—	538	2	no	—
364	#41	—	538	1	no	—
365	#41	—	538	1	no	—
366	#41	—	538	1	no	—
367	#41	—	538	1	no	—
368	#41	C₂²×D₂₆₉	1	2	yes	—
369	#41	C₂²×D₂₆₉	1	2	yes	complemented
370	C₂×Q₈	—	269	1	no	—
371	C₂×Q₈	—	269	4	no	—
372	C₂×Q₈	—	269	1	no	—
373	C₂×Q₈	—	269	1	no	—
374	C₂×Q₈	—	538	2	no	—
375	C₂×Q₈	—	538	2	no	—
376	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic, complemented
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	269	2	no	—
382	C₂×D₄	—	269	4	no	—
383	C₂×D₄	—	269	8	no	—
384	C₂×D₄	—	269	1	no	—
385	C₂×D₄	—	269	1	no	—
386	C₂×D₄	—	269	1	no	—
387	C₂×D₄	—	538	2	no	—
388	C₂×D₄	—	538	4	no	—
389	C₂×D₄	—	538	2	no	—
390	C₂×D₄	—	538	4	no	—
391	C₂×D₄	—	538	4	no	—
392	C₂×D₄	—	538	4	no	—
393	C₂×D₄	—	538	2	no	—
394	C₂×D₄	—	538	4	no	—
395	C₂×D₄	—	538	8	no	—
396	C₂×D₄	—	538	8	no	—
397	C₂×D₄	—	538	4	no	—
398	C₂×D₄	—	538	2	no	—
399	C₂×D₄	—	538	1	no	—
400	C₂×D₄	—	538	1	no	—
401	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
402	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
403	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
404	C₂×D₄	C₂²×D₂₆₉	1	1	yes	characteristic, complemented
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	269	2	no	—
407	C₂²×C₄	—	538	4	no	—
408	C₂²×C₄	—	538	2	no	—
409	C₂²×C₄	—	538	1	no	—
410	C₂²×C₄	—	538	1	no	—
411	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
412	Q₁₆	—	269	4	no	—
413	Q₁₆	—	269	4	no	—
414	SD₁₆	—	269	4	no	—
415	SD₁₆	—	269	8	no	—
416	SD₁₆	—	269	4	no	—
417	SD₁₆	—	269	4	no	—
418	SD₁₆	—	269	4	no	—
419	SD₁₆	C₂²×D₂₆₉	1	8	yes	complemented
420	D₈	—	269	8	no	—
421	D₈	—	269	4	no	—
422	D₈	—	269	4	no	—
423	D₈	C₂²×D₂₆₉	1	8	yes	complemented
424	M₁₆	—	269	2	no	—
425	M₁₆	—	269	2	no	—
426	M₁₆	—	269	8	no	—
427	M₁₆	C₂²×D₂₆₉	1	2	yes	—
428	M₁₆	C₂²×D₂₆₉	1	2	yes	complemented
429	C₂×C₈	—	269	2	no	—
430	C₂×C₈	—	269	8	no	—
431	C₂×C₈	C₂²×D₂₆₉	1	2	yes	complemented
432	C₂³	—	2	2	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	8	2	no	—
437	C₂³	—	269	2	no	—
438	C₂³	—	538	4	no	—
439	C₂³	—	538	8	no	—
440	C₂³	—	538	1	no	—
441	C₂³	—	538	1	no	—
442	C₂³	—	2152	4	no	—
443	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
444	Q₈	—	2	1	no	—
445	Q₈	—	269	2	no	—
446	Q₈	—	269	2	no	—
447	Q₈	—	269	2	no	—
448	Q₈	—	269	2	no	—
449	Q₈	—	538	4	no	—
450	Q₈	—	538	1	no	—
451	Q₈	—	538	1	no	—
452	Q₈	C₂³×D₂₆₉	1	2	yes	—
453	D₄	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	269	4	no	—
460	D₄	—	269	4	no	—
461	D₄	—	269	2	no	—
462	D₄	—	269	2	no	—
463	D₄	—	269	2	no	—
464	D₄	—	269	2	no	—
465	D₄	—	538	4	no	—
466	D₄	—	538	8	no	—
467	D₄	—	538	4	no	—
468	D₄	—	538	8	no	—
469	D₄	—	538	4	no	—
470	D₄	—	538	2	no	—
471	D₄	—	538	2	no	—
472	D₄	—	538	2	no	—
473	D₄	—	538	2	no	—
474	D₄	—	538	1	no	—
475	D₄	—	538	1	no	—
476	D₄	C₂³×D₂₆₉	1	2	yes	—
477	D₄	C₂³×D₂₆₉	1	4	yes	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	269	2	no	—
482	C₂×C₄	—	269	4	no	—
483	C₂×C₄	—	269	4	no	—
484	C₂×C₄	—	538	4	no	—
485	C₂×C₄	—	538	4	no	—
486	C₂×C₄	—	538	8	no	—
487	C₂×C₄	—	538	4	no	—
488	C₂×C₄	—	538	1	no	—
489	C₂×C₄	—	538	1	no	—
490	C₂×C₄	—	538	2	no	—
491	C₂×C₄	—	538	2	no	—
492	C₂×C₄	—	538	2	no	—
493	C₂×C₄	—	538	2	no	—
494	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
495	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
496	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
497	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
498	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
499	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
500	C₈	—	269	4	no	—
501	C₈	C₂³×D₂₆₉	1	4	yes	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	269	4	no	—
510	C₂²	—	538	2	no	—
511	C₂²	—	538	2	no	—
512	C₂²	—	1076	2	no	—
513	C₂²	—	1076	2	no	—
514	C₂²	—	2152	1	no	—
515	C₂²	—	2152	8	no	—
516	C₂²	—	2152	4	no	—
517	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
518	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
519	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
520	C₄	—	2	2	no	—
521	C₄	—	269	4	no	—
522	C₄	—	538	2	no	—
523	C₄	—	538	2	no	—
524	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
525	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
526	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
527	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	2	1	no	—
531	C₂	—	4	4	no	—
532	C₂	—	4	2	no	—
533	C₂	—	538	4	no	—
534	C₂	—	1076	2	no	—
535	C₂	—	1076	2	no	—
536	C₂	C₂²×D₄×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₂₆₉
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂×AGL(1,ℤ/8)×C₂₆₉	C₂
socle	C₂×C₂₆₉	C₂³×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₂₆₉
hypocenter	C₂₆₉	#3585
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #31771441	C₂	semidirect
2. C₂×#22575498	C₂	semidirect
3. #31771415	C₂	semidirect
4. #31771411	C₂	semidirect
5. #31771409	C₂	semidirect
6. #31771406	C₂	semidirect
7. #31771405	C₂	semidirect
8. #31771403	C₂	semidirect
9. #31771401	C₂	semidirect
10. C₂×#22575479	C₂	semidirect
11. AGL(1,ℤ/8)×D₂₆₉	C₂	semidirect
12. #31771371	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₂₆₉	C₂	semidirect
14. #22575501	C₂²	semidirect
16. #22575500	C₂²	semidirect
18. #22575499	C₂²	semidirect
22. #22575498	C₂²	semidirect
23. #22575498	C₂²	semidirect
24. #22575498	C₂²	semidirect
27. C₂×#16864402	C₂²	semidirect
28. C₂×#16864402	C₂²	semidirect
30. C₂×#16864401	C₂²	semidirect
32. #41×D₂₆₉	C₂²	semidirect
35. C₂×D₄×D₂₆₉	C₂²	semidirect
36. #22575486	C₂²	semidirect
37. #22575486	C₂²	semidirect
38. #22575484	C₂²	semidirect
39. #22575483	C₂²	semidirect
40. #22575482	C₂²	semidirect
41. #22575481	C₂²	semidirect
42. #22575481	C₂²	semidirect
43. #22575480	C₂²	semidirect
44. #22575479	C₂²	semidirect
45. #22575479	C₂²	semidirect
46. #22575479	C₂²	semidirect
47. #22575478	C₂²	semidirect
48. #22575477	C₂²	semidirect
49. #22575477	C₂²	semidirect
50. #22575475	C₂²	semidirect
51. #22575474	C₂²	semidirect
52. #22575473	C₂²	semidirect
53. #22575472	C₂²	semidirect
54. C₂×#16864390	C₂²	semidirect
55. SD₁₆×D₂₆₉	C₂²	semidirect
56. C₂×#16864387	C₂²	semidirect
57. D₈×D₂₆₉	C₂²	semidirect
58. #22575461	C₂²	semidirect
59. #22575459	C₂²	semidirect
60. C₂×#16864386	C₂²	semidirect
61. M₁₆×D₂₆₉	C₂²	semidirect
62. C₂×#41×C₂₆₉	C₂²	semidirect
63. C₂²×D₄×C₂₆₉	C₂²	semidirect
64. AGL(1,ℤ/8)×C₂₆₉	C₂²	semidirect
65. AGL(1,ℤ/8)×C₂₆₉	C₂²	semidirect
66. C₂×SD₁₆×C₂₆₉	C₂²	semidirect
67. C₂×D₈×C₂₆₉	C₂²	semidirect
68. C₂×M₁₆×C₂₆₉	C₂²	semidirect
71. #16864403	C₂³	semidirect
87. #16864402	C₂³	semidirect
95. #16864401	C₂³	semidirect
99. C₂×Dic₅₃₈	C₂³	semidirect
101. Q₈×D₂₆₉	C₂³	semidirect
105. C₂×D₁₀₇₆	C₂³	semidirect
121. D₄×D₂₆₉	C₂³	semidirect
127. Dic₁₀₇₆	C₂³	semidirect
128. #16864392	C₂³	semidirect
129. #16864391	C₂³	semidirect
130. #16864391	C₂³	semidirect
131. #16864390	C₂³	semidirect
132. #16864390	C₂³	semidirect
133. #16864389	C₂³	semidirect
134. D₂₁₅₂	C₂³	semidirect
135. #16864387	C₂³	semidirect
136. #16864387	C₂³	semidirect
138. #16864386	C₂³	semidirect
139. #16864385	C₂³	semidirect
140. C₂×#14524130	C₂³	semidirect
141. C₈×D₂₆₉	C₂³	semidirect
146. #41×C₂₆₉	C₂³	semidirect
147. C₂×Q₈×C₂₆₉	C₂³	semidirect
155. C₂×D₄×C₂₆₉	C₂³	semidirect
158. SD₁₆×C₂₆₉	C₂³	semidirect
159. D₈×C₂₆₉	C₂³	semidirect
161. M₁₆×C₂₆₉	C₂³	semidirect
162. C₂×C₈×C₂₆₉	C₂³	semidirect
268. C₂₆₉	#3585	semidirect
281. C₂×AGL(1,ℤ/8)	D₂₆₉	semidirect
302. C₂×#41	C₂×D₂₆₉	semidirect
306. C₂²×D₄	C₂×D₂₆₉	semidirect
321. AGL(1,ℤ/8)	C₂×D₂₆₉	semidirect
322. AGL(1,ℤ/8)	C₂×D₂₆₉	semidirect
329. C₂×SD₁₆	C₂×D₂₆₉	semidirect
332. C₂×D₈	C₂×D₂₆₉	semidirect
337. C₂×M₁₆	C₂×D₂₆₉	semidirect
369. #41	C₂²×D₂₆₉	semidirect
376. C₂×Q₈	C₂²×D₂₆₉	semidirect
404. C₂×D₄	C₂²×D₂₆₉	semidirect
419. SD₁₆	C₂²×D₂₆₉	semidirect
423. D₈	C₂²×D₂₆₉	semidirect
428. M₁₆	C₂²×D₂₆₉	semidirect
431. C₂×C₈	C₂²×D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771441	2
C₂×#22575498	2
#31771415	2
#31771411	2
#31771409	2
#31771406	2
#31771405	2
#31771403	2
#31771401	2
C₂×#22575479	2
AGL(1,ℤ/8)×D₂₆₉	2
#31771371	2
C₂×AGL(1,ℤ/8)×C₂₆₉	2
#3585	269