Subgroups — #30862286

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569664	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863516	C₂	1	2	yes	maximal, complemented
3	#22568922	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561128	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560857	C₂	1	2	yes	maximal, complemented
7	#22560536	C₂	1	2	yes	maximal, complemented
8	#22560229	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560076	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560055	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	2	yes	—
13	C₂×#14524049	C₂²	1	2	yes	—
14	#16863946	C₂²	1	1	yes	characteristic
15	#16863618	C₄	1	2	yes	complemented
16	#16863611	C₄	1	2	yes	complemented
17	#16863606	C₄	1	1	yes	characteristic, complemented
18	#16863601	C₄	1	1	yes	characteristic, complemented
19	#16863596	C₄	1	1	yes	characteristic, complemented
20	#16863591	C₄	1	1	yes	characteristic, complemented
21	#16863516	—	2	8	no	—
22	#16863475	—	2	1	no	—
23	#16863475	—	2	1	no	—
24	C₈×D₂₆₈	—	2	1	no	—
25	C₈×D₂₆₈	—	2	1	no	—
26	C₂×#14523978	C₂²	1	2	yes	—
27	#16863420	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×#14523970	C₂²	1	2	yes	—
30	C₂×#14523969	C₂²	1	2	yes	—
31	C₂×#14523969	C₂²	1	2	yes	—
32	#16863390	C₂²	1	2	yes	—
33	#16863362	C₂²	1	2	yes	—
34	C₂×#14523953	C₂²	1	2	yes	—
35	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
36	#16863160	C₂²	1	2	yes	—
37	#16863155	C₂²	1	2	yes	—
38	#16863150	C₂²	1	2	yes	—
39	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
41	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
42	#405×C₆₇	—	2	1	no	—
43	#405×C₆₇	—	2	1	no	—
44	#405×C₆₇	C₂²	1	1	yes	characteristic
45	#405×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	—	2	1	no	—
47	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	—	2	1	no	—
49	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#11761628	C₂³	1	2	yes	—
51	#14524063	—	2	4	no	—
52	#14524063	—	2	4	no	—
53	C₂×#11761618	C₂×C₄	1	2	yes	—
54	C₂×#11761616	C₂×C₄	1	2	yes	—
55	C₂×#11761615	C₂×C₄	1	2	yes	—
56	C₂×#11761613	C₂×C₄	1	2	yes	—
57	#14524049	—	2	4	no	—
58	#14524049	—	2	4	no	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₂×#11761611	—	2	2	no	—
61	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
62	C₂×C₈×D₆₇	—	2	2	no	—
63	#14524043	C₂×C₄	1	1	yes	characteristic
64	#14524040	C₂×C₄	1	1	yes	characteristic
65	#14524031	C₂×C₄	1	2	yes	—
66	#14524017	—	2	1	no	—
67	#14524017	—	2	1	no	—
68	C₄×D₂₆₈	—	2	1	no	—
69	C₄×D₂₆₈	—	2	1	no	—
70	C₂×C₄⋊Dic₆₇	—	2	1	no	—
71	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
72	#14523978	—	2	4	no	—
73	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
74	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
75	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
78	#14523970	—	2	4	no	—
79	#14523969	—	2	4	no	—
80	#14523969	—	2	4	no	—
81	#14523966	C₂×C₄	1	2	yes	—
82	#14523966	C₂×C₄	1	2	yes	—
83	#14523963	C₂×C₄	1	2	yes	—
84	#14523963	C₂×C₄	1	2	yes	—
85	(C₂×D₆₇)⋊C₈	—	2	2	no	—
86	(C₂×D₆₇)⋊C₈	—	2	2	no	—
87	#14523953	—	2	4	no	—
88	C₆₇⋊#95	—	2	1	no	—
89	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
90	#131×C₆₇	—	2	2	no	—
91	#131×C₆₇	—	2	2	no	—
92	#131×C₆₇	—	2	2	no	—
93	#131×C₆₇	D₄	1	4	yes	complemented
94	C₂×M₁₆×C₆₇	—	2	1	no	—
95	C₂×M₁₆×C₆₇	—	2	1	no	—
96	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	—	2	1	no	—
99	C₂²×C₈×C₆₇	—	2	1	no	—
100	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₆₇	—	2	1	no	—
105	C₄×D₄×C₆₇	—	2	1	no	—
106	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#117×C₆₇	—	2	1	no	—
109	#117×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	#97×C₆₇	—	2	1	no	—
113	#97×C₆₇	—	2	1	no	—
114	#97×C₆₇	—	2	1	no	—
115	C₄×C₈×C₆₇	—	2	1	no	—
116	C₄×C₈×C₆₇	—	2	1	no	—
117	C₄×C₈×C₆₇	—	2	1	no	—
118	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
119	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
120	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
121	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
122	#11761628	—	2	2	no	—
123	#11761628	—	2	2	no	—
124	#11761628	—	2	2	no	—
125	#11761628	—	2	2	no	—
126	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
127	C₂×D₂₆₈	—	2	1	no	—
128	C₂×D₂₆₈	—	2	1	no	—
129	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
130	C₂×C₂²⋊D₆₇	—	2	2	no	—
131	C₂²×Dic₆₇	—	2	2	no	—
132	C₂×C₄×D₆₇	—	2	2	no	—
133	#11761618	—	2	4	no	—
134	#11761616	—	2	4	no	—
135	#11761615	—	2	4	no	—
136	#11761613	—	2	4	no	—
137	#11761612	—	2	2	no	—
138	#11761612	—	2	2	no	—
139	#11761611	—	4	4	no	—
140	C₂×C₆₇⋊C₈	—	2	2	no	—
141	C₂×C₆₇⋊C₈	—	2	2	no	—
142	C₂×C₆₇⋊C₈	—	2	2	no	—
143	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
145	C₈×D₆₇	—	4	4	no	—
146	C₄⋊Dic₆₇	—	2	1	no	—
147	C₄⋊Dic₆₇	—	2	1	no	—
148	C₄⋊Dic₆₇	—	4	2	no	—
149	(C₂×D₆₇)⋊C₄	—	2	2	no	—
150	(C₂×D₆₇)⋊C₄	—	2	2	no	—
151	C₄×Dic₆₇	—	2	2	no	—
152	C₄×Dic₆₇	—	2	2	no	—
153	#41×C₆₇	—	2	1	no	—
154	#41×C₆₇	—	2	1	no	—
155	#41×C₆₇	C₂×D₄	1	2	yes	—
156	#41×C₆₇	#41	1	2	yes	—
157	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂×D₄×C₆₇	—	2	1	no	—
159	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	M₁₆×C₆₇	—	2	4	no	—
167	M₁₆×C₆₇	—	2	1	no	—
168	M₁₆×C₆₇	—	2	1	no	—
169	M₁₆×C₆₇	—	2	4	no	—
170	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
171	M₁₆×C₆₇	#41	1	2	yes	—
172	C₂×C₈×C₆₇	—	2	4	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	1	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	4	no	—
178	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₆₇	#41	1	2	yes	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
188	C₂²⋊C₄×C₆₇	—	2	1	no	—
189	C₂²⋊C₄×C₆₇	—	2	1	no	—
190	C₂²⋊C₄×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂²×D₆₇	—	2	2	no	—
201	Q₈×C₆₇	—	2	1	no	—
202	Q₈×C₆₇	C₄×D₄	1	2	yes	—
203	D₄×C₆₇	—	2	2	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	2	1	no	—
206	D₄×C₆₇	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
226	C₂²⋊D₆₇	—	4	2	no	—
227	C₂²⋊D₆₇	—	4	2	no	—
228	C₂×Dic₆₇	—	2	2	no	—
229	C₂×Dic₆₇	—	4	2	no	—
230	C₂×Dic₆₇	—	4	2	no	—
231	C₂×Dic₆₇	—	4	2	no	—
232	D₂₆₈	—	2	2	no	—
233	D₂₆₈	—	2	2	no	—
234	D₂₆₈	—	4	2	no	—
235	D₂₆₈	—	4	2	no	—
236	C₄×D₆₇	—	4	2	no	—
237	C₄×D₆₇	—	4	2	no	—
238	Dic₁₃₄	—	2	2	no	—
239	Dic₁₃₄	—	2	2	no	—
240	C₈×C₆₇	—	2	2	no	—
241	C₈×C₆₇	—	2	1	no	—
242	C₈×C₆₇	—	2	2	no	—
243	C₈×C₆₇	—	2	1	no	—
244	C₈×C₆₇	#121	1	2	yes	—
245	C₈×C₆₇	#121	1	2	yes	—
246	C₆₇⋊C₈	—	2	2	no	—
247	C₆₇⋊C₈	—	2	2	no	—
248	C₆₇⋊C₈	—	4	2	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	—	4	1	no	—
254	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₆₇	#529	1	1	yes	characteristic
256	C₂²×C₆₇	#532	1	1	yes	characteristic
257	C₂×D₆₇	—	4	2	no	—
258	C₂×D₆₇	—	4	2	no	—
259	C₂×D₆₇	—	4	2	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	4	1	no	—
265	C₄×C₆₇	—	4	1	no	—
266	C₄×C₆₇	#444	1	2	yes	complemented
267	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
268	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
269	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
270	C₄×C₆₇	#525	1	1	yes	characteristic
271	Dic₆₇	—	4	2	no	—
272	Dic₆₇	—	8	2	no	—
273	#21518	—	67	1	no	maximal, Sylow
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	4	2	no	—
277	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
278	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
279	C₂×C₆₇	#2955	1	1	yes	characteristic
280	D₆₇	—	8	2	no	—
281	#3035	—	67	1	no	—
282	C₂×#444	—	67	2	no	—
283	#2931	—	67	1	no	—
284	C₄×#131	D₆₇	1	1	yes	characteristic, complemented
285	#1980	—	67	1	no	—
286	#1950	—	67	1	no	—
287	#1929	—	67	2	no	—
288	#1876	—	67	2	no	—
289	#1806	—	67	1	no	—
290	#1770	—	67	1	no	—
291	#1765	—	67	2	no	—
292	C₆₇	#21518	1	1	yes	characteristic, complemented, Sylow
293	C₂×#135	—	67	2	no	—
294	C₂×#131	—	67	2	no	—
295	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
296	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
297	#532	—	67	1	no	—
298	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
299	#464	—	67	2	no	—
300	#463	—	67	2	no	—
301	#462	—	67	1	no	—
302	#461	—	67	1	no	—
303	#460	—	67	1	no	—
304	#459	—	67	1	no	—
305	#444	—	134	8	no	—
306	#436	—	134	1	no	—
307	#436	—	134	1	no	—
308	C₈×D₄	—	134	1	no	—
309	C₈×D₄	—	134	1	no	—
310	C₂×#108	—	67	2	no	—
311	#423	—	67	1	no	—
312	C₂×C₄⋊C₈	—	67	1	no	—
313	C₂×#104	—	67	2	no	—
314	C₂×#104	—	67	2	no	—
315	C₂×#104	—	67	2	no	—
316	#416	—	67	2	no	—
317	#408	—	67	2	no	—
318	#405	—	2	1	no	—
319	#405	—	2	1	no	—
320	#405	C₂×D₆₇	1	1	yes	characteristic
321	#405	C₂×D₆₇	1	1	yes	characteristic
322	C₄×M₁₆	—	2	1	no	—
323	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
324	C₂×#97	—	67	2	no	—
325	C₂×C₄×C₈	—	2	1	no	—
326	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
327	C₄×C₄⋊C₄	—	67	1	no	—
328	#341	—	67	2	no	—
329	#339	—	67	2	no	—
330	#336	—	67	2	no	—
331	C₂×#41	—	67	2	no	—
332	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
333	#135	—	134	4	no	—
334	#135	—	134	4	no	—
335	C₂×Q₁₆	—	67	2	no	—
336	C₂×SD₁₆	—	67	2	no	—
337	C₂×SD₁₆	—	67	2	no	—
338	C₂×D₈	—	67	2	no	—
339	#131	—	2	2	no	—
340	#131	—	2	2	no	—
341	#131	—	2	2	no	—
342	#131	—	134	4	no	—
343	#131	—	134	4	no	—
344	#131	D₂₆₈	1	4	yes	complemented
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	1	no	—
347	C₂×M₁₆	—	67	2	no	—
348	C₂×M₁₆	—	134	2	no	—
349	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
350	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
351	C₂²×C₈	—	2	1	no	—
352	C₂²×C₈	—	2	1	no	—
353	C₂²×C₈	—	67	2	no	—
354	C₂²×C₈	—	134	2	no	—
355	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₄⋊Q₈	—	67	1	no	—
358	#127	—	67	1	no	—
359	#124	—	67	2	no	—
360	#121	—	134	1	no	—
361	#121	—	134	1	no	—
362	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
363	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	134	1	no	—
367	C₄×D₄	—	134	1	no	—
368	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
369	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
370	#117	—	2	1	no	—
371	#117	C₂²×D₆₇	1	1	yes	characteristic
372	C₂×C₄⋊C₄	—	134	1	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	—	67	2	no	—
375	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
376	#108	—	134	4	no	—
377	C₄⋊C₈	—	67	1	no	—
378	C₄⋊C₈	—	67	1	no	—
379	C₄⋊C₈	—	67	2	no	—
380	C₄⋊C₈	—	134	1	no	—
381	C₄⋊C₈	—	134	1	no	—
382	#104	—	134	4	no	—
383	#104	—	134	4	no	—
384	#104	—	134	4	no	—
385	#103	—	67	2	no	—
386	#103	—	67	2	no	—
387	#102	—	67	2	no	—
388	#102	—	67	2	no	—
389	C₂²⋊C₈	—	134	2	no	—
390	C₂²⋊C₈	—	134	2	no	—
391	#97	—	2	1	no	—
392	#97	—	2	1	no	—
393	#97	—	2	1	no	—
394	#97	—	134	4	no	—
395	C₄×C₈	—	2	1	no	—
396	C₄×C₈	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
399	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
400	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
401	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
402	#95	—	134	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	134	2	no	—
406	#41	—	134	2	no	—
407	#41	—	134	2	no	—
408	#41	—	134	2	no	—
409	#41	C₂×D₂₆₈	1	2	yes	—
410	#41	#11761628	1	2	yes	—
411	C₂×Q₈	—	67	2	no	—
412	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	67	2	no	—
415	C₂×D₄	—	134	2	no	—
416	C₂×D₄	—	134	1	no	—
417	C₂×D₄	—	134	1	no	—
418	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	134	2	no	—
423	C₂²×C₄	—	134	2	no	—
424	C₂²×C₄	#11761628	1	1	yes	characteristic
425	C₂²×C₄	#11761628	1	1	yes	characteristic
426	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
427	Q₁₆	—	134	4	no	—
428	SD₁₆	—	134	4	no	—
429	SD₁₆	—	134	4	no	—
430	D₈	—	134	4	no	—
431	M₁₆	—	2	4	no	—
432	M₁₆	—	2	1	no	—
433	M₁₆	—	2	1	no	—
434	M₁₆	—	2	4	no	—
435	M₁₆	—	268	4	no	—
436	M₁₆	—	134	2	no	—
437	M₁₆	—	134	2	no	—
438	M₁₆	C₂×D₂₆₈	1	2	yes	—
439	M₁₆	#11761628	1	2	yes	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	4	no	—
446	C₂×C₈	—	268	4	no	—
447	C₂×C₈	—	67	2	no	—
448	C₂×C₈	—	67	2	no	—
449	C₂×C₈	—	134	2	no	—
450	C₂×C₈	—	134	2	no	—
451	C₂×C₈	—	134	2	no	—
452	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
453	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
454	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
455	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
456	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
457	C₂×C₈	#11761628	1	2	yes	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	268	2	no	—
461	C₄⋊C₄	—	134	1	no	—
462	C₄⋊C₄	—	134	1	no	—
463	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
464	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	134	2	no	—
469	C₂²⋊C₄	—	134	2	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	134	2	no	—
474	C₄²	—	134	2	no	—
475	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
476	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
477	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
478	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
479	C₂³	—	2	1	no	—
480	C₂³	—	134	2	no	—
481	C₂³	C₂×#11761628	1	1	yes	characteristic
482	Q₈	—	2	1	no	—
483	Q₈	—	134	2	no	—
484	Q₈	—	134	2	no	—
485	Q₈	C₄×D₂₆₈	1	2	yes	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	1	no	—
489	D₄	—	268	2	no	—
490	D₄	—	268	2	no	—
491	D₄	—	268	2	no	—
492	D₄	—	268	2	no	—
493	D₄	—	134	2	no	—
494	D₄	—	134	2	no	—
495	D₄	C₄×D₂₆₈	1	2	yes	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	268	2	no	—
506	C₂×C₄	—	268	2	no	—
507	C₂×C₄	—	268	2	no	—
508	C₂×C₄	—	268	2	no	—
509	C₂×C₄	—	268	2	no	—
510	C₂×C₄	—	134	2	no	—
511	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
512	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
513	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
514	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
518	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
519	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
520	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
521	C₈	—	2	2	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	2	no	—
524	C₈	—	2	1	no	—
525	C₈	—	268	2	no	—
526	C₈	—	134	2	no	—
527	C₈	—	134	2	no	—
528	C₈	#14524017	1	2	yes	—
529	C₈	#14524017	1	2	yes	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	268	2	no	—
536	C₂²	—	268	2	no	—
537	C₂²	—	268	2	no	—
538	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
539	C₂²	#16863930	1	1	yes	characteristic
540	C₂²	#16863946	1	1	yes	characteristic
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	268	2	no	—
548	C₄	—	536	2	no	—
549	C₄	#16863516	1	2	yes	complemented
550	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
551	C₄	C₄×#11761628	1	1	yes	characteristic
552	C₄	C₂×#14524017	1	1	yes	characteristic
553	C₄	#16863907	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	536	2	no	—
558	C₂	C₄×#14524017	1	1	yes	characteristic
559	C₂	C₂×#16863516	1	1	yes	characteristic
560	C₂	#22569110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569664	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#21518
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#16863516	C₂	semidirect
3. #22568922	C₂	semidirect
4. #22561128	C₂	semidirect
5. #22560984	C₂	semidirect
6. #22560857	C₂	semidirect
7. #22560536	C₂	semidirect
8. #22560229	C₂	semidirect
9. #22560076	C₂	semidirect
10. #22560055	C₂	semidirect
11. C₄×#131×C₆₇	C₂	semidirect
15. #16863618	C₄	semidirect
16. #16863611	C₄	semidirect
17. #16863606	C₄	semidirect
18. #16863601	C₄	semidirect
19. #16863596	C₄	semidirect
20. #16863591	C₄	semidirect
93. #131×C₆₇	D₄	semidirect
266. C₄×C₆₇	#444	semidirect
284. C₄×#131	D₆₇	semidirect
292. C₆₇	#21518	semidirect
344. #131	D₂₆₈	semidirect
549. C₄	#16863516	semidirect

Maximal subgroups

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

subgroup	index
#22569664	2
C₂×#16863516	2
#22568922	2
#22561128	2
#22560984	2
#22560857	2
#22560536	2
#22560229	2
#22560076	2
#22560055	2
C₄×#131×C₆₇	2
#21518	67