Subgroups — #30864013

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570552	C₂	1	1	yes	maximal, characteristic
2	#22569141	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568919	C₂	1	1	yes	maximal, characteristic
4	#22561188	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561039	C₂	1	1	yes	maximal, characteristic
6	#22560922	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560921	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560918	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560917	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560348	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560048	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560046	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560044	C₂	1	1	yes	maximal, characteristic, complemented
14	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524049	C₂²	1	1	yes	characteristic
17	#16863949	C₂²	1	1	yes	characteristic
18	#16863942	C₂²	1	1	yes	characteristic
19	#16863763	C₄	1	2	yes	complemented
20	#16863759	C₄	1	2	yes	complemented
21	#16863757	C₄	1	2	yes	complemented
22	#16863754	C₄	1	2	yes	complemented
23	#16863524	—	2	2	no	—
24	C₈×Dic₁₃₄	—	2	2	no	—
25	#16863472	—	2	2	no	—
26	C₈×C₂²⋊D₆₇	—	2	2	no	—
27	#16863424	C₂²	1	1	yes	characteristic
28	#16863423	C₂²	1	1	yes	characteristic
29	#16863419	C₂²	1	1	yes	characteristic
30	#16863419	C₂²	1	1	yes	characteristic
31	C₂×#14523971	C₂²	1	1	yes	characteristic
32	C₂×#14523970	C₂²	1	1	yes	characteristic
33	C₂×#14523969	C₂²	1	1	yes	characteristic
34	#16863393	C₂²	1	1	yes	characteristic
35	#16863392	C₂²	1	1	yes	characteristic
36	#16863390	C₂²	1	1	yes	characteristic
37	#16863364	C₂²	1	1	yes	characteristic
38	#16863363	C₂²	1	1	yes	characteristic
39	#16863351	C₂²	1	1	yes	characteristic
40	C₂×#14523954	C₂²	1	1	yes	characteristic
41	C₂×#14523953	C₂²	1	1	yes	characteristic
42	C₂×#14523953	C₂²	1	1	yes	characteristic
43	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
45	#16863242	C₂²	1	1	yes	characteristic
46	#16863161	C₂²	1	1	yes	characteristic
47	#16863160	C₂²	1	1	yes	characteristic
48	#16863160	C₂²	1	1	yes	characteristic
49	#16863157	C₂²	1	1	yes	characteristic
50	#16863156	C₂²	1	1	yes	characteristic
51	#16863155	C₂²	1	1	yes	characteristic
52	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
54	#416×C₆₇	C₂²	1	1	yes	characteristic
55	#405×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
57	#341×C₆₇	C₂²	1	1	yes	characteristic
58	#339×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×#11761628	C₂³	1	1	yes	characteristic
60	#14524049	—	2	2	no	—
61	#14524049	—	2	2	no	—
62	C₂×#11761612	C₂³	1	1	yes	characteristic
63	C₂×#11761612	C₂³	1	1	yes	characteristic
64	C₂×#11761611	—	2	1	no	—
65	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈×D₆₇	—	2	1	no	—
68	#14524042	C₂×C₄	1	2	yes	—
69	#14524041	C₂×C₄	1	1	yes	characteristic
70	#14524039	C₂×C₄	1	1	yes	characteristic
71	#14524034	C₂×C₄	1	2	yes	—
72	#14524032	C₂×C₄	1	1	yes	characteristic
73	#14524030	C₂×C₄	1	1	yes	characteristic
74	#14524020	—	2	2	no	—
75	#14524015	—	2	2	no	—
76	C₄×Dic₁₃₄	—	2	2	no	—
77	C₄×C₂²⋊D₆₇	—	2	2	no	—
78	#14524002	C₂³	1	1	yes	characteristic
79	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
82	Dic₆₇⋊C₈	—	2	2	no	—
83	Dic₆₇⋊C₈	—	2	2	no	—
84	Dic₆₇⋊C₈	—	2	2	no	—
85	#14523971	—	2	2	no	—
86	#14523970	—	2	2	no	—
87	#14523969	—	2	2	no	—
88	#14523968	C₂×C₄	1	2	yes	—
89	#14523967	C₂×C₄	1	2	yes	—
90	#14523966	C₂×C₄	1	2	yes	—
91	#14523965	C₂×C₄	1	2	yes	—
92	#14523964	C₂×C₄	1	2	yes	—
93	#14523963	C₂×C₄	1	2	yes	—
94	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
95	(C₂×D₆₇)⋊C₈	—	2	2	no	—
96	#14523954	—	2	2	no	—
97	#14523953	—	2	2	no	—
98	#14523953	—	2	1	no	—
99	#14523953	C₂×C₄	1	2	yes	—
100	#14523953	C₂×C₄	1	2	yes	—
101	C₈×Dic₆₇	—	2	1	no	—
102	C₈×Dic₆₇	—	2	2	no	—
103	C₈×Dic₆₇	D₄	1	2	yes	—
104	C₈×Dic₆₇	D₄	1	2	yes	—
105	C₆₇⋊#95	—	2	2	no	—
106	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
107	#131×C₆₇	—	2	2	no	—
108	#131×C₆₇	—	2	2	no	—
109	C₂×M₁₆×C₆₇	—	2	1	no	—
110	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₆₇	—	2	1	no	—
113	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
115	#117×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	#104×C₆₇	—	2	2	no	—
118	#103×C₆₇	C₂×C₄	1	2	yes	—
119	#102×C₆₇	C₂×C₄	1	2	yes	—
120	#97×C₆₇	—	2	1	no	—
121	C₄×C₈×C₆₇	—	2	1	no	—
122	C₄×C₈×C₆₇	—	2	2	no	—
123	#11761628	—	2	1	no	—
124	#11761628	—	2	1	no	—
125	#11761628	—	2	1	no	—
126	#11761628	—	2	1	no	—
127	C₂×Dic₁₃₄	—	2	2	no	—
128	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₂²⋊D₆₇	—	2	1	no	—
131	C₂×C₂²⋊D₆₇	—	2	2	no	—
132	C₂²×Dic₆₇	—	2	1	no	—
133	C₂²×Dic₆₇	#41	1	2	yes	—
134	C₂×C₄×D₆₇	—	2	1	no	—
135	#11761612	—	2	1	no	—
136	#11761612	—	2	1	no	—
137	#11761612	—	2	1	no	—
138	#11761612	—	2	1	no	—
139	#11761611	—	4	2	no	—
140	C₂×C₆₇⋊C₈	—	2	1	no	—
141	C₂×C₆₇⋊C₈	—	2	1	no	—
142	C₂×C₆₇⋊C₈	—	2	1	no	—
143	C₂×C₆₇⋊C₈	—	2	1	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
147	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
148	C₈×D₆₇	—	4	2	no	—
149	C₄⋊Dic₆₇	—	2	2	no	—
150	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
151	Dic₆₇⋊C₄	—	2	2	no	—
152	Dic₆₇⋊C₄	—	2	2	no	—
153	Dic₆₇⋊C₄	—	2	2	no	—
154	C₂²⋊Dic₆₇	—	2	1	no	—
155	C₂²⋊Dic₆₇	—	2	2	no	—
156	(C₂×D₆₇)⋊C₄	—	2	2	no	—
157	C₄×Dic₆₇	—	2	1	no	—
158	C₄×Dic₆₇	—	2	1	no	—
159	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
162	C₄×Dic₆₇	#41	1	2	yes	—
163	#41×C₆₇	—	2	1	no	—
164	#41×C₆₇	—	2	1	no	—
165	#41×C₆₇	—	2	1	no	—
166	#41×C₆₇	—	2	1	no	—
167	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₂×D₄×C₆₇	—	2	1	no	—
169	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	M₁₆×C₆₇	—	2	1	no	—
174	M₁₆×C₆₇	—	2	1	no	—
175	M₁₆×C₆₇	—	2	1	no	—
176	M₁₆×C₆₇	—	2	1	no	—
177	M₁₆×C₆₇	—	4	2	no	—
178	C₂×C₈×C₆₇	—	2	1	no	—
179	C₂×C₈×C₆₇	—	2	1	no	—
180	C₂×C₈×C₆₇	—	2	1	no	—
181	C₂×C₈×C₆₇	—	2	1	no	—
182	C₂×C₈×C₆₇	—	4	2	no	—
183	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
184	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₆₇	#41	1	2	yes	—
187	C₂×C₈×C₆₇	#41	1	2	yes	—
188	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
189	C₂²⋊C₄×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	#41	1	2	yes	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
195	C₂²×D₆₇	—	2	1	no	—
196	Q₈×C₆₇	—	2	1	no	—
197	Q₈×C₆₇	—	2	1	no	—
198	D₄×C₆₇	—	2	1	no	—
199	D₄×C₆₇	—	2	1	no	—
200	D₄×C₆₇	—	4	1	no	—
201	D₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
214	C₂²⋊D₆₇	—	4	4	no	—
215	C₂²⋊D₆₇	—	4	1	no	—
216	C₂²⋊D₆₇	—	4	1	no	—
217	C₂×Dic₆₇	—	2	2	no	—
218	C₂×Dic₆₇	—	2	1	no	—
219	C₂×Dic₆₇	—	2	2	no	—
220	C₂×Dic₆₇	—	4	2	no	—
221	C₂×Dic₆₇	—	4	2	no	—
222	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
223	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
224	D₂₆₈	—	2	1	no	—
225	D₂₆₈	—	2	1	no	—
226	C₄×D₆₇	—	4	1	no	—
227	C₄×D₆₇	—	4	1	no	—
228	Dic₁₃₄	—	2	1	no	—
229	Dic₁₃₄	—	2	1	no	—
230	Dic₁₃₄	—	4	4	no	—
231	C₈×C₆₇	—	2	2	no	—
232	C₈×C₆₇	—	4	1	no	—
233	C₈×C₆₇	—	4	1	no	—
234	C₈×C₆₇	#124	1	2	yes	—
235	C₈×C₆₇	#124	1	2	yes	—
236	C₆₇⋊C₈	—	2	2	no	—
237	C₆₇⋊C₈	—	2	2	no	—
238	C₆₇⋊C₈	—	4	1	no	—
239	C₆₇⋊C₈	—	4	1	no	—
240	C₂²×C₆₇	—	2	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	1	no	—
243	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₆₇	#529	1	1	yes	characteristic
245	C₂²×C₆₇	#529	1	1	yes	characteristic
246	C₂×D₆₇	—	4	2	no	—
247	C₂×D₆₇	—	4	1	no	—
248	C₄×C₆₇	—	4	2	no	—
249	C₄×C₆₇	—	4	2	no	—
250	C₄×C₆₇	—	4	1	no	—
251	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
252	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
253	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
254	C₄×C₆₇	#532	1	1	yes	characteristic
255	Dic₆₇	—	2	2	no	—
256	Dic₆₇	—	2	2	no	—
257	Dic₆₇	—	4	2	no	—
258	Dic₆₇	—	4	2	no	—
259	Dic₆₇	—	4	1	no	—
260	#21661	—	67	1	no	maximal, Sylow
261	C₂×C₆₇	—	2	1	no	—
262	C₂×C₆₇	—	2	1	no	—
263	C₂×C₆₇	—	8	1	no	—
264	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
265	C₂×C₆₇	#2955	1	2	yes	—
266	D₆₇	—	8	1	no	—
267	#3135	—	67	1	no	—
268	#2960	—	67	1	no	—
269	#2931	—	67	1	no	—
270	#1995	—	67	1	no	—
271	#1963	—	67	1	no	—
272	#1938	—	67	1	no	—
273	#1938	—	67	1	no	—
274	#1938	—	67	1	no	—
275	#1938	—	67	1	no	—
276	#1838	—	67	1	no	—
277	#1764	—	67	1	no	—
278	#1764	—	67	1	no	—
279	#1764	—	67	1	no	—
280	#1764	D₆₇	1	1	yes	characteristic, complemented
281	#1726	—	67	1	no	—
282	C₆₇	#21661	1	1	yes	characteristic, complemented, Sylow
283	C₂×#131	—	67	1	no	—
284	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
285	#533	—	67	1	no	—
286	#532	—	67	1	no	—
287	#491	—	67	2	no	—
288	#490	—	67	2	no	—
289	#489	—	67	2	no	—
290	#489	—	67	2	no	—
291	#446	—	134	2	no	—
292	C₈×Q₈	—	134	2	no	—
293	#436	—	134	2	no	—
294	C₈×D₄	—	134	2	no	—
295	#424	—	67	1	no	—
296	#424	—	67	1	no	—
297	#423	—	67	1	no	—
298	#423	—	67	1	no	—
299	C₂×#104	—	67	1	no	—
300	C₂×#104	—	67	1	no	—
301	C₂×#104	—	67	1	no	—
302	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
303	#416	—	67	1	no	—
304	#416	—	67	1	no	—
305	#416	—	67	1	no	—
306	#416	C₂×D₆₇	1	1	yes	characteristic
307	#408	—	67	1	no	—
308	#408	—	67	1	no	—
309	#405	—	67	1	no	—
310	#405	C₂×D₆₇	1	1	yes	characteristic
311	C₂×#97	—	67	1	no	—
312	C₂×#97	—	67	1	no	—
313	C₂×#97	—	67	1	no	—
314	C₂×C₄×C₈	—	67	1	no	—
315	C₂×C₄×C₈	—	67	1	no	—
316	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
317	#376	—	67	1	no	—
318	#341	—	67	1	no	—
319	#341	—	67	1	no	—
320	#341	—	67	1	no	—
321	#341	C₂×D₆₇	1	1	yes	characteristic
322	#339	—	67	1	no	—
323	#339	—	67	1	no	—
324	#339	—	67	1	no	—
325	#339	C₂×D₆₇	1	1	yes	characteristic
326	C₂×#41	—	67	1	no	—
327	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
328	#131	—	2	2	no	—
329	#131	—	2	2	no	—
330	#131	—	134	2	no	—
331	#131	—	134	2	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	67	1	no	—
334	C₂×M₁₆	—	67	1	no	—
335	C₂×M₁₆	—	134	1	no	—
336	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
337	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	—	67	1	no	—
340	C₂²×C₈	—	67	1	no	—
341	C₂²×C₈	—	134	1	no	—
342	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
343	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
344	C₄⋊Q₈	—	67	2	no	—
345	C₄⋊Q₈	—	67	1	no	—
346	#127	—	67	1	no	—
347	#125	—	67	2	no	—
348	#124	—	67	1	no	—
349	#124	—	67	1	no	—
350	C₂²⋊Q₈	—	134	2	no	—
351	#121	—	134	2	no	—
352	C₄×Q₈	—	134	2	no	—
353	C₄×D₄	—	134	2	no	—
354	#117	—	67	1	no	—
355	#117	C₂²×D₆₇	1	1	yes	characteristic
356	C₂×C₄²	—	67	1	no	—
357	C₂×C₄²	—	67	1	no	—
358	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
359	C₄⋊C₈	—	134	2	no	—
360	C₄⋊C₈	—	134	2	no	—
361	C₄⋊C₈	—	134	2	no	—
362	C₄⋊C₈	—	134	2	no	—
363	#104	—	2	2	no	—
364	#104	—	134	2	no	—
365	#104	—	134	2	no	—
366	#104	—	134	2	no	—
367	#103	—	67	2	no	—
368	#103	—	67	2	no	—
369	#103	—	67	2	no	—
370	#103	C₄×D₆₇	1	2	yes	—
371	#102	—	67	2	no	—
372	#102	—	67	2	no	—
373	#102	—	67	2	no	—
374	#102	C₄×D₆₇	1	2	yes	—
375	C₂²⋊C₈	—	134	2	no	—
376	C₂²⋊C₈	—	134	2	no	—
377	#97	—	2	1	no	—
378	#97	—	67	2	no	—
379	#97	—	67	2	no	—
380	#97	—	134	2	no	—
381	#97	—	134	1	no	—
382	#97	—	134	2	no	—
383	C₄×C₈	—	2	1	no	—
384	C₄×C₈	—	2	2	no	—
385	C₄×C₈	—	67	2	no	—
386	C₄×C₈	—	67	2	no	—
387	C₄×C₈	—	134	1	no	—
388	C₄×C₈	—	134	2	no	—
389	#95	—	134	2	no	—
390	#41	—	2	1	no	—
391	#41	—	2	1	no	—
392	#41	—	2	1	no	—
393	#41	—	2	1	no	—
394	#41	—	134	1	no	—
395	#41	—	134	1	no	—
396	#41	—	134	1	no	—
397	#41	—	134	1	no	—
398	C₂×Q₈	—	67	1	no	—
399	C₂×Q₈	—	134	2	no	—
400	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	67	1	no	—
403	C₂×D₄	—	134	2	no	—
404	C₂×D₄	—	134	1	no	—
405	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	67	2	no	—
409	C₂²×C₄	—	134	1	no	—
410	C₂²×C₄	—	134	1	no	—
411	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
412	M₁₆	—	2	1	no	—
413	M₁₆	—	2	1	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	2	1	no	—
416	M₁₆	—	4	2	no	—
417	M₁₆	—	268	2	no	—
418	M₁₆	—	134	1	no	—
419	M₁₆	—	134	1	no	—
420	M₁₆	—	134	1	no	—
421	M₁₆	—	134	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	—	268	2	no	—
428	C₂×C₈	—	67	2	no	—
429	C₂×C₈	—	67	2	no	—
430	C₂×C₈	—	134	1	no	—
431	C₂×C₈	—	134	1	no	—
432	C₂×C₈	—	134	1	no	—
433	C₂×C₈	—	134	1	no	—
434	C₂×C₈	—	134	1	no	—
435	C₂×C₈	—	134	1	no	—
436	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
437	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
438	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
439	C₂×C₈	#11761627	1	2	yes	—
440	C₂×C₈	#11761628	1	2	yes	—
441	C₄⋊C₄	—	67	2	no	—
442	C₄⋊C₄	—	134	2	no	—
443	C₄⋊C₄	—	134	2	no	—
444	C₄⋊C₄	—	134	2	no	—
445	C₄⋊C₄	—	134	2	no	—
446	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
447	C₂²⋊C₄	—	2	1	no	—
448	C₂²⋊C₄	—	134	2	no	—
449	C₂²⋊C₄	—	134	2	no	—
450	C₂²⋊C₄	—	134	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	67	1	no	—
454	C₄²	—	67	1	no	—
455	C₄²	—	67	2	no	—
456	C₄²	—	67	2	no	—
457	C₄²	—	134	1	no	—
458	C₄²	—	134	1	no	—
459	C₄²	#11761628	1	2	yes	—
460	C₂³	—	2	1	no	—
461	C₂³	—	134	1	no	—
462	C₂³	#41×D₆₇	1	1	yes	characteristic
463	Q₈	—	2	1	no	—
464	Q₈	—	2	1	no	—
465	Q₈	—	268	4	no	—
466	Q₈	—	134	1	no	—
467	Q₈	—	134	1	no	—
468	D₄	—	2	1	no	—
469	D₄	—	2	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	1	no	—
472	D₄	—	268	1	no	—
473	D₄	—	268	1	no	—
474	D₄	—	268	4	no	—
475	D₄	—	134	1	no	—
476	D₄	—	134	1	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	268	2	no	—
484	C₂×C₄	—	268	1	no	—
485	C₂×C₄	—	268	1	no	—
486	C₂×C₄	—	268	2	no	—
487	C₂×C₄	—	67	2	no	—
488	C₂×C₄	—	67	2	no	—
489	C₂×C₄	—	134	2	no	—
490	C₂×C₄	—	134	1	no	—
491	C₂×C₄	—	134	2	no	—
492	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
493	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
494	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
495	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
496	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
497	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
498	C₈	—	2	2	no	—
499	C₈	—	4	1	no	—
500	C₈	—	4	1	no	—
501	C₈	—	268	1	no	—
502	C₈	—	268	1	no	—
503	C₈	—	134	2	no	—
504	C₈	—	134	2	no	—
505	C₈	#14524029	1	2	yes	—
506	C₈	#14524029	1	2	yes	—
507	C₂²	—	2	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	268	2	no	—
511	C₂²	—	268	1	no	—
512	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
513	C₂²	#16863924	1	1	yes	characteristic
514	C₂²	#16863926	1	1	yes	characteristic
515	C₄	—	4	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	1	no	—
518	C₄	—	268	2	no	—
519	C₄	—	268	2	no	—
520	C₄	—	268	1	no	—
521	C₄	—	134	2	no	—
522	C₄	—	134	2	no	—
523	C₄	C₄×#11761627	1	1	yes	characteristic
524	C₄	C₄×#11761628	1	1	yes	characteristic
525	C₄	C₂×#14524029	1	1	yes	characteristic
526	C₄	#16863943	1	1	yes	characteristic
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	8	1	no	—
530	C₂	—	536	1	no	—
531	C₂	C₄×#14524029	1	1	yes	characteristic
532	C₂	#22569107	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1764	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863942	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1764	D₆₇
hypocenter	C₆₇	#21661
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1764

Complements

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

N	H ≅ G/N	type
2. #22569141	C₂	semidirect
4. #22561188	C₂	semidirect
6. #22560922	C₂	semidirect
7. #22560921	C₂	semidirect
8. #22560918	C₂	semidirect
9. #22560917	C₂	semidirect
10. #22560348	C₂	semidirect
11. #22560048	C₂	semidirect
12. #22560046	C₂	semidirect
13. #22560044	C₂	semidirect
14. #22559895	C₂	semidirect
15. C₆₇×#1764	C₂	semidirect
19. #16863763	C₄	semidirect
20. #16863759	C₄	semidirect
21. #16863757	C₄	semidirect
22. #16863754	C₄	semidirect
280. #1764	D₆₇	semidirect
282. C₆₇	#21661	semidirect

Maximal subgroups

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

subgroup	index
#22570552	2
#22569141	2
#22568919	2
#22561188	2
#22561039	2
#22560922	2
#22560921	2
#22560918	2
#22560917	2
#22560348	2
#22560048	2
#22560046	2
#22560044	2
#22559895	2
C₆₇×#1764	2
#21661	67