Subgroups — #30860217

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568977	C₂	1	1	yes	maximal, characteristic
2	#22568852	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568849	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561096	C₂	1	1	yes	maximal, characteristic
5	#22561094	C₂	1	1	yes	maximal, characteristic
6	#22560956	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560954	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560915	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560913	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560911	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560110	C₂	1	1	yes	maximal, characteristic, complemented
12	#104×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560013	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1937	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524063	C₂²	1	1	yes	characteristic
17	#41×Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863864	C₂²	1	1	yes	characteristic
19	#16863488	C₄	1	2	yes	complemented
20	#16863487	C₄	1	2	yes	complemented
21	#16863481	C₄	1	2	yes	complemented
22	#16863476	C₄	1	2	yes	complemented
23	#16863456	—	2	2	no	—
24	#16863454	—	2	2	no	—
25	#16863448	—	2	2	no	—
26	#16863446	—	2	2	no	—
27	C₂×#14523978	C₂²	1	1	yes	characteristic
28	C₂×#14523978	C₂²	1	1	yes	characteristic
29	#16863432	C₂²	1	1	yes	characteristic
30	#16863424	C₂²	1	1	yes	characteristic
31	#16863423	C₂²	1	1	yes	characteristic
32	#16863422	C₂²	1	1	yes	characteristic
33	C₂×#14523971	C₂²	1	1	yes	characteristic
34	C₂×#14523970	C₂²	1	1	yes	characteristic
35	C₂×#14523969	C₂²	1	1	yes	characteristic
36	#16863393	C₂²	1	1	yes	characteristic
37	#16863364	C₂²	1	1	yes	characteristic
38	#16863363	C₂²	1	1	yes	characteristic
39	#16863362	C₂²	1	1	yes	characteristic
40	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
41	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
43	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
47	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
48	#16863161	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	#424×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
54	#416×C₆₇	C₂²	1	1	yes	characteristic
55	#408×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	#14524063	—	2	2	no	—
61	#14524063	—	2	2	no	—
62	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
63	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
64	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
65	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
66	C₂×#11761612	C₂³	1	1	yes	characteristic
67	C₂×#11761612	C₂³	1	1	yes	characteristic
68	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
71	#14524008	C₂×C₄	1	2	yes	—
72	D₄×Dic₆₇	—	2	2	no	—
73	D₄×Dic₆₇	C₂×C₄	1	2	yes	—
74	#14524005	C₂×C₄	1	2	yes	—
75	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
76	#14524002	—	2	1	no	—
77	#14524002	C₂³	1	1	yes	characteristic
78	#14524000	—	2	1	no	—
79	C₂×C₄×Dic₆₇	—	2	2	no	—
80	C₂×C₄×Dic₆₇	—	2	1	no	—
81	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
83	C₄²×D₆₇	—	2	1	no	—
84	#14523978	—	2	2	no	—
85	#14523978	—	2	2	no	—
86	#14523976	C₂×C₄	1	2	yes	—
87	#14523974	C₂×C₄	1	2	yes	—
88	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
89	Dic₆₇⋊C₈	—	2	2	no	—
90	Dic₆₇⋊C₈	—	2	2	no	—
91	#14523971	—	2	2	no	—
92	#14523970	—	2	2	no	—
93	#14523969	—	2	2	no	—
94	#14523968	C₂×C₄	1	2	yes	—
95	#14523965	C₂×C₄	1	2	yes	—
96	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
97	(C₂×D₆₇)⋊C₈	—	2	2	no	—
98	(C₂×D₆₇)⋊C₈	—	2	2	no	—
99	#14523953	—	2	1	no	—
100	#14523953	—	2	1	no	—
101	C₄×C₆₇⋊C₈	—	2	2	no	—
102	C₈×Dic₆₇	—	2	2	no	—
103	C₈×Dic₆₇	—	2	1	no	—
104	C₈×Dic₆₇	—	2	1	no	—
105	C₈×Dic₆₇	—	2	2	no	—
106	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
107	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
108	C₈×Dic₆₇	D₄	1	2	yes	—
109	C₈×Dic₆₇	D₄	1	2	yes	—
110	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
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	C₄⋊C₈×C₆₇	—	2	2	no	—
118	#104×C₆₇	—	2	2	no	—
119	#103×C₆₇	C₂×C₄	1	2	yes	—
120	#102×C₆₇	C₂×C₄	1	2	yes	—
121	C₂²⋊C₈×C₆₇	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₂²⋊D₆₇	—	2	1	no	—
130	C₂²×Dic₆₇	—	2	1	no	—
131	C₂²×Dic₆₇	—	2	2	no	—
132	C₂²×Dic₆₇	#41	1	2	yes	—
133	C₂×C₄×D₆₇	—	2	1	no	—
134	C₂×C₄×D₆₇	—	2	2	no	—
135	#11761618	—	2	2	no	—
136	#11761616	—	2	2	no	—
137	#11761615	—	2	2	no	—
138	#11761613	—	2	2	no	—
139	#11761612	—	2	1	no	—
140	#11761612	—	2	1	no	—
141	#11761612	—	2	1	no	—
142	#11761612	—	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	—	2	1	no	—
147	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
150	C₂×C₆₇⋊C₈	#41	1	2	yes	—
151	C₂×C₆₇⋊C₈	#41	1	2	yes	—
152	C₄⋊Dic₆₇	—	2	2	no	—
153	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
154	Dic₆₇⋊C₄	—	2	2	no	—
155	C₂²⋊Dic₆₇	—	2	1	no	—
156	C₂²⋊Dic₆₇	—	2	2	no	—
157	(C₂×D₆₇)⋊C₄	—	2	2	no	—
158	C₄×Dic₆₇	—	2	8	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	C₄×Dic₆₇	—	2	2	no	—
162	C₄×Dic₆₇	—	2	2	no	—
163	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
164	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
165	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
166	C₄×Dic₆₇	#41	1	2	yes	—
167	#41×C₆₇	—	2	1	no	—
168	#41×C₆₇	—	2	1	no	—
169	#41×C₆₇	—	2	1	no	—
170	#41×C₆₇	—	2	1	no	—
171	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂×D₄×C₆₇	—	2	1	no	—
173	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	M₁₆×C₆₇	—	2	1	no	—
178	M₁₆×C₆₇	—	2	1	no	—
179	M₁₆×C₆₇	—	2	1	no	—
180	M₁₆×C₆₇	—	2	1	no	—
181	C₂×C₈×C₆₇	—	2	1	no	—
182	C₂×C₈×C₆₇	—	2	1	no	—
183	C₂×C₈×C₆₇	—	2	1	no	—
184	C₂×C₈×C₆₇	—	2	1	no	—
185	C₂×C₈×C₆₇	—	2	1	no	—
186	C₂×C₈×C₆₇	—	2	1	no	—
187	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
188	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
190	C₂²⋊C₄×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	#41	1	2	yes	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂²×D₆₇	—	2	1	no	—
197	Q₈×C₆₇	—	2	1	no	—
198	Q₈×C₆₇	—	2	1	no	—
199	D₄×C₆₇	—	2	1	no	—
200	D₄×C₆₇	—	2	1	no	—
201	D₄×C₆₇	—	4	1	no	—
202	D₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂²×D₄	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	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₂²⋊D₆₇	—	4	1	no	—
216	C₂²⋊D₆₇	—	4	1	no	—
217	C₂×Dic₆₇	—	2	4	no	—
218	C₂×Dic₆₇	—	2	2	no	—
219	C₂×Dic₆₇	—	2	4	no	—
220	C₂×Dic₆₇	—	2	2	no	—
221	C₂×Dic₆₇	—	2	1	no	—
222	C₂×Dic₆₇	—	4	4	no	—
223	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
224	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
225	D₂₆₈	—	2	1	no	—
226	D₂₆₈	—	2	1	no	—
227	C₄×D₆₇	—	4	1	no	—
228	C₄×D₆₇	—	4	1	no	—
229	C₄×D₆₇	—	4	4	no	—
230	Dic₁₃₄	—	2	1	no	—
231	Dic₁₃₄	—	2	1	no	—
232	C₈×C₆₇	—	2	2	no	—
233	C₈×C₆₇	—	2	2	no	—
234	C₈×C₆₇	—	4	1	no	—
235	C₈×C₆₇	—	4	1	no	—
236	C₆₇⋊C₈	—	2	2	no	—
237	C₆₇⋊C₈	—	4	1	no	—
238	C₆₇⋊C₈	—	4	1	no	—
239	C₆₇⋊C₈	#117	1	2	yes	—
240	C₆₇⋊C₈	#117	1	2	yes	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	1	no	—
244	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₆₇	#529	1	1	yes	characteristic
246	C₂²×C₆₇	#529	1	1	yes	characteristic
247	C₂×D₆₇	—	4	2	no	—
248	C₂×D₆₇	—	4	1	no	—
249	C₄×C₆₇	—	2	4	no	—
250	C₄×C₆₇	—	4	2	no	—
251	C₄×C₆₇	—	4	1	no	—
252	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
253	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
254	C₄×C₆₇	#532	1	1	yes	characteristic
255	C₄×C₆₇	#532	1	1	yes	characteristic
256	Dic₆₇	—	2	4	no	—
257	Dic₆₇	—	2	2	no	—
258	Dic₆₇	—	4	2	no	—
259	Dic₆₇	—	4	1	no	—
260	Dic₆₇	#443	1	4	yes	—
261	#21292	—	67	1	no	maximal, Sylow
262	C₂×C₆₇	—	2	1	no	—
263	C₂×C₆₇	—	2	1	no	—
264	C₂×C₆₇	—	8	1	no	—
265	C₂×C₆₇	#2293	1	1	yes	characteristic
266	C₂×C₆₇	C₂×#443	1	2	yes	—
267	D₆₇	—	8	1	no	—
268	C₄×#135	—	67	1	no	—
269	#2922	—	67	1	no	—
270	#2922	—	67	1	no	—
271	#1974	—	67	1	no	—
272	#1974	—	67	1	no	—
273	#1945	—	67	1	no	—
274	#1945	—	67	1	no	—
275	#1937	—	67	1	no	—
276	#1937	—	67	1	no	—
277	#1937	—	67	1	no	—
278	#1937	D₆₇	1	1	yes	characteristic, complemented
279	C₄×#108	—	67	1	no	—
280	C₄×#104	—	67	1	no	—
281	C₄×#104	—	67	1	no	—
282	C₄²×C₈	—	67	1	no	—
283	C₆₇	#21292	1	1	yes	characteristic, complemented, Sylow
284	C₂×#135	—	67	1	no	—
285	C₄×#41	—	67	1	no	—
286	C₄×#41	—	67	1	no	—
287	C₄×Q₁₆	—	67	2	no	—
288	C₄×SD₁₆	—	67	2	no	—
289	C₄×SD₁₆	—	67	2	no	—
290	C₄×D₈	—	67	2	no	—
291	#433	—	134	2	no	—
292	#433	—	134	2	no	—
293	#431	—	134	2	no	—
294	#431	—	134	2	no	—
295	C₂×#108	—	67	1	no	—
296	C₂×#108	—	67	1	no	—
297	#427	—	67	1	no	—
298	#424	—	67	1	no	—
299	#424	—	67	1	no	—
300	#424	—	67	1	no	—
301	#424	C₂×D₆₇	1	1	yes	characteristic
302	C₂×#104	—	67	1	no	—
303	C₂×#104	—	67	1	no	—
304	C₂×#104	—	67	1	no	—
305	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
306	#416	—	67	1	no	—
307	#416	C₂×D₆₇	1	1	yes	characteristic
308	#408	—	67	1	no	—
309	#408	—	67	1	no	—
310	#408	—	67	1	no	—
311	#408	C₂×D₆₇	1	1	yes	characteristic
312	C₄×M₁₆	—	67	1	no	—
313	C₄×M₁₆	—	67	1	no	—
314	C₂×C₄×C₈	—	67	1	no	—
315	C₂×C₄×C₈	—	67	1	no	—
316	C₂×C₄×C₈	—	67	1	no	—
317	C₂×C₄×C₈	—	67	1	no	—
318	C₂×C₄×C₈	—	67	1	no	—
319	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
320	C₄³	—	67	1	no	—
321	#341	—	67	1	no	—
322	#341	C₂×D₆₇	1	1	yes	characteristic
323	#339	—	67	1	no	—
324	#339	—	67	1	no	—
325	#339	—	67	1	no	—
326	#339	C₂×D₆₇	1	1	yes	characteristic
327	C₂×#41	—	67	1	no	—
328	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
329	#135	—	134	2	no	—
330	#135	—	134	2	no	—
331	C₂×Q₁₆	—	67	1	no	—
332	C₂×SD₁₆	—	67	1	no	—
333	C₂×SD₁₆	—	67	1	no	—
334	C₂×D₈	—	67	1	no	—
335	C₂×M₁₆	—	67	1	no	—
336	C₂×M₁₆	—	67	1	no	—
337	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
338	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
339	C₂²×C₈	—	67	1	no	—
340	C₂²×C₈	—	67	1	no	—
341	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
342	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
343	C₄×Q₈	—	67	2	no	—
344	C₄×Q₈	—	67	2	no	—
345	C₄×D₄	—	67	2	no	—
346	C₄×D₄	—	67	2	no	—
347	C₄×D₄	—	134	2	no	—
348	C₄×D₄	—	134	2	no	—
349	#117	—	67	1	no	—
350	#117	—	134	1	no	—
351	#117	—	134	1	no	—
352	#117	C₂²×D₆₇	1	1	yes	characteristic
353	C₂×C₄²	—	67	1	no	—
354	C₂×C₄²	—	67	1	no	—
355	C₂×C₄²	—	134	1	no	—
356	C₂×C₄²	—	134	1	no	—
357	C₂×C₄²	—	134	2	no	—
358	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
359	#108	—	134	2	no	—
360	#108	—	134	2	no	—
361	#107	—	67	2	no	—
362	#106	—	67	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	134	2	no	—
365	C₄⋊C₈	—	134	2	no	—
366	C₄⋊C₈	—	134	2	no	—
367	#104	—	2	2	no	—
368	#104	—	134	2	no	—
369	#104	—	134	2	no	—
370	#104	—	134	2	no	—
371	#103	—	67	2	no	—
372	#103	C₄×D₆₇	1	2	yes	—
373	#102	—	67	2	no	—
374	#102	C₄×D₆₇	1	2	yes	—
375	C₂²⋊C₈	—	2	2	no	—
376	C₂²⋊C₈	—	134	2	no	—
377	C₂²⋊C₈	—	134	2	no	—
378	C₂²⋊C₈	—	134	2	no	—
379	#97	—	134	1	no	—
380	#97	—	134	1	no	—
381	C₄×C₈	—	2	2	no	—
382	C₄×C₈	—	67	2	no	—
383	C₄×C₈	—	67	2	no	—
384	C₄×C₈	—	67	2	no	—
385	C₄×C₈	—	67	2	no	—
386	C₄×C₈	—	134	2	no	—
387	C₄×C₈	—	134	2	no	—
388	C₄×C₈	—	134	1	no	—
389	C₄×C₈	—	134	1	no	—
390	C₄×C₈	—	134	2	no	—
391	#41	—	2	1	no	—
392	#41	—	2	1	no	—
393	#41	—	2	1	no	—
394	#41	—	2	1	no	—
395	#41	—	134	1	no	—
396	#41	—	134	1	no	—
397	#41	—	134	1	no	—
398	#41	—	134	1	no	—
399	C₂×Q₈	—	67	1	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	1	no	—
404	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	67	2	no	—
408	C₂²×C₄	—	134	2	no	—
409	C₂²×C₄	—	134	2	no	—
410	C₂²×C₄	—	134	1	no	—
411	C₂²×C₄	—	134	1	no	—
412	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
413	Q₁₆	—	134	2	no	—
414	SD₁₆	—	134	2	no	—
415	SD₁₆	—	134	2	no	—
416	D₈	—	134	2	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	1	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	134	1	no	—
422	M₁₆	—	134	1	no	—
423	M₁₆	—	134	1	no	—
424	M₁₆	—	134	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	67	1	no	—
432	C₂×C₈	—	67	1	no	—
433	C₂×C₈	—	67	2	no	—
434	C₂×C₈	—	67	2	no	—
435	C₂×C₈	—	67	2	no	—
436	C₂×C₈	—	134	1	no	—
437	C₂×C₈	—	134	1	no	—
438	C₂×C₈	—	134	1	no	—
439	C₂×C₈	—	134	1	no	—
440	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
441	C₂×C₈	D₄×D₆₇	1	2	yes	—
442	C₄⋊C₄	—	67	2	no	—
443	C₄⋊C₄	—	134	2	no	—
444	C₄⋊C₄	—	134	2	no	—
445	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
446	C₂²⋊C₄	—	2	1	no	—
447	C₂²⋊C₄	—	134	2	no	—
448	C₂²⋊C₄	—	134	2	no	—
449	C₂²⋊C₄	—	134	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	67	2	no	—
453	C₄²	—	67	1	no	—
454	C₄²	—	67	1	no	—
455	C₄²	—	67	2	no	—
456	C₄²	—	134	2	no	—
457	C₄²	—	134	2	no	—
458	C₄²	—	134	8	no	—
459	C₄²	—	134	1	no	—
460	C₄²	—	134	1	no	—
461	C₄²	#11761627	1	2	yes	—
462	C₂³	—	2	1	no	—
463	C₂³	—	134	1	no	—
464	C₂³	#41×D₆₇	1	1	yes	characteristic
465	Q₈	—	2	1	no	—
466	Q₈	—	2	1	no	—
467	Q₈	—	134	1	no	—
468	Q₈	—	134	1	no	—
469	D₄	—	2	1	no	—
470	D₄	—	2	1	no	—
471	D₄	—	4	1	no	—
472	D₄	—	4	1	no	—
473	D₄	—	268	1	no	—
474	D₄	—	268	1	no	—
475	D₄	—	134	1	no	—
476	D₄	—	134	1	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	268	4	no	—
484	C₂×C₄	—	268	4	no	—
485	C₂×C₄	—	268	1	no	—
486	C₂×C₄	—	268	1	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	4	no	—
492	C₂×C₄	—	134	4	no	—
493	C₂×C₄	—	134	2	no	—
494	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
495	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
496	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
497	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
498	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
499	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	2	2	no	—
502	C₈	—	4	1	no	—
503	C₈	—	4	1	no	—
504	C₈	—	268	1	no	—
505	C₈	—	268	1	no	—
506	C₈	—	67	2	no	—
507	C₈	—	67	2	no	—
508	C₈	—	134	2	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	268	1	no	—
513	C₂²	—	268	2	no	—
514	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
515	C₂²	#16863923	1	1	yes	characteristic
516	C₂²	#16863929	1	1	yes	characteristic
517	C₄	—	2	4	no	—
518	C₄	—	4	1	no	—
519	C₄	—	4	2	no	—
520	C₄	—	268	1	no	—
521	C₄	—	268	2	no	—
522	C₄	—	67	4	no	—
523	C₄	—	134	4	no	—
524	C₄	—	134	2	no	—
525	C₄	#117×D₆₇	1	1	yes	characteristic
526	C₄	C₄×#11761627	1	1	yes	characteristic
527	C₄	#16863941	1	1	yes	characteristic
528	C₄	#16863945	1	1	yes	characteristic
529	C₂	—	2	1	no	—
530	C₂	—	2	1	no	—
531	C₂	—	8	1	no	—
532	C₂	—	536	1	no	—
533	C₂	#22562669	1	1	yes	characteristic
534	C₂	#443×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1937	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1937	D₆₇
hypocenter	C₆₇	#21292
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1937

Complements

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

N	H ≅ G/N	type
2. #22568852	C₂	semidirect
3. #22568849	C₂	semidirect
6. #22560956	C₂	semidirect
7. #22560954	C₂	semidirect
8. #22560915	C₂	semidirect
9. #22560913	C₂	semidirect
10. #22560911	C₂	semidirect
11. #22560110	C₂	semidirect
12. #104×Dic₆₇	C₂	semidirect
13. #22560013	C₂	semidirect
14. C₄×C₈×Dic₆₇	C₂	semidirect
15. C₆₇×#1937	C₂	semidirect
19. #16863488	C₄	semidirect
20. #16863487	C₄	semidirect
21. #16863481	C₄	semidirect
22. #16863476	C₄	semidirect
278. #1937	D₆₇	semidirect
283. C₆₇	#21292	semidirect

Maximal subgroups

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

subgroup	index
#22568977	2
#22568852	2
#22568849	2
#22561096	2
#22561094	2
#22560956	2
#22560954	2
#22560915	2
#22560913	2
#22560911	2
#22560110	2
#104×Dic₆₇	2
#22560013	2
C₄×C₈×Dic₆₇	2
C₆₇×#1937	2
#21292	67