Subgroups — #30859049

Order 17152.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524049	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560854	C₂	1	2	yes	maximal, complemented
4	#22560805	C₂	1	2	yes	maximal, complemented
5	#22560539	C₂	1	2	yes	maximal
6	#22560483	C₂	1	2	yes	maximal, complemented
7	C₄×#14523978	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14523969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523970	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559954	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	2	yes	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
16	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
17	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
22	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
23	C₂×#14523978	C₂²	1	2	yes	—
24	#16863432	C₂²	1	1	yes	characteristic
25	C₂×#14523970	C₂²	1	2	yes	—
26	C₂×#14523969	C₂²	1	2	yes	—
27	#16863392	C₂²	1	1	yes	characteristic
28	#16863390	C₂²	1	1	yes	characteristic
29	#16863351	C₂²	1	2	yes	—
30	#16863350	—	2	2	no	—
31	C₄×#11761612	C₂²	1	1	yes	characteristic
32	C₄×#11761611	—	2	1	no	—
33	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	—	2	1	no	—
35	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
36	#16863160	C₂²	1	2	yes	—
37	#16863157	C₂²	1	2	yes	—
38	#16863155	C₂²	1	2	yes	—
39	#16863150	C₂²	1	2	yes	—
40	#16863150	C₂²	1	2	yes	—
41	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
42	#433×C₆₇	—	2	2	no	—
43	#431×C₆₇	—	2	1	no	—
44	#431×C₆₇	—	2	1	no	—
45	#424×C₆₇	C₂²	1	2	yes	—
46	#408×C₆₇	C₂²	1	2	yes	—
47	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	#14524063	—	2	4	no	—
51	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
55	#14524049	—	2	4	no	—
56	#14524049	—	2	4	no	—
57	C₂×#11761612	C₂³	1	2	yes	—
58	C₂×#11761611	—	2	2	no	—
59	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₆₇	—	2	2	no	—
62	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₂²⋊D₆₇	—	2	1	no	—
67	C₄×C₂²⋊D₆₇	—	2	1	no	—
68	#14524002	C₂³	1	1	yes	characteristic
69	#14524000	—	2	1	no	—
70	C₂×C₄×Dic₆₇	—	2	1	no	—
71	C₂×C₄×Dic₆₇	—	2	1	no	—
72	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
73	C₄²×D₆₇	—	2	1	no	—
74	#14523978	—	2	4	no	—
75	#14523976	C₂×C₄	1	1	yes	characteristic
76	#14523976	C₂×C₄	1	1	yes	characteristic
77	#14523974	C₂×C₄	1	1	yes	characteristic
78	#14523974	C₂×C₄	1	1	yes	characteristic
79	#14523970	—	2	4	no	—
80	#14523969	—	2	4	no	—
81	#14523967	C₂×C₄	1	1	yes	characteristic
82	#14523967	C₂×C₄	1	1	yes	characteristic
83	#14523966	C₂×C₄	1	1	yes	characteristic
84	#14523966	C₂×C₄	1	1	yes	characteristic
85	#14523964	C₂×C₄	1	1	yes	characteristic
86	#14523964	C₂×C₄	1	1	yes	characteristic
87	#14523963	C₂×C₄	1	1	yes	characteristic
88	#14523963	C₂×C₄	1	1	yes	characteristic
89	#14523954	—	2	1	no	—
90	#14523953	—	2	2	no	—
91	C₄×C₆₇⋊C₈	—	2	1	no	—
92	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
93	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₆₇	—	2	2	no	—
97	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
99	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
100	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₆₇	—	2	1	no	—
103	C₄×D₄×C₆₇	—	2	1	no	—
104	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	#117×C₆₇	—	2	1	no	—
107	#117×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₆₇	—	2	2	no	—
111	C₄⋊C₈×C₆₇	—	2	2	no	—
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	2	2	no	—
114	#97×C₆₇	—	2	1	no	—
115	C₄×C₈×C₆₇	—	2	1	no	—
116	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
117	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
118	C₄×C₈×C₆₇	D₄	1	2	yes	—
119	#11761628	—	2	2	no	—
120	#11761628	#41	1	4	yes	—
121	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₂²⋊D₆₇	—	2	1	no	—
124	C₂²×Dic₆₇	—	2	2	no	—
125	C₂×C₄×D₆₇	—	2	1	no	—
126	C₂×C₄×D₆₇	—	2	1	no	—
127	C₂×C₄×D₆₇	—	2	1	no	—
128	#11761618	—	2	2	no	—
129	#11761616	—	2	2	no	—
130	#11761615	—	2	2	no	—
131	#11761613	—	2	2	no	—
132	#11761612	—	2	2	no	—
133	#11761612	#41	1	4	yes	—
134	#11761611	—	2	8	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
142	C₈×D₆₇	—	2	8	no	—
143	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
145	Dic₆₇⋊C₄	—	2	1	no	—
146	Dic₆₇⋊C₄	—	2	1	no	—
147	C₂²⋊Dic₆₇	—	2	1	no	—
148	(C₂×D₆₇)⋊C₄	—	2	1	no	—
149	(C₂×D₆₇)⋊C₄	—	2	1	no	—
150	C₄×Dic₆₇	—	2	4	no	—
151	C₄×Dic₆₇	—	2	1	no	—
152	C₄×Dic₆₇	—	2	1	no	—
153	C₄×Dic₆₇	—	2	1	no	—
154	C₄×Dic₆₇	—	2	2	no	—
155	C₄×Dic₆₇	—	2	2	no	—
156	C₄×Dic₆₇	—	4	2	no	—
157	#41×C₆₇	—	2	2	no	—
158	#41×C₆₇	—	2	2	no	—
159	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂×D₄×C₆₇	—	2	1	no	—
161	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	1	no	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
174	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	1	no	—
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²⋊C₄×C₆₇	—	2	1	no	—
180	C₂²⋊C₄×C₆₇	—	2	1	no	—
181	C₂²⋊C₄×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	—	2	1	no	—
184	C₄²×C₆₇	—	2	1	no	—
185	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₂²×D₆₇	—	2	1	no	—
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	—	2	1	no	—
194	D₄×C₆₇	—	2	1	no	—
195	D₄×C₆₇	—	2	1	no	—
196	D₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
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	2	yes	—
212	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
213	C₂²⋊D₆₇	—	2	4	no	—
214	C₂×Dic₆₇	—	2	1	no	—
215	C₂×Dic₆₇	—	2	4	no	—
216	C₂×Dic₆₇	—	2	1	no	—
217	C₂×Dic₆₇	—	2	1	no	—
218	C₂×Dic₆₇	—	4	2	no	—
219	C₂×Dic₆₇	—	4	2	no	—
220	D₂₆₈	—	2	1	no	—
221	D₂₆₈	#117	1	2	yes	—
222	C₄×D₆₇	—	2	4	no	—
223	C₄×D₆₇	—	4	2	no	—
224	C₄×D₆₇	—	4	2	no	—
225	Dic₁₃₄	—	2	1	no	—
226	Dic₁₃₄	#117	1	2	yes	—
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₈×C₆₇	—	4	2	no	—
230	C₆₇⋊C₈	—	2	1	no	—
231	C₆₇⋊C₈	—	2	1	no	—
232	C₆₇⋊C₈	—	2	1	no	—
233	C₆₇⋊C₈	—	2	1	no	—
234	C₆₇⋊C₈	—	2	4	no	—
235	C₂²×C₆₇	—	2	1	no	—
236	C₂²×C₆₇	—	2	1	no	—
237	C₂²×C₆₇	—	4	1	no	—
238	C₂²×C₆₇	—	4	1	no	—
239	C₂²×C₆₇	—	4	1	no	—
240	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₆₇	#529	1	2	yes	—
242	C₂×D₆₇	—	2	2	no	—
243	C₂×D₆₇	—	4	1	no	—
244	C₂×D₆₇	—	4	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	4	1	no	—
248	C₄×C₆₇	—	4	1	no	—
249	C₄×C₆₇	—	4	1	no	—
250	C₄×C₆₇	#443	1	2	yes	—
251	C₄×C₆₇	#444	1	2	yes	—
252	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
253	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
254	C₄×C₆₇	#525	1	2	yes	—
255	Dic₆₇	—	2	2	no	—
256	Dic₆₇	—	4	4	no	—
257	Dic₆₇	—	4	1	no	—
258	Dic₆₇	—	4	1	no	—
259	#21137	—	67	1	no	maximal, Sylow
260	C₂×C₆₇	—	2	2	no	—
261	C₂×C₆₇	—	8	1	no	—
262	C₂×C₆₇	#2290	1	1	yes	characteristic
263	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
264	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
265	D₆₇	—	4	2	no	—
266	C₄×#135	—	67	1	no	—
267	#2922	D₆₇	1	1	yes	characteristic, complemented
268	C₄×#131	—	67	1	no	—
269	#1929	—	67	2	no	—
270	#1922	—	67	2	no	—
271	#1876	—	67	2	no	—
272	#1866	—	67	2	no	—
273	C₄×#108	—	67	1	no	—
274	C₄×#104	—	67	1	no	—
275	C₄×#104	—	67	1	no	—
276	#1744	—	67	1	no	—
277	C₆₇	#21137	1	1	yes	characteristic, complemented, Sylow
278	C₂×#135	—	67	1	no	—
279	C₂×#131	—	67	2	no	—
280	C₄×#41	—	67	1	no	—
281	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
282	C₄×Q₁₆	—	67	1	no	—
283	C₄×Q₁₆	—	67	1	no	—
284	C₄×SD₁₆	—	67	1	no	—
285	C₄×SD₁₆	—	67	1	no	—
286	C₄×SD₁₆	—	67	1	no	—
287	C₄×SD₁₆	—	67	1	no	—
288	C₄×D₈	—	67	1	no	—
289	C₄×D₈	—	67	1	no	—
290	#433	—	2	2	no	—
291	#431	—	2	1	no	—
292	#431	—	2	1	no	—
293	C₂×#108	—	67	2	no	—
294	#427	—	67	1	no	—
295	#424	C₂×D₆₇	1	2	yes	—
296	C₂×#104	—	67	2	no	—
297	C₂×#104	—	67	2	no	—
298	#416	—	67	1	no	—
299	#416	—	67	1	no	—
300	#408	C₂×D₆₇	1	2	yes	—
301	#405	—	67	2	no	—
302	#405	—	134	2	no	—
303	C₄×M₁₆	—	67	1	no	—
304	C₄×M₁₆	—	134	1	no	—
305	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
306	C₂×C₄×C₈	—	67	1	no	—
307	C₂×C₄×C₈	—	134	1	no	—
308	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
309	C₄³	—	67	1	no	—
310	#341	—	67	2	no	—
311	#339	—	67	2	no	—
312	#339	—	67	2	no	—
313	#336	—	67	2	no	—
314	#336	—	67	2	no	—
315	C₂×#41	—	67	1	no	—
316	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
317	#135	—	134	4	no	—
318	C₂×Q₁₆	—	67	1	no	—
319	C₂×SD₁₆	—	67	1	no	—
320	C₂×SD₁₆	—	67	1	no	—
321	C₂×D₈	—	67	1	no	—
322	#131	—	134	4	no	—
323	#131	—	134	4	no	—
324	C₂×M₁₆	—	67	2	no	—
325	C₂×M₁₆	—	134	2	no	—
326	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
327	C₂²×C₈	—	67	1	no	—
328	C₂²×C₈	—	67	1	no	—
329	C₂²×C₈	—	134	2	no	—
330	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
331	C₄×Q₈	—	67	1	no	—
332	C₄×Q₈	—	67	1	no	—
333	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
334	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	67	1	no	—
338	C₄×D₄	—	67	1	no	—
339	C₄×D₄	—	134	1	no	—
340	C₄×D₄	—	134	1	no	—
341	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
342	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
343	#117	—	2	1	no	—
344	#117	—	67	1	no	—
345	#117	—	134	1	no	—
346	#117	C₂²×D₆₇	1	1	yes	characteristic
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	67	2	no	—
349	C₂×C₄²	—	134	1	no	—
350	C₂×C₄²	—	134	1	no	—
351	C₂×C₄²	—	134	1	no	—
352	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
353	#108	—	134	4	no	—
354	#107	—	67	1	no	—
355	#107	—	67	1	no	—
356	#106	—	67	1	no	—
357	#106	—	67	1	no	—
358	C₄⋊C₈	—	2	2	no	—
359	C₄⋊C₈	—	2	2	no	—
360	#104	—	134	4	no	—
361	#104	—	134	4	no	—
362	#103	—	67	1	no	—
363	#103	—	67	1	no	—
364	#103	—	67	1	no	—
365	#103	—	67	1	no	—
366	#102	—	67	1	no	—
367	#102	—	67	1	no	—
368	#102	—	67	1	no	—
369	#102	—	67	1	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	#97	—	2	1	no	—
373	#97	—	134	2	no	—
374	#97	—	134	1	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	67	1	no	—
377	C₄×C₈	—	67	1	no	—
378	C₄×C₈	—	67	1	no	—
379	C₄×C₈	—	67	1	no	—
380	C₄×C₈	—	134	2	no	—
381	C₄×C₈	—	134	1	no	—
382	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
383	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
384	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	67	4	no	—
388	#41	—	134	2	no	—
389	C₂×Q₈	—	67	1	no	—
390	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	67	1	no	—
393	C₂×D₄	—	134	1	no	—
394	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	134	2	no	—
399	C₂²×C₄	—	134	1	no	—
400	C₂²×C₄	—	134	1	no	—
401	C₂²×C₄	—	134	1	no	—
402	C₂²×C₄	#11761628	1	1	yes	characteristic
403	C₂²×C₄	#11761628	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
405	Q₁₆	—	134	2	no	—
406	SD₁₆	—	134	2	no	—
407	SD₁₆	—	134	2	no	—
408	D₈	—	134	2	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	67	4	no	—
412	M₁₆	—	134	8	no	—
413	M₁₆	—	134	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	67	1	no	—
418	C₂×C₈	—	67	1	no	—
419	C₂×C₈	—	67	1	no	—
420	C₂×C₈	—	67	1	no	—
421	C₂×C₈	—	134	8	no	—
422	C₂×C₈	—	134	2	no	—
423	C₂×C₈	—	134	2	no	—
424	C₂×C₈	—	134	2	no	—
425	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
426	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	67	1	no	—
430	C₄⋊C₄	—	67	1	no	—
431	C₄⋊C₄	—	134	1	no	—
432	C₄⋊C₄	—	134	1	no	—
433	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
434	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	—	134	1	no	—
439	C₂²⋊C₄	—	134	1	no	—
440	C₂²⋊C₄	—	134	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	268	2	no	—
445	C₄²	—	134	4	no	—
446	C₄²	—	134	2	no	—
447	C₄²	—	134	2	no	—
448	C₄²	—	134	1	no	—
449	C₄²	—	134	1	no	—
450	C₄²	—	134	1	no	—
451	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
452	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
453	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
454	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
455	C₂³	—	2	1	no	—
456	C₂³	—	134	1	no	—
457	C₂³	C₂×#11761628	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	—	2	1	no	—
460	Q₈	—	67	2	no	—
461	Q₈	—	134	1	no	—
462	D₄	—	2	1	no	—
463	D₄	—	2	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	67	2	no	—
466	D₄	—	134	4	no	—
467	D₄	—	134	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	268	2	no	—
477	C₂×C₄	—	268	2	no	—
478	C₂×C₄	—	268	2	no	—
479	C₂×C₄	—	268	2	no	—
480	C₂×C₄	—	134	4	no	—
481	C₂×C₄	—	134	1	no	—
482	C₂×C₄	—	134	1	no	—
483	C₂×C₄	—	134	1	no	—
484	C₂×C₄	—	134	4	no	—
485	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
486	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
487	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
488	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
491	C₂×C₄	#41×D₆₇	1	2	yes	—
492	C₂×C₄	#41×D₆₇	1	2	yes	—
493	C₈	—	2	2	no	—
494	C₈	—	2	2	no	—
495	C₈	—	4	2	no	—
496	C₈	—	134	1	no	—
497	C₈	—	134	1	no	—
498	C₈	—	134	1	no	—
499	C₈	—	134	1	no	—
500	C₈	—	134	4	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	268	1	no	—
507	C₂²	—	268	1	no	—
508	C₂²	—	134	2	no	—
509	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
510	C₂²	#16863924	1	2	yes	—
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	4	1	no	—
516	C₄	—	268	4	no	—
517	C₄	—	268	1	no	—
518	C₄	—	268	1	no	—
519	C₄	—	134	2	no	—
520	C₄	#16863511	1	2	yes	—
521	C₄	#16863516	1	2	yes	—
522	C₄	#117×D₆₇	1	1	yes	characteristic
523	C₄	#16863864	1	1	yes	characteristic
524	C₄	#16863903	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	8	1	no	—
527	C₂	—	268	2	no	—
528	C₂	#22562646	1	1	yes	characteristic
529	C₂	C₂×#16863511	1	1	yes	characteristic
530	C₂	C₂×#16863516	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2922	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2922	D₆₇
hypocenter	C₆₇	#21137
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2922

Complements

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

N	H ≅ G/N	type
2. C₄×#14524049	C₂	semidirect
3. #22560854	C₂	semidirect
4. #22560805	C₂	semidirect
6. #22560483	C₂	semidirect
7. C₄×#14523978	C₂	semidirect
8. C₄×#14523969	C₂	semidirect
9. C₄×#14523970	C₂	semidirect
10. #22559954	C₂	semidirect
11. C₆₇×#2922	C₂	semidirect
15. C₄×#11761618	C₄	semidirect
16. C₄×#11761618	C₄	semidirect
17. C₄×#11761615	C₄	semidirect
18. C₄×#11761615	C₄	semidirect
19. C₄×#11761616	C₄	semidirect
20. C₄×#11761616	C₄	semidirect
21. C₄×#11761613	C₄	semidirect
22. C₄×#11761613	C₄	semidirect
267. #2922	D₆₇	semidirect
277. C₆₇	#21137	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524063	2
C₄×#14524049	2
#22560854	2
#22560805	2
#22560539	2
#22560483	2
C₄×#14523978	2
C₄×#14523969	2
C₄×#14523970	2
#22559954	2
C₆₇×#2922	2
#21137	67