Subgroups — #30859053

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×#14524065	C₂	1	1	yes	maximal, characteristic
2	#22568854	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524051	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560856	C₂	1	2	yes	maximal, complemented
5	#22560807	C₂	1	2	yes	maximal, complemented
6	#22560537	C₂	1	2	yes	maximal
7	#22560481	C₂	1	2	yes	maximal, complemented
8	C₄×#14523969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523969	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559954	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	C₂×#14524051	C₂²	1	2	yes	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	C₄×#11761628	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic, complemented
17	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761617	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761617	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761617	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761617	C₄	1	1	yes	characteristic, complemented
22	C₄×D₅₃₆	C₄	1	1	yes	characteristic, complemented
23	C₄×D₅₃₆	C₄	1	1	yes	characteristic, complemented
24	#16863458	—	2	2	no	—
25	#16863447	—	2	1	no	—
26	#16863447	—	2	1	no	—
27	#16863434	C₂²	1	1	yes	characteristic
28	#16863424	C₂²	1	2	yes	—
29	C₂×#14523969	C₂²	1	2	yes	—
30	C₂×#14523969	C₂²	1	2	yes	—
31	#16863393	C₂²	1	1	yes	characteristic
32	#16863393	C₂²	1	1	yes	characteristic
33	#16863364	C₂²	1	2	yes	—
34	#16863354	C₂²	1	2	yes	—
35	#16863353	—	2	2	no	—
36	C₄×#11761611	—	2	1	no	—
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₈×D₆₇	—	2	1	no	—
39	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
40	#16863157	C₂²	1	2	yes	—
41	#16863157	C₂²	1	2	yes	—
42	#16863150	C₂²	1	2	yes	—
43	#16863150	C₂²	1	2	yes	—
44	C₂×#108×C₆₇	C₂²	1	2	yes	—
45	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
48	#341×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	C₂×#11761628	C₂³	1	1	yes	characteristic
51	#14524065	—	2	4	no	—
52	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761617	C₂×C₄	1	1	yes	characteristic
55	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
56	#14524051	—	2	4	no	—
57	#14524051	—	2	4	no	—
58	C₂×#11761611	—	2	2	no	—
59	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₂×C₈×D₆₇	—	2	2	no	—
61	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
68	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	C₄×C₂²⋊D₆₇	—	2	1	no	—
71	C₄×C₂²⋊D₆₇	—	2	1	no	—
72	C₄×C₂²⋊D₆₇	—	2	1	no	—
73	#14524002	C₂³	1	1	yes	characteristic
74	#14524002	C₂³	1	1	yes	characteristic
75	#14524000	—	2	1	no	—
76	#14524000	—	2	1	no	—
77	C₂×C₄×Dic₆₇	—	2	1	no	—
78	C₂×C₄×Dic₆₇	—	2	1	no	—
79	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
80	C₄²×D₆₇	—	2	1	no	—
81	C₄²×D₆₇	—	2	1	no	—
82	#14523977	C₂×C₄	1	1	yes	characteristic
83	#14523977	C₂×C₄	1	1	yes	characteristic
84	#14523975	C₂×C₄	1	1	yes	characteristic
85	#14523975	C₂×C₄	1	1	yes	characteristic
86	Dic₆₇⋊C₈	—	2	2	no	—
87	Dic₆₇⋊C₈	—	2	2	no	—
88	#14523969	—	2	4	no	—
89	#14523969	—	2	4	no	—
90	#14523968	C₂×C₄	1	1	yes	characteristic
91	#14523968	C₂×C₄	1	1	yes	characteristic
92	#14523968	C₂×C₄	1	1	yes	characteristic
93	#14523968	C₂×C₄	1	1	yes	characteristic
94	#14523965	C₂×C₄	1	1	yes	characteristic
95	#14523965	C₂×C₄	1	1	yes	characteristic
96	#14523965	C₂×C₄	1	1	yes	characteristic
97	#14523965	C₂×C₄	1	1	yes	characteristic
98	(C₂×D₆₇)⋊C₈	—	2	2	no	—
99	(C₂×D₆₇)⋊C₈	—	2	2	no	—
100	#14523953	—	2	2	no	—
101	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
102	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
103	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
104	C₈×Dic₆₇	—	2	2	no	—
105	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
106	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
107	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	#108×C₆₇	—	2	4	no	—
111	#97×C₆₇	—	2	1	no	—
112	#97×C₆₇	—	2	1	no	—
113	C₄×C₈×C₆₇	—	2	1	no	—
114	C₄×C₈×C₆₇	—	2	1	no	—
115	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
119	#11761628	—	2	2	no	—
120	#11761628	—	2	2	no	—
121	#11761628	—	2	2	no	—
122	#11761628	#41	1	4	yes	—
123	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₂²⋊D₆₇	—	2	1	no	—
128	C₂×C₂²⋊D₆₇	—	2	1	no	—
129	C₂²×Dic₆₇	—	2	2	no	—
130	C₂×C₄×D₆₇	—	2	1	no	—
131	C₂×C₄×D₆₇	—	2	1	no	—
132	C₂×C₄×D₆₇	—	2	1	no	—
133	C₂×C₄×D₆₇	—	2	1	no	—
134	C₂×C₄×D₆₇	—	2	1	no	—
135	C₂×C₄×D₆₇	—	2	1	no	—
136	Dic₂₆₈	—	2	2	no	—
137	#11761617	—	2	2	no	—
138	#11761617	—	2	2	no	—
139	D₅₃₆	—	2	2	no	—
140	#11761611	—	2	8	no	—
141	C₂×C₆₇⋊C₈	—	2	2	no	—
142	C₂×C₆₇⋊C₈	—	2	2	no	—
143	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
144	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
145	C₈×D₆₇	—	2	8	no	—
146	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
150	Dic₆₇⋊C₄	—	2	1	no	—
151	Dic₆₇⋊C₄	—	2	1	no	—
152	Dic₆₇⋊C₄	—	2	1	no	—
153	Dic₆₇⋊C₄	—	2	1	no	—
154	C₂²⋊Dic₆₇	—	2	1	no	—
155	C₂²⋊Dic₆₇	—	2	1	no	—
156	(C₂×D₆₇)⋊C₄	—	2	1	no	—
157	(C₂×D₆₇)⋊C₄	—	2	1	no	—
158	(C₂×D₆₇)⋊C₄	—	2	1	no	—
159	(C₂×D₆₇)⋊C₄	—	2	1	no	—
160	C₄×Dic₆₇	—	2	4	no	—
161	C₄×Dic₆₇	—	2	1	no	—
162	C₄×Dic₆₇	—	2	1	no	—
163	C₄×Dic₆₇	—	2	1	no	—
164	C₄×Dic₆₇	—	2	1	no	—
165	C₄×Dic₆₇	—	2	1	no	—
166	C₄×Dic₆₇	—	2	1	no	—
167	C₄×Dic₆₇	—	2	2	no	—
168	C₄×Dic₆₇	—	2	2	no	—
169	C₄×Dic₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	M₁₆×C₆₇	—	2	2	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	#41	1	4	yes	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	2	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
190	C₂²×D₆₇	—	2	1	no	—
191	C₂²×D₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
201	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
202	C₂²⋊D₆₇	—	2	4	no	—
203	C₂²⋊D₆₇	—	4	2	no	—
204	C₂×Dic₆₇	—	2	1	no	—
205	C₂×Dic₆₇	—	2	4	no	—
206	C₂×Dic₆₇	—	2	1	no	—
207	C₂×Dic₆₇	—	2	1	no	—
208	C₂×Dic₆₇	—	2	1	no	—
209	C₂×Dic₆₇	—	2	1	no	—
210	C₂×Dic₆₇	—	2	1	no	—
211	C₂×Dic₆₇	—	4	2	no	—
212	C₂×Dic₆₇	—	4	2	no	—
213	D₂₆₈	—	2	1	no	—
214	D₂₆₈	—	2	1	no	—
215	D₂₆₈	—	2	1	no	—
216	D₂₆₈	#117	1	2	yes	—
217	C₄×D₆₇	—	2	4	no	—
218	C₄×D₆₇	—	4	2	no	—
219	C₄×D₆₇	—	4	2	no	—
220	C₄×D₆₇	—	4	2	no	—
221	C₄×D₆₇	—	4	2	no	—
222	C₄×D₆₇	—	4	2	no	—
223	Dic₁₃₄	—	2	1	no	—
224	Dic₁₃₄	—	2	1	no	—
225	Dic₁₃₄	—	2	1	no	—
226	Dic₁₃₄	#117	1	2	yes	—
227	C₈×C₆₇	—	2	1	no	—
228	C₈×C₆₇	—	2	1	no	—
229	C₈×C₆₇	—	2	1	no	—
230	C₈×C₆₇	—	2	1	no	—
231	C₈×C₆₇	—	2	4	no	—
232	C₈×C₆₇	—	4	2	no	—
233	C₆₇⋊C₈	—	2	2	no	—
234	C₆₇⋊C₈	—	2	2	no	—
235	C₂²×C₆₇	—	2	1	no	—
236	C₂²×C₆₇	—	2	1	no	—
237	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
238	C₂²×C₆₇	#529	1	2	yes	—
239	C₂×D₆₇	—	2	2	no	—
240	C₂×D₆₇	—	4	1	no	—
241	C₂×D₆₇	—	4	1	no	—
242	C₂×D₆₇	—	4	1	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	#443	1	2	yes	—
248	C₄×C₆₇	#444	1	2	yes	—
249	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
250	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
251	C₄×C₆₇	#525	1	2	yes	—
252	Dic₆₇	—	2	2	no	—
253	Dic₆₇	—	4	4	no	—
254	Dic₆₇	—	4	1	no	—
255	Dic₆₇	—	4	1	no	—
256	Dic₆₇	—	4	1	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	#2290	1	1	yes	characteristic
262	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
263	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
264	D₆₇	—	4	2	no	—
265	D₆₇	—	8	1	no	—
266	C₄×#135	—	67	1	no	—
267	#2922	—	67	1	no	—
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	D₆₇	1	1	yes	characteristic, complemented
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	—	67	1	no	—
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	—	134	2	no	—
291	#431	—	134	1	no	—
292	#431	—	134	1	no	—
293	C₂×#108	C₂×D₆₇	1	2	yes	—
294	#427	—	67	1	no	—
295	#424	—	67	2	no	—
296	C₂×#104	—	67	2	no	—
297	C₂×#104	—	67	2	no	—
298	#416	—	67	1	no	—
299	#416	—	67	1	no	—
300	#408	—	67	2	no	—
301	#405	—	67	2	no	—
302	#405	—	134	2	no	—
303	C₄×M₁₆	—	134	1	no	—
304	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
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	C₂×D₆₇	1	2	yes	—
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	—	67	1	no	—
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	—	134	2	no	—
325	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
326	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
327	C₂²×C₈	—	67	2	no	—
328	C₂²×C₈	—	134	2	no	—
329	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
331	C₄×Q₈	—	67	1	no	—
332	C₄×Q₈	—	67	1	no	—
333	C₄×Q₈	—	67	1	no	—
334	C₄×Q₈	—	67	1	no	—
335	C₄×D₄	—	67	1	no	—
336	C₄×D₄	—	67	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	—	134	1	no	—
342	C₄×D₄	—	134	1	no	—
343	#117	—	67	1	no	—
344	#117	—	67	1	no	—
345	#117	—	134	1	no	—
346	#117	—	134	1	no	—
347	C₂×C₄²	—	67	2	no	—
348	C₂×C₄²	—	134	1	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	—	2	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	—	134	2	no	—
359	C₄⋊C₈	—	134	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	—	134	2	no	—
371	C₂²⋊C₈	—	134	2	no	—
372	#97	—	2	1	no	—
373	#97	—	2	1	no	—
374	#97	—	134	2	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	2	1	no	—
377	C₄×C₈	—	67	2	no	—
378	C₄×C₈	—	67	1	no	—
379	C₄×C₈	—	67	1	no	—
380	C₄×C₈	—	134	2	no	—
381	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
382	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
383	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
384	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
385	#41	—	67	4	no	—
386	#41	—	134	2	no	—
387	#41	—	134	2	no	—
388	#41	—	134	2	no	—
389	C₂×Q₈	—	67	1	no	—
390	C₂×Q₈	—	67	1	no	—
391	C₂×D₄	—	67	1	no	—
392	C₂×D₄	—	67	1	no	—
393	C₂×D₄	—	134	1	no	—
394	C₂×D₄	—	134	1	no	—
395	C₂²×C₄	—	134	2	no	—
396	C₂²×C₄	—	134	1	no	—
397	C₂²×C₄	—	134	1	no	—
398	C₂²×C₄	—	134	1	no	—
399	C₂²×C₄	—	134	1	no	—
400	C₂²×C₄	—	134	1	no	—
401	C₂²×C₄	—	134	1	no	—
402	C₂²×C₄	#11761629	1	1	yes	characteristic
403	C₂²×C₄	#11761629	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	—	2	2	no	—
412	M₁₆	—	134	8	no	—
413	M₁₆	#11761628	1	4	yes	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	67	2	no	—
419	C₂×C₈	—	67	2	no	—
420	C₂×C₈	—	134	8	no	—
421	C₂×C₈	—	134	2	no	—
422	C₂×C₈	—	134	2	no	—
423	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
424	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
425	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
426	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
427	C₄⋊C₄	—	67	1	no	—
428	C₄⋊C₄	—	67	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	—	134	1	no	—
434	C₄⋊C₄	—	134	1	no	—
435	C₂²⋊C₄	—	134	1	no	—
436	C₂²⋊C₄	—	134	1	no	—
437	C₂²⋊C₄	—	134	1	no	—
438	C₂²⋊C₄	—	134	1	no	—
439	C₂²⋊C₄	—	134	1	no	—
440	C₂²⋊C₄	—	134	1	no	—
441	C₄²	—	268	2	no	—
442	C₄²	—	134	4	no	—
443	C₄²	—	134	2	no	—
444	C₄²	—	134	2	no	—
445	C₄²	—	134	1	no	—
446	C₄²	—	134	1	no	—
447	C₄²	—	134	1	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	D₄×D₆₇	1	1	yes	characteristic
454	C₄²	D₄×D₆₇	1	1	yes	characteristic
455	C₂³	—	134	1	no	—
456	C₂³	—	134	1	no	—
457	C₂³	C₂×#11761629	1	1	yes	characteristic
458	Q₈	—	67	2	no	—
459	Q₈	—	134	1	no	—
460	Q₈	—	134	1	no	—
461	Q₈	—	134	1	no	—
462	D₄	—	268	2	no	—
463	D₄	—	67	2	no	—
464	D₄	—	134	4	no	—
465	D₄	—	134	1	no	—
466	D₄	—	134	1	no	—
467	D₄	—	134	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	268	2	no	—
471	C₂×C₄	—	268	2	no	—
472	C₂×C₄	—	268	2	no	—
473	C₂×C₄	—	268	2	no	—
474	C₂×C₄	—	268	2	no	—
475	C₂×C₄	—	268	2	no	—
476	C₂×C₄	—	268	2	no	—
477	C₂×C₄	—	134	4	no	—
478	C₂×C₄	—	134	1	no	—
479	C₂×C₄	—	134	1	no	—
480	C₂×C₄	—	134	1	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	#14524005	1	1	yes	characteristic
486	C₂×C₄	#14524005	1	1	yes	characteristic
487	C₂×C₄	#14524005	1	1	yes	characteristic
488	C₂×C₄	#14524005	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
491	C₂×C₄	#41×D₆₇	1	2	yes	—
492	C₂×C₄	C₂×#11761628	1	2	yes	—
493	C₈	—	2	1	no	—
494	C₈	—	2	1	no	—
495	C₈	—	2	1	no	—
496	C₈	—	2	1	no	—
497	C₈	—	2	4	no	—
498	C₈	—	4	2	no	—
499	C₈	—	134	2	no	—
500	C₈	—	134	2	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	268	1	no	—
504	C₂²	—	268	1	no	—
505	C₂²	—	268	1	no	—
506	C₂²	—	268	1	no	—
507	C₂²	—	268	1	no	—
508	C₂²	—	134	2	no	—
509	C₂²	C₂×#14524005	1	1	yes	characteristic
510	C₂²	#16863933	1	2	yes	—
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	268	4	no	—
514	C₄	—	268	1	no	—
515	C₄	—	268	1	no	—
516	C₄	—	268	1	no	—
517	C₄	—	268	1	no	—
518	C₄	—	268	1	no	—
519	C₄	—	134	2	no	—
520	C₄	#16863512	1	2	yes	—
521	C₄	#16863517	1	2	yes	—
522	C₄	#117×D₆₇	1	1	yes	characteristic
523	C₄	C₄×#11761628	1	1	yes	characteristic
524	C₄	#16863905	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	268	2	no	—
527	C₂	—	536	1	no	—
528	C₂	#22562650	1	1	yes	characteristic
529	C₂	C₂×#16863512	1	1	yes	characteristic
530	C₂	C₂×#16863517	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#21137
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
2. #22568854	C₂	semidirect
3. C₄×#14524051	C₂	semidirect
4. #22560856	C₂	semidirect
5. #22560807	C₂	semidirect
7. #22560481	C₂	semidirect
8. C₄×#14523969	C₂	semidirect
9. C₄×#14523969	C₂	semidirect
10. #22559954	C₂	semidirect
11. C₄×#108×C₆₇	C₂	semidirect
16. C₄×Dic₂₆₈	C₄	semidirect
17. C₄×Dic₂₆₈	C₄	semidirect
18. C₄×#11761617	C₄	semidirect
19. C₄×#11761617	C₄	semidirect
20. C₄×#11761617	C₄	semidirect
21. C₄×#11761617	C₄	semidirect
22. C₄×D₅₃₆	C₄	semidirect
23. C₄×D₅₃₆	C₄	semidirect
273. C₄×#108	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×#14524065	2
#22568854	2
C₄×#14524051	2
#22560856	2
#22560807	2
#22560537	2
#22560481	2
C₄×#14523969	2
#22559954	2
C₄×#108×C₆₇	2
#21137	67