Subgroups — #30825364

Order 17152.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11761628	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
4	#22559061	C₂	1	2	yes	maximal, complemented
5	#22559054	C₂	1	2	yes	maximal, complemented
6	#22559047	C₂	1	2	yes	maximal, complemented
7	#22559040	C₂	1	2	yes	maximal, complemented
8	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558592	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558592	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	#16863526	C₂²	1	2	yes	—
16	C₈×Dic₁₃₄	C₂²	1	2	yes	—
17	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
18	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
19	C₄×#11761617	C₄	1	1	yes	characteristic
20	C₄×#11761617	C₄	1	1	yes	characteristic
21	C₄×#11761617	C₂²	1	1	yes	characteristic
22	C₄×#11761617	C₂²	1	1	yes	characteristic
23	C₄×D₅₃₆	C₄	1	1	yes	characteristic
24	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
25	#16863475	C₂²	1	2	yes	—
26	#16863467	—	2	2	no	—
27	C₈×D₂₆₈	C₂²	1	2	yes	—
28	C₈×C₂²⋊D₆₇	—	2	2	no	—
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863447	—	2	1	no	—
32	#16863447	—	2	1	no	—
33	#16863434	C₄	1	1	yes	characteristic
34	#16863393	C₄	1	1	yes	characteristic
35	#16863393	C₄	1	1	yes	characteristic
36	C₄×#11761611	—	2	1	no	—
37	C₄×C₈×D₆₇	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	—
39	#16863147	C₂²	1	2	yes	—
40	#16863128	C₂²	1	2	yes	—
41	#16863128	C₂²	1	2	yes	—
42	#16863125	C₂²	1	2	yes	—
43	#16863125	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	C₂×#11761628	C₂×C₄	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	C₂×#11761611	—	2	2	no	—
57	C₂×C₈×D₆₇	—	2	2	no	—
58	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
62	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₆₇	—	2	1	no	—
67	C₄×C₂²⋊D₆₇	—	2	1	no	—
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	#14524002	C₂×C₄	1	1	yes	characteristic
71	#14524002	C₂×C₄	1	1	yes	characteristic
72	#14524000	—	2	1	no	—
73	#14524000	—	2	1	no	—
74	C₄²×D₆₇	—	2	1	no	—
75	C₄²×D₆₇	—	2	1	no	—
76	#14523977	C₂×C₄	1	1	yes	characteristic
77	#14523977	C₂×C₄	1	1	yes	characteristic
78	#14523975	C₂×C₄	1	1	yes	characteristic
79	#14523975	C₂×C₄	1	1	yes	characteristic
80	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
81	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
82	Dic₆₇⋊C₈	—	2	2	no	—
83	Dic₆₇⋊C₈	—	2	2	no	—
84	#14523968	C₂×C₄	1	1	yes	characteristic
85	#14523968	C₂×C₄	1	1	yes	characteristic
86	#14523968	C₂×C₄	1	1	yes	characteristic
87	#14523968	C₂×C₄	1	1	yes	characteristic
88	#14523965	C₂×C₄	1	1	yes	characteristic
89	#14523965	C₂×C₄	1	1	yes	characteristic
90	#14523965	C₂×C₄	1	1	yes	characteristic
91	#14523965	C₂×C₄	1	1	yes	characteristic
92	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
93	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
94	(C₂×D₆₇)⋊C₈	—	2	2	no	—
95	(C₂×D₆₇)⋊C₈	—	2	2	no	—
96	#14523953	—	2	2	no	—
97	C₄×C₆₇⋊C₈	—	2	1	no	—
98	C₄×C₆₇⋊C₈	—	2	1	no	—
99	C₈×Dic₆₇	—	2	2	no	—
100	C₂²×C₈×C₆₇	—	2	1	no	—
101	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	D₄	1	2	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₄×C₈×C₆₇	—	2	1	no	—
106	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
109	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
110	C₄×C₈×C₆₇	C₂³	1	2	yes	—
111	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	2	yes	—
114	#11761628	—	2	2	no	—
115	#11761628	—	2	2	no	—
116	#11761628	—	2	2	no	—
117	#11761628	M₁₆	1	4	yes	complemented
118	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₂²⋊D₆₇	—	2	1	no	—
123	C₂×C₂²⋊D₆₇	—	2	1	no	—
124	C₂×C₄×D₆₇	—	2	1	no	—
125	C₂×C₄×D₆₇	—	2	1	no	—
126	C₂×C₄×D₆₇	—	2	1	no	—
127	C₂×C₄×D₆₇	—	2	1	no	—
128	C₂×C₄×D₆₇	—	2	1	no	—
129	C₂×C₄×D₆₇	—	2	1	no	—
130	Dic₂₆₈	—	2	2	no	—
131	#11761617	—	2	2	no	—
132	#11761617	—	2	2	no	—
133	D₅₃₆	—	2	2	no	—
134	#11761611	—	4	4	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₈×D₆₇	—	4	4	no	—
138	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
142	Dic₆₇⋊C₄	—	2	1	no	—
143	Dic₆₇⋊C₄	—	2	1	no	—
144	Dic₆₇⋊C₄	—	2	1	no	—
145	Dic₆₇⋊C₄	—	2	1	no	—
146	C₂²⋊Dic₆₇	—	2	1	no	—
147	C₂²⋊Dic₆₇	—	2	1	no	—
148	(C₂×D₆₇)⋊C₄	—	2	1	no	—
149	(C₂×D₆₇)⋊C₄	—	2	1	no	—
150	(C₂×D₆₇)⋊C₄	—	2	1	no	—
151	(C₂×D₆₇)⋊C₄	—	2	1	no	—
152	C₄×Dic₆₇	—	2	1	no	—
153	C₄×Dic₆₇	—	2	1	no	—
154	C₄×Dic₆₇	—	2	1	no	—
155	C₄×Dic₆₇	—	2	1	no	—
156	C₄×Dic₆₇	—	2	1	no	—
157	C₄×Dic₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	4	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	4	2	no	—
166	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	#41	1	1	yes	characteristic
175	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₆₇	—	2	1	no	—
178	C₂²×D₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₆₇	#131	1	2	yes	—
186	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂²⋊D₆₇	—	2	4	no	—
190	C₂²⋊D₆₇	—	4	2	no	—
191	C₂×Dic₆₇	—	2	1	no	—
192	C₂×Dic₆₇	—	2	1	no	—
193	C₂×Dic₆₇	—	2	1	no	—
194	C₂×Dic₆₇	—	2	1	no	—
195	C₂×Dic₆₇	—	2	1	no	—
196	C₂×Dic₆₇	—	2	1	no	—
197	D₂₆₈	—	2	1	no	—
198	D₂₆₈	—	2	1	no	—
199	D₂₆₈	—	2	1	no	—
200	D₂₆₈	C₂×M₁₆	1	2	yes	—
201	C₄×D₆₇	—	2	4	no	—
202	C₄×D₆₇	—	4	2	no	—
203	C₄×D₆₇	—	4	2	no	—
204	C₄×D₆₇	—	4	2	no	—
205	C₄×D₆₇	—	4	2	no	—
206	C₄×D₆₇	—	4	2	no	—
207	Dic₁₃₄	—	2	1	no	—
208	Dic₁₃₄	—	2	1	no	—
209	Dic₁₃₄	—	2	1	no	—
210	Dic₁₃₄	C₂×M₁₆	1	2	yes	—
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	2	1	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₈×C₆₇	#135	1	4	yes	complemented
218	C₆₇⋊C₈	—	4	2	no	—
219	C₆₇⋊C₈	—	4	2	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	1	no	—
222	C₂²×C₆₇	#435	1	2	yes	—
223	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×D₆₇	—	2	2	no	—
225	C₂×D₆₇	—	4	1	no	—
226	C₂×D₆₇	—	4	1	no	—
227	C₂×D₆₇	—	4	1	no	—
228	C₂×D₆₇	—	4	1	no	—
229	C₂×D₆₇	—	4	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	#435	1	2	yes	—
233	C₄×C₆₇	#444	1	2	yes	—
234	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
235	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
236	C₄×C₆₇	C₂×#135	1	2	yes	—
237	Dic₆₇	—	2	2	no	—
238	Dic₆₇	—	4	1	no	—
239	Dic₆₇	—	4	1	no	—
240	Dic₆₇	—	4	1	no	—
241	Dic₆₇	—	4	1	no	—
242	Dic₆₇	—	4	1	no	—
243	#17863	—	67	1	no	maximal, Sylow
244	C₂×C₆₇	—	2	2	no	—
245	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
246	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
247	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
248	D₆₇	—	4	2	no	—
249	D₆₇	—	8	1	no	—
250	#2965	—	67	1	no	—
251	C₈×#41	—	67	1	no	—
252	C₄×#135	—	67	1	no	—
253	#1584	—	67	2	no	—
254	#1583	—	67	2	no	—
255	#1582	—	67	2	no	—
256	#1581	—	67	2	no	—
257	#1564	—	67	1	no	—
258	#1476	—	67	1	no	—
259	#1476	—	67	1	no	—
260	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
261	C₆₇	#17863	1	1	yes	characteristic, complemented, Sylow
262	C₂×#135	—	67	1	no	—
263	C₄×#41	—	67	1	no	—
264	C₄×#41	—	67	1	no	—
265	#446	—	67	2	no	—
266	C₈×Q₈	—	67	2	no	—
267	C₄×Q₁₆	—	67	1	no	—
268	C₄×Q₁₆	—	67	1	no	—
269	C₄×SD₁₆	—	67	1	no	—
270	C₄×SD₁₆	—	67	1	no	—
271	C₄×SD₁₆	—	67	1	no	—
272	C₄×SD₁₆	—	67	1	no	—
273	C₄×D₈	—	67	1	no	—
274	C₄×D₈	—	67	1	no	—
275	#436	—	67	2	no	—
276	#435	—	134	2	no	—
277	C₈×D₄	—	67	2	no	—
278	C₈×D₄	—	134	2	no	—
279	#431	—	67	1	no	—
280	#431	—	67	1	no	—
281	#431	—	134	1	no	—
282	#431	—	134	1	no	—
283	#427	—	67	1	no	—
284	#416	—	67	1	no	—
285	#416	—	67	1	no	—
286	C₄×M₁₆	—	134	1	no	—
287	C₂×C₄×C₈	—	134	1	no	—
288	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
291	#335	—	67	2	no	—
292	#334	—	67	2	no	—
293	Q₈⋊C₈	—	67	2	no	—
294	Q₈⋊C₈	—	67	2	no	—
295	D₄⋊C₈	—	67	2	no	—
296	D₄⋊C₈	—	67	2	no	—
297	#322	C₂×D₆₇	1	2	yes	—
298	#322	C₂×D₆₇	1	2	yes	—
299	C₂×#41	—	67	1	no	—
300	C₂×#41	—	67	1	no	—
301	#135	—	134	4	no	—
302	C₂×Q₁₆	—	67	1	no	—
303	C₂×SD₁₆	—	67	1	no	—
304	C₂×SD₁₆	—	67	1	no	—
305	C₂×D₈	—	67	1	no	—
306	C₂×M₁₆	—	134	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	134	2	no	—
309	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
310	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
311	C₂²×C₈	D₂₆₈	1	2	yes	—
312	C₄×Q₈	—	67	1	no	—
313	C₄×Q₈	—	67	1	no	—
314	C₄×Q₈	—	67	1	no	—
315	C₄×Q₈	—	67	1	no	—
316	C₄×D₄	—	67	1	no	—
317	C₄×D₄	—	67	1	no	—
318	C₄×D₄	—	67	1	no	—
319	C₄×D₄	—	67	1	no	—
320	C₄×D₄	—	134	1	no	—
321	C₄×D₄	—	134	1	no	—
322	C₄×D₄	—	134	1	no	—
323	C₄×D₄	—	134	1	no	—
324	#117	—	67	1	no	—
325	#117	—	67	1	no	—
326	#117	—	134	1	no	—
327	#117	—	134	1	no	—
328	C₂×C₄²	—	134	1	no	—
329	C₂×C₄²	—	134	1	no	—
330	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
331	#107	—	67	1	no	—
332	#107	—	67	1	no	—
333	#106	—	67	1	no	—
334	#106	—	67	1	no	—
335	C₄⋊C₈	—	67	2	no	—
336	C₄⋊C₈	—	67	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	C₄⋊C₈	—	134	2	no	—
339	#103	—	67	1	no	—
340	#103	—	67	1	no	—
341	#103	—	67	1	no	—
342	#103	—	67	1	no	—
343	#102	—	67	1	no	—
344	#102	—	67	1	no	—
345	#102	—	67	1	no	—
346	#102	—	67	1	no	—
347	C₂²⋊C₈	—	134	1	no	—
348	C₂²⋊C₈	—	134	1	no	—
349	C₂²⋊C₈	—	134	2	no	—
350	C₂²⋊C₈	—	134	2	no	—
351	#97	—	134	2	no	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	134	2	no	—
354	C₄×C₈	—	134	1	no	—
355	C₄×C₈	—	134	1	no	—
356	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
357	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
358	C₄×C₈	D₂₆₈	1	1	yes	characteristic
359	C₄×C₈	D₂₆₈	1	1	yes	characteristic
360	C₄×C₈	C₂²×D₆₇	1	2	yes	—
361	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
362	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
363	C₄×C₈	C₂²×D₆₇	1	2	yes	—
364	#41	—	67	4	no	—
365	#41	—	134	2	no	—
366	#41	—	134	2	no	—
367	#41	—	134	2	no	—
368	C₂×Q₈	—	67	1	no	—
369	C₂×Q₈	—	67	1	no	—
370	C₂×D₄	—	67	1	no	—
371	C₂×D₄	—	67	1	no	—
372	C₂×D₄	—	134	1	no	—
373	C₂×D₄	—	134	1	no	—
374	C₂²×C₄	—	134	1	no	—
375	C₂²×C₄	—	134	1	no	—
376	C₂²×C₄	—	134	1	no	—
377	C₂²×C₄	—	134	1	no	—
378	C₂²×C₄	—	134	1	no	—
379	C₂²×C₄	—	134	1	no	—
380	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
382	C₂²×C₄	#11761628	1	1	yes	characteristic
383	Q₁₆	—	134	2	no	—
384	SD₁₆	—	134	2	no	—
385	SD₁₆	—	134	2	no	—
386	D₈	—	134	2	no	—
387	M₁₆	—	268	4	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	—	268	4	no	—
394	C₂×C₈	—	134	2	no	—
395	C₂×C₈	—	134	2	no	—
396	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
397	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
398	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
399	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
400	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
401	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
402	C₄⋊C₄	—	67	1	no	—
403	C₄⋊C₄	—	67	1	no	—
404	C₄⋊C₄	—	67	1	no	—
405	C₄⋊C₄	—	67	1	no	—
406	C₄⋊C₄	—	134	1	no	—
407	C₄⋊C₄	—	134	1	no	—
408	C₄⋊C₄	—	134	1	no	—
409	C₄⋊C₄	—	134	1	no	—
410	C₂²⋊C₄	—	134	1	no	—
411	C₂²⋊C₄	—	134	1	no	—
412	C₂²⋊C₄	—	134	1	no	—
413	C₂²⋊C₄	—	134	1	no	—
414	C₂²⋊C₄	—	134	1	no	—
415	C₂²⋊C₄	—	134	1	no	—
416	C₄²	—	134	1	no	—
417	C₄²	—	134	1	no	—
418	C₄²	—	134	1	no	—
419	C₄²	—	134	1	no	—
420	C₄²	—	134	1	no	—
421	C₄²	—	134	1	no	—
422	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
423	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
424	C₄²	#11761628	1	1	yes	characteristic
425	C₄²	C₂³×D₆₇	1	1	yes	characteristic
426	C₂³	—	134	1	no	—
427	C₂³	—	134	1	no	—
428	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
429	Q₈	—	67	2	no	—
430	Q₈	—	134	1	no	—
431	Q₈	—	134	1	no	—
432	Q₈	—	134	1	no	—
433	D₄	—	268	2	no	—
434	D₄	—	67	2	no	—
435	D₄	—	134	4	no	—
436	D₄	—	134	1	no	—
437	D₄	—	134	1	no	—
438	D₄	—	134	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	268	2	no	—
442	C₂×C₄	—	268	2	no	—
443	C₂×C₄	—	268	2	no	—
444	C₂×C₄	—	268	2	no	—
445	C₂×C₄	—	268	2	no	—
446	C₂×C₄	—	134	1	no	—
447	C₂×C₄	—	134	1	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	—	134	1	no	—
450	C₂×C₄	—	134	1	no	—
451	C₂×C₄	—	134	1	no	—
452	C₂×C₄	—	134	4	no	—
453	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
454	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
455	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
456	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
457	C₂×C₄	#14524051	1	2	yes	—
458	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
460	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	2	no	—
466	C₈	—	4	2	no	—
467	C₈	—	268	2	no	—
468	C₈	—	268	2	no	—
469	C₈	#14524065	1	4	yes	complemented
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	268	1	no	—
473	C₂²	—	268	1	no	—
474	C₂²	—	268	1	no	—
475	C₂²	—	268	1	no	—
476	C₂²	—	268	1	no	—
477	C₂²	—	134	2	no	—
478	C₂²	#16863469	1	2	yes	—
479	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	268	1	no	—
483	C₄	—	268	1	no	—
484	C₄	—	268	1	no	—
485	C₄	—	268	1	no	—
486	C₄	—	268	1	no	—
487	C₄	—	134	2	no	—
488	C₄	#16863469	1	2	yes	—
489	C₄	#16863519	1	2	yes	—
490	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
491	C₄	C₂×#14524051	1	1	yes	characteristic
492	C₄	C₂×#14524065	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	268	2	no	—
495	C₂	—	536	1	no	—
496	C₂	C₂×#16863469	1	1	yes	characteristic
497	C₂	C₄×#14524065	1	1	yes	characteristic
498	C₂	C₂×#16863519	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#17863
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. #22569175	C₂	semidirect
2. C₈×#11761628	C₂	semidirect
4. #22559061	C₂	semidirect
5. #22559054	C₂	semidirect
6. #22559047	C₂	semidirect
7. #22559040	C₂	semidirect
8. #22558957	C₂	semidirect
9. #22558592	C₂	semidirect
10. #22558592	C₂	semidirect
11. C₂×#322×C₆₇	C₂	semidirect
117. #11761628	M₁₆	semidirect
217. C₈×C₆₇	#135	semidirect
260. C₂×#322	D₆₇	semidirect
261. C₆₇	#17863	semidirect
469. C₈	#14524065	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
C₈×#11761628	2
C₄×#14524065	2
#22559061	2
#22559054	2
#22559047	2
#22559040	2
#22558957	2
#22558592	2
C₂×#322×C₆₇	2
#17863	67