Subgroups — #30827577

Order 17152.

652 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569335	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
4	#22559115	C₂	1	2	yes	maximal, complemented
5	#22559048	C₂	1	2	yes	maximal, complemented
6	#22559042	C₂	1	2	yes	maximal, complemented
7	#22559030	C₂	1	2	yes	maximal, complemented
8	#22558964	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558649	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1489	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863527	C₂²	1	2	yes	—
15	#16863523	C₂²	1	2	yes	—
16	C₄×#11761618	C₄	1	1	yes	characteristic
17	C₄×#11761618	C₂²	1	1	yes	characteristic
18	C₄×#11761615	C₄	1	1	yes	characteristic
19	C₄×#11761615	C₂²	1	1	yes	characteristic
20	C₄×#11761616	C₄	1	1	yes	characteristic
21	C₄×#11761616	C₂²	1	1	yes	characteristic
22	C₄×#11761613	C₄	1	1	yes	characteristic
23	C₄×#11761613	C₂²	1	1	yes	characteristic
24	#16863468	—	2	2	no	—
25	#16863465	—	2	2	no	—
26	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
27	#16863461	C₂²	1	2	yes	—
28	#16863453	—	2	1	no	—
29	#16863452	—	2	1	no	—
30	#16863449	C₂²	1	1	yes	characteristic
31	#16863432	C₄	1	1	yes	characteristic
32	#16863420	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊C₈×D₆₇	—	2	1	no	—
35	#16863392	C₄	1	1	yes	characteristic
36	#16863390	C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#16863146	C₂²	1	2	yes	—
39	#16863146	C₂²	1	2	yes	—
40	#16863128	C₂²	1	2	yes	—
41	#16863125	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	—
43	#16863118	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
48	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	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	C₂×#11761612	—	2	1	no	—
56	C₂²×C₆₇⋊C₈	—	2	2	no	—
57	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₇	—	2	2	no	—
60	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
62	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
64	C₄×C₂²⋊D₆₇	—	2	1	no	—
65	C₄×C₂²⋊D₆₇	—	2	1	no	—
66	#14524002	C₂×C₄	1	1	yes	characteristic
67	#14524000	—	2	1	no	—
68	C₄²×D₆₇	—	2	1	no	—
69	#14523976	C₂×C₄	1	1	yes	characteristic
70	#14523976	C₂×C₄	1	1	yes	characteristic
71	#14523974	C₂×C₄	1	1	yes	characteristic
72	#14523974	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
74	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
75	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
77	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	#14523967	C₂×C₄	1	1	yes	characteristic
80	#14523967	C₂×C₄	1	1	yes	characteristic
81	#14523966	C₂×C₄	1	1	yes	characteristic
82	#14523966	C₂×C₄	1	1	yes	characteristic
83	#14523964	C₂×C₄	1	1	yes	characteristic
84	#14523964	C₂×C₄	1	1	yes	characteristic
85	#14523963	C₂×C₄	1	1	yes	characteristic
86	#14523963	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
88	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
89	(C₂×D₆₇)⋊C₈	—	2	2	no	—
90	#14523954	—	2	2	no	—
91	#14523953	—	2	2	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	C₂³	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
96	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
97	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂×M₁₆×C₆₇	—	2	1	no	—
99	C₂×M₁₆×C₆₇	—	2	1	no	—
100	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₄×Q₈×C₆₇	C₂³	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	C₂³	1	1	yes	characteristic
106	#117×C₆₇	—	2	1	no	—
107	#117×C₆₇	C₂×C₄	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	1	no	—
111	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
112	#97×C₆₇	—	2	1	no	—
113	C₄×C₈×C₆₇	C₂³	1	2	yes	—
114	#11761628	—	2	2	no	—
115	#11761628	—	4	1	no	—
116	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₂²⋊D₆₇	—	2	1	no	—
119	C₂×C₄×D₆₇	—	2	1	no	—
120	C₂×C₄×D₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	#11761618	—	2	2	no	—
123	#11761616	—	2	2	no	—
124	#11761615	—	2	2	no	—
125	#11761613	M₁₆	1	4	yes	complemented
126	#11761612	—	8	1	no	—
127	C₂×C₆₇⋊C₈	—	2	2	no	—
128	C₂×C₆₇⋊C₈	—	2	2	no	—
129	C₂×C₆₇⋊C₈	—	4	1	no	—
130	C₂×C₆₇⋊C₈	—	8	2	no	—
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
136	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
137	C₈×D₆₇	—	8	2	no	—
138	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
140	Dic₆₇⋊C₄	—	2	1	no	—
141	Dic₆₇⋊C₄	—	2	1	no	—
142	C₂²⋊Dic₆₇	—	2	1	no	—
143	(C₂×D₆₇)⋊C₄	—	2	1	no	—
144	(C₂×D₆₇)⋊C₄	—	2	1	no	—
145	C₄×Dic₆₇	—	2	1	no	—
146	C₄×Dic₆₇	—	2	1	no	—
147	C₄×Dic₆₇	—	2	1	no	—
148	#41×C₆₇	—	2	2	no	—
149	#41×C₆₇	—	4	1	no	—
150	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
159	M₁₆×C₆₇	—	4	2	no	—
160	M₁₆×C₆₇	—	8	1	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²⋊C₄×C₆₇	—	2	1	no	—
168	C₂²⋊C₄×C₆₇	—	2	1	no	—
169	C₂²⋊C₄×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	#41	1	1	yes	characteristic
176	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
179	C₂²×D₆₇	—	2	1	no	—
180	Q₈×C₆₇	—	2	1	no	—
181	Q₈×C₆₇	—	2	1	no	—
182	D₄×C₆₇	—	2	1	no	—
183	D₄×C₆₇	—	4	2	no	—
184	D₄×C₆₇	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₂²⋊D₆₇	—	4	2	no	—
204	C₂×Dic₆₇	—	2	1	no	—
205	C₂×Dic₆₇	—	2	1	no	—
206	C₂×Dic₆₇	—	2	1	no	—
207	D₂₆₈	—	2	1	no	—
208	D₂₆₈	C₂×M₁₆	1	2	yes	—
209	C₄×D₆₇	—	4	2	no	—
210	C₄×D₆₇	—	4	1	no	—
211	C₄×D₆₇	—	4	1	no	—
212	C₄×D₆₇	—	4	2	no	—
213	Dic₁₃₄	—	2	1	no	—
214	Dic₁₃₄	—	2	1	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₆₇⋊C₈	—	2	1	no	—
218	C₆₇⋊C₈	—	2	1	no	—
219	C₆₇⋊C₈	—	2	1	no	—
220	C₆₇⋊C₈	—	2	2	no	—
221	C₆₇⋊C₈	—	2	2	no	—
222	C₆₇⋊C₈	—	4	2	no	—
223	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	—	4	1	no	—
230	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂×D₆₇	—	2	2	no	—
232	C₂×D₆₇	—	4	1	no	—
233	C₂×D₆₇	—	4	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	—	4	1	no	—
239	C₄×C₆₇	—	4	1	no	—
240	C₄×C₆₇	#444	1	2	yes	—
241	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
242	C₄×C₆₇	#568	1	1	yes	characteristic
243	C₄×C₆₇	#577	1	1	yes	characteristic
244	C₄×C₆₇	#577	1	1	yes	characteristic
245	Dic₆₇	—	4	1	no	—
246	Dic₆₇	—	4	1	no	—
247	Dic₆₇	—	4	1	no	—
248	#18065	—	67	1	no	maximal, Sylow
249	C₂×C₆₇	—	4	1	no	—
250	C₂×C₆₇	—	4	2	no	—
251	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
252	C₂×C₆₇	#2952	1	1	yes	characteristic
253	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
254	D₆₇	—	4	2	no	—
255	#2990	—	67	1	no	—
256	#2990	—	67	1	no	—
257	C₄×#135	—	67	1	no	—
258	#1593	—	67	2	no	—
259	#1582	—	67	2	no	—
260	#1582	—	67	2	no	—
261	#1580	—	67	2	no	—
262	#1566	—	67	1	no	—
263	#1489	—	67	1	no	—
264	#1489	D₆₇	1	1	yes	characteristic, complemented
265	#1451	—	67	1	no	—
266	C₆₇	#18065	1	1	yes	characteristic, complemented, Sylow
267	C₂×#135	—	67	1	no	—
268	C₄×#41	—	67	1	no	—
269	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
270	#446	—	67	2	no	—
271	#446	—	67	2	no	—
272	C₄×Q₁₆	—	67	1	no	—
273	C₄×Q₁₆	—	67	1	no	—
274	C₄×SD₁₆	—	67	1	no	—
275	C₄×SD₁₆	—	67	1	no	—
276	C₄×SD₁₆	—	67	1	no	—
277	C₄×SD₁₆	—	67	1	no	—
278	C₄×D₈	—	67	1	no	—
279	C₄×D₈	—	67	1	no	—
280	#435	—	134	2	no	—
281	#435	—	134	2	no	—
282	C₈×D₄	—	67	2	no	—
283	C₈×D₄	—	67	2	no	—
284	#432	—	134	1	no	—
285	#432	—	134	1	no	—
286	#431	—	67	1	no	—
287	#427	—	67	1	no	—
288	#423	—	67	1	no	—
289	#423	C₂×D₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₈	—	134	1	no	—
291	C₂×C₄⋊C₈	—	134	1	no	—
292	#416	—	67	1	no	—
293	#416	—	67	1	no	—
294	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
295	C₂×C₄×C₈	—	67	1	no	—
296	#334	—	67	2	no	—
297	#334	—	67	2	no	—
298	Q₈⋊C₈	—	67	2	no	—
299	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
300	D₄⋊C₈	—	67	2	no	—
301	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
302	#322	—	67	2	no	—
303	#322	—	67	2	no	—
304	C₂×#41	—	67	1	no	—
305	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
306	#135	—	134	4	no	—
307	C₂×Q₁₆	—	67	1	no	—
308	C₂×SD₁₆	—	67	1	no	—
309	C₂×SD₁₆	—	67	1	no	—
310	C₂×D₈	—	67	1	no	—
311	C₂×M₁₆	—	2	1	no	—
312	C₂×M₁₆	—	2	1	no	—
313	C₂×M₁₆	—	134	1	no	—
314	C₂²×C₈	—	67	1	no	—
315	C₂²×C₈	—	67	1	no	—
316	C₂²×C₈	—	134	2	no	—
317	C₂²×C₈	—	134	2	no	—
318	C₄×Q₈	—	67	1	no	—
319	C₄×Q₈	—	67	1	no	—
320	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
321	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	67	1	no	—
325	C₄×D₄	—	67	1	no	—
326	C₄×D₄	—	134	1	no	—
327	C₄×D₄	—	134	1	no	—
328	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
329	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	—	67	1	no	—
332	#117	—	134	1	no	—
333	#117	C₄×D₆₇	1	1	yes	characteristic
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	134	1	no	—
336	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
337	#107	—	67	1	no	—
338	#107	—	67	1	no	—
339	#106	—	67	1	no	—
340	#106	—	67	1	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	67	1	no	—
343	C₄⋊C₈	—	67	1	no	—
344	C₄⋊C₈	—	67	2	no	—
345	C₄⋊C₈	—	134	2	no	—
346	C₄⋊C₈	—	134	2	no	—
347	C₄⋊C₈	—	134	1	no	—
348	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
349	#103	—	67	1	no	—
350	#103	—	67	1	no	—
351	#103	—	67	1	no	—
352	#103	—	67	1	no	—
353	#102	—	67	1	no	—
354	#102	—	67	1	no	—
355	#102	—	67	1	no	—
356	#102	—	67	1	no	—
357	C₂²⋊C₈	—	67	2	no	—
358	C₂²⋊C₈	—	134	2	no	—
359	C₂²⋊C₈	—	134	2	no	—
360	#97	—	2	1	no	—
361	#97	—	134	2	no	—
362	#97	—	134	2	no	—
363	C₄×C₈	—	67	1	no	—
364	C₄×C₈	—	67	1	no	—
365	C₄×C₈	—	67	1	no	—
366	C₄×C₈	—	67	1	no	—
367	C₄×C₈	—	67	2	no	—
368	C₄×C₈	C₂²×D₆₇	1	2	yes	—
369	#41	—	2	2	no	—
370	#41	—	4	1	no	—
371	#41	—	268	1	no	—
372	#41	—	134	2	no	—
373	C₂×Q₈	—	67	1	no	—
374	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	67	1	no	—
377	C₂×D₄	—	134	1	no	—
378	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	134	1	no	—
383	C₂²×C₄	—	134	1	no	—
384	C₂²×C₄	—	134	1	no	—
385	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
386	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
387	C₂²×C₄	#11761627	1	1	yes	characteristic
388	Q₁₆	—	134	2	no	—
389	SD₁₆	—	134	2	no	—
390	SD₁₆	—	134	2	no	—
391	D₈	—	67	4	no	—
392	M₁₆	—	4	2	no	—
393	M₁₆	—	8	1	no	—
394	M₁₆	—	536	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	268	1	no	—
398	C₂×C₈	—	536	2	no	—
399	C₂×C₈	—	536	2	no	—
400	C₂×C₈	—	67	1	no	—
401	C₂×C₈	—	67	1	no	—
402	C₂×C₈	—	67	1	no	—
403	C₂×C₈	—	67	1	no	—
404	C₂×C₈	—	67	2	no	—
405	C₂×C₈	—	67	2	no	—
406	C₂×C₈	—	134	2	no	—
407	C₂×C₈	—	134	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	67	1	no	—
411	C₄⋊C₄	—	67	1	no	—
412	C₄⋊C₄	—	134	1	no	—
413	C₄⋊C₄	—	134	1	no	—
414	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₂²⋊C₄	—	134	1	no	—
420	C₂²⋊C₄	—	134	1	no	—
421	C₂²⋊C₄	—	134	1	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	134	1	no	—
426	C₄²	—	134	1	no	—
427	C₄²	—	134	1	no	—
428	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
429	C₄²	D₄×D₆₇	1	1	yes	characteristic
430	C₄²	#11761627	1	1	yes	characteristic
431	C₄²	C₂³×D₆₇	1	1	yes	characteristic
432	C₂³	—	2	1	no	—
433	C₂³	—	134	1	no	—
434	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
435	Q₈	—	2	1	no	—
436	Q₈	—	2	1	no	—
437	Q₈	—	134	1	no	—
438	Q₈	—	134	1	no	—
439	D₄	—	2	1	no	—
440	D₄	—	4	2	no	—
441	D₄	—	268	2	no	—
442	D₄	—	67	2	no	—
443	D₄	—	134	1	no	—
444	D₄	M₁₆×D₆₇	1	2	yes	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	268	2	no	—
458	C₂×C₄	—	268	2	no	—
459	C₂×C₄	—	268	1	no	—
460	C₂×C₄	—	268	1	no	—
461	C₂×C₄	—	134	1	no	—
462	C₂×C₄	—	134	1	no	—
463	C₂×C₄	—	134	1	no	—
464	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
465	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
466	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
468	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
470	C₈	—	4	2	no	—
471	C₈	—	4	2	no	—
472	C₈	—	268	2	no	—
473	C₈	—	67	2	no	—
474	C₈	—	134	1	no	—
475	C₈	—	134	1	no	—
476	C₈	—	134	1	no	—
477	C₈	—	134	2	no	—
478	C₈	—	134	2	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	268	1	no	—
486	C₂²	—	268	1	no	—
487	C₂²	—	134	2	no	—
488	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	268	1	no	—
496	C₄	—	268	1	no	—
497	C₄	—	268	1	no	—
498	C₄	#16863515	1	2	yes	—
499	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
500	C₄	#16864162	1	1	yes	characteristic
501	C₄	#16864205	1	1	yes	characteristic
502	C₄	#16864206	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	4	2	no	—
505	C₂	—	268	2	no	—
506	C₂	#22568941	1	1	yes	characteristic
507	C₂	#22569088	1	1	yes	characteristic
508	C₂	C₂×#16863515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1489	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1489	D₆₇
hypocenter	C₆₇	#18065
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1489

Complements

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

N	H ≅ G/N	type
1. #22569335	C₂	semidirect
2. #22569334	C₂	semidirect
4. #22559115	C₂	semidirect
5. #22559048	C₂	semidirect
6. #22559042	C₂	semidirect
7. #22559030	C₂	semidirect
8. #22558964	C₂	semidirect
9. #22558649	C₂	semidirect
10. #22558496	C₂	semidirect
11. C₆₇×#1489	C₂	semidirect
125. #11761613	M₁₆	semidirect
264. #1489	D₆₇	semidirect
266. C₆₇	#18065	semidirect

Maximal subgroups

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

subgroup	index
#22569335	2
#22569334	2
C₄×#14524063	2
#22559115	2
#22559048	2
#22559042	2
#22559030	2
#22558964	2
#22558649	2
#22558496	2
C₆₇×#1489	2
#18065	67