Subgroups — #30825542

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569322	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569317	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
4	#22559089	C₂	1	2	yes	maximal, complemented
5	#22559068	C₂	1	2	yes	maximal, complemented
6	#22559061	C₂	1	2	yes	maximal, complemented
7	#22559040	C₂	1	2	yes	maximal, complemented
8	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558597	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558597	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1452	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	#16863464	—	2	2	no	—
27	C₈×D₂₆₈	C₂²	1	2	yes	—
28	(C₂²⋊D₆₇)⋊C₈	—	2	2	no	—
29	#16863452	—	2	1	no	—
30	#16863448	—	2	1	no	—
31	#16863434	C₄	1	1	yes	characteristic
32	#16863418	—	2	1	no	—
33	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×D₆₇	—	2	1	no	—
36	#16863393	C₄	1	1	yes	characteristic
37	#16863393	C₄	1	1	yes	characteristic
38	#16863149	C₂²	1	2	yes	—
39	#16863149	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	#431×C₆₇	C₂²	1	1	yes	characteristic
45	#431×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	C₂²×C₆₇⋊C₈	—	2	1	no	—
58	C₂²×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₈×D₆₇	—	2	2	no	—
60	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
64	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	C₄×C₂²⋊D₆₇	—	2	1	no	—
71	C₄×C₂²⋊D₆₇	—	2	1	no	—
72	#14524002	C₂×C₄	1	1	yes	characteristic
73	#14524002	C₂×C₄	1	1	yes	characteristic
74	#14524000	—	2	1	no	—
75	#14524000	—	2	1	no	—
76	C₄²×D₆₇	—	2	1	no	—
77	C₄²×D₆₇	—	2	1	no	—
78	#14523977	C₂×C₄	1	1	yes	characteristic
79	#14523977	C₂×C₄	1	1	yes	characteristic
80	#14523975	C₂×C₄	1	1	yes	characteristic
81	#14523975	C₂×C₄	1	1	yes	characteristic
82	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
83	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
84	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
85	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
86	Dic₆₇⋊C₈	—	2	2	no	—
87	Dic₆₇⋊C₈	—	2	2	no	—
88	#14523968	C₂×C₄	1	1	yes	characteristic
89	#14523968	C₂×C₄	1	1	yes	characteristic
90	#14523968	C₂×C₄	1	1	yes	characteristic
91	#14523968	C₂×C₄	1	1	yes	characteristic
92	#14523965	C₂×C₄	1	1	yes	characteristic
93	#14523965	C₂×C₄	1	1	yes	characteristic
94	#14523965	C₂×C₄	1	1	yes	characteristic
95	#14523965	C₂×C₄	1	1	yes	characteristic
96	(C₂×D₆₇)⋊C₈	—	2	2	no	—
97	(C₂×D₆₇)⋊C₈	—	2	2	no	—
98	#14523953	—	2	2	no	—
99	C₈×Dic₆₇	—	2	2	no	—
100	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₆₇	—	2	1	no	—
104	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
106	C₂²⋊C₈×C₆₇	—	2	1	no	—
107	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
108	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈×C₆₇	C₂×C₄	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	—	8	2	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	8	1	no	—
138	C₂×C₆₇⋊C₈	—	8	1	no	—
139	C₈×D₆₇	—	8	2	no	—
140	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
144	Dic₆₇⋊C₄	—	2	1	no	—
145	Dic₆₇⋊C₄	—	2	1	no	—
146	Dic₆₇⋊C₄	—	2	1	no	—
147	Dic₆₇⋊C₄	—	2	1	no	—
148	C₂²⋊Dic₆₇	—	2	1	no	—
149	C₂²⋊Dic₆₇	—	2	1	no	—
150	(C₂×D₆₇)⋊C₄	—	2	1	no	—
151	(C₂×D₆₇)⋊C₄	—	2	1	no	—
152	(C₂×D₆₇)⋊C₄	—	2	1	no	—
153	(C₂×D₆₇)⋊C₄	—	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	C₄×Dic₆₇	—	2	1	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	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	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
186	C₂×C₄×C₆₇	#131	1	2	yes	—
187	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂²⋊D₆₇	—	2	4	no	—
191	C₂²⋊D₆₇	—	4	2	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	C₂×Dic₆₇	—	2	1	no	—
198	D₂₆₈	—	2	1	no	—
199	D₂₆₈	—	2	1	no	—
200	D₂₆₈	—	2	1	no	—
201	D₂₆₈	C₂×M₁₆	1	2	yes	—
202	C₄×D₆₇	—	2	4	no	—
203	C₄×D₆₇	—	4	1	no	—
204	C₄×D₆₇	—	4	1	no	—
205	C₄×D₆₇	—	4	1	no	—
206	C₄×D₆₇	—	4	1	no	—
207	C₄×D₆₇	—	4	2	no	—
208	C₄×D₆₇	—	8	1	no	—
209	C₄×D₆₇	—	8	1	no	—
210	Dic₁₃₄	—	2	1	no	—
211	Dic₁₃₄	—	2	1	no	—
212	Dic₁₃₄	—	2	1	no	—
213	Dic₁₃₄	C₂×M₁₆	1	2	yes	—
214	C₈×C₆₇	—	2	1	no	—
215	C₈×C₆₇	—	2	1	no	—
216	C₈×C₆₇	—	2	1	no	—
217	C₈×C₆₇	—	2	1	no	—
218	C₈×C₆₇	—	2	2	no	—
219	C₈×C₆₇	—	2	2	no	—
220	C₈×C₆₇	—	4	2	no	—
221	C₆₇⋊C₈	—	4	2	no	—
222	C₆₇⋊C₈	—	4	2	no	—
223	C₂²×C₆₇	—	2	1	no	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	#435	1	2	yes	—
226	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂×D₆₇	—	2	2	no	—
228	C₂×D₆₇	—	4	1	no	—
229	C₂×D₆₇	—	4	1	no	—
230	C₂×D₆₇	—	4	1	no	—
231	C₂×D₆₇	—	4	1	no	—
232	C₂×D₆₇	—	4	1	no	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	#435	1	2	yes	—
237	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
238	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
239	C₄×C₆₇	#576	1	1	yes	characteristic
240	C₄×C₆₇	#578	1	1	yes	characteristic
241	Dic₆₇	—	2	2	no	—
242	Dic₆₇	—	4	1	no	—
243	Dic₆₇	—	4	1	no	—
244	Dic₆₇	—	4	1	no	—
245	Dic₆₇	—	4	1	no	—
246	Dic₆₇	—	4	1	no	—
247	#17881	—	67	1	no	maximal, Sylow
248	C₂×C₆₇	—	2	2	no	—
249	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
250	C₂×C₆₇	#2951	1	1	yes	characteristic
251	C₂×C₆₇	#2957	1	1	yes	characteristic
252	D₆₇	—	4	2	no	—
253	D₆₇	—	8	1	no	—
254	#2989	—	67	1	no	—
255	#2988	—	67	1	no	—
256	C₄×#135	—	67	1	no	—
257	#1588	—	67	2	no	—
258	#1585	—	67	2	no	—
259	#1584	—	67	2	no	—
260	#1581	—	67	2	no	—
261	C₂×#335	—	67	1	no	—
262	#1477	—	67	1	no	—
263	#1477	—	67	1	no	—
264	#1452	D₆₇	1	1	yes	characteristic, complemented
265	C₆₇	#17881	1	1	yes	characteristic, complemented, Sylow
266	C₂×#135	—	67	1	no	—
267	C₄×#41	—	67	1	no	—
268	C₄×#41	—	67	1	no	—
269	#446	—	67	2	no	—
270	C₈×Q₈	—	67	2	no	—
271	C₄×Q₁₆	—	67	1	no	—
272	C₄×Q₁₆	—	67	1	no	—
273	C₄×SD₁₆	—	67	1	no	—
274	C₄×SD₁₆	—	67	1	no	—
275	C₄×SD₁₆	—	67	1	no	—
276	C₄×SD₁₆	—	67	1	no	—
277	C₄×D₈	—	67	1	no	—
278	C₄×D₈	—	67	1	no	—
279	#436	—	67	2	no	—
280	#435	—	134	2	no	—
281	C₈×D₄	—	67	2	no	—
282	C₈×D₄	—	134	2	no	—
283	#432	—	134	1	no	—
284	#431	—	134	1	no	—
285	#431	C₂×D₆₇	1	1	yes	characteristic
286	#431	C₂×D₆₇	1	1	yes	characteristic
287	#427	—	67	1	no	—
288	#423	—	134	1	no	—
289	C₂×C₄⋊C₈	—	67	1	no	—
290	C₂×C₄⋊C₈	—	67	1	no	—
291	C₂×C₄⋊C₈	—	134	1	no	—
292	#416	—	67	1	no	—
293	#416	—	67	1	no	—
294	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
295	#335	—	67	2	no	—
296	#335	—	67	2	no	—
297	Q₈⋊C₈	—	67	2	no	—
298	Q₈⋊C₈	—	67	2	no	—
299	D₄⋊C₈	—	67	2	no	—
300	D₄⋊C₈	—	67	2	no	—
301	#322	C₂×D₆₇	1	2	yes	—
302	#322	C₂×D₆₇	1	2	yes	—
303	C₂×#41	—	67	1	no	—
304	C₂×#41	—	67	1	no	—
305	#135	—	134	4	no	—
306	C₂×Q₁₆	—	67	1	no	—
307	C₂×SD₁₆	—	67	1	no	—
308	C₂×SD₁₆	—	67	1	no	—
309	C₂×D₈	—	67	1	no	—
310	C₂×M₁₆	—	134	2	no	—
311	C₂²×C₈	—	134	1	no	—
312	C₂²×C₈	—	134	1	no	—
313	C₂²×C₈	—	134	2	no	—
314	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
315	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
316	C₄×Q₈	—	67	1	no	—
317	C₄×Q₈	—	67	1	no	—
318	C₄×Q₈	—	67	1	no	—
319	C₄×Q₈	—	67	1	no	—
320	C₄×D₄	—	67	1	no	—
321	C₄×D₄	—	67	1	no	—
322	C₄×D₄	—	67	1	no	—
323	C₄×D₄	—	67	1	no	—
324	C₄×D₄	—	134	1	no	—
325	C₄×D₄	—	134	1	no	—
326	C₄×D₄	—	134	1	no	—
327	C₄×D₄	—	134	1	no	—
328	#117	—	67	1	no	—
329	#117	—	67	1	no	—
330	#117	—	134	1	no	—
331	#117	—	134	1	no	—
332	C₂×C₄²	—	134	1	no	—
333	C₂×C₄²	—	134	1	no	—
334	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
335	#107	—	67	1	no	—
336	#107	—	67	1	no	—
337	#106	—	67	1	no	—
338	#106	—	67	1	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	67	2	no	—
341	C₄⋊C₈	—	67	2	no	—
342	C₄⋊C₈	—	134	2	no	—
343	C₄⋊C₈	—	134	2	no	—
344	C₄⋊C₈	—	134	1	no	—
345	C₄⋊C₈	—	134	1	no	—
346	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
347	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
348	#103	—	67	1	no	—
349	#103	—	67	1	no	—
350	#103	—	67	1	no	—
351	#103	—	67	1	no	—
352	#102	—	67	1	no	—
353	#102	—	67	1	no	—
354	#102	—	67	1	no	—
355	#102	—	67	1	no	—
356	C₂²⋊C₈	—	2	1	no	—
357	C₂²⋊C₈	—	134	2	no	—
358	C₂²⋊C₈	—	134	2	no	—
359	C₂²⋊C₈	D₂₆₈	1	2	yes	—
360	#97	—	134	2	no	—
361	C₄×C₈	—	134	2	no	—
362	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
363	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
364	C₄×C₈	C₂²×D₆₇	1	2	yes	—
365	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
366	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
367	C₄×C₈	C₂²×D₆₇	1	2	yes	—
368	#41	—	67	4	no	—
369	#41	—	134	2	no	—
370	#41	—	134	2	no	—
371	#41	—	134	2	no	—
372	C₂×Q₈	—	67	1	no	—
373	C₂×Q₈	—	67	1	no	—
374	C₂×D₄	—	67	1	no	—
375	C₂×D₄	—	67	1	no	—
376	C₂×D₄	—	134	1	no	—
377	C₂×D₄	—	134	1	no	—
378	C₂²×C₄	—	134	1	no	—
379	C₂²×C₄	—	134	1	no	—
380	C₂²×C₄	—	134	1	no	—
381	C₂²×C₄	—	134	1	no	—
382	C₂²×C₄	—	134	1	no	—
383	C₂²×C₄	—	134	1	no	—
384	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
386	C₂²×C₄	#11761628	1	1	yes	characteristic
387	Q₁₆	—	134	2	no	—
388	SD₁₆	—	134	2	no	—
389	SD₁₆	—	134	2	no	—
390	D₈	—	134	2	no	—
391	M₁₆	—	536	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	536	2	no	—
396	C₂×C₈	—	536	1	no	—
397	C₂×C₈	—	536	1	no	—
398	C₂×C₈	—	134	2	no	—
399	C₂×C₈	—	134	2	no	—
400	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
401	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
402	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
403	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
404	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
405	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
406	C₄⋊C₄	—	67	1	no	—
407	C₄⋊C₄	—	67	1	no	—
408	C₄⋊C₄	—	67	1	no	—
409	C₄⋊C₄	—	67	1	no	—
410	C₄⋊C₄	—	134	1	no	—
411	C₄⋊C₄	—	134	1	no	—
412	C₄⋊C₄	—	134	1	no	—
413	C₄⋊C₄	—	134	1	no	—
414	C₂²⋊C₄	—	134	1	no	—
415	C₂²⋊C₄	—	134	1	no	—
416	C₂²⋊C₄	—	134	1	no	—
417	C₂²⋊C₄	—	134	1	no	—
418	C₂²⋊C₄	—	134	1	no	—
419	C₂²⋊C₄	—	134	1	no	—
420	C₄²	—	134	1	no	—
421	C₄²	—	134	1	no	—
422	C₄²	—	134	1	no	—
423	C₄²	—	134	1	no	—
424	C₄²	—	134	1	no	—
425	C₄²	—	134	1	no	—
426	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
427	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
428	C₄²	#11761628	1	1	yes	characteristic
429	C₄²	C₂³×D₆₇	1	1	yes	characteristic
430	C₂³	—	134	1	no	—
431	C₂³	—	134	1	no	—
432	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
433	Q₈	—	67	2	no	—
434	Q₈	—	134	1	no	—
435	Q₈	—	134	1	no	—
436	Q₈	—	134	1	no	—
437	D₄	—	268	2	no	—
438	D₄	—	67	2	no	—
439	D₄	—	134	4	no	—
440	D₄	—	134	1	no	—
441	D₄	—	134	1	no	—
442	D₄	—	134	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	268	1	no	—
447	C₂×C₄	—	268	1	no	—
448	C₂×C₄	—	268	1	no	—
449	C₂×C₄	—	268	1	no	—
450	C₂×C₄	—	268	2	no	—
451	C₂×C₄	—	536	1	no	—
452	C₂×C₄	—	536	1	no	—
453	C₂×C₄	—	134	1	no	—
454	C₂×C₄	—	134	1	no	—
455	C₂×C₄	—	134	1	no	—
456	C₂×C₄	—	134	1	no	—
457	C₂×C₄	—	134	1	no	—
458	C₂×C₄	—	134	1	no	—
459	C₂×C₄	—	134	4	no	—
460	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
461	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
462	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
463	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
464	C₂×C₄	#14524051	1	2	yes	—
465	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
467	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
468	C₈	—	2	1	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	4	2	no	—
475	C₈	—	268	2	no	—
476	C₈	—	268	2	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	268	1	no	—
480	C₂²	—	268	1	no	—
481	C₂²	—	268	1	no	—
482	C₂²	—	268	1	no	—
483	C₂²	—	268	1	no	—
484	C₂²	—	134	2	no	—
485	C₂²	#16863469	1	2	yes	—
486	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	268	1	no	—
491	C₄	—	268	1	no	—
492	C₄	—	268	1	no	—
493	C₄	—	268	1	no	—
494	C₄	—	268	1	no	—
495	C₄	—	134	2	no	—
496	C₄	#16863469	1	2	yes	—
497	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
498	C₄	C₂×#14524051	1	1	yes	characteristic
499	C₄	#16864201	1	1	yes	characteristic
500	C₄	#16864213	1	1	yes	characteristic
501	C₂	—	2	2	no	—
502	C₂	—	268	2	no	—
503	C₂	—	536	1	no	—
504	C₂	C₂×#16863469	1	1	yes	characteristic
505	C₂	#22569086	1	1	yes	characteristic
506	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#17881
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
1. #22569322	C₂	semidirect
2. #22569317	C₂	semidirect
4. #22559089	C₂	semidirect
5. #22559068	C₂	semidirect
6. #22559061	C₂	semidirect
7. #22559040	C₂	semidirect
8. C₂×#16863149	C₂	semidirect
9. #22558597	C₂	semidirect
10. #22558597	C₂	semidirect
11. C₆₇×#1452	C₂	semidirect
117. #11761628	M₁₆	semidirect
264. #1452	D₆₇	semidirect
265. C₆₇	#17881	semidirect

Maximal subgroups

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

subgroup	index
#22569322	2
#22569317	2
C₄×#14524065	2
#22559089	2
#22559068	2
#22559061	2
#22559040	2
C₂×#16863149	2
#22558597	2
C₆₇×#1452	2
#17881	67