Subgroups — #30752014

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561016	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863140	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558093	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₆₇⋊C₈	—	2	1	no	—
9	D₄×C₆₇⋊C₈	—	2	2	no	—
10	D₄×C₆₇⋊C₈	—	2	2	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14523964	—	2	1	no	—
14	C₂×#14523964	—	2	1	no	—
15	C₂×#14523964	—	2	1	no	—
16	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863266	C₂²	1	1	yes	characteristic
19	#16863158	—	2	1	no	—
20	#16863140	—	2	2	no	—
21	C₂²×D₈×C₆₇	—	2	1	no	—
22	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	#459×C₆₇	—	2	2	no	—
25	#459×C₆₇	C₄	1	4	yes	complemented
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	—	2	4	no	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	C₂²×C₆₇⋊C₈	—	4	2	no	—
32	C₂²×C₆₇⋊C₈	—	4	2	no	—
33	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	—	2	1	no	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
41	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
42	#14523964	—	4	2	no	—
43	#14523964	—	4	2	no	—
44	#14523964	—	4	2	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
46	C₄×C₆₇⋊C₈	—	2	1	no	—
47	C₄×C₆₇⋊C₈	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	4	2	no	—
55	C₂×D₈×C₆₇	—	4	2	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	#127×C₆₇	—	2	1	no	—
58	#127×C₆₇	—	2	1	no	—
59	#127×C₆₇	—	2	1	no	—
60	#127×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×D₄×C₆₇	—	2	1	no	—
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₄×D₄×C₆₇	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
70	#102×C₆₇	—	2	4	no	—
71	C₂²×Dic₆₇	—	4	1	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	2	1	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	1	no	—
77	C₂×C₆₇⋊C₈	—	2	1	no	—
78	C₂×C₆₇⋊C₈	—	4	8	no	—
79	C₂×C₆₇⋊C₈	—	4	1	no	—
80	C₂×C₆₇⋊C₈	—	4	1	no	—
81	C₂×C₆₇⋊C₈	—	4	1	no	—
82	C₂×C₆₇⋊C₈	—	4	1	no	—
83	C₂×C₆₇⋊C₈	—	4	1	no	—
84	C₂×C₆₇⋊C₈	D₈	1	2	yes	complemented
85	C₄⋊Dic₆₇	—	4	1	no	—
86	C₄⋊Dic₆₇	—	4	1	no	—
87	C₄⋊Dic₆₇	—	4	1	no	—
88	C₄⋊Dic₆₇	—	4	1	no	—
89	C₄⋊Dic₆₇	—	8	1	no	—
90	C₂⁴×C₆₇	—	2	1	no	—
91	C₂⁴×C₆₇	—	4	1	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	4	1	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	4	1	no	—
104	C₂×D₄×C₆₇	—	4	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	4	1	no	—
109	C₂×D₄×C₆₇	—	8	1	no	—
110	C₂×D₄×C₆₇	—	8	1	no	—
111	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
112	C₂×D₄×C₆₇	D₈	1	2	yes	—
113	C₂×D₄×C₆₇	SD₁₆	1	2	yes	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	4	1	no	—
118	C₂²×C₄×C₆₇	—	4	1	no	—
119	C₂²×C₄×C₆₇	—	4	1	no	—
120	C₂²×C₄×C₆₇	—	4	1	no	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
124	D₈×C₆₇	—	4	8	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	4	1	no	—
127	C₂×C₈×C₆₇	—	4	1	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	#41	1	1	yes	characteristic
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	4	1	no	—
140	C₂³×C₆₇	—	4	1	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	1	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	8	1	no	—
145	C₂³×C₆₇	—	8	1	no	—
146	C₂³×C₆₇	—	8	1	no	—
147	C₂³×C₆₇	—	8	1	no	—
148	C₂³×C₆₇	—	8	1	no	—
149	C₂³×C₆₇	—	8	1	no	—
150	C₂³×C₆₇	#121	1	1	yes	characteristic
151	D₄×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	2	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	8	2	no	—
159	D₄×C₆₇	—	8	2	no	—
160	D₄×C₆₇	—	8	1	no	—
161	D₄×C₆₇	—	8	1	no	—
162	D₄×C₆₇	—	8	1	no	—
163	D₄×C₆₇	—	8	1	no	—
164	D₄×C₆₇	#102	1	4	yes	complemented
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
178	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
184	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
185	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
186	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×Dic₆₇	—	8	1	no	—
189	C₂×Dic₆₇	—	8	1	no	—
190	C₂×Dic₆₇	—	8	1	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₆₇⋊C₈	—	2	2	no	—
193	C₆₇⋊C₈	—	4	2	no	—
194	C₆₇⋊C₈	—	4	1	no	—
195	C₆₇⋊C₈	—	8	1	no	—
196	C₂²×C₆₇	—	2	4	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	8	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	—	16	1	no	—
207	C₂²×C₆₇	—	16	1	no	—
208	C₂²×C₆₇	—	16	1	no	—
209	C₂²×C₆₇	—	16	1	no	—
210	C₂²×C₆₇	#386	1	1	yes	characteristic
211	C₂²×C₆₇	#459	1	1	yes	characteristic
212	C₂²×C₆₇	#463	1	1	yes	characteristic
213	C₂²×C₆₇	#465	1	1	yes	characteristic
214	C₂²×C₆₇	#466	1	1	yes	characteristic
215	C₂²×C₆₇	#471	1	1	yes	characteristic
216	C₂²×C₆₇	#472	1	1	yes	characteristic
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
221	C₄×C₆₇	#416	1	1	yes	characteristic
222	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
223	C₄×C₆₇	#442	1	1	yes	characteristic
224	C₄×C₆₇	#447	1	1	yes	characteristic
225	C₄×C₆₇	#449	1	1	yes	characteristic
226	C₄×C₆₇	#465	1	1	yes	characteristic
227	C₄×C₆₇	#466	1	1	yes	characteristic
228	Dic₆₇	—	16	1	no	—
229	#9221	—	67	1	no	maximal, Sylow
230	C₂×C₆₇	—	4	4	no	—
231	C₂×C₆₇	—	16	2	no	—
232	C₂×C₆₇	#1665	1	1	yes	characteristic
233	C₂×C₆₇	#1667	1	1	yes	characteristic
234	C₂×C₆₇	#1677	1	1	yes	characteristic
235	C₂×C₆₇	#1681	1	1	yes	characteristic
236	C₂×C₆₇	#1879	1	1	yes	characteristic
237	C₂×C₆₇	#1893	1	1	yes	characteristic
238	C₂×C₆₇	#1907	1	1	yes	characteristic
239	C₂×#459	D₆₇	1	1	yes	characteristic
240	C₂×C₈×D₄	—	67	1	no	—
241	#1957	—	67	1	no	—
242	#1802	—	67	1	no	—
243	C₂×#331	—	67	1	no	—
244	#1296	—	67	1	no	—
245	#1277	—	67	1	no	—
246	C₆₇	#9221	1	1	yes	characteristic, complemented, Sylow
247	C₂²×D₈	—	2	1	no	—
248	C₂³×C₈	—	134	1	no	—
249	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
250	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
251	#459	—	2	2	no	—
252	#459	Dic₆₇	1	4	yes	complemented
253	C₈×D₄	—	134	2	no	—
254	C₈×D₄	—	134	2	no	—
255	C₂×C₄⋊C₈	—	67	1	no	—
256	C₂×C₄⋊C₈	—	67	1	no	—
257	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
258	C₂×#102	—	2	1	no	—
259	C₂×#102	—	134	1	no	—
260	C₂×#102	—	134	1	no	—
261	C₂×#102	—	134	1	no	—
262	C₂×C₂²⋊C₈	—	134	1	no	—
263	C₂×C₄×C₈	—	67	1	no	—
264	#384	—	67	1	no	—
265	#340	—	134	1	no	—
266	#331	—	134	2	no	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
271	C₂³×C₄	—	2	1	no	—
272	C₂×D₈	—	2	4	no	—
273	C₂×D₈	—	4	2	no	—
274	C₂×D₈	—	4	2	no	—
275	C₂²×C₈	—	2	1	no	—
276	C₂²×C₈	—	268	2	no	—
277	C₂²×C₈	—	268	2	no	—
278	C₂²×C₈	—	67	1	no	—
279	C₂²×C₈	—	67	1	no	—
280	C₂²×C₈	—	134	1	no	—
281	C₂²×C₈	—	134	4	no	—
282	C₂²×C₈	—	134	1	no	—
283	#127	—	2	1	no	—
284	#127	—	2	1	no	—
285	#127	—	2	1	no	—
286	#127	C₂×Dic₆₇	1	2	yes	—
287	C₄×D₄	—	2	1	no	—
288	C₄×D₄	—	2	1	no	—
289	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
290	C₄×D₄	C₂²⋊D₆₇	1	2	yes	—
291	C₂×C₄⋊C₄	—	134	1	no	—
292	C₂×C₄⋊C₄	—	134	1	no	—
293	C₂×C₄⋊C₄	—	134	1	no	—
294	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	—	67	2	no	—
299	C₄⋊C₈	—	134	1	no	—
300	C₄⋊C₈	—	134	1	no	—
301	C₄⋊C₈	—	134	1	no	—
302	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
303	#102	—	2	4	no	—
304	#102	—	268	2	no	—
305	#102	—	268	2	no	—
306	#102	—	268	2	no	—
307	C₂²⋊C₈	—	134	4	no	—
308	C₄×C₈	—	134	1	no	—
309	C₄×C₈	—	134	1	no	—
310	C₂⁴	—	2	1	no	—
311	C₂⁴	—	4	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	1	no	—
322	C₂×D₄	—	4	1	no	—
323	C₂×D₄	—	4	1	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂×D₄	—	8	1	no	—
330	C₂×D₄	—	8	1	no	—
331	C₂×D₄	C₂²⋊Dic₆₇	1	2	yes	—
332	C₂×D₄	#11761613	1	2	yes	—
333	C₂×D₄	#11761616	1	2	yes	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	1	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	268	1	no	—
342	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
344	C₂²×C₄	#11761627	1	1	yes	characteristic
345	D₈	—	4	8	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	4	1	no	—
348	C₂×C₈	—	4	1	no	—
349	C₂×C₈	—	268	1	no	—
350	C₂×C₈	—	268	1	no	—
351	C₂×C₈	—	268	8	no	—
352	C₂×C₈	—	268	1	no	—
353	C₂×C₈	—	268	1	no	—
354	C₂×C₈	—	268	1	no	—
355	C₂×C₈	—	67	2	no	—
356	C₂×C₈	—	134	1	no	—
357	C₂×C₈	—	134	2	no	—
358	C₂×C₈	—	134	1	no	—
359	C₂×C₈	—	134	1	no	—
360	C₂×C₈	—	134	1	no	—
361	C₂×C₈	—	134	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	268	1	no	—
364	C₄⋊C₄	—	268	1	no	—
365	C₄⋊C₄	—	268	1	no	—
366	C₄⋊C₄	—	268	1	no	—
367	C₄⋊C₄	—	536	1	no	—
368	C₄⋊C₄	C₂²⋊Dic₆₇	1	2	yes	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	2	2	no	—
371	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
372	C₄²	D₄×D₆₇	1	1	yes	characteristic
373	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
374	C₄²	#11761627	1	1	yes	characteristic
375	C₂³	—	2	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	1	no	—
380	C₂³	—	4	1	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	8	1	no	—
384	C₂³	—	8	1	no	—
385	C₂³	—	8	1	no	—
386	C₂³	—	8	1	no	—
387	C₂³	—	8	1	no	—
388	C₂³	—	8	1	no	—
389	C₂³	#14524015	1	1	yes	characteristic
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	2	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	8	2	no	—
399	D₄	—	8	1	no	—
400	D₄	—	8	1	no	—
401	D₄	—	8	1	no	—
402	D₄	—	8	1	no	—
403	D₄	#14523964	1	4	yes	complemented
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	536	1	no	—
414	C₂×C₄	—	536	1	no	—
415	C₂×C₄	—	536	1	no	—
416	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
417	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
418	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
419	C₂×C₄	#14524011	1	1	yes	characteristic
420	C₂×C₄	#14524015	1	1	yes	characteristic
421	C₂×C₄	#14524024	1	1	yes	characteristic
422	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
424	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
425	C₂×C₄	#14524062	1	1	yes	characteristic
426	C₂×C₄	#14524063	1	1	yes	characteristic
427	C₂×C₄	#14524063	1	1	yes	characteristic
428	C₂×C₄	#14524067	1	1	yes	characteristic
429	C₂×C₄	#14524067	1	1	yes	characteristic
430	C₈	—	4	2	no	—
431	C₈	—	268	2	no	—
432	C₈	—	268	1	no	—
433	C₈	—	536	1	no	—
434	C₈	—	134	2	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	16	1	no	—
446	C₂²	—	16	1	no	—
447	C₂²	—	16	1	no	—
448	C₂²	—	16	1	no	—
449	C₂²	#16863276	1	1	yes	characteristic
450	C₂²	#16863589	1	1	yes	characteristic
451	C₂²	#16863609	1	1	yes	characteristic
452	C₂²	#16863623	1	1	yes	characteristic
453	C₂²	#16863630	1	1	yes	characteristic
454	C₂²	#16863657	1	1	yes	characteristic
455	C₂²	#16863664	1	1	yes	characteristic
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	2	no	—
459	C₄	—	1072	1	no	—
460	C₄	C₂×#14523964	1	1	yes	characteristic
461	C₄	#16863392	1	1	yes	characteristic
462	C₄	D₈×Dic₆₇	1	1	yes	characteristic
463	C₄	#16863508	1	1	yes	characteristic
464	C₄	#16863528	1	1	yes	characteristic
465	C₄	#16863538	1	1	yes	characteristic
466	C₄	#16863623	1	1	yes	characteristic
467	C₄	#16863630	1	1	yes	characteristic
468	C₂	—	4	4	no	—
469	C₂	—	16	2	no	—
470	C₂	#22559548	1	1	yes	characteristic
471	C₂	#22559562	1	1	yes	characteristic
472	C₂	#22559616	1	1	yes	characteristic
473	C₂	#22559644	1	1	yes	characteristic
474	C₂	#22560559	1	1	yes	characteristic
475	C₂	#22560638	1	1	yes	characteristic
476	C₂	#22560719	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524015
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting, prime radical	C₂×#459×C₆₇	C₂
socle	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#459	D₆₇
hypocenter	C₆₇	#9221
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#459

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₆₇⋊C₈	C₂	semidirect
2. #22561016	C₂	semidirect
3. #22560209	C₂	semidirect
4. C₂×#16863140	C₂	semidirect
5. #22558093	C₂	semidirect
6. #22558043	C₂	semidirect
25. #459×C₆₇	C₄	semidirect
84. C₂×C₆₇⋊C₈	D₈	semidirect
164. D₄×C₆₇	#102	semidirect
246. C₆₇	#9221	semidirect
252. #459	Dic₆₇	semidirect
403. D₄	#14523964	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₆₇⋊C₈	2
#22561016	2
#22560209	2
C₂×#16863140	2
#22558093	2
#22558043	2
C₂×#459×C₆₇	2
#9221	67