Subgroups — #30750419

Order 17152.

584 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761617	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863475	C₂	1	1	yes	maximal, characteristic
3	#22561145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863125	C₂	1	1	yes	maximal, characteristic
6	#22558044	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761617	C₄	1	1	yes	characteristic
9	C₂²×#11761611	—	2	1	no	—
10	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
12	C₄×#11761617	—	2	2	no	—
13	C₄×#11761617	—	2	2	no	—
14	#16863475	—	2	2	no	—
15	#16863475	—	2	2	no	—
16	C₂×#14523975	C₄	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×#14523968	C₄	1	1	yes	characteristic
20	C₂×#14523965	C₄	1	1	yes	characteristic
21	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
22	#16863150	—	2	1	no	—
23	#16863150	—	2	1	no	—
24	#16863128	—	2	2	no	—
25	#16863125	—	2	2	no	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₆₇	—	2	1	no	—
32	C₂×#11761617	—	2	2	no	—
33	C₂×#11761617	—	2	2	no	—
34	C₂×#11761617	—	2	2	no	—
35	C₂×#11761611	—	2	4	no	—
36	C₂×#11761611	—	4	2	no	—
37	C₂×#11761611	—	4	2	no	—
38	C₂²×C₆₇⋊C₈	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	—	2	1	no	—
40	C₄×Dic₁₃₄	—	2	1	no	—
41	C₄×Dic₁₃₄	—	2	1	no	—
42	C₄×Dic₁₃₄	—	2	1	no	—
43	C₄×Dic₁₃₄	—	2	1	no	—
44	C₄×D₂₆₈	—	2	1	no	—
45	C₄×D₂₆₈	—	2	1	no	—
46	C₄×D₂₆₈	D₄	1	2	yes	—
47	C₄×D₂₆₈	D₄	1	2	yes	—
48	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₆₇⋊C₄	—	2	1	no	—
51	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
52	C₂×C₄×Dic₆₇	—	2	1	no	—
53	#14523975	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
58	#14523968	—	2	2	no	—
59	#14523965	—	2	2	no	—
60	(C₂×D₆₇)⋊C₈	—	2	4	no	—
61	C₂²×C₈×C₆₇	—	2	1	no	—
62	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	—	2	1	no	—
70	C₄×C₈×C₆₇	—	2	1	no	—
71	C₄×C₈×C₆₇	—	2	1	no	—
72	C₄×C₈×C₆₇	—	2	1	no	—
73	C₂³×D₆₇	—	2	1	no	—
74	C₂×Dic₁₃₄	—	2	1	no	—
75	C₂×Dic₁₃₄	—	2	1	no	—
76	C₂×Dic₁₃₄	—	2	1	no	—
77	C₂×Dic₁₃₄	—	2	1	no	—
78	C₂×Dic₁₃₄	—	2	1	no	—
79	C₂×Dic₁₃₄	M₁₆	1	2	yes	complemented
80	C₂×D₂₆₈	—	2	1	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×D₂₆₈	—	2	1	no	—
83	C₂×D₂₆₈	—	2	1	no	—
84	C₂×D₂₆₈	—	2	1	no	—
85	C₂×D₂₆₈	C₂²⋊C₄	1	2	yes	—
86	C₂²×Dic₆₇	—	2	1	no	—
87	C₂²×Dic₆₇	—	2	1	no	—
88	C₂²×Dic₆₇	—	2	1	no	—
89	C₂×C₄×D₆₇	—	2	2	no	—
90	C₂×C₄×D₆₇	—	2	2	no	—
91	C₂×C₄×D₆₇	—	4	1	no	—
92	C₂×C₄×D₆₇	—	4	1	no	—
93	C₂×C₄×D₆₇	—	4	1	no	—
94	C₂×C₄×D₆₇	—	4	1	no	—
95	#11761617	—	2	8	no	—
96	#11761611	—	4	8	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₂×C₆₇⋊C₈	—	4	1	no	—
99	C₂×C₆₇⋊C₈	—	4	1	no	—
100	C₂×C₆₇⋊C₈	—	4	1	no	—
101	C₂×C₆₇⋊C₈	—	4	1	no	—
102	C₂×C₆₇⋊C₈	—	4	1	no	—
103	C₄⋊Dic₆₇	—	2	1	no	—
104	C₄⋊Dic₆₇	—	2	1	no	—
105	C₄⋊Dic₆₇	—	2	1	no	—
106	C₄⋊Dic₆₇	C₂²⋊C₄	1	2	yes	—
107	Dic₆₇⋊C₄	—	4	1	no	—
108	Dic₆₇⋊C₄	—	4	1	no	—
109	(C₂×D₆₇)⋊C₄	—	2	2	no	—
110	(C₂×D₆₇)⋊C₄	—	2	2	no	—
111	C₄×Dic₆₇	—	4	1	no	—
112	C₄×Dic₆₇	—	4	1	no	—
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	4	1	no	—
128	C₂×C₈×C₆₇	—	4	1	no	—
129	C₂×C₈×C₆₇	M₁₆	1	2	yes	complemented
130	C₂×C₈×C₆₇	SD₁₆	1	2	yes	complemented
131	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	#41	1	1	yes	characteristic
135	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
136	C₂²×D₆₇	—	2	2	no	—
137	C₂²×D₆₇	—	4	1	no	—
138	C₂²×D₆₇	—	4	1	no	—
139	C₂²×D₆₇	—	4	1	no	—
140	C₂²×D₆₇	—	4	1	no	—
141	C₂²×D₆₇	—	4	1	no	—
142	C₂²×D₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
157	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
159	C₂×Dic₆₇	—	2	2	no	—
160	C₂×Dic₆₇	—	2	2	no	—
161	C₂×Dic₆₇	—	4	1	no	—
162	C₂×Dic₆₇	—	4	1	no	—
163	C₂×Dic₆₇	—	4	1	no	—
164	C₂×Dic₆₇	—	4	1	no	—
165	C₂×Dic₆₇	—	4	1	no	—
166	C₂×Dic₆₇	—	4	1	no	—
167	C₂×Dic₆₇	—	4	1	no	—
168	D₂₆₈	—	2	2	no	—
169	D₂₆₈	—	2	2	no	—
170	D₂₆₈	—	4	1	no	—
171	D₂₆₈	—	4	1	no	—
172	C₄×D₆₇	—	4	2	no	—
173	C₄×D₆₇	—	4	2	no	—
174	C₄×D₆₇	—	4	2	no	—
175	C₄×D₆₇	—	4	2	no	—
176	Dic₁₃₄	—	2	2	no	—
177	Dic₁₃₄	—	2	2	no	—
178	Dic₁₃₄	—	4	1	no	—
179	Dic₁₃₄	—	4	1	no	—
180	C₈×C₆₇	—	2	2	no	—
181	C₈×C₆₇	—	2	2	no	—
182	C₈×C₆₇	—	4	2	no	—
183	C₈×C₆₇	—	4	1	no	—
184	C₈×C₆₇	—	4	1	no	—
185	C₆₇⋊C₈	—	4	2	no	—
186	C₆₇⋊C₈	—	8	1	no	—
187	C₂²×C₆₇	#386	1	1	yes	characteristic
188	C₂²×C₆₇	#435	1	1	yes	characteristic
189	C₂²×C₆₇	#436	1	1	yes	characteristic
190	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
191	C₂²×C₆₇	#440	1	1	yes	characteristic
192	C₂²×C₆₇	#443	1	1	yes	characteristic
193	C₂²×C₆₇	#444	1	1	yes	characteristic
194	C₂×D₆₇	—	4	2	no	—
195	C₂×D₆₇	—	4	2	no	—
196	C₂×D₆₇	—	4	2	no	—
197	C₂×D₆₇	—	8	1	no	—
198	C₂×D₆₇	—	8	1	no	—
199	C₂×D₆₇	—	8	1	no	—
200	C₂×D₆₇	—	8	1	no	—
201	C₄×C₆₇	—	2	1	no	—
202	C₄×C₆₇	—	2	1	no	—
203	C₄×C₆₇	#407	1	1	yes	characteristic
204	C₄×C₆₇	#408	1	1	yes	characteristic
205	C₄×C₆₇	#435	1	1	yes	characteristic
206	C₄×C₆₇	#436	1	1	yes	characteristic
207	C₄×C₆₇	#450	1	1	yes	characteristic
208	C₄×C₆₇	#452	1	1	yes	characteristic
209	C₄×C₆₇	#461	1	1	yes	characteristic
210	C₄×C₆₇	#463	1	1	yes	characteristic
211	Dic₆₇	—	4	2	no	—
212	Dic₆₇	—	4	2	no	—
213	Dic₆₇	—	8	1	no	—
214	#8949	—	67	1	no	maximal, Sylow
215	C₂×C₆₇	#1583	1	1	yes	characteristic
216	C₂×C₆₇	#1588	1	1	yes	characteristic
217	C₂×C₆₇	#1590	1	1	yes	characteristic
218	C₂×C₆₇	#1592	1	1	yes	characteristic
219	C₂×C₆₇	#1870	1	1	yes	characteristic
220	C₂×C₆₇	#1876	1	1	yes	characteristic
221	C₂×C₆₇	#1877	1	1	yes	characteristic
222	D₆₇	—	8	2	no	—
223	C₂×C₄×SD₁₆	—	67	1	no	—
224	C₂×#436	—	67	1	no	—
225	#1984	—	67	1	no	—
226	C₂×Q₈⋊C₈	—	67	1	no	—
227	C₂×D₄⋊C₈	—	67	1	no	—
228	#1277	—	67	1	no	—
229	#1273	D₆₇	1	1	yes	characteristic, complemented
230	C₆₇	#8949	1	1	yes	characteristic, complemented, Sylow
231	C₂²×SD₁₆	—	67	1	no	—
232	C₂²×M₁₆	—	134	1	no	—
233	C₂×C₄×Q₈	—	67	1	no	—
234	C₂×C₄×D₄	—	67	1	no	—
235	C₄×SD₁₆	—	134	2	no	—
236	C₄×SD₁₆	—	134	2	no	—
237	#436	—	134	2	no	—
238	#436	—	134	2	no	—
239	C₂×#106	—	67	1	no	—
240	C₂×C₄⋊C₈	—	67	1	no	—
241	C₂×C₄⋊C₈	—	67	1	no	—
242	C₂×#103	—	67	1	no	—
243	C₂×#102	—	67	1	no	—
244	C₂×C₂²⋊C₈	—	134	1	no	—
245	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
248	#336	—	134	1	no	—
249	#336	—	134	1	no	—
250	Q₈⋊C₈	—	134	2	no	—
251	D₄⋊C₈	—	134	2	no	—
252	C₂²×Q₈	—	67	1	no	—
253	C₂²×D₄	—	67	1	no	—
254	C₂³×C₄	—	134	1	no	—
255	C₂×SD₁₆	—	134	2	no	—
256	C₂×SD₁₆	—	134	2	no	—
257	C₂×SD₁₆	—	134	2	no	—
258	C₂×M₁₆	—	268	2	no	—
259	C₂×M₁₆	—	268	2	no	—
260	C₂×M₁₆	—	134	4	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	134	1	no	—
263	C₂²×C₈	—	134	1	no	—
264	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
265	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
266	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
267	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
268	C₄×Q₈	—	134	1	no	—
269	C₄×Q₈	—	134	1	no	—
270	C₄×Q₈	—	134	1	no	—
271	C₄×Q₈	—	134	1	no	—
272	C₄×D₄	—	67	2	no	—
273	C₄×D₄	—	67	2	no	—
274	C₄×D₄	—	134	1	no	—
275	C₄×D₄	—	134	1	no	—
276	C₂×C₄⋊C₄	—	67	1	no	—
277	C₂×C₄⋊C₄	—	67	1	no	—
278	C₂×C₄⋊C₄	—	134	1	no	—
279	C₂×C₂²⋊C₄	—	134	1	no	—
280	C₂×C₄²	—	134	1	no	—
281	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
282	#106	—	134	2	no	—
283	C₄⋊C₈	—	67	2	no	—
284	C₄⋊C₈	—	134	1	no	—
285	C₄⋊C₈	—	134	1	no	—
286	C₄⋊C₈	—	134	1	no	—
287	#103	—	134	2	no	—
288	#102	—	134	2	no	—
289	C₂²⋊C₈	—	134	4	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₂⁴	—	134	1	no	—
297	C₂×Q₈	—	67	2	no	—
298	C₂×Q₈	—	134	1	no	—
299	C₂×Q₈	—	134	1	no	—
300	C₂×Q₈	—	134	1	no	—
301	C₂×Q₈	—	134	1	no	—
302	C₂×Q₈	—	134	1	no	—
303	C₂×D₄	—	67	2	no	—
304	C₂×D₄	—	134	1	no	—
305	C₂×D₄	—	134	1	no	—
306	C₂×D₄	—	134	1	no	—
307	C₂×D₄	—	134	1	no	—
308	C₂×D₄	—	134	1	no	—
309	C₂²×C₄	—	268	1	no	—
310	C₂²×C₄	—	268	1	no	—
311	C₂²×C₄	—	268	1	no	—
312	C₂²×C₄	—	268	1	no	—
313	C₂²×C₄	—	134	1	no	—
314	C₂²×C₄	—	134	1	no	—
315	C₂²×C₄	—	134	1	no	—
316	C₂²×C₄	—	134	2	no	—
317	C₂²×C₄	—	134	2	no	—
318	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
320	C₂²×C₄	#11761628	1	1	yes	characteristic
321	SD₁₆	—	134	8	no	—
322	M₁₆	—	268	8	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	268	1	no	—
337	C₂×C₈	—	268	1	no	—
338	C₂×C₈	—	268	1	no	—
339	C₂×C₈	—	268	1	no	—
340	C₂×C₈	—	268	1	no	—
341	C₂×C₈	—	134	2	no	—
342	C₂×C₈	#11761611	1	2	yes	complemented
343	C₂×C₈	#11761617	1	2	yes	complemented
344	C₄⋊C₄	—	268	1	no	—
345	C₄⋊C₄	—	268	1	no	—
346	C₄⋊C₄	—	67	2	no	—
347	C₄⋊C₄	—	134	1	no	—
348	C₄⋊C₄	—	134	1	no	—
349	C₄⋊C₄	—	134	1	no	—
350	C₂²⋊C₄	—	134	2	no	—
351	C₂²⋊C₄	—	134	2	no	—
352	C₄²	—	268	1	no	—
353	C₄²	—	268	1	no	—
354	C₄²	D₄×D₆₇	1	1	yes	characteristic
355	C₄²	D₄×D₆₇	1	1	yes	characteristic
356	C₄²	D₄×D₆₇	1	1	yes	characteristic
357	C₄²	#11761629	1	1	yes	characteristic
358	C₂³	—	268	1	no	—
359	C₂³	—	268	1	no	—
360	C₂³	—	268	1	no	—
361	C₂³	—	268	1	no	—
362	C₂³	—	268	1	no	—
363	C₂³	—	268	1	no	—
364	C₂³	—	134	2	no	—
365	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
366	Q₈	—	268	1	no	—
367	Q₈	—	268	1	no	—
368	Q₈	—	134	2	no	—
369	Q₈	—	134	2	no	—
370	D₄	—	268	1	no	—
371	D₄	—	268	1	no	—
372	D₄	—	134	2	no	—
373	D₄	—	134	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	268	1	no	—
377	C₂×C₄	—	268	1	no	—
378	C₂×C₄	—	268	1	no	—
379	C₂×C₄	—	268	1	no	—
380	C₂×C₄	—	268	2	no	—
381	C₂×C₄	—	268	2	no	—
382	C₂×C₄	—	268	2	no	—
383	C₂×C₄	—	268	2	no	—
384	C₂×C₄	—	268	1	no	—
385	C₂×C₄	—	268	1	no	—
386	C₂×C₄	—	268	1	no	—
387	C₂×C₄	—	134	2	no	—
388	C₂×C₄	—	134	2	no	—
389	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
390	C₂×C₄	#14524005	1	1	yes	characteristic
391	C₂×C₄	#14524013	1	1	yes	characteristic
392	C₂×C₄	#14524016	1	1	yes	characteristic
393	C₂×C₄	#14524018	1	1	yes	characteristic
394	C₂×C₄	#14524028	1	1	yes	characteristic
395	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
397	C₂×C₄	#14524049	1	1	yes	characteristic
398	C₂×C₄	#14524051	1	1	yes	characteristic
399	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
400	C₂×C₄	#14524065	1	1	yes	characteristic
401	C₂×C₄	#14524071	1	1	yes	characteristic
402	C₂×C₄	#14524076	1	1	yes	characteristic
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	1	no	—
407	C₈	—	4	1	no	—
408	C₈	—	268	2	no	—
409	C₈	—	536	1	no	—
410	C₂²	—	268	2	no	—
411	C₂²	—	268	2	no	—
412	C₂²	—	268	2	no	—
413	C₂²	—	536	1	no	—
414	C₂²	—	536	1	no	—
415	C₂²	—	536	1	no	—
416	C₂²	—	536	1	no	—
417	C₂²	#16863274	1	1	yes	characteristic
418	C₂²	#16863469	1	1	yes	characteristic
419	C₂²	#16863475	1	1	yes	characteristic
420	C₂²	C₄×#11761617	1	1	yes	characteristic
421	C₂²	#16863498	1	1	yes	characteristic
422	C₂²	#16863514	1	1	yes	characteristic
423	C₂²	#16863519	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	268	2	no	—
427	C₄	—	268	2	no	—
428	C₄	—	536	1	no	—
429	C₄	#16863358	1	1	yes	characteristic
430	C₄	#16863361	1	1	yes	characteristic
431	C₄	#16863466	1	1	yes	characteristic
432	C₄	#16863473	1	1	yes	characteristic
433	C₄	#16863549	1	1	yes	characteristic
434	C₄	#16863560	1	1	yes	characteristic
435	C₄	#16863602	1	1	yes	characteristic
436	C₄	#16863612	1	1	yes	characteristic
437	C₂	—	536	2	no	—
438	C₂	#22559054	1	1	yes	characteristic
439	C₂	#22559089	1	1	yes	characteristic
440	C₂	#22559101	1	1	yes	characteristic
441	C₂	#22559113	1	1	yes	characteristic
442	C₂	#22560505	1	1	yes	characteristic
443	C₂	#22560535	1	1	yes	characteristic
444	C₂	#22560542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1273	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#8949
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1273

Complements

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

N	H ≅ G/N	type
3. #22561145	C₂	semidirect
4. C₂×#16863128	C₂	semidirect
6. #22558044	C₂	semidirect
7. C₆₇×#1273	C₂	semidirect
79. C₂×Dic₁₃₄	M₁₆	semidirect
129. C₂×C₈×C₆₇	M₁₆	semidirect
130. C₂×C₈×C₆₇	SD₁₆	semidirect
229. #1273	D₆₇	semidirect
230. C₆₇	#8949	semidirect
342. C₂×C₈	#11761611	semidirect
343. C₂×C₈	#11761617	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11761617	2
C₂×#16863475	2
#22561145	2
C₂×#16863128	2
C₂×#16863125	2
#22558044	2
C₆₇×#1273	2
#8949	67