Subgroups — #30827634

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570769	C₂	1	1	yes	maximal, characteristic
2	#22568955	C₂	1	1	yes	maximal, characteristic
3	#22559109	C₂	1	2	yes	maximal, complemented
4	#22559087	C₂	1	2	yes	maximal, complemented
5	#22559064	C₂	1	2	yes	maximal, complemented
6	#22559019	C₂	1	2	yes	maximal, complemented
7	C₂×#16863148	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558612	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558607	C₂	1	1	yes	maximal, characteristic
10	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2962	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	—	2	1	no	—
13	#16863946	C₂²	1	1	yes	characteristic
14	#16863792	C₄	1	1	yes	characteristic
15	#16863770	C₄	1	1	yes	characteristic
16	#16863746	C₄	1	1	yes	characteristic
17	#16863744	C₄	1	1	yes	characteristic
18	#16863627	C₄	1	2	yes	—
19	#16863627	C₄	1	2	yes	—
20	C₄×#11761615	C₂²	1	2	yes	—
21	C₄×#11761615	C₂²	1	2	yes	—
22	#16863475	C₂²	1	2	yes	—
23	#16863469	—	2	2	no	—
24	C₈×D₂₆₈	C₂²	1	2	yes	—
25	#16863449	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	#16863364	—	2	1	no	—
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	#16863148	C₂²	1	1	yes	characteristic
30	#16863148	C₂²	1	1	yes	characteristic
31	#16863148	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	1	yes	characteristic
33	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
34	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	#16863123	C₂²	1	2	yes	—
36	#16863123	C₂²	1	2	yes	—
37	#16863118	C₂²	1	2	yes	—
38	#16863118	C₂²	1	2	yes	—
39	#533×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	—	2	2	no	—
41	#446×C₆₇	C₂²	1	2	yes	—
42	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
43	#424×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#405×C₆₇	—	2	1	no	—
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	—	2	1	no	—
48	C₂×#11761615	C₂×C₄	1	2	yes	—
49	C₂×#11761615	C₂×C₄	1	2	yes	—
50	#14524051	—	4	4	no	—
51	C₂×#11761611	—	2	2	no	—
52	C₂²×C₆₇⋊C₈	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₆₇	—	2	2	no	—
56	#14524043	C₂×C₄	1	1	yes	characteristic
57	#14524040	C₂×C₄	1	1	yes	characteristic
58	#14524031	—	2	1	no	—
59	#14524017	—	2	1	no	—
60	#14524017	C₂×C₄	1	2	yes	—
61	C₄×D₂₆₈	—	2	1	no	—
62	C₄×D₂₆₈	C₂³	1	2	yes	—
63	#14523974	C₂×C₄	1	2	yes	—
64	#14523974	C₂×C₄	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
66	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
69	#14523967	C₂×C₄	1	2	yes	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523963	C₂×C₄	1	2	yes	—
72	#14523963	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
82	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	—	2	1	no	—
84	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
86	#125×C₆₇	—	2	1	no	—
87	C₂²⋊Q₈×C₆₇	—	2	1	no	—
88	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₆₇	—	2	1	no	—
90	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
91	#117×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	—	2	2	no	—
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	C₄⋊C₈×C₆₇	—	2	1	no	—
96	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#97×C₆₇	—	2	2	no	—
99	#97×C₆₇	D₄	1	2	yes	—
100	C₄×C₈×C₆₇	—	2	2	no	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	#11761628	—	4	2	no	—
103	#11761628	—	4	2	no	—
104	C₂×Dic₁₃₄	—	2	1	no	—
105	C₂×D₂₆₈	—	2	1	no	—
106	C₂×D₂₆₈	—	2	1	no	—
107	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
108	C₂×C₂²⋊D₆₇	—	2	2	no	—
109	C₂×C₄×D₆₇	—	2	2	no	—
110	#11761615	—	2	4	no	—
111	#11761615	—	4	2	no	—
112	#11761611	—	4	4	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	4	1	no	—
118	C₂×C₆₇⋊C₈	—	4	2	no	—
119	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
123	C₈×D₆₇	—	4	4	no	—
124	C₄⋊Dic₆₇	—	2	1	no	—
125	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
126	(C₂×D₆₇)⋊C₄	—	2	2	no	—
127	(C₂×D₆₇)⋊C₄	—	2	2	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
133	M₁₆×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	4	2	no	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₆₇	#41	1	1	yes	characteristic
151	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂²×D₆₇	—	2	2	no	—
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	2	2	no	—
156	Q₈×C₆₇	—	8	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₆₇	#137	1	2	yes	—
167	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂²⋊D₆₇	—	4	4	no	—
171	C₂×Dic₆₇	—	2	2	no	—
172	D₂₆₈	—	2	2	no	—
173	D₂₆₈	—	2	2	no	—
174	D₂₆₈	—	4	2	no	—
175	D₂₆₈	—	8	1	no	—
176	C₄×D₆₇	—	4	2	no	—
177	C₄×D₆₇	—	4	2	no	—
178	Dic₁₃₄	—	4	2	no	—
179	C₈×C₆₇	—	2	2	no	—
180	C₈×C₆₇	—	2	2	no	—
181	C₈×C₆₇	—	4	1	no	—
182	C₈×C₆₇	—	4	1	no	—
183	C₆₇⋊C₈	—	2	1	no	—
184	C₆₇⋊C₈	—	2	1	no	—
185	C₆₇⋊C₈	—	2	1	no	—
186	C₆₇⋊C₈	—	4	2	no	—
187	C₆₇⋊C₈	—	4	1	no	—
188	C₆₇⋊C₈	—	4	1	no	—
189	C₆₇⋊C₈	#131	1	2	yes	complemented
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	#440	1	2	yes	—
193	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×D₆₇	—	4	2	no	—
195	C₂×D₆₇	—	4	2	no	—
196	C₂×D₆₇	—	4	2	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	2	1	no	—
200	C₄×C₆₇	—	4	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	#440	1	2	yes	—
203	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
204	C₄×C₆₇	#568	1	1	yes	characteristic
205	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
207	Dic₆₇	—	4	2	no	—
208	#18071	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	—	2	2	no	—
210	C₂×C₆₇	#2935	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
212	C₂×C₆₇	#2957	1	1	yes	characteristic
213	D₆₇	—	8	2	no	—
214	#3167	—	67	1	no	—
215	#2962	D₆₇	1	1	yes	characteristic, complemented
216	#2935	—	67	1	no	—
217	#1592	—	67	2	no	—
218	#1588	—	67	2	no	—
219	#1585	—	67	2	no	—
220	SD₁₆⋊C₈	—	67	2	no	—
221	C₂×#335	—	67	1	no	—
222	#1481	—	67	1	no	—
223	#1480	—	67	1	no	—
224	#1451	—	67	1	no	—
225	C₆₇	#18071	1	1	yes	characteristic, complemented, Sylow
226	C₂×#131	—	134	1	no	—
227	#533	C₂×D₆₇	1	1	yes	characteristic
228	#532	—	67	1	no	—
229	#498	—	67	1	no	—
230	#492	—	67	1	no	—
231	#487	—	67	1	no	—
232	#486	—	67	1	no	—
233	#465	—	67	2	no	—
234	#465	—	67	2	no	—
235	#446	—	2	2	no	—
236	#446	C₂×D₆₇	1	2	yes	—
237	C₈×Q₈	C₂×D₆₇	1	2	yes	—
238	C₄×SD₁₆	—	67	2	no	—
239	C₄×SD₁₆	—	67	2	no	—
240	#436	—	67	2	no	—
241	#435	—	134	2	no	—
242	C₈×D₄	—	67	2	no	—
243	#431	—	67	1	no	—
244	#424	—	2	1	no	—
245	C₂×C₄⋊C₈	—	67	1	no	—
246	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
247	#408	—	134	1	no	—
248	#405	—	2	1	no	—
249	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
250	C₂×C₄×C₈	—	67	1	no	—
251	#335	—	67	1	no	—
252	#335	—	67	1	no	—
253	#335	—	67	1	no	—
254	#335	—	67	1	no	—
255	Q₈⋊C₈	—	67	2	no	—
256	Q₈⋊C₈	—	67	2	no	—
257	D₄⋊C₈	—	67	2	no	—
258	D₄⋊C₈	—	67	2	no	—
259	#322	—	67	2	no	—
260	#322	—	67	2	no	—
261	C₂×#41	—	134	1	no	—
262	C₂×SD₁₆	—	67	2	no	—
263	C₂×SD₁₆	—	67	2	no	—
264	#131	—	268	4	no	—
265	C₂×M₁₆	—	134	2	no	—
266	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
267	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	—	67	1	no	—
270	C₂²×C₈	—	67	1	no	—
271	C₂²×C₈	—	134	1	no	—
272	C₂²×C₈	—	134	2	no	—
273	C₄⋊Q₈	—	67	1	no	—
274	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
275	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
276	#127	—	67	1	no	—
277	#125	—	2	1	no	—
278	#124	—	134	1	no	—
279	C₂²⋊Q₈	—	2	1	no	—
280	C₂²⋊Q₈	C₄×D₆₇	1	2	yes	—
281	#121	—	67	2	no	—
282	#121	—	134	1	no	—
283	C₄×Q₈	—	2	1	no	—
284	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
285	C₄×D₄	—	67	2	no	—
286	C₄×D₄	—	134	1	no	—
287	#117	—	2	1	no	—
288	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
289	#106	—	67	2	no	—
290	#106	—	67	2	no	—
291	C₄⋊C₈	—	2	2	no	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	67	1	no	—
295	C₄⋊C₈	—	67	1	no	—
296	C₄⋊C₈	—	134	1	no	—
297	C₄⋊C₈	—	134	1	no	—
298	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
299	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
300	#103	—	67	2	no	—
301	#103	—	67	2	no	—
302	#102	—	67	2	no	—
303	#102	—	67	2	no	—
304	C₂²⋊C₈	—	134	1	no	—
305	C₂²⋊C₈	—	134	2	no	—
306	C₂²⋊C₈	—	134	2	no	—
307	#97	—	2	2	no	—
308	#97	C₂²⋊D₆₇	1	2	yes	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	67	1	no	—
311	C₄×C₈	—	67	1	no	—
312	C₄×C₈	—	67	1	no	—
313	C₄×C₈	—	67	1	no	—
314	C₄×C₈	—	67	2	no	—
315	C₄×C₈	C₂²×D₆₇	1	2	yes	—
316	#41	—	268	2	no	—
317	#41	—	268	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	134	1	no	—
320	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
321	C₂×D₄	—	67	2	no	—
322	C₂×D₄	—	134	2	no	—
323	C₂×D₄	—	134	1	no	—
324	C₂×D₄	—	134	1	no	—
325	C₂²×C₄	—	134	2	no	—
326	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
328	C₂²×C₄	#11761628	1	1	yes	characteristic
329	SD₁₆	—	268	2	no	—
330	SD₁₆	—	134	4	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	268	4	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	268	4	no	—
338	C₂×C₈	—	268	2	no	—
339	C₂×C₈	—	268	1	no	—
340	C₂×C₈	—	67	1	no	—
341	C₂×C₈	—	67	1	no	—
342	C₂×C₈	—	67	1	no	—
343	C₂×C₈	—	67	1	no	—
344	C₂×C₈	—	134	1	no	—
345	C₂×C₈	—	134	1	no	—
346	C₂×C₈	—	134	2	no	—
347	C₂×C₈	—	134	2	no	—
348	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
349	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	67	2	no	—
356	C₄⋊C₄	—	134	1	no	—
357	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	134	2	no	—
360	C₂²⋊C₄	—	134	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
363	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
364	C₄²	#11761628	1	1	yes	characteristic
365	C₄²	C₂³×D₆₇	1	1	yes	characteristic
366	C₂³	—	134	2	no	—
367	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	8	1	no	—
371	Q₈	—	268	2	no	—
372	D₄	—	268	4	no	—
373	D₄	—	268	2	no	—
374	D₄	—	536	1	no	—
375	D₄	—	134	2	no	—
376	D₄	—	134	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	268	2	no	—
383	C₂×C₄	—	268	2	no	—
384	C₂×C₄	—	134	2	no	—
385	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
386	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
387	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
388	C₂×C₄	#14524068	1	2	yes	—
389	C₂×C₄	#14524074	1	2	yes	—
390	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
393	C₈	—	2	2	no	—
394	C₈	—	2	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	268	2	no	—
398	C₈	—	268	1	no	—
399	C₈	—	268	1	no	—
400	C₈	—	67	2	no	—
401	C₈	—	134	1	no	—
402	C₈	—	134	1	no	—
403	C₈	—	134	1	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	268	2	no	—
407	C₂²	—	268	2	no	—
408	C₂²	—	268	2	no	—
409	C₂²	#16863494	1	2	yes	—
410	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	268	2	no	—
417	C₄	#16863494	1	2	yes	—
418	C₄	#131×D₆₇	1	1	yes	characteristic
419	C₄	#16864161	1	1	yes	characteristic
420	C₄	C₂×#14524068	1	1	yes	characteristic
421	C₄	C₂×#14524074	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	536	2	no	—
424	C₂	#22568951	1	1	yes	characteristic
425	C₂	C₂×#16863494	1	1	yes	characteristic
426	C₂	#22569124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2962	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2962	D₆₇
hypocenter	C₆₇	#18071
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #2962

Complements

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

N	H ≅ G/N	type
3. #22559109	C₂	semidirect
4. #22559087	C₂	semidirect
5. #22559064	C₂	semidirect
6. #22559019	C₂	semidirect
7. C₂×#16863148	C₂	semidirect
8. #22558612	C₂	semidirect
10. #22558496	C₂	semidirect
11. C₆₇×#2962	C₂	semidirect
189. C₆₇⋊C₈	#131	semidirect
215. #2962	D₆₇	semidirect
225. C₆₇	#18071	semidirect

Maximal subgroups

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

subgroup	index
#22570769	2
#22568955	2
#22559109	2
#22559087	2
#22559064	2
#22559019	2
C₂×#16863148	2
#22558612	2
#22558607	2
#22558496	2
C₆₇×#2962	2
#18071	67