Subgroups — #30827601

Order 17152.

654 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	#22570768	C₂	1	1	yes	maximal, characteristic
2	#22569156	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568955	C₂	1	1	yes	maximal, characteristic
4	#22559101	C₂	1	2	yes	maximal, complemented
5	#22559054	C₂	1	2	yes	maximal, complemented
6	#22559047	C₂	1	2	yes	maximal, complemented
7	C₈×#11761617	C₂	1	2	yes	maximal, complemented
8	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558613	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558609	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	—	2	1	no	—
13	#16863953	C₂²	1	1	yes	characteristic
14	#16863946	C₂²	1	1	yes	characteristic
15	#16863794	C₄	1	1	yes	characteristic
16	#16863769	C₄	1	1	yes	characteristic
17	#16863747	C₄	1	1	yes	characteristic
18	#16863743	C₄	1	1	yes	characteristic
19	#16863625	C₄	1	2	yes	—
20	#16863625	C₄	1	2	yes	—
21	#16863526	C₂²	1	2	yes	—
22	#16863524	—	2	2	no	—
23	C₈×Dic₁₃₄	C₂²	1	2	yes	—
24	C₄×#11761617	C₂²	1	2	yes	—
25	C₄×#11761617	C₂²	1	2	yes	—
26	#16863475	C₂²	1	2	yes	—
27	#16863469	—	2	2	no	—
28	C₈×D₂₆₈	C₂²	1	2	yes	—
29	#16863424	—	2	1	no	—
30	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
32	#16863364	—	2	1	no	—
33	#16863354	—	2	1	no	—
34	#16863147	C₂²	1	1	yes	characteristic
35	#16863147	C₂²	1	1	yes	characteristic
36	#16863147	C₂²	1	1	yes	characteristic
37	#16863147	C₂²	1	1	yes	characteristic
38	#16863128	C₂²	1	2	yes	—
39	#16863128	C₂²	1	2	yes	—
40	#16863125	C₂²	1	2	yes	—
41	#16863125	C₂²	1	2	yes	—
42	#431×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#322×C₆₇	C₂²	1	2	yes	—
46	C₈²×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761628	—	2	1	no	—
48	C₂×#11761617	C₂×C₄	1	2	yes	—
49	C₂×#11761617	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	—	2	1	no	—
54	C₂×C₈×D₆₇	—	2	2	no	—
55	#14524043	C₂×C₄	1	1	yes	characteristic
56	#14524043	C₂×C₄	1	1	yes	characteristic
57	#14524043	C₂×C₄	1	1	yes	characteristic
58	#14524040	C₂×C₄	1	1	yes	characteristic
59	#14524035	—	2	1	no	—
60	#14524031	—	2	1	no	—
61	#14524022	—	2	1	no	—
62	#14524022	C₂×C₄	1	2	yes	—
63	#14524017	—	2	1	no	—
64	#14524017	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃₄	—	2	1	no	—
66	C₄×Dic₁₃₄	C₂³	1	2	yes	—
67	C₄×D₂₆₈	—	2	1	no	—
68	C₄×D₂₆₈	C₂³	1	2	yes	—
69	#14524002	—	2	1	no	—
70	#14523975	C₂×C₄	1	2	yes	—
71	#14523975	C₂×C₄	1	2	yes	—
72	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
73	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
74	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
77	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
78	Dic₆₇⋊C₈	—	2	2	no	—
79	Dic₆₇⋊C₈	—	2	2	no	—
80	#14523968	C₂×C₄	1	2	yes	—
81	#14523968	C₂×C₄	1	2	yes	—
82	#14523965	C₂×C₄	1	2	yes	—
83	#14523965	C₂×C₄	1	2	yes	—
84	(C₂×D₆₇)⋊C₈	—	2	2	no	—
85	(C₂×D₆₇)⋊C₈	—	2	2	no	—
86	#14523953	—	2	2	no	—
87	C₈×Dic₆₇	—	2	2	no	—
88	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
89	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	—	2	1	no	—
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	#97×C₆₇	D₄	1	2	yes	—
96	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₄×C₈×C₆₇	C₂³	1	2	yes	—
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
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×Dic₁₃₄	—	2	1	no	—
106	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
107	C₂×D₂₆₈	—	2	1	no	—
108	C₂×D₂₆₈	—	2	1	no	—
109	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
110	C₂×C₂²⋊D₆₇	—	2	2	no	—
111	C₂×C₄×D₆₇	—	2	2	no	—
112	#11761617	—	2	4	no	—
113	#11761617	—	4	2	no	—
114	#11761611	—	4	4	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	1	no	—
117	C₂×C₆₇⋊C₈	—	2	1	no	—
118	C₂×C₆₇⋊C₈	—	2	1	no	—
119	C₂×C₆₇⋊C₈	—	4	2	no	—
120	C₂×C₆₇⋊C₈	—	4	2	no	—
121	C₈×D₆₇	—	4	4	no	—
122	C₄⋊Dic₆₇	—	2	1	no	—
123	C₄⋊Dic₆₇	—	2	1	no	—
124	C₄⋊Dic₆₇	—	2	1	no	—
125	C₄⋊Dic₆₇	—	2	1	no	—
126	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
127	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
128	Dic₆₇⋊C₄	—	2	2	no	—
129	Dic₆₇⋊C₄	—	2	2	no	—
130	C₂²⋊Dic₆₇	—	2	2	no	—
131	(C₂×D₆₇)⋊C₄	—	2	2	no	—
132	(C₂×D₆₇)⋊C₄	—	2	2	no	—
133	C₄×Dic₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
137	M₁₆×C₆₇	—	2	2	no	—
138	M₁₆×C₆₇	SD₁₆	1	4	yes	complemented
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	4	1	no	—
142	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
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	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	#135	1	2	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₂²⋊D₆₇	—	4	4	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	2	no	—
168	C₂×Dic₆₇	—	2	2	no	—
169	D₂₆₈	—	2	2	no	—
170	D₂₆₈	—	2	2	no	—
171	D₂₆₈	—	4	2	no	—
172	D₂₆₈	—	8	1	no	—
173	C₄×D₆₇	—	4	2	no	—
174	C₄×D₆₇	—	4	2	no	—
175	Dic₁₃₄	—	2	2	no	—
176	Dic₁₃₄	—	2	2	no	—
177	Dic₁₃₄	—	4	2	no	—
178	Dic₁₃₄	—	8	1	no	—
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	2	1	no	—
181	C₈×C₆₇	—	2	1	no	—
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₈×C₆₇	#131	1	2	yes	complemented
185	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
186	C₆₇⋊C₈	—	4	1	no	—
187	C₆₇⋊C₈	—	4	1	no	—
188	C₆₇⋊C₈	—	4	1	no	—
189	C₆₇⋊C₈	—	4	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	C₄×SD₁₆	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	C₄×SD₁₆	1	2	yes	—
201	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
202	C₄×C₆₇	#568	1	1	yes	characteristic
203	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
204	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
205	Dic₆₇	—	4	2	no	—
206	Dic₆₇	—	4	2	no	—
207	Dic₆₇	—	4	2	no	—
208	#18067	—	67	1	no	maximal, Sylow
209	C₂×C₆₇	—	2	2	no	—
210	C₂×C₆₇	#2935	1	1	yes	characteristic
211	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
212	C₂×C₆₇	#2957	1	1	yes	characteristic
213	D₆₇	—	8	2	no	—
214	#3167	—	67	1	no	—
215	#2962	—	67	1	no	—
216	#2935	—	67	1	no	—
217	#1590	—	67	2	no	—
218	#1583	—	67	2	no	—
219	#1582	—	67	2	no	—
220	C₈×SD₁₆	—	67	2	no	—
221	C₂×#334	—	67	1	no	—
222	#1481	—	67	1	no	—
223	#1480	—	67	1	no	—
224	C₈×M₁₆	D₆₇	1	1	yes	characteristic, complemented
225	C₆₇	#18067	1	1	yes	characteristic, complemented, Sylow
226	C₂×#131	—	134	1	no	—
227	#533	—	67	1	no	—
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	—	67	2	no	—
236	#446	—	134	2	no	—
237	C₈×Q₈	—	67	2	no	—
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	C₂×D₆₇	1	1	yes	characteristic
244	#424	—	134	1	no	—
245	C₂×C₄⋊C₈	—	67	1	no	—
246	C₂×C₄⋊C₈	—	67	1	no	—
247	#408	—	134	1	no	—
248	#405	—	134	1	no	—
249	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
250	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
251	#334	—	67	1	no	—
252	#334	—	67	1	no	—
253	#334	—	67	1	no	—
254	#334	—	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	C₂×D₆₇	1	2	yes	—
260	C₈²	C₂×D₆₇	1	2	yes	—
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	D₂₆₈	1	1	yes	characteristic
267	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
268	C₂²×C₈	—	134	1	no	—
269	C₂²×C₈	—	134	1	no	—
270	C₂²×C₈	—	134	2	no	—
271	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
273	C₄⋊Q₈	—	67	1	no	—
274	C₄⋊Q₈	—	67	1	no	—
275	C₄⋊Q₈	—	67	1	no	—
276	#127	—	67	1	no	—
277	#125	—	134	1	no	—
278	#124	—	134	1	no	—
279	C₂²⋊Q₈	—	67	2	no	—
280	C₂²⋊Q₈	—	134	1	no	—
281	#121	—	67	2	no	—
282	#121	—	134	1	no	—
283	C₄×Q₈	—	67	2	no	—
284	C₄×Q₈	—	134	1	no	—
285	C₄×D₄	—	67	2	no	—
286	C₄×D₄	—	134	1	no	—
287	#117	—	134	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	1	no	—
292	C₄⋊C₈	—	67	1	no	—
293	C₄⋊C₈	—	67	1	no	—
294	C₄⋊C₈	—	67	1	no	—
295	C₄⋊C₈	—	67	1	no	—
296	C₄⋊C₈	—	134	2	no	—
297	C₄⋊C₈	—	134	2	no	—
298	C₄⋊C₈	—	134	1	no	—
299	C₄⋊C₈	—	134	1	no	—
300	#103	—	67	2	no	—
301	#103	—	67	2	no	—
302	#102	—	67	2	no	—
303	#102	—	67	2	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	134	2	no	—
306	C₂²⋊C₈	—	134	2	no	—
307	#97	—	134	2	no	—
308	#97	D₂₆₈	1	2	yes	—
309	C₄×C₈	—	134	2	no	—
310	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
311	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
312	C₄×C₈	C₂²×D₆₇	1	2	yes	—
313	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
315	C₄×C₈	C₂²×D₆₇	1	2	yes	—
316	#41	—	268	2	no	—
317	#41	—	268	2	no	—
318	C₂×Q₈	—	67	2	no	—
319	C₂×Q₈	—	134	1	no	—
320	C₂×Q₈	—	134	1	no	—
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×D₂₆₈	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	—	268	4	no	—
333	M₁₆	#11761617	1	4	yes	complemented
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	268	2	no	—
338	C₂×C₈	—	268	2	no	—
339	C₂×C₈	—	268	4	no	—
340	C₂×C₈	—	134	1	no	—
341	C₂×C₈	—	134	1	no	—
342	C₂×C₈	—	134	1	no	—
343	C₂×C₈	—	134	1	no	—
344	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
347	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
349	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
350	C₄⋊C₄	—	67	2	no	—
351	C₄⋊C₄	—	67	2	no	—
352	C₄⋊C₄	—	134	1	no	—
353	C₄⋊C₄	—	134	1	no	—
354	C₄⋊C₄	—	134	1	no	—
355	C₄⋊C₄	—	134	2	no	—
356	C₄⋊C₄	—	134	1	no	—
357	C₄⋊C₄	—	134	2	no	—
358	C₂²⋊C₄	—	134	2	no	—
359	C₂²⋊C₄	—	134	2	no	—
360	C₂²⋊C₄	—	134	2	no	—
361	C₄²	—	134	2	no	—
362	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
363	C₄²	C₂×D₂₆₈	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×D₂₆₈	1	1	yes	characteristic
368	Q₈	—	268	2	no	—
369	Q₈	—	536	1	no	—
370	Q₈	—	134	2	no	—
371	Q₈	—	134	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	—	4	1	no	—
380	C₂×C₄	—	268	2	no	—
381	C₂×C₄	—	268	2	no	—
382	C₂×C₄	—	134	2	no	—
383	C₂×C₄	—	134	2	no	—
384	C₂×C₄	—	134	2	no	—
385	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
386	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
387	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
388	C₂×C₄	C₂×#11761617	1	2	yes	—
389	C₂×C₄	#14524065	1	2	yes	—
390	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
392	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	2	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	268	1	no	—
399	C₈	—	268	1	no	—
400	C₈	—	268	1	no	—
401	C₈	—	268	1	no	—
402	C₈	#14524051	1	2	yes	complemented
403	C₈	C₂×#11761617	1	4	yes	—
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	C₄×#11761617	1	2	yes	—
410	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	268	2	no	—
415	C₄	—	268	2	no	—
416	C₄	—	268	2	no	—
417	C₄	C₄×#11761617	1	2	yes	—
418	C₄	C₂×#14524051	1	1	yes	characteristic
419	C₄	#16864162	1	1	yes	characteristic
420	C₄	C₂²×#11761617	1	1	yes	characteristic
421	C₄	C₂×#14524065	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	536	2	no	—
424	C₂	#22568955	1	1	yes	characteristic
425	C₂	C₂×C₄×#11761617	1	1	yes	characteristic
426	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	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#18067
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22569156	C₂	semidirect
4. #22559101	C₂	semidirect
5. #22559054	C₂	semidirect
6. #22559047	C₂	semidirect
7. C₈×#11761617	C₂	semidirect
8. C₂×#16863147	C₂	semidirect
9. #22558613	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
138. M₁₆×C₆₇	SD₁₆	semidirect
184. C₈×C₆₇	#131	semidirect
224. C₈×M₁₆	D₆₇	semidirect
225. C₆₇	#18067	semidirect
333. M₁₆	#11761617	semidirect
402. C₈	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22570768	2
#22569156	2
#22568955	2
#22559101	2
#22559054	2
#22559047	2
C₈×#11761617	2
C₂×#16863147	2
#22558613	2
#22558609	2
C₈×M₁₆×C₆₇	2
#18067	67