Subgroups — C₄×#16863666 #30862711

Order 17152.

946 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863666	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524068	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561119	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560994	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560871	C₂	1	2	yes	maximal, complemented
7	#22560717	C₂	1	2	yes	maximal, complemented
8	#22560273	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523959	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524068	C₂²	1	2	yes	—
13	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₆₈	C₂²	1	2	yes	—
15	#16863666	C₄	1	8	yes	direct product factor
16	#16863506	—	2	2	no	—
17	#16863494	—	2	2	no	—
18	C₄×#11761615	—	2	2	no	—
19	C₄×D₅₃₆	—	2	1	no	—
20	C₄×D₅₃₆	—	2	1	no	—
21	C₄×D₅₃₆	—	2	1	no	—
22	C₄×D₅₃₆	—	2	1	no	—
23	C₄×#11761613	—	2	2	no	—
24	C₂×#14523979	C₂²	1	1	yes	characteristic
25	C₂×#14523979	C₂²	1	1	yes	characteristic
26	#16863439	C₂²	1	2	yes	—
27	C₂×#14523977	C₂²	1	1	yes	characteristic
28	#16863400	C₂²	1	2	yes	—
29	#16863397	C₂²	1	2	yes	—
30	C₂×#14523965	C₂²	1	2	yes	—
31	C₂×#14523959	C₂²	1	2	yes	—
32	C₂×#14523959	C₂²	1	2	yes	—
33	C₄×#11761612	C₂²	1	2	yes	—
34	#16863164	C₂²	1	2	yes	—
35	#16863164	C₂²	1	2	yes	—
36	#16863161	C₂²	1	2	yes	—
37	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#405×C₆₇	—	2	1	no	—
41	#405×C₆₇	—	2	1	no	—
42	#405×C₆₇	C₂²	1	1	yes	characteristic
43	#405×C₆₇	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₆₇	—	2	1	no	—
45	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	—	2	1	no	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂²×D₂₆₈	C₂³	1	2	yes	—
49	C₂²×C₄×D₆₇	—	2	2	no	—
50	#14524068	—	2	4	no	—
51	#14524068	C₂×C₄	1	8	yes	complemented
52	C₂×#11761615	—	2	2	no	—
53	C₂×D₅₃₆	—	2	1	no	—
54	C₂×D₅₃₆	—	2	1	no	—
55	C₂×D₅₃₆	—	2	2	no	—
56	C₂×D₅₃₆	C₂×C₄	1	4	yes	complemented
57	C₂×#11761613	—	2	2	no	—
58	C₂×#11761612	C₂³	1	2	yes	—
59	C₂×#11761612	C₂³	1	2	yes	—
60	C₄×D₂₆₈	—	2	2	no	—
61	C₄×D₂₆₈	—	2	2	no	—
62	C₄×D₂₆₈	—	2	2	no	—
63	C₄×D₂₆₈	—	2	2	no	—
64	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
65	C₂×(C₂×D₆₇)⋊C₄	—	2	2	no	—
66	#14523979	—	2	2	no	—
67	#14523979	C₂×C₄	1	4	yes	complemented
68	#14523977	—	2	1	no	—
69	#14523977	—	2	1	no	—
70	#14523976	—	2	2	no	—
71	#14523974	—	2	2	no	—
72	#14523967	—	2	2	no	—
73	#14523965	—	2	2	no	—
74	#14523965	—	2	2	no	—
75	#14523964	—	2	2	no	—
76	#14523963	—	2	2	no	—
77	#14523963	—	2	2	no	—
78	#14523959	—	2	4	no	—
79	#14523959	C₂×C₄	1	8	yes	complemented
80	#14523954	—	2	2	no	—
81	C₄×C₆₇⋊C₈	—	2	2	no	—
82	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
83	#131×C₆₇	—	2	2	no	—
84	#131×C₆₇	—	2	2	no	—
85	#131×C₆₇	—	2	2	no	—
86	#131×C₆₇	C₂×C₄	1	4	yes	complemented
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×M₁₆×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	—	2	1	no	—
92	C₂²×C₈×C₆₇	—	2	1	no	—
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
97	C₄×D₄×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
100	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	—	2	1	no	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	#97×C₆₇	—	2	1	no	—
106	#97×C₆₇	—	2	1	no	—
107	#97×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	—	2	1	no	—
111	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₂³×D₆₇	—	2	2	no	—
116	C₂×D₂₆₈	—	2	2	no	—
117	C₂×D₂₆₈	—	2	2	no	—
118	C₂×D₂₆₈	—	2	2	no	—
119	C₂×D₂₆₈	—	2	2	no	—
120	C₂×D₂₆₈	—	2	2	no	—
121	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
122	C₂²×Dic₆₇	—	2	2	no	—
123	C₂×C₄×D₆₇	—	4	2	no	—
124	C₂×C₄×D₆₇	—	4	2	no	—
125	C₂×C₄×D₆₇	—	8	4	no	—
126	#11761615	—	2	8	no	—
127	D₅₃₆	—	2	4	no	—
128	D₅₃₆	—	2	4	no	—
129	#11761613	—	2	8	no	—
130	#11761612	—	2	2	no	—
131	#11761612	—	2	2	no	—
132	#11761612	—	2	2	no	—
133	#11761612	C₂²×C₄	1	4	yes	—
134	C₂×C₆₇⋊C₈	—	2	2	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₄⋊Dic₆₇	—	2	2	no	—
137	C₄⋊Dic₆₇	—	2	2	no	—
138	(C₂×D₆₇)⋊C₄	—	4	2	no	—
139	(C₂×D₆₇)⋊C₄	—	4	2	no	—
140	#41×C₆₇	—	2	1	no	—
141	#41×C₆₇	—	2	1	no	—
142	#41×C₆₇	C₂²×C₄	1	2	yes	—
143	#41×C₆₇	#41	1	2	yes	—
144	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
153	M₁₆×C₆₇	—	2	4	no	—
154	M₁₆×C₆₇	—	2	1	no	—
155	M₁₆×C₆₇	—	2	1	no	—
156	M₁₆×C₆₇	—	2	4	no	—
157	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
158	M₁₆×C₆₇	#41	1	2	yes	—
159	C₂×C₈×C₆₇	—	2	4	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	4	no	—
165	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
166	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂×C₈×C₆₇	#41	1	2	yes	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²⋊C₄×C₆₇	—	2	1	no	—
176	C₂²⋊C₄×C₆₇	—	2	1	no	—
177	C₂²⋊C₄×C₆₇	—	2	1	no	—
178	C₄²×C₆₇	—	2	1	no	—
179	C₄²×C₆₇	—	2	1	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂³×C₆₇	—	2	1	no	—
186	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂²×D₆₇	—	2	4	no	—
188	C₂²×D₆₇	—	4	2	no	—
189	C₂²×D₆₇	—	4	2	no	—
190	C₂²×D₆₇	—	8	2	no	—
191	C₂²×D₆₇	—	8	2	no	—
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	C₄×D₄	1	2	yes	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	2	2	no	—
196	D₄×C₆₇	—	2	1	no	—
197	D₄×C₆₇	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
208	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
209	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
210	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
217	C₂×Dic₆₇	—	4	2	no	—
218	C₂×Dic₆₇	—	8	2	no	—
219	D₂₆₈	—	2	4	no	—
220	D₂₆₈	—	2	4	no	—
221	D₂₆₈	—	4	2	no	—
222	D₂₆₈	—	4	2	no	—
223	C₄×D₆₇	—	8	4	no	—
224	C₄×D₆₇	—	8	4	no	—
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	—	2	1	no	—
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	1	no	—
229	C₈×C₆₇	C₄×D₄	1	2	yes	—
230	C₈×C₆₇	C₄×D₄	1	2	yes	—
231	C₆₇⋊C₈	—	2	4	no	—
232	C₆₇⋊C₈	—	4	2	no	—
233	C₂²×C₆₇	—	2	1	no	—
234	C₂²×C₆₇	—	2	1	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	4	1	no	—
237	C₂²×C₆₇	—	4	1	no	—
238	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
239	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
240	C₂²×C₆₇	#529	1	1	yes	characteristic
241	C₂×D₆₇	—	4	4	no	—
242	C₂×D₆₇	—	8	4	no	—
243	C₂×D₆₇	—	8	2	no	—
244	C₂×D₆₇	—	8	2	no	—
245	C₂×D₆₇	—	8	4	no	—
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	2	1	no	—
248	C₄×C₆₇	—	2	1	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	4	1	no	—
251	C₄×C₆₇	—	4	1	no	—
252	C₄×C₆₇	#472	1	2	yes	complemented
253	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
254	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₆₇	#525	1	1	yes	characteristic
256	C₄×C₆₇	#532	1	1	yes	characteristic
257	Dic₆₇	—	8	2	no	—
258	C₄×#472	—	67	1	no	maximal, Sylow
259	C₂×C₆₇	—	2	1	no	—
260	C₂×C₆₇	—	2	1	no	—
261	C₂×C₆₇	—	4	2	no	—
262	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
263	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
264	C₂×C₆₇	#3067	1	1	yes	characteristic
265	D₆₇	—	8	4	no	—
266	C₂×#472	—	67	1	no	—
267	C₄×AGL(1,ℤ/8)	—	67	2	no	—
268	C₂×C₄×D₈	—	67	1	no	—
269	C₄×#131	D₆₇	1	1	yes	characteristic, complemented
270	#1978	—	67	1	no	—
271	#1952	—	67	1	no	—
272	#1931	—	67	2	no	—
273	#1907	—	67	2	no	—
274	#1816	—	67	1	no	—
275	C₄×#108	—	67	1	no	—
276	C₄×#100	—	67	2	no	—
277	C₆₇	C₄×#472	1	1	yes	characteristic, complemented, Sylow
278	C₂×AGL(1,ℤ/8)	—	67	2	no	—
279	C₂²×D₈	—	67	1	no	—
280	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
281	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
282	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
283	C₂×C₄×D₄	—	67	2	no	—
284	#472	—	67	8	no	—
285	#442	—	134	2	no	—
286	#440	—	134	2	no	—
287	C₄×SD₁₆	—	134	2	no	—
288	C₄×D₈	—	134	2	no	—
289	C₄×D₈	—	134	1	no	—
290	C₄×D₈	—	134	1	no	—
291	C₄×D₈	—	134	1	no	—
292	C₄×D₈	—	134	1	no	—
293	C₂×#108	—	67	1	no	—
294	C₂×#108	—	67	1	no	—
295	#428	—	67	2	no	—
296	C₂×#107	—	67	1	no	—
297	#418	—	67	2	no	—
298	#417	—	67	2	no	—
299	C₂×#102	—	67	2	no	—
300	C₂×#100	—	67	2	no	—
301	C₂×#100	—	67	2	no	—
302	#405	—	2	1	no	—
303	#405	—	2	1	no	—
304	#405	C₂×D₆₇	1	1	yes	characteristic
305	#405	C₂×D₆₇	1	1	yes	characteristic
306	C₄×M₁₆	—	2	1	no	—
307	C₄×M₁₆	—	67	2	no	—
308	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
309	C₂×C₄×C₈	—	2	1	no	—
310	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
311	#343	—	67	2	no	—
312	#343	—	67	2	no	—
313	#341	—	67	2	no	—
314	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
315	C₂²×D₄	—	67	2	no	—
316	C₂³×C₄	—	134	2	no	—
317	AGL(1,ℤ/8)	—	67	8	no	—
318	AGL(1,ℤ/8)	—	134	4	no	—
319	C₂×SD₁₆	—	134	2	no	—
320	C₂×D₈	—	67	4	no	—
321	C₂×D₈	—	134	2	no	—
322	C₂×D₈	—	134	1	no	—
323	C₂×D₈	—	134	1	no	—
324	C₂×D₈	—	134	2	no	—
325	#131	—	2	2	no	—
326	#131	—	2	2	no	—
327	#131	—	2	2	no	—
328	#131	C₄×D₆₇	1	4	yes	complemented
329	C₂×M₁₆	—	2	1	no	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂×M₁₆	—	67	2	no	—
332	C₂×M₁₆	—	67	2	no	—
333	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
334	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
338	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
339	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
340	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	134	2	no	—
344	C₄×D₄	—	134	2	no	—
345	C₄×D₄	—	134	2	no	—
346	C₄×D₄	—	134	2	no	—
347	C₄×D₄	D₂₆₈	1	1	yes	characteristic
348	C₄×D₄	D₂₆₈	1	1	yes	characteristic
349	#117	—	2	1	no	—
350	#117	C₂²×D₆₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	—	67	2	no	—
352	C₂×C₂²⋊C₄	—	134	2	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
355	#108	—	67	4	no	—
356	#108	—	134	2	no	—
357	#107	—	134	1	no	—
358	#107	—	134	1	no	—
359	#107	—	134	2	no	—
360	#106	—	134	2	no	—
361	#103	—	134	2	no	—
362	#102	—	134	2	no	—
363	#102	—	134	2	no	—
364	#102	—	134	2	no	—
365	#102	—	134	2	no	—
366	#102	—	134	2	no	—
367	#100	—	67	8	no	—
368	#100	—	134	4	no	—
369	#97	—	2	1	no	—
370	#97	—	2	1	no	—
371	#97	—	2	1	no	—
372	#97	—	134	2	no	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	134	2	no	—
377	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
378	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
379	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
380	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
381	C₂⁴	—	134	2	no	—
382	#41	—	2	1	no	—
383	#41	—	2	1	no	—
384	#41	C₂×C₄×D₆₇	1	2	yes	—
385	#41	#11761628	1	2	yes	—
386	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	67	4	no	—
389	C₂×D₄	—	134	2	no	—
390	C₂×D₄	—	134	2	no	—
391	C₂×D₄	—	134	2	no	—
392	C₂×D₄	—	134	2	no	—
393	C₂×D₄	—	134	2	no	—
394	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	268	2	no	—
399	C₂²×C₄	—	268	2	no	—
400	C₂²×C₄	—	536	4	no	—
401	C₂²×C₄	—	134	2	no	—
402	C₂²×C₄	#11761628	1	1	yes	characteristic
403	C₂²×C₄	#11761628	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
405	SD₁₆	—	134	8	no	—
406	D₈	—	134	8	no	—
407	D₈	—	134	4	no	—
408	D₈	—	134	4	no	—
409	M₁₆	—	2	4	no	—
410	M₁₆	—	2	1	no	—
411	M₁₆	—	2	1	no	—
412	M₁₆	—	2	4	no	—
413	M₁₆	—	67	4	no	—
414	M₁₆	—	134	2	no	—
415	M₁₆	—	134	2	no	—
416	M₁₆	—	134	2	no	—
417	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
418	M₁₆	#11761628	1	2	yes	—
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	4	no	—
425	C₂×C₈	—	134	2	no	—
426	C₂×C₈	—	134	2	no	—
427	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
428	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
429	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
430	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
431	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
432	C₂×C₈	#11761628	1	2	yes	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	134	2	no	—
436	C₄⋊C₄	—	134	2	no	—
437	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
438	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
439	C₂²⋊C₄	—	2	1	no	—
440	C₂²⋊C₄	—	2	1	no	—
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	268	2	no	—
443	C₂²⋊C₄	—	268	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
448	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
449	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
450	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
451	C₂³	—	2	1	no	—
452	C₂³	—	268	2	no	—
453	C₂³	—	268	2	no	—
454	C₂³	—	536	2	no	—
455	C₂³	—	536	2	no	—
456	C₂³	—	134	4	no	—
457	C₂³	C₂×#11761628	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	C₄×D₂₆₈	1	2	yes	—
460	D₄	—	2	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	1	no	—
463	D₄	—	268	2	no	—
464	D₄	—	268	2	no	—
465	D₄	—	134	4	no	—
466	D₄	—	134	4	no	—
467	D₄	C₄×D₂₆₈	1	2	yes	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	268	2	no	—
478	C₂×C₄	—	536	4	no	—
479	C₂×C₄	—	536	4	no	—
480	C₂×C₄	—	536	2	no	—
481	C₂×C₄	#14524017	1	1	yes	characteristic
482	C₂×C₄	#14524017	1	1	yes	characteristic
483	C₂×C₄	#14524017	1	1	yes	characteristic
484	C₂×C₄	#14524017	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
488	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
489	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
490	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	1	no	—
493	C₈	—	2	2	no	—
494	C₈	—	2	1	no	—
495	C₈	—	268	2	no	—
496	C₈	—	134	4	no	—
497	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
498	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	268	4	no	—
505	C₂²	—	536	4	no	—
506	C₂²	—	536	4	no	—
507	C₂²	—	536	2	no	—
508	C₂²	—	536	2	no	—
509	C₂²	C₄×#11761628	1	1	yes	characteristic
510	C₂²	C₂×#14524017	1	1	yes	characteristic
511	C₂²	#16863930	1	1	yes	characteristic
512	C₄	—	2	2	no	—
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	536	2	no	—
519	C₄	#16863666	1	2	yes	direct product factor
520	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
521	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
522	C₄	#16863907	1	1	yes	characteristic
523	C₄	#16863946	1	1	yes	characteristic
524	C₂	—	2	1	no	—
525	C₂	—	2	1	no	—
526	C₂	—	4	2	no	—
527	C₂	—	536	4	no	—
528	C₂	C₄×#14524017	1	1	yes	characteristic
529	C₂	C₂×#16863666	1	1	yes	characteristic
530	C₂	#22569929	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#16863666	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	C₄×#472
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#14524068	C₂	semidirect
3. C₂×C₄×D₅₃₆	C₂	semidirect
4. #22561119	C₂	semidirect
5. #22560994	C₂	semidirect
6. #22560871	C₂	semidirect
7. #22560717	C₂	semidirect
8. #22560273	C₂	semidirect
9. C₄×#14523979	C₂	semidirect
10. C₄×#14523959	C₂	semidirect
11. C₄×#131×C₆₇	C₂	semidirect
15. #16863666	C₄	direct (N × H)
51. #14524068	C₂×C₄	semidirect
56. C₂×D₅₃₆	C₂×C₄	semidirect
67. #14523979	C₂×C₄	semidirect
79. #14523959	C₂×C₄	semidirect
86. #131×C₆₇	C₂×C₄	semidirect
252. C₄×C₆₇	#472	semidirect
269. C₄×#131	D₆₇	semidirect
277. C₆₇	C₄×#472	semidirect
328. #131	C₄×D₆₇	semidirect
519. C₄	#16863666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863666	2
C₄×#14524068	2
C₂×C₄×D₅₃₆	2
#22561119	2
#22560994	2
#22560871	2
#22560717	2
#22560273	2
C₄×#14523979	2
C₄×#14523959	2
C₄×#131×C₆₇	2
C₄×#472	67

Subgroups — C4×#16863666 #30862711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863666 #30862711