Subgroups — #30859051

Order 17152.

800 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	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
2	#22568854	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524050	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560855	C₂	1	2	yes	maximal, complemented
5	#22560806	C₂	1	2	yes	maximal, complemented
6	#22560534	C₂	1	2	yes	maximal
7	#22560482	C₂	1	2	yes	maximal, complemented
8	C₄×#14523978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523971	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559955	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	C₂×#14524050	C₂²	1	2	yes	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
16	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
17	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
22	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
23	#16863458	—	2	2	no	—
24	#16863447	—	2	1	no	—
25	#16863447	—	2	1	no	—
26	C₂×#14523978	C₂²	1	2	yes	—
27	#16863432	C₂²	1	1	yes	characteristic
28	#16863424	C₂²	1	2	yes	—
29	C₂×#14523971	C₂²	1	2	yes	—
30	#16863392	C₂²	1	1	yes	characteristic
31	#16863390	C₂²	1	1	yes	characteristic
32	#16863364	C₂²	1	2	yes	—
33	#16863352	—	2	2	no	—
34	#16863352	C₂²	1	2	yes	—
35	C₄×#11761612	—	2	1	no	—
36	C₄×#11761612	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
38	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#16863160	C₂²	1	2	yes	—
41	#16863156	C₂²	1	2	yes	—
42	#16863151	C₂²	1	2	yes	—
43	#16863151	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#104×C₆₇	C₂²	1	2	yes	—
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₄³×C₆₇	C₂²	1	1	yes	characteristic
48	#339×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	#14524063	—	2	4	no	—
51	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
55	#14524050	—	2	4	no	—
56	#14524050	—	2	4	no	—
57	C₂×#11761612	—	2	2	no	—
58	C₂×#11761612	C₂³	1	2	yes	—
59	C₂²×C₆₇⋊C₈	—	2	2	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
63	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₂²⋊D₆₇	—	2	1	no	—
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	#14524002	C₂³	1	1	yes	characteristic
70	#14524000	—	2	1	no	—
71	C₄²×D₆₇	—	2	1	no	—
72	#14523978	—	2	4	no	—
73	#14523976	C₂×C₄	1	1	yes	characteristic
74	#14523976	C₂×C₄	1	1	yes	characteristic
75	#14523974	C₂×C₄	1	1	yes	characteristic
76	#14523974	C₂×C₄	1	1	yes	characteristic
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	#14523971	—	2	4	no	—
80	#14523967	C₂×C₄	1	1	yes	characteristic
81	#14523967	C₂×C₄	1	1	yes	characteristic
82	#14523966	C₂×C₄	1	1	yes	characteristic
83	#14523966	C₂×C₄	1	1	yes	characteristic
84	#14523964	C₂×C₄	1	1	yes	characteristic
85	#14523964	C₂×C₄	1	1	yes	characteristic
86	#14523963	C₂×C₄	1	1	yes	characteristic
87	#14523963	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₆₇)⋊C₈	—	2	2	no	—
89	(C₂×D₆₇)⋊C₈	—	2	2	no	—
90	#14523954	—	2	2	no	—
91	#14523954	—	2	1	no	—
92	C₄×C₆₇⋊C₈	—	2	2	no	—
93	C₄×C₆₇⋊C₈	—	2	1	no	—
94	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
98	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
99	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
100	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
101	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₆₇	—	2	1	no	—
106	C₄×D₄×C₆₇	—	2	1	no	—
107	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
109	#117×C₆₇	—	2	1	no	—
110	#117×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	—	2	1	no	—
112	C₂×C₄²×C₆₇	—	2	1	no	—
113	C₂×C₄²×C₆₇	—	2	1	no	—
114	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	#104×C₆₇	—	2	4	no	—
117	#97×C₆₇	—	2	1	no	—
118	C₄×C₈×C₆₇	—	2	1	no	—
119	#11761628	—	2	2	no	—
120	#11761628	—	2	2	no	—
121	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₂²⋊D₆₇	—	2	1	no	—
124	C₂×C₄×D₆₇	—	2	1	no	—
125	C₂×C₄×D₆₇	—	2	1	no	—
126	C₂×C₄×D₆₇	—	2	1	no	—
127	#11761618	—	2	2	no	—
128	#11761616	—	2	2	no	—
129	#11761615	—	2	2	no	—
130	#11761613	—	2	2	no	—
131	#11761612	—	2	8	no	—
132	#11761612	—	2	2	no	—
133	#11761612	#41	1	4	yes	—
134	C₂×C₆₇⋊C₈	—	2	8	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	—	2	2	no	—
139	C₂×C₆₇⋊C₈	—	2	2	no	—
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
145	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
146	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
148	Dic₆₇⋊C₄	—	2	1	no	—
149	Dic₆₇⋊C₄	—	2	1	no	—
150	C₂²⋊Dic₆₇	—	2	1	no	—
151	(C₂×D₆₇)⋊C₄	—	2	1	no	—
152	(C₂×D₆₇)⋊C₄	—	2	1	no	—
153	C₄×Dic₆₇	—	2	1	no	—
154	C₄×Dic₆₇	—	2	1	no	—
155	C₄×Dic₆₇	—	2	1	no	—
156	#41×C₆₇	—	2	2	no	—
157	#41×C₆₇	#41	1	4	yes	—
158	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	1	no	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₆₇	C₂²×C₄	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	4	no	—
179	C₄²×C₆₇	—	2	1	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	2	no	—
183	C₄²×C₆₇	—	2	2	no	—
184	C₄²×C₆₇	—	4	2	no	—
185	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₂²×D₆₇	—	2	1	no	—
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	#117	1	2	yes	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	2	1	no	—
196	D₄×C₆₇	#117	1	2	yes	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
216	C₂²⋊D₆₇	—	4	2	no	—
217	C₂×Dic₆₇	—	2	1	no	—
218	C₂×Dic₆₇	—	2	1	no	—
219	C₂×Dic₆₇	—	2	1	no	—
220	D₂₆₈	—	2	1	no	—
221	D₂₆₈	—	2	1	no	—
222	C₄×D₆₇	—	4	2	no	—
223	C₄×D₆₇	—	4	2	no	—
224	C₄×D₆₇	—	4	2	no	—
225	Dic₁₃₄	—	2	1	no	—
226	Dic₁₃₄	—	2	1	no	—
227	C₈×C₆₇	—	4	2	no	—
228	C₆₇⋊C₈	—	2	1	no	—
229	C₆₇⋊C₈	—	2	1	no	—
230	C₆₇⋊C₈	—	2	1	no	—
231	C₆₇⋊C₈	—	2	1	no	—
232	C₆₇⋊C₈	—	2	2	no	—
233	C₆₇⋊C₈	—	2	4	no	—
234	C₆₇⋊C₈	—	2	2	no	—
235	C₂²×C₆₇	—	2	1	no	—
236	C₂²×C₆₇	—	2	1	no	—
237	C₂²×C₆₇	—	2	2	no	—
238	C₂²×C₆₇	—	4	1	no	—
239	C₂²×C₆₇	—	4	1	no	—
240	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₆₇	#529	1	2	yes	—
242	C₂×D₆₇	—	4	1	no	—
243	C₂×D₆₇	—	4	1	no	—
244	C₂×D₆₇	—	4	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	4	no	—
249	C₄×C₆₇	—	4	1	no	—
250	C₄×C₆₇	—	4	1	no	—
251	C₄×C₆₇	#443	1	2	yes	—
252	C₄×C₆₇	#444	1	2	yes	—
253	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
254	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
255	C₄×C₆₇	#525	1	2	yes	—
256	Dic₆₇	—	4	1	no	—
257	Dic₆₇	—	4	1	no	—
258	Dic₆₇	—	4	1	no	—
259	#21137	—	67	1	no	maximal, Sylow
260	C₂×C₆₇	—	2	2	no	—
261	C₂×C₆₇	—	4	2	no	—
262	C₂×C₆₇	#2290	1	1	yes	characteristic
263	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
264	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
265	D₆₇	—	8	1	no	—
266	C₄×#135	—	67	1	no	—
267	#2922	—	67	1	no	—
268	C₄×#131	—	67	1	no	—
269	#1929	—	67	2	no	—
270	#1922	—	67	2	no	—
271	#1876	—	67	2	no	—
272	#1866	—	67	2	no	—
273	C₄×#108	—	67	1	no	—
274	C₄×#104	—	67	1	no	—
275	C₄×#104	D₆₇	1	1	yes	characteristic, complemented
276	#1744	—	67	1	no	—
277	C₆₇	#21137	1	1	yes	characteristic, complemented, Sylow
278	C₂×#135	—	67	1	no	—
279	C₂×#131	—	67	2	no	—
280	C₄×#41	—	67	1	no	—
281	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
282	C₄×Q₁₆	—	67	1	no	—
283	C₄×Q₁₆	—	67	1	no	—
284	C₄×SD₁₆	—	67	1	no	—
285	C₄×SD₁₆	—	67	1	no	—
286	C₄×SD₁₆	—	67	1	no	—
287	C₄×SD₁₆	—	67	1	no	—
288	C₄×D₈	—	67	1	no	—
289	C₄×D₈	—	67	1	no	—
290	#433	—	134	2	no	—
291	#431	—	134	1	no	—
292	#431	—	134	1	no	—
293	C₂×#108	—	67	2	no	—
294	#427	—	67	1	no	—
295	#424	—	67	2	no	—
296	C₂×#104	—	67	2	no	—
297	C₂×#104	C₂×D₆₇	1	2	yes	—
298	#416	—	67	1	no	—
299	#416	—	67	1	no	—
300	#408	—	67	2	no	—
301	#405	—	67	2	no	—
302	#405	—	134	2	no	—
303	C₄×M₁₆	—	67	1	no	—
304	C₄×M₁₆	—	134	1	no	—
305	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
306	C₂×C₄×C₈	—	67	1	no	—
307	C₂×C₄×C₈	—	67	1	no	—
308	C₂×C₄×C₈	—	134	1	no	—
309	C₄³	C₂×D₆₇	1	1	yes	characteristic
310	#341	—	67	2	no	—
311	#339	—	67	2	no	—
312	#339	C₂×D₆₇	1	2	yes	—
313	#336	—	67	2	no	—
314	#336	—	67	2	no	—
315	C₂×#41	—	67	1	no	—
316	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
317	#135	—	134	4	no	—
318	C₂×Q₁₆	—	67	1	no	—
319	C₂×SD₁₆	—	67	1	no	—
320	C₂×SD₁₆	—	67	1	no	—
321	C₂×D₈	—	67	1	no	—
322	#131	—	134	4	no	—
323	#131	—	134	4	no	—
324	C₂×M₁₆	—	67	2	no	—
325	C₂×M₁₆	—	134	2	no	—
326	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
327	C₂²×C₈	—	67	2	no	—
328	C₂²×C₈	—	67	1	no	—
329	C₂²×C₈	—	67	1	no	—
330	C₂²×C₈	—	134	2	no	—
331	C₄×Q₈	—	67	1	no	—
332	C₄×Q₈	—	67	1	no	—
333	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
334	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	67	1	no	—
338	C₄×D₄	—	67	1	no	—
339	C₄×D₄	—	134	1	no	—
340	C₄×D₄	—	134	1	no	—
341	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
342	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
343	#117	—	2	1	no	—
344	#117	—	67	1	no	—
345	#117	—	134	1	no	—
346	#117	C₂²×D₆₇	1	1	yes	characteristic
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	134	1	no	—
351	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
352	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
353	#108	—	134	4	no	—
354	#107	—	67	1	no	—
355	#107	—	67	1	no	—
356	#106	—	67	1	no	—
357	#106	—	67	1	no	—
358	C₄⋊C₈	—	134	2	no	—
359	C₄⋊C₈	—	134	2	no	—
360	#104	—	2	4	no	—
361	#104	—	134	4	no	—
362	#103	—	67	1	no	—
363	#103	—	67	1	no	—
364	#103	—	67	1	no	—
365	#103	—	67	1	no	—
366	#102	—	67	1	no	—
367	#102	—	67	1	no	—
368	#102	—	67	1	no	—
369	#102	—	67	1	no	—
370	C₂²⋊C₈	—	134	2	no	—
371	C₂²⋊C₈	—	134	2	no	—
372	#97	—	2	1	no	—
373	#97	—	134	2	no	—
374	#97	—	134	1	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	67	1	no	—
377	C₄×C₈	—	67	1	no	—
378	C₄×C₈	—	67	1	no	—
379	C₄×C₈	—	67	1	no	—
380	C₄×C₈	—	67	2	no	—
381	C₄×C₈	—	67	1	no	—
382	C₄×C₈	—	67	1	no	—
383	C₄×C₈	—	134	2	no	—
384	C₄×C₈	—	134	1	no	—
385	#41	—	2	2	no	—
386	#41	—	134	2	no	—
387	#41	—	134	2	no	—
388	#41	#11761628	1	4	yes	—
389	C₂×Q₈	—	67	1	no	—
390	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	67	1	no	—
393	C₂×D₄	—	134	1	no	—
394	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	134	1	no	—
400	C₂²×C₄	—	134	1	no	—
401	C₂²×C₄	—	134	1	no	—
402	C₂²×C₄	#11761627	1	1	yes	characteristic
403	C₂²×C₄	#11761627	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
405	Q₁₆	—	134	2	no	—
406	SD₁₆	—	134	2	no	—
407	SD₁₆	—	134	2	no	—
408	D₈	—	134	2	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	67	4	no	—
412	M₁₆	—	134	8	no	—
413	M₁₆	—	134	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	67	1	no	—
416	C₂×C₈	—	67	1	no	—
417	C₂×C₈	—	67	1	no	—
418	C₂×C₈	—	67	1	no	—
419	C₂×C₈	—	67	2	no	—
420	C₂×C₈	—	67	2	no	—
421	C₂×C₈	—	134	8	no	—
422	C₂×C₈	—	134	2	no	—
423	C₂×C₈	—	134	2	no	—
424	C₂×C₈	—	134	2	no	—
425	C₂×C₈	—	134	2	no	—
426	C₂×C₈	—	134	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	67	1	no	—
430	C₄⋊C₄	—	67	1	no	—
431	C₄⋊C₄	—	134	1	no	—
432	C₄⋊C₄	—	134	1	no	—
433	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
434	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	2	1	no	—
438	C₂²⋊C₄	—	134	1	no	—
439	C₂²⋊C₄	—	134	1	no	—
440	C₂²⋊C₄	—	134	1	no	—
441	C₄²	—	2	4	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	4	2	no	—
448	C₄²	—	134	1	no	—
449	C₄²	—	134	1	no	—
450	C₄²	—	134	1	no	—
451	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
452	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
453	C₄²	D₄×D₆₇	1	1	yes	characteristic
454	C₄²	D₄×D₆₇	1	1	yes	characteristic
455	C₂³	—	2	1	no	—
456	C₂³	—	134	1	no	—
457	C₂³	C₂×#11761627	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	—	134	1	no	—
460	Q₈	—	134	1	no	—
461	Q₈	#14524000	1	2	yes	—
462	D₄	—	2	4	no	—
463	D₄	—	2	1	no	—
464	D₄	—	268	2	no	—
465	D₄	—	134	1	no	—
466	D₄	—	134	1	no	—
467	D₄	#14524000	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	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	268	2	no	—
480	C₂×C₄	—	268	2	no	—
481	C₂×C₄	—	268	2	no	—
482	C₂×C₄	—	134	1	no	—
483	C₂×C₄	—	134	1	no	—
484	C₂×C₄	—	134	1	no	—
485	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
486	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
487	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
488	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
489	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
491	C₂×C₄	#41×D₆₇	1	2	yes	—
492	C₂×C₄	C₂×#11761628	1	2	yes	—
493	C₈	—	4	2	no	—
494	C₈	—	134	1	no	—
495	C₈	—	134	1	no	—
496	C₈	—	134	1	no	—
497	C₈	—	134	1	no	—
498	C₈	—	134	2	no	—
499	C₈	—	134	2	no	—
500	C₈	—	134	4	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	268	1	no	—
507	C₂²	—	268	1	no	—
508	C₂²	—	268	1	no	—
509	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
510	C₂²	#16863926	1	2	yes	—
511	C₄	—	2	2	no	—
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	1	no	—
516	C₄	—	4	1	no	—
517	C₄	—	268	1	no	—
518	C₄	—	268	1	no	—
519	C₄	—	268	1	no	—
520	C₄	#16863510	1	2	yes	—
521	C₄	#16863515	1	2	yes	—
522	C₄	C₂×#14524000	1	1	yes	characteristic
523	C₄	#16863864	1	1	yes	characteristic
524	C₄	#16863905	1	2	yes	—
525	C₂	—	2	2	no	—
526	C₂	—	4	2	no	—
527	C₂	—	536	1	no	—
528	C₂	#22562648	1	1	yes	characteristic
529	C₂	C₂×#16863510	1	1	yes	characteristic
530	C₂	C₂×#16863515	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#104×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#104	D₆₇
hypocenter	C₆₇	#21137
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#104

Complements

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

N	H ≅ G/N	type
2. #22568854	C₂	semidirect
3. C₄×#14524050	C₂	semidirect
4. #22560855	C₂	semidirect
5. #22560806	C₂	semidirect
7. #22560482	C₂	semidirect
8. C₄×#14523978	C₂	semidirect
9. C₄×#14523971	C₂	semidirect
10. #22559955	C₂	semidirect
11. C₄×#104×C₆₇	C₂	semidirect
15. C₄×#11761618	C₄	semidirect
16. C₄×#11761618	C₄	semidirect
17. C₄×#11761615	C₄	semidirect
18. C₄×#11761615	C₄	semidirect
19. C₄×#11761616	C₄	semidirect
20. C₄×#11761616	C₄	semidirect
21. C₄×#11761613	C₄	semidirect
22. C₄×#11761613	C₄	semidirect
275. C₄×#104	D₆₇	semidirect
277. C₆₇	#21137	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524063	2
#22568854	2
C₄×#14524050	2
#22560855	2
#22560806	2
#22560534	2
#22560482	2
C₄×#14523978	2
C₄×#14523971	2
#22559955	2
C₄×#104×C₆₇	2
#21137	67