Subgroups — #30862279

Order 17152.

910 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569664	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863515	C₂	1	2	yes	maximal, complemented
3	#22568920	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#14524050	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561127	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560855	C₂	1	2	yes	maximal, complemented
8	#22560534	C₂	1	2	yes	maximal, complemented
9	#22560078	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560050	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	2	yes	—
13	C₂×#14524051	C₂²	1	2	yes	—
14	C₂×#14524050	C₂²	1	1	yes	characteristic
15	C₂×#14524050	C₂²	1	1	yes	characteristic
16	#16863946	C₂²	1	1	yes	characteristic
17	#16863618	C₄	1	2	yes	complemented
18	#16863611	C₄	1	2	yes	complemented
19	#16863606	C₄	1	1	yes	characteristic, complemented
20	#16863601	C₄	1	1	yes	characteristic, complemented
21	#16863596	C₄	1	1	yes	characteristic, complemented
22	#16863591	C₄	1	1	yes	characteristic, complemented
23	#16863515	—	2	8	no	—
24	#16863473	—	2	1	no	—
25	#16863473	—	2	1	no	—
26	#16863461	—	2	1	no	—
27	#16863461	—	2	1	no	—
28	C₂×#14523978	C₂²	1	2	yes	—
29	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂×#14523971	C₂²	1	2	yes	—
31	C₂×#14523971	C₂²	1	2	yes	—
32	#16863390	C₂²	1	2	yes	—
33	#16863364	C₂²	1	2	yes	—
34	#16863352	—	2	1	no	—
35	#16863352	—	2	1	no	—
36	#16863352	C₂²	1	1	yes	characteristic
37	#16863352	C₂²	1	1	yes	characteristic
38	C₄×#11761612	—	2	1	no	—
39	C₄×#11761612	C₂²	1	1	yes	characteristic
40	C₂×#14523954	C₂²	1	2	yes	—
41	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
42	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
43	#16863160	C₂²	1	2	yes	—
44	#16863151	C₂²	1	2	yes	—
45	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
46	#423×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#104×C₆₇	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
49	#339×C₆₇	C₂²	1	2	yes	—
50	C₂×#11761628	C₂³	1	2	yes	—
51	#14524063	—	2	4	no	—
52	#14524063	—	2	4	no	—
53	C₂×#11761618	C₂×C₄	1	2	yes	—
54	C₂×#11761616	C₂×C₄	1	2	yes	—
55	C₂×#11761615	C₂×C₄	1	2	yes	—
56	C₂×#11761613	C₂×C₄	1	2	yes	—
57	#14524051	—	2	4	no	—
58	#14524051	—	2	4	no	—
59	#14524050	—	2	2	no	—
60	#14524050	—	2	2	no	—
61	#14524050	—	2	2	no	—
62	#14524050	D₄	1	4	yes	complemented
63	C₂×#11761612	—	2	1	no	—
64	C₂×#11761612	—	2	1	no	—
65	C₂×#11761612	C₂³	1	1	yes	characteristic
66	C₂×#11761612	C₂³	1	1	yes	characteristic
67	C₂×#11761611	—	2	2	no	—
68	C₂²×C₆₇⋊C₈	—	2	1	no	—
69	C₂²×C₆₇⋊C₈	—	2	1	no	—
70	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
71	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈×D₆₇	—	2	2	no	—
74	#14524043	C₂×C₄	1	1	yes	characteristic
75	#14524040	C₂×C₄	1	1	yes	characteristic
76	#14524031	C₂×C₄	1	2	yes	—
77	#14524017	—	2	1	no	—
78	#14524017	—	2	1	no	—
79	C₄×D₂₆₈	—	2	1	no	—
80	C₄×D₂₆₈	—	2	1	no	—
81	#14523978	—	2	4	no	—
82	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
83	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
84	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
85	#14523971	—	2	4	no	—
86	#14523971	—	2	4	no	—
87	#14523966	C₂×C₄	1	2	yes	—
88	#14523966	C₂×C₄	1	2	yes	—
89	#14523963	C₂×C₄	1	2	yes	—
90	#14523963	C₂×C₄	1	2	yes	—
91	(C₂×D₆₇)⋊C₈	—	2	2	no	—
92	(C₂×D₆₇)⋊C₈	—	2	2	no	—
93	#14523954	—	2	4	no	—
94	#14523954	—	2	1	no	—
95	#14523954	—	2	1	no	—
96	#14523954	—	2	1	no	—
97	C₄×C₆₇⋊C₈	—	2	1	no	—
98	C₄×C₆₇⋊C₈	—	2	1	no	—
99	C₄×C₆₇⋊C₈	—	2	1	no	—
100	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
101	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
102	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
103	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
104	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
106	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₆₇	—	2	1	no	—
109	C₄×D₄×C₆₇	—	2	1	no	—
110	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
112	#117×C₆₇	—	2	1	no	—
113	#117×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄²×C₆₇	—	2	1	no	—
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₆₇	—	2	1	no	—
119	C₄⋊C₈×C₆₇	—	2	1	no	—
120	#104×C₆₇	—	2	4	no	—
121	#95×C₆₇	—	2	1	no	—
122	#11761628	—	2	2	no	—
123	#11761628	—	2	2	no	—
124	#11761628	—	2	2	no	—
125	#11761628	—	2	2	no	—
126	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
127	C₂×D₂₆₈	—	2	1	no	—
128	C₂×D₂₆₈	—	2	1	no	—
129	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
130	C₂×C₂²⋊D₆₇	—	2	2	no	—
131	C₂×C₄×D₆₇	—	2	2	no	—
132	#11761618	—	2	4	no	—
133	#11761616	—	2	4	no	—
134	#11761615	—	2	4	no	—
135	#11761613	—	2	4	no	—
136	#11761612	—	2	4	no	—
137	#11761612	—	2	1	no	—
138	#11761612	—	2	1	no	—
139	#11761612	—	2	4	no	—
140	#11761612	C₂×D₄	1	2	yes	—
141	#11761612	#41	1	2	yes	—
142	#11761611	—	4	4	no	—
143	C₂×C₆₇⋊C₈	—	2	4	no	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	—	2	2	no	—
147	C₂×C₆₇⋊C₈	—	2	1	no	—
148	C₂×C₆₇⋊C₈	—	2	1	no	—
149	C₂×C₆₇⋊C₈	—	2	2	no	—
150	C₂×C₆₇⋊C₈	—	2	4	no	—
151	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
152	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
153	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
154	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
155	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
156	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
157	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
158	C₂×C₆₇⋊C₈	#41	1	2	yes	—
159	C₈×D₆₇	—	4	4	no	—
160	C₄⋊Dic₆₇	—	2	1	no	—
161	C₄⋊Dic₆₇	—	2	1	no	—
162	(C₂×D₆₇)⋊C₄	—	2	2	no	—
163	(C₂×D₆₇)⋊C₄	—	2	2	no	—
164	#41×C₆₇	—	2	1	no	—
165	#41×C₆₇	—	2	1	no	—
166	#41×C₆₇	C₂×D₄	1	2	yes	—
167	#41×C₆₇	#41	1	2	yes	—
168	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂×D₄×C₆₇	—	2	1	no	—
170	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	M₁₆×C₆₇	—	2	2	no	—
179	M₁₆×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	4	2	no	—
184	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₂²⋊C₄×C₆₇	—	2	1	no	—
187	C₂²⋊C₄×C₆₇	—	2	1	no	—
188	C₂²⋊C₄×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂²×D₆₇	—	2	2	no	—
201	Q₈×C₆₇	—	2	1	no	—
202	Q₈×C₆₇	C₄×D₄	1	2	yes	—
203	D₄×C₆₇	—	2	2	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	2	1	no	—
206	D₄×C₆₇	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
229	C₂²⋊D₆₇	—	4	2	no	—
230	C₂²⋊D₆₇	—	4	2	no	—
231	C₂×Dic₆₇	—	2	2	no	—
232	D₂₆₈	—	2	2	no	—
233	D₂₆₈	—	2	2	no	—
234	D₂₆₈	—	4	2	no	—
235	D₂₆₈	—	4	2	no	—
236	C₄×D₆₇	—	4	2	no	—
237	C₄×D₆₇	—	4	2	no	—
238	Dic₁₃₄	—	2	2	no	—
239	Dic₁₃₄	—	2	2	no	—
240	C₈×C₆₇	—	4	2	no	—
241	C₆₇⋊C₈	—	2	2	no	—
242	C₆₇⋊C₈	—	2	2	no	—
243	C₆₇⋊C₈	—	2	2	no	—
244	C₆₇⋊C₈	—	2	1	no	—
245	C₆₇⋊C₈	—	2	2	no	—
246	C₆₇⋊C₈	—	2	1	no	—
247	C₆₇⋊C₈	#121	1	2	yes	—
248	C₆₇⋊C₈	#121	1	2	yes	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	—	4	1	no	—
254	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₂²×C₆₇	#529	1	1	yes	characteristic
256	C₂²×C₆₇	#532	1	1	yes	characteristic
257	C₂×D₆₇	—	4	2	no	—
258	C₂×D₆₇	—	4	2	no	—
259	C₂×D₆₇	—	4	2	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	2	1	no	—
262	C₄×C₆₇	—	2	1	no	—
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	4	1	no	—
265	C₄×C₆₇	—	4	1	no	—
266	C₄×C₆₇	—	8	2	no	—
267	C₄×C₆₇	#444	1	2	yes	complemented
268	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
270	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
271	C₄×C₆₇	#525	1	1	yes	characteristic
272	Dic₆₇	—	4	2	no	—
273	#21518	—	67	1	no	maximal, Sylow
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	4	2	no	—
277	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
278	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
279	C₂×C₆₇	#2955	1	1	yes	characteristic
280	D₆₇	—	8	2	no	—
281	#3035	—	67	1	no	—
282	C₂×#444	—	67	2	no	—
283	#2931	—	67	1	no	—
284	C₄×#131	—	67	1	no	—
285	#1980	—	67	1	no	—
286	#1950	—	67	1	no	—
287	#1929	—	67	2	no	—
288	#1876	—	67	2	no	—
289	#1806	D₆₇	1	1	yes	characteristic, complemented
290	#1770	—	67	1	no	—
291	#1765	—	67	2	no	—
292	C₆₇	#21518	1	1	yes	characteristic, complemented, Sylow
293	C₂×#135	—	67	2	no	—
294	C₂×#131	—	67	1	no	—
295	C₂×#131	—	67	1	no	—
296	C₂×#131	—	67	2	no	—
297	#532	—	67	1	no	—
298	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
299	#464	—	67	2	no	—
300	#463	—	67	2	no	—
301	#462	—	67	1	no	—
302	#461	—	67	1	no	—
303	#460	—	67	1	no	—
304	#459	—	67	1	no	—
305	#444	—	134	8	no	—
306	#436	—	134	1	no	—
307	#436	—	134	1	no	—
308	C₈×D₄	—	134	1	no	—
309	C₈×D₄	—	134	1	no	—
310	C₂×#108	—	67	2	no	—
311	#423	C₂×D₆₇	1	1	yes	characteristic
312	C₂×C₄⋊C₈	—	67	1	no	—
313	C₂×#104	—	67	2	no	—
314	C₂×#104	—	67	2	no	—
315	C₂×#104	C₂×D₆₇	1	2	yes	—
316	#416	—	67	2	no	—
317	#408	—	67	2	no	—
318	#405	—	67	1	no	—
319	#405	—	67	1	no	—
320	#405	—	134	1	no	—
321	#405	—	134	1	no	—
322	C₄×M₁₆	—	67	1	no	—
323	C₄×M₁₆	—	134	1	no	—
324	C₂×#97	—	67	2	no	—
325	C₂×C₄×C₈	—	67	1	no	—
326	C₂×C₄×C₈	—	134	1	no	—
327	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
328	#341	—	67	2	no	—
329	#339	C₂×D₆₇	1	2	yes	—
330	#336	—	67	2	no	—
331	C₂×#41	—	67	2	no	—
332	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
333	#135	—	134	4	no	—
334	#135	—	134	4	no	—
335	C₂×Q₁₆	—	67	2	no	—
336	C₂×SD₁₆	—	67	2	no	—
337	C₂×SD₁₆	—	67	2	no	—
338	C₂×D₈	—	67	2	no	—
339	#131	—	67	4	no	—
340	#131	—	134	4	no	—
341	#131	—	134	4	no	—
342	#131	—	134	2	no	—
343	#131	—	134	2	no	—
344	#131	—	134	2	no	—
345	C₂×M₁₆	—	67	1	no	—
346	C₂×M₁₆	—	67	1	no	—
347	C₂×M₁₆	—	134	2	no	—
348	C₂×M₁₆	—	134	1	no	—
349	C₂×M₁₆	—	134	1	no	—
350	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
351	C₂²×C₈	—	67	1	no	—
352	C₂²×C₈	—	67	1	no	—
353	C₂²×C₈	—	67	2	no	—
354	C₂²×C₈	—	134	2	no	—
355	C₂²×C₈	—	134	1	no	—
356	C₂²×C₈	—	134	1	no	—
357	C₄⋊Q₈	—	67	1	no	—
358	#127	—	67	1	no	—
359	#124	—	67	2	no	—
360	#121	—	134	1	no	—
361	#121	—	134	1	no	—
362	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
363	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	134	1	no	—
367	C₄×D₄	—	134	1	no	—
368	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
369	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
370	#117	—	2	1	no	—
371	#117	C₂²×D₆₇	1	1	yes	characteristic
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
375	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
376	#108	—	134	4	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	67	1	no	—
380	C₄⋊C₈	—	67	1	no	—
381	C₄⋊C₈	—	67	2	no	—
382	#104	—	2	4	no	—
383	#104	—	134	4	no	—
384	#104	—	134	4	no	—
385	#103	—	67	2	no	—
386	#103	—	67	2	no	—
387	#102	—	67	2	no	—
388	#102	—	67	2	no	—
389	C₂²⋊C₈	—	134	2	no	—
390	C₂²⋊C₈	—	134	2	no	—
391	#97	—	134	4	no	—
392	#97	—	134	1	no	—
393	#97	—	134	1	no	—
394	#97	—	134	1	no	—
395	C₄×C₈	—	67	1	no	—
396	C₄×C₈	—	67	1	no	—
397	C₄×C₈	—	67	1	no	—
398	C₄×C₈	—	67	1	no	—
399	C₄×C₈	—	134	1	no	—
400	C₄×C₈	—	134	1	no	—
401	C₄×C₈	—	134	1	no	—
402	#95	—	2	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	134	2	no	—
406	#41	—	134	2	no	—
407	#41	—	134	2	no	—
408	#41	—	134	2	no	—
409	#41	D₄×D₆₇	1	2	yes	—
410	#41	#11761629	1	2	yes	—
411	C₂×Q₈	—	67	2	no	—
412	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	67	2	no	—
415	C₂×D₄	—	134	2	no	—
416	C₂×D₄	—	134	1	no	—
417	C₂×D₄	—	134	1	no	—
418	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	134	2	no	—
424	C₂²×C₄	#11761627	1	1	yes	characteristic
425	C₂²×C₄	#11761627	1	1	yes	characteristic
426	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
427	Q₁₆	—	134	4	no	—
428	SD₁₆	—	134	4	no	—
429	SD₁₆	—	134	4	no	—
430	D₈	—	134	4	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	268	4	no	—
434	M₁₆	—	67	2	no	—
435	M₁₆	—	67	2	no	—
436	M₁₆	—	134	4	no	—
437	M₁₆	—	134	4	no	—
438	M₁₆	—	134	1	no	—
439	M₁₆	—	134	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	268	4	no	—
442	C₂×C₈	—	67	2	no	—
443	C₂×C₈	—	67	1	no	—
444	C₂×C₈	—	67	1	no	—
445	C₂×C₈	—	67	2	no	—
446	C₂×C₈	—	67	2	no	—
447	C₂×C₈	—	67	2	no	—
448	C₂×C₈	—	67	1	no	—
449	C₂×C₈	—	67	1	no	—
450	C₂×C₈	—	134	4	no	—
451	C₂×C₈	—	134	4	no	—
452	C₂×C₈	—	134	2	no	—
453	C₂×C₈	—	134	1	no	—
454	C₂×C₈	—	134	1	no	—
455	C₂×C₈	—	134	1	no	—
456	C₂×C₈	—	134	1	no	—
457	C₂×C₈	—	134	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	2	no	—
461	C₄⋊C₄	—	134	1	no	—
462	C₄⋊C₄	—	134	1	no	—
463	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
464	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	134	2	no	—
469	C₂²⋊C₄	—	134	2	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	2	no	—
475	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
476	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
477	C₄²	D₄×D₆₇	1	1	yes	characteristic
478	C₄²	D₄×D₆₇	1	1	yes	characteristic
479	C₂³	—	2	1	no	—
480	C₂³	—	134	2	no	—
481	C₂³	C₂×#11761627	1	1	yes	characteristic
482	Q₈	—	2	1	no	—
483	Q₈	—	134	2	no	—
484	Q₈	—	134	2	no	—
485	Q₈	#14524005	1	2	yes	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	1	no	—
489	D₄	—	268	2	no	—
490	D₄	—	268	2	no	—
491	D₄	—	268	2	no	—
492	D₄	—	268	2	no	—
493	D₄	—	134	2	no	—
494	D₄	—	134	2	no	—
495	D₄	#14524005	1	2	yes	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	268	2	no	—
509	C₂×C₄	—	268	2	no	—
510	C₂×C₄	—	134	2	no	—
511	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
512	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
513	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
514	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
515	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
518	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
519	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
521	C₈	—	4	2	no	—
522	C₈	—	67	2	no	—
523	C₈	—	67	2	no	—
524	C₈	—	134	2	no	—
525	C₈	—	134	2	no	—
526	C₈	—	134	2	no	—
527	C₈	—	134	1	no	—
528	C₈	—	134	1	no	—
529	C₈	—	134	2	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	268	2	no	—
536	C₂²	—	268	2	no	—
537	C₂²	—	268	2	no	—
538	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
539	C₂²	#16863932	1	1	yes	characteristic
540	C₂²	#16863941	1	1	yes	characteristic
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	8	2	no	—
548	C₄	—	268	2	no	—
549	C₄	#16863515	1	2	yes	complemented
550	C₄	C₂×#14524005	1	1	yes	characteristic
551	C₄	#16863864	1	1	yes	characteristic
552	C₄	#121×D₆₇	1	1	yes	characteristic
553	C₄	#16863909	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	536	2	no	—
558	C₂	#22562715	1	1	yes	characteristic
559	C₂	C₂×#16863515	1	1	yes	characteristic
560	C₂	#22569108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1806	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569664	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1806	D₆₇
hypocenter	C₆₇	#21518
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1806

Complements

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

N	H ≅ G/N	type
2. C₂×#16863515	C₂	semidirect
3. #22568920	C₂	semidirect
4. C₄×#14524050	C₂	semidirect
5. #22561127	C₂	semidirect
6. #22560982	C₂	semidirect
7. #22560855	C₂	semidirect
8. #22560534	C₂	semidirect
9. #22560078	C₂	semidirect
10. #22560050	C₂	semidirect
11. C₆₇×#1806	C₂	semidirect
17. #16863618	C₄	semidirect
18. #16863611	C₄	semidirect
19. #16863606	C₄	semidirect
20. #16863601	C₄	semidirect
21. #16863596	C₄	semidirect
22. #16863591	C₄	semidirect
62. #14524050	D₄	semidirect
267. C₄×C₆₇	#444	semidirect
289. #1806	D₆₇	semidirect
292. C₆₇	#21518	semidirect
549. C₄	#16863515	semidirect

Maximal subgroups

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

subgroup	index
#22569664	2
C₂×#16863515	2
#22568920	2
C₄×#14524050	2
#22561127	2
#22560982	2
#22560855	2
#22560534	2
#22560078	2
#22560050	2
C₆₇×#1806	2
#21518	67