Subgroups — #30827540

Order 17152.

668 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569298	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569246	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11761615	C₂	1	1	yes	maximal, characteristic
4	#22559114	C₂	1	2	yes	maximal, complemented
5	#22559049	C₂	1	2	yes	maximal, complemented
6	#22559048	C₂	1	2	yes	maximal, complemented
7	#22559024	C₂	1	2	yes	maximal, complemented
8	#22558970	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558653	C₂	1	1	yes	maximal, characteristic
10	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1491	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761615	C₄	1	1	yes	characteristic
13	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
14	#16863527	—	2	2	no	—
15	#16863527	C₂²	1	2	yes	complemented
16	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	complemented
17	C₄×#11761615	C₄	1	1	yes	characteristic
18	C₄×#11761615	C₄	1	1	yes	characteristic
19	C₄×#11761615	C₄	1	1	yes	characteristic
20	C₄×#11761615	C₄	1	1	yes	characteristic
21	C₄×#11761615	C₂²	1	1	yes	characteristic
22	C₄×#11761615	C₂²	1	1	yes	characteristic
23	C₄×#11761615	C₂²	1	1	yes	characteristic
24	C₄×#11761615	C₂²	1	1	yes	characteristic
25	#16863473	C₂²	1	2	yes	—
26	#16863469	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863453	—	2	1	no	—
29	#16863453	C₂²	1	1	yes	characteristic
30	#16863449	—	2	1	no	—
31	#16863449	C₂²	1	1	yes	characteristic
32	C₂×#14523974	C₄	1	1	yes	characteristic
33	C₂×#14523967	C₄	1	1	yes	characteristic
34	C₂×#14523963	C₄	1	1	yes	characteristic
35	#16863358	—	2	1	no	—
36	C₂²⋊C₈×D₆₇	—	2	1	no	—
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#16863148	C₂²	1	2	yes	complemented
39	#16863146	C₂²	1	2	yes	complemented
40	#16863125	C₂²	1	2	yes	—
41	#16863125	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	complemented
43	#16863118	C₂²	1	2	yes	complemented
44	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
45	#432×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
48	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₆₇	—	2	1	no	—
51	C₂×#11761615	—	2	4	no	—
52	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761611	—	2	2	no	—
57	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₇	—	2	2	no	—
60	C₄×D₂₆₈	—	2	1	no	—
61	C₄×D₂₆₈	—	2	1	no	—
62	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
64	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
65	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
68	#14523974	C₂×C₄	1	1	yes	characteristic
69	#14523974	C₂×C₄	1	1	yes	characteristic
70	#14523974	C₂×C₄	1	1	yes	characteristic
71	#14523974	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
73	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
74	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
75	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
76	#14523967	C₂×C₄	1	1	yes	characteristic
77	#14523967	C₂×C₄	1	1	yes	characteristic
78	#14523967	C₂×C₄	1	1	yes	characteristic
79	#14523967	C₂×C₄	1	1	yes	characteristic
80	#14523963	C₂×C₄	1	1	yes	characteristic
81	#14523963	C₂×C₄	1	1	yes	characteristic
82	#14523963	C₂×C₄	1	1	yes	characteristic
83	#14523963	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
85	C₂²⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
86	C₂²⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
87	(C₂×D₆₇)⋊C₈	—	2	2	no	—
88	(C₂×D₆₇)⋊C₈	—	2	2	no	—
89	#14523954	—	2	2	no	—
90	#14523954	—	2	1	no	—
91	C₄×C₆₇⋊C₈	—	2	2	no	—
92	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
93	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
96	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
97	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂×M₁₆×C₆₇	—	2	1	no	—
99	C₄×Q₈×C₆₇	—	2	1	no	—
100	C₄×Q₈×C₆₇	—	2	1	no	—
101	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
106	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
110	C₂²⋊C₈×C₆₇	—	2	1	no	—
111	#97×C₆₇	—	2	1	no	—
112	#97×C₆₇	—	2	1	no	—
113	C₄×C₈×C₆₇	C₂³	1	2	yes	—
114	C₂³×D₆₇	—	2	1	no	—
115	C₂×D₂₆₈	—	2	2	no	—
116	C₂×D₂₆₈	—	2	1	no	—
117	C₂×D₂₆₈	—	4	1	no	—
118	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×Dic₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	C₂×C₄×D₆₇	—	2	1	no	—
123	C₂×C₄×D₆₇	—	2	1	no	—
124	C₂×C₄×D₆₇	—	2	1	no	—
125	C₂×C₄×D₆₇	—	8	2	no	—
126	#11761615	—	2	2	no	—
127	#11761615	—	2	2	no	—
128	#11761615	—	2	2	no	—
129	#11761615	M₁₆	1	4	yes	complemented
130	#11761611	—	8	2	no	—
131	C₂×C₆₇⋊C₈	—	2	2	no	—
132	C₂×C₆₇⋊C₈	—	2	2	no	—
133	C₂×C₆₇⋊C₈	—	4	1	no	—
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
139	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
140	C₈×D₆₇	—	8	2	no	—
141	C₄⋊Dic₆₇	—	2	1	no	—
142	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
144	(C₂×D₆₇)⋊C₄	—	2	1	no	—
145	(C₂×D₆₇)⋊C₄	—	2	1	no	—
146	(C₂×D₆₇)⋊C₄	—	2	1	no	—
147	(C₂×D₆₇)⋊C₄	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	1	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	4	1	no	—
151	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
158	M₁₆×C₆₇	—	4	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	—	2	1	no	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	#41	1	1	yes	characteristic
175	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₆₇	—	2	1	no	—
178	C₂²×D₆₇	—	2	1	no	—
179	C₂²×D₆₇	—	4	2	no	—
180	C₂²×D₆₇	—	4	2	no	—
181	C₂²×D₆₇	—	8	1	no	—
182	Q₈×C₆₇	—	2	1	no	—
183	Q₈×C₆₇	—	2	1	no	—
184	Q₈×C₆₇	—	2	1	no	—
185	Q₈×C₆₇	—	4	2	no	—
186	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₂×Dic₆₇	—	2	1	no	—
204	C₂×Dic₆₇	—	2	1	no	—
205	C₂×Dic₆₇	—	8	1	no	—
206	D₂₆₈	—	2	1	no	—
207	D₂₆₈	—	2	1	no	—
208	D₂₆₈	—	2	1	no	—
209	D₂₆₈	—	4	2	no	—
210	D₂₆₈	C₂×M₁₆	1	2	yes	—
211	C₄×D₆₇	—	4	2	no	—
212	C₄×D₆₇	—	4	2	no	—
213	C₄×D₆₇	—	8	1	no	—
214	C₄×D₆₇	—	8	1	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₆₇⋊C₈	—	2	1	no	—
218	C₆₇⋊C₈	—	2	1	no	—
219	C₆₇⋊C₈	—	2	1	no	—
220	C₆₇⋊C₈	—	2	2	no	—
221	C₆₇⋊C₈	—	2	2	no	—
222	C₆₇⋊C₈	—	4	2	no	—
223	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	#444	1	2	yes	—
227	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂×D₆₇	—	2	2	no	—
229	C₂×D₆₇	—	4	1	no	—
230	C₂×D₆₇	—	4	1	no	—
231	C₂×D₆₇	—	4	1	no	—
232	C₂×D₆₇	—	4	1	no	—
233	C₂×D₆₇	—	4	1	no	—
234	C₂×D₆₇	—	8	4	no	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	—	2	1	no	—
240	C₄×C₆₇	—	4	1	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	—	4	1	no	—
243	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
244	C₄×C₆₇	#568	1	1	yes	characteristic
245	C₄×C₆₇	#577	1	2	yes	—
246	Dic₆₇	—	4	1	no	—
247	Dic₆₇	—	4	1	no	—
248	#18060	—	67	1	no	maximal, Sylow
249	C₂×C₆₇	—	2	2	no	—
250	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
251	C₂×C₆₇	#2949	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
253	D₆₇	—	4	2	no	—
254	D₆₇	—	8	1	no	—
255	#2984	—	67	1	no	—
256	#2974	—	67	1	no	—
257	C₂×C₄×SD₁₆	—	67	1	no	—
258	#1592	—	67	2	no	—
259	#1583	—	67	2	no	—
260	#1582	—	67	2	no	—
261	SD₁₆⋊C₈	—	67	2	no	—
262	#1568	—	67	1	no	—
263	#1491	D₆₇	1	1	yes	characteristic, complemented
264	#1490	—	67	1	no	—
265	#1451	—	67	1	no	—
266	C₆₇	#18060	1	1	yes	characteristic, complemented, Sylow
267	C₂²×SD₁₆	—	67	1	no	—
268	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
269	C₂×C₄×D₄	—	67	1	no	—
270	#446	—	67	2	no	—
271	#446	—	134	2	no	—
272	C₈×Q₈	—	67	2	no	—
273	C₄×SD₁₆	—	67	1	no	—
274	C₄×SD₁₆	—	67	1	no	—
275	C₄×SD₁₆	—	67	1	no	—
276	C₄×SD₁₆	—	67	1	no	—
277	C₄×SD₁₆	—	67	1	no	—
278	C₄×SD₁₆	—	67	1	no	—
279	C₄×SD₁₆	—	67	1	no	—
280	C₄×SD₁₆	—	67	1	no	—
281	#436	—	67	2	no	—
282	#435	—	134	2	no	—
283	C₈×D₄	—	67	2	no	—
284	#432	—	67	1	no	—
285	#432	—	134	1	no	—
286	#432	C₂×D₆₇	1	1	yes	characteristic
287	#431	—	67	1	no	—
288	#431	—	134	1	no	—
289	C₂×#106	—	67	1	no	—
290	C₂×#103	—	67	1	no	—
291	C₂×#102	—	67	1	no	—
292	#407	—	134	1	no	—
293	C₂×C₂²⋊C₈	—	134	1	no	—
294	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
295	C₂×C₄×C₈	—	67	1	no	—
296	#335	—	67	2	no	—
297	#334	—	67	2	no	—
298	Q₈⋊C₈	C₂×D₆₇	1	2	yes	complemented
299	Q₈⋊C₈	C₂×D₆₇	1	2	yes	complemented
300	D₄⋊C₈	—	67	2	no	—
301	D₄⋊C₈	—	67	2	no	—
302	#322	—	67	2	no	—
303	#322	—	67	2	no	—
304	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
305	C₂²×D₄	—	67	1	no	—
306	C₂³×C₄	—	134	1	no	—
307	C₂×SD₁₆	—	67	1	no	—
308	C₂×SD₁₆	—	67	1	no	—
309	C₂×SD₁₆	—	67	1	no	—
310	C₂×SD₁₆	—	67	1	no	—
311	C₂×SD₁₆	—	134	4	no	—
312	C₂×M₁₆	—	2	1	no	—
313	C₂×M₁₆	—	134	2	no	—
314	C₂²×C₈	—	67	1	no	—
315	C₂²×C₈	—	67	1	no	—
316	C₂²×C₈	—	134	2	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
320	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
321	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
322	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
323	C₄×D₄	—	67	1	no	—
324	C₄×D₄	—	67	1	no	—
325	C₄×D₄	—	67	1	no	—
326	C₄×D₄	—	67	1	no	—
327	C₄×D₄	—	134	1	no	—
328	C₄×D₄	—	134	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	67	1	no	—
331	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	134	1	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
335	#106	—	67	1	no	—
336	#106	—	67	1	no	—
337	#106	—	67	1	no	—
338	#106	—	67	1	no	—
339	C₄⋊C₈	—	67	2	no	—
340	C₄⋊C₈	—	67	2	no	—
341	C₄⋊C₈	—	134	2	no	—
342	C₄⋊C₈	—	134	2	no	—
343	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
344	#103	—	67	1	no	—
345	#103	—	67	1	no	—
346	#103	—	67	1	no	—
347	#103	—	67	1	no	—
348	#102	—	67	1	no	—
349	#102	—	67	1	no	—
350	#102	—	67	1	no	—
351	#102	—	67	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	67	1	no	—
354	C₂²⋊C₈	—	67	1	no	—
355	C₂²⋊C₈	—	134	2	no	—
356	C₂²⋊C₈	—	134	2	no	—
357	C₂²⋊C₈	—	134	1	no	—
358	#97	—	2	1	no	—
359	#97	—	2	1	no	—
360	#97	—	134	2	no	—
361	#97	—	134	1	no	—
362	C₄×C₈	—	67	1	no	—
363	C₄×C₈	—	67	1	no	—
364	C₄×C₈	—	67	1	no	—
365	C₄×C₈	—	67	1	no	—
366	C₄×C₈	—	67	2	no	—
367	C₄×C₈	—	134	2	no	—
368	C₄×C₈	C₂²×D₆₇	1	2	yes	—
369	C₂⁴	—	134	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
374	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₂×D₄	—	268	1	no	—
376	C₂×D₄	—	67	1	no	—
377	C₂×D₄	—	67	1	no	—
378	C₂×D₄	—	134	2	no	—
379	C₂×D₄	—	134	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	536	2	no	—
383	C₂²×C₄	—	134	1	no	—
384	C₂²×C₄	—	134	1	no	—
385	C₂²×C₄	—	134	1	no	—
386	C₂²×C₄	—	134	1	no	—
387	C₂²×C₄	—	134	1	no	—
388	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
389	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
390	C₂²×C₄	#11761627	1	1	yes	characteristic
391	SD₁₆	—	67	4	no	—
392	SD₁₆	—	134	2	no	—
393	SD₁₆	—	134	2	no	—
394	SD₁₆	—	134	2	no	—
395	M₁₆	—	4	2	no	—
396	M₁₆	—	536	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	268	1	no	—
400	C₂×C₈	—	536	2	no	—
401	C₂×C₈	—	67	1	no	—
402	C₂×C₈	—	67	1	no	—
403	C₂×C₈	—	67	1	no	—
404	C₂×C₈	—	67	1	no	—
405	C₂×C₈	—	67	2	no	—
406	C₂×C₈	—	67	2	no	—
407	C₂×C₈	—	134	2	no	—
408	C₂×C₈	—	134	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	67	1	no	—
415	C₄⋊C₄	—	67	1	no	—
416	C₄⋊C₄	—	134	1	no	—
417	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
418	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
419	C₂²⋊C₄	—	134	1	no	—
420	C₂²⋊C₄	—	134	1	no	—
421	C₂²⋊C₄	—	134	1	no	—
422	C₂²⋊C₄	—	134	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
428	C₄²	D₄×D₆₇	1	1	yes	characteristic
429	C₄²	#11761627	1	1	yes	characteristic
430	C₄²	C₂³×D₆₇	1	1	yes	characteristic
431	C₂³	—	268	2	no	—
432	C₂³	—	268	2	no	—
433	C₂³	—	536	1	no	—
434	C₂³	—	134	1	no	—
435	C₂³	—	134	1	no	—
436	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
437	Q₈	—	2	1	no	—
438	Q₈	—	2	1	no	—
439	Q₈	—	2	1	no	—
440	Q₈	—	4	2	no	—
441	Q₈	M₁₆×D₆₇	1	2	yes	—
442	D₄	—	268	2	no	—
443	D₄	—	67	2	no	—
444	D₄	—	134	1	no	—
445	D₄	—	134	1	no	—
446	D₄	—	134	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	268	2	no	—
458	C₂×C₄	—	268	2	no	—
459	C₂×C₄	—	536	1	no	—
460	C₂×C₄	—	536	1	no	—
461	C₂×C₄	—	536	1	no	—
462	C₂×C₄	—	134	1	no	—
463	C₂×C₄	—	134	1	no	—
464	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
465	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
466	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
468	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
470	C₈	—	4	2	no	—
471	C₈	—	4	2	no	—
472	C₈	—	268	2	no	—
473	C₈	—	67	2	no	—
474	C₈	—	134	1	no	—
475	C₈	—	134	1	no	—
476	C₈	—	134	1	no	—
477	C₈	—	134	2	no	—
478	C₈	—	134	2	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	268	1	no	—
482	C₂²	—	268	1	no	—
483	C₂²	—	268	1	no	—
484	C₂²	—	268	1	no	—
485	C₂²	—	268	1	no	—
486	C₂²	—	536	4	no	—
487	C₂²	—	134	2	no	—
488	C₂²	#16863515	1	2	yes	—
489	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	—	268	1	no	—
499	C₄	—	268	1	no	—
500	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
501	C₄	#16864162	1	1	yes	characteristic
502	C₄	#16864205	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	268	2	no	—
505	C₂	—	536	1	no	—
506	C₂	#22568941	1	1	yes	characteristic
507	C₂	#22569064	1	1	yes	characteristic
508	C₂	C₂×#16863515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1491	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1491	D₆₇
hypocenter	C₆₇	#18060
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1491

Complements

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

N	H ≅ G/N	type
1. #22569298	C₂	semidirect
4. #22559114	C₂	semidirect
5. #22559049	C₂	semidirect
6. #22559048	C₂	semidirect
7. #22559024	C₂	semidirect
8. #22558970	C₂	semidirect
10. #22558496	C₂	semidirect
11. C₆₇×#1491	C₂	semidirect
15. #16863527	C₂²	semidirect
16. Q₈×C₆₇⋊C₈	C₂²	semidirect
38. #16863148	C₂²	semidirect
39. #16863146	C₂²	semidirect
42. #16863118	C₂²	semidirect
43. #16863118	C₂²	semidirect
47. Q₈⋊C₈×C₆₇	C₂²	semidirect
48. Q₈⋊C₈×C₆₇	C₂²	semidirect
129. #11761615	M₁₆	semidirect
263. #1491	D₆₇	semidirect
266. C₆₇	#18060	semidirect
298. Q₈⋊C₈	C₂×D₆₇	semidirect
299. Q₈⋊C₈	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569298	2
#22569246	2
C₂×C₄×#11761615	2
#22559114	2
#22559049	2
#22559048	2
#22559024	2
#22558970	2
#22558653	2
#22558496	2
C₆₇×#1491	2
#18060	67