Subgroups — #30856733

Order 17152.

1312 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864162	C₂	1	1	yes	maximal, characteristic
2	#22568772	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863412	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863412	C₂	1	1	yes	maximal, characteristic
5	#22560446	C₂	1	8	yes	maximal, complemented
6	C₂×#16863168	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559920	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11761628	C₂²	1	1	yes	characteristic
10	#16864162	—	2	2	no	—
11	#16864162	—	2	2	no	—
12	#16864162	—	2	2	no	—
13	#16864162	C₂²	1	4	yes	—
14	C₂×#14524051	—	2	2	no	—
15	C₂×#14524049	—	2	2	no	—
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×D₆₇	—	2	2	no	—
18	C₂×#14524002	C₂²	1	1	yes	characteristic
19	C₂×#14524002	C₂²	1	1	yes	characteristic
20	#16863424	—	2	2	no	—
21	#16863422	—	2	2	no	—
22	#16863419	C₂²	1	4	yes	complemented
23	#16863412	—	2	2	no	—
24	#16863412	C₄	1	4	yes	complemented
25	#16863412	C₄	1	4	yes	complemented
26	#16863412	C₂²	1	4	yes	—
27	C₂×#14523969	—	2	2	no	—
28	C₂×#14523969	C₂²	1	4	yes	—
29	#16863168	C₂²	1	4	yes	complemented
30	#16863157	C₂²	1	4	yes	complemented
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	#430×C₆₇	—	2	2	no	—
35	#430×C₆₇	C₂²	1	4	yes	complemented
36	C₂×#108×C₆₇	—	2	2	no	—
37	C₂×#11761628	—	2	1	no	—
38	C₂×#11761628	—	2	1	no	—
39	C₂×#11761628	—	2	1	no	—
40	C₂×#11761628	—	2	1	no	—
41	C₂×#11761628	—	2	1	no	—
42	C₂×#11761628	—	2	1	no	—
43	C₂×#11761628	C₂×C₄	1	2	yes	—
44	C₂×#11761628	C₂×C₄	1	2	yes	—
45	C₂×#11761628	D₄	1	2	yes	—
46	C₂×#11761628	D₄	1	2	yes	—
47	C₂×#11761628	Q₈	1	2	yes	—
48	C₂×#11761628	C₂³	1	2	yes	—
49	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
50	C₂²×D₂₆₈	Q₈	1	1	yes	characteristic
51	C₂²×C₂²⋊D₆₇	—	2	1	no	—
52	C₂²×C₄×D₆₇	—	2	1	no	—
53	#14524051	—	2	8	no	—
54	#14524051	—	2	8	no	—
55	#14524049	—	2	8	no	—
56	#14524049	—	2	8	no	—
57	C₂×#11761612	—	2	1	no	—
58	C₂×#11761612	—	2	1	no	—
59	C₂×#11761612	—	2	1	no	—
60	C₂×#11761612	—	2	1	no	—
61	C₂×#11761612	D₄	1	2	yes	—
62	C₂×#11761612	C₂³	1	2	yes	—
63	C₂×#11761611	—	2	4	no	—
64	M₁₆×D₆₇	—	2	8	no	—
65	M₁₆×D₆₇	—	2	8	no	—
66	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
67	C₂×C₈×D₆₇	—	2	4	no	—
68	#14524002	—	2	1	no	—
69	#14524002	—	2	1	no	—
70	#14524002	—	2	1	no	—
71	#14524002	—	2	1	no	—
72	#14524002	—	2	1	no	—
73	#14524002	C₂×C₄	1	2	yes	—
74	#14524002	C₂×C₄	1	2	yes	—
75	#14524002	C₂³	1	2	yes	—
76	C₂×C₄⋊Dic₆₇	—	2	1	no	—
77	C₂×C₄⋊Dic₆₇	—	2	1	no	—
78	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
79	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
80	C₂×C₄×Dic₆₇	—	2	1	no	—
81	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
82	Dic₆₇⋊C₈	—	2	8	no	—
83	#14523969	—	2	8	no	—
84	#14523969	C₂×C₄	1	16	yes	complemented
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×M₁₆×C₆₇	—	2	1	no	—
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×M₁₆×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	—	2	1	no	—
90	C₂×M₁₆×C₆₇	—	2	1	no	—
91	C₂×M₁₆×C₆₇	—	2	1	no	—
92	C₂×M₁₆×C₆₇	—	2	1	no	—
93	C₂×M₁₆×C₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	—	2	1	no	—
95	C₂×M₁₆×C₆₇	—	2	1	no	—
96	C₂×M₁₆×C₆₇	—	2	1	no	—
97	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
98	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
99	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
100	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
101	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
104	C₂²×C₈×C₆₇	—	2	1	no	—
105	C₂²×C₈×C₆₇	—	2	1	no	—
106	C₂²×C₈×C₆₇	—	2	1	no	—
107	#108×C₆₇	—	2	8	no	—
108	C₂³×D₆₇	—	2	1	no	—
109	#11761628	—	2	2	no	—
110	#11761628	—	2	2	no	—
111	#11761628	—	2	2	no	—
112	#11761628	—	2	2	no	—
113	#11761628	—	2	2	no	—
114	#11761628	—	2	2	no	—
115	#11761628	—	2	2	no	—
116	#11761628	—	2	2	no	—
117	#11761628	—	2	2	no	—
118	#11761628	—	2	2	no	—
119	#11761628	—	2	2	no	—
120	#11761628	—	2	2	no	—
121	#11761628	C₄⋊C₄	1	4	yes	—
122	#11761628	C₄⋊C₄	1	4	yes	—
123	#11761628	C₂²×C₄	1	4	yes	—
124	#11761628	C₂×D₄	1	4	yes	—
125	C₂×Dic₁₃₄	—	2	1	no	—
126	C₂×Dic₁₃₄	—	2	1	no	—
127	C₂×Dic₁₃₄	—	2	1	no	—
128	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
129	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
130	C₂×Dic₁₃₄	C₂×Q₈	1	2	yes	—
131	C₂×D₂₆₈	—	2	1	no	—
132	C₂×D₂₆₈	—	2	1	no	—
133	C₂×D₂₆₈	—	2	1	no	—
134	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
135	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
136	C₂×D₂₆₈	C₂×Q₈	1	2	yes	—
137	C₂×C₂²⋊D₆₇	—	2	2	no	—
138	C₂×C₂²⋊D₆₇	—	2	2	no	—
139	C₂×C₂²⋊D₆₇	—	2	2	no	—
140	C₂×C₂²⋊D₆₇	—	2	2	no	—
141	C₂×C₂²⋊D₆₇	—	2	2	no	—
142	C₂×C₂²⋊D₆₇	—	2	2	no	—
143	C₂²×Dic₆₇	—	2	1	no	—
144	C₂²×Dic₆₇	—	2	2	no	—
145	C₂²×Dic₆₇	—	4	1	no	—
146	C₂×C₄×D₆₇	—	2	2	no	—
147	C₂×C₄×D₆₇	—	2	2	no	—
148	C₂×C₄×D₆₇	—	2	2	no	—
149	C₂×C₄×D₆₇	—	2	2	no	—
150	C₂×C₄×D₆₇	—	2	2	no	—
151	C₂×C₄×D₆₇	—	2	2	no	—
152	#11761612	—	2	2	no	—
153	#11761612	—	2	2	no	—
154	#11761612	—	2	2	no	—
155	#11761612	—	2	2	no	—
156	#11761611	—	2	16	no	—
157	C₂×C₆₇⋊C₈	—	2	2	no	—
158	C₂×C₆₇⋊C₈	—	2	2	no	—
159	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
160	C₈×D₆₇	—	2	16	no	—
161	C₄⋊Dic₆₇	—	2	2	no	—
162	C₄⋊Dic₆₇	—	2	2	no	—
163	C₄⋊Dic₆₇	—	2	2	no	—
164	C₄⋊Dic₆₇	—	4	1	no	—
165	C₂²⋊Dic₆₇	—	2	2	no	—
166	C₂²⋊Dic₆₇	—	2	2	no	—
167	C₂²⋊Dic₆₇	—	2	2	no	—
168	C₂²⋊Dic₆₇	—	4	1	no	—
169	C₄×Dic₆₇	—	2	2	no	—
170	C₄×Dic₆₇	—	2	2	no	—
171	C₄×Dic₆₇	—	4	1	no	—
172	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
173	C₂⁴×C₆₇	#41	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	M₁₆×C₆₇	—	2	2	no	—
189	M₁₆×C₆₇	—	2	2	no	—
190	M₁₆×C₆₇	—	2	2	no	—
191	M₁₆×C₆₇	—	2	2	no	—
192	M₁₆×C₆₇	—	2	2	no	—
193	M₁₆×C₆₇	—	2	2	no	—
194	M₁₆×C₆₇	—	2	2	no	—
195	M₁₆×C₆₇	—	2	2	no	—
196	M₁₆×C₆₇	—	4	1	no	—
197	M₁₆×C₆₇	—	4	1	no	—
198	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
199	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
200	C₂×C₈×C₆₇	—	2	2	no	—
201	C₂×C₈×C₆₇	—	2	2	no	—
202	C₂×C₈×C₆₇	—	2	2	no	—
203	C₂×C₈×C₆₇	—	2	2	no	—
204	C₂×C₈×C₆₇	—	2	2	no	—
205	C₂×C₈×C₆₇	—	2	2	no	—
206	C₂×C₈×C₆₇	—	2	2	no	—
207	C₂×C₈×C₆₇	—	4	1	no	—
208	C₂×C₈×C₆₇	—	4	1	no	—
209	C₂³×C₆₇	—	8	1	no	—
210	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
211	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
213	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
214	C₂³×C₆₇	#123	1	1	yes	characteristic
215	C₂³×C₆₇	#123	1	1	yes	characteristic
216	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
217	C₂²×D₆₇	—	2	2	no	—
218	C₂²×D₆₇	—	2	2	no	—
219	C₂²×D₆₇	—	2	2	no	—
220	C₂²×D₆₇	—	8	2	no	—
221	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
232	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
233	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
234	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
241	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
242	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
249	C₂²⋊D₆₇	—	2	4	no	—
250	C₂²⋊D₆₇	—	2	4	no	—
251	C₂²⋊D₆₇	—	2	4	no	—
252	C₂²⋊D₆₇	—	2	4	no	—
253	C₂×Dic₆₇	—	2	2	no	—
254	C₂×Dic₆₇	—	2	2	no	—
255	C₂×Dic₆₇	—	2	2	no	—
256	C₂×Dic₆₇	—	2	4	no	—
257	C₂×Dic₆₇	—	4	2	no	—
258	C₂×Dic₆₇	—	8	2	no	—
259	C₂×Dic₆₇	—	8	2	no	—
260	D₂₆₈	—	2	2	no	—
261	D₂₆₈	—	2	2	no	—
262	D₂₆₈	—	2	2	no	—
263	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
264	C₄×D₆₇	—	2	4	no	—
265	C₄×D₆₇	—	2	4	no	—
266	C₄×D₆₇	—	2	4	no	—
267	C₄×D₆₇	—	2	4	no	—
268	Dic₁₃₄	—	2	2	no	—
269	Dic₁₃₄	—	2	2	no	—
270	Dic₁₃₄	—	2	2	no	—
271	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
272	C₈×C₆₇	—	2	4	no	—
273	C₈×C₆₇	—	2	4	no	—
274	C₈×C₆₇	—	4	2	no	—
275	C₆₇⋊C₈	—	2	4	no	—
276	C₂²×C₆₇	—	4	1	no	—
277	C₂²×C₆₇	—	4	1	no	—
278	C₂²×C₆₇	—	4	1	no	—
279	C₂²×C₆₇	—	4	1	no	—
280	C₂²×C₆₇	—	4	1	no	—
281	C₂²×C₆₇	—	4	1	no	—
282	C₂²×C₆₇	—	4	1	no	—
283	C₂²×C₆₇	#385	1	1	yes	characteristic
284	C₂²×C₆₇	#385	1	1	yes	characteristic
285	C₂²×C₆₇	#385	1	1	yes	characteristic
286	C₂²×C₆₇	#385	1	1	yes	characteristic
287	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
290	C₂×D₆₇	—	2	4	no	—
291	C₂×D₆₇	—	4	4	no	—
292	C₂×D₆₇	—	8	2	no	—
293	C₂×D₆₇	—	8	2	no	—
294	C₄×C₆₇	#385	1	1	yes	characteristic
295	C₄×C₆₇	#385	1	1	yes	characteristic
296	C₄×C₆₇	#385	1	1	yes	characteristic
297	C₄×C₆₇	#385	1	1	yes	characteristic
298	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
299	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
300	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
301	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
302	Dic₆₇	—	2	4	no	—
303	Dic₆₇	—	4	4	no	—
304	Dic₆₇	—	8	2	no	—
305	#20867	—	67	1	no	maximal, Sylow
306	C₂×C₆₇	—	2	1	no	—
307	C₂×C₆₇	—	2	1	no	—
308	C₂×C₆₇	—	2	1	no	—
309	C₂×C₆₇	—	2	1	no	—
310	C₂×C₆₇	—	2	1	no	—
311	C₂×C₆₇	—	2	1	no	—
312	C₂×C₆₇	—	2	1	no	—
313	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
314	D₆₇	—	4	4	no	—
315	C₂×#568	—	67	1	no	—
316	C₂×#430	D₆₇	1	1	yes	characteristic, complemented
317	#2905	—	67	1	no	—
318	C₂×#421	—	67	1	no	—
319	C₂×#421	—	67	1	no	—
320	#1861	—	67	8	no	—
321	C₂×#344	—	67	1	no	—
322	#1733	—	67	1	no	—
323	C₆₇	#20867	1	1	yes	characteristic, complemented, Sylow
324	C₂²×#41	—	67	1	no	—
325	#568	—	67	4	no	—
326	#568	—	134	2	no	—
327	#568	—	134	2	no	—
328	#568	—	134	2	no	—
329	C₂×#131	—	134	2	no	—
330	C₂×#131	—	134	2	no	—
331	C₂²×M₁₆	—	67	1	no	—
332	C₂²×M₁₆	—	134	2	no	—
333	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
334	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
335	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
336	C₂×#117	—	67	1	no	—
337	C₂×#117	—	67	1	no	—
338	#430	—	2	2	no	—
339	#430	C₂×D₆₇	1	4	yes	complemented
340	C₂×#108	—	2	2	no	—
341	#424	—	134	2	no	—
342	#424	—	134	2	no	—
343	#423	—	67	4	no	—
344	#421	—	67	4	no	—
345	#421	—	67	4	no	—
346	#421	—	67	4	no	—
347	#421	—	134	2	no	—
348	C₂×#104	—	67	4	no	—
349	C₂×#104	—	134	2	no	—
350	#344	—	67	4	no	—
351	#339	—	67	4	no	—
352	C₂×#41	—	67	2	no	—
353	C₂×#41	—	67	2	no	—
354	C₂×#41	—	67	2	no	—
355	C₂×#41	—	67	2	no	—
356	C₂×#41	—	67	2	no	—
357	C₂×#41	—	67	2	no	—
358	C₂×#41	—	134	1	no	—
359	C₂×#41	—	134	1	no	—
360	C₂×#41	—	134	1	no	—
361	C₂×#41	—	134	1	no	—
362	C₂×#41	—	134	1	no	—
363	C₂×#41	—	134	1	no	—
364	C₂²×Q₈	—	67	1	no	—
365	C₂²×D₄	—	67	1	no	—
366	C₂²×D₄	—	134	1	no	—
367	C₂³×C₄	—	134	1	no	—
368	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
369	#131	—	134	8	no	—
370	#131	—	134	8	no	—
371	#131	—	134	8	no	—
372	#131	—	134	8	no	—
373	C₂×M₁₆	—	2	1	no	—
374	C₂×M₁₆	—	2	1	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	1	no	—
383	C₂×M₁₆	—	2	1	no	—
384	C₂×M₁₆	—	67	2	no	—
385	C₂×M₁₆	—	67	2	no	—
386	C₂×M₁₆	—	134	4	no	—
387	C₂×M₁₆	—	134	8	no	—
388	C₂×M₁₆	—	134	1	no	—
389	C₂×M₁₆	—	134	1	no	—
390	C₂×M₁₆	—	134	1	no	—
391	C₂×M₁₆	—	134	1	no	—
392	C₂×M₁₆	—	134	8	no	—
393	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
394	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
395	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
396	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
397	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
398	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
399	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
400	C₂²×C₈	—	2	1	no	—
401	C₂²×C₈	—	2	1	no	—
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	—	67	2	no	—
404	C₂²×C₈	—	134	4	no	—
405	#117	—	67	2	no	—
406	#117	—	67	2	no	—
407	#117	—	67	2	no	—
408	#117	—	134	1	no	—
409	#117	—	134	1	no	—
410	#117	—	134	1	no	—
411	#117	—	134	1	no	—
412	#117	—	134	1	no	—
413	C₂×C₄⋊C₄	—	134	1	no	—
414	C₂×C₄⋊C₄	—	134	1	no	—
415	C₂×C₂²⋊C₄	—	134	1	no	—
416	C₂×C₂²⋊C₄	—	134	1	no	—
417	C₂×C₄²	—	67	2	no	—
418	C₂×C₄²	—	134	1	no	—
419	#108	—	2	8	no	—
420	C₄⋊C₈	—	134	8	no	—
421	#104	—	67	16	no	—
422	#104	—	134	8	no	—
423	C₂⁴	—	134	1	no	—
424	C₂⁴	#11761629	1	1	yes	characteristic
425	#41	—	67	4	no	—
426	#41	—	67	4	no	—
427	#41	—	67	4	no	—
428	#41	—	67	4	no	—
429	#41	—	134	2	no	—
430	#41	—	134	2	no	—
431	#41	—	134	2	no	—
432	#41	—	134	2	no	—
433	#41	—	134	2	no	—
434	#41	—	134	2	no	—
435	#41	—	134	2	no	—
436	#41	—	134	2	no	—
437	#41	—	134	2	no	—
438	#41	—	134	2	no	—
439	#41	—	134	2	no	—
440	#41	—	134	2	no	—
441	C₂×Q₈	—	67	2	no	—
442	C₂×Q₈	—	67	2	no	—
443	C₂×Q₈	—	67	2	no	—
444	C₂×Q₈	—	134	1	no	—
445	C₂×Q₈	—	134	1	no	—
446	C₂×Q₈	—	134	1	no	—
447	C₂×D₄	—	67	2	no	—
448	C₂×D₄	—	67	2	no	—
449	C₂×D₄	—	67	2	no	—
450	C₂×D₄	—	134	2	no	—
451	C₂×D₄	—	134	2	no	—
452	C₂×D₄	—	134	2	no	—
453	C₂×D₄	—	134	2	no	—
454	C₂×D₄	—	134	2	no	—
455	C₂×D₄	—	134	2	no	—
456	C₂×D₄	—	134	1	no	—
457	C₂×D₄	—	134	1	no	—
458	C₂×D₄	—	134	1	no	—
459	C₂²×C₄	—	268	1	no	—
460	C₂²×C₄	—	134	2	no	—
461	C₂²×C₄	—	134	1	no	—
462	C₂²×C₄	—	134	2	no	—
463	C₂²×C₄	—	134	2	no	—
464	C₂²×C₄	—	134	2	no	—
465	C₂²×C₄	—	134	2	no	—
466	C₂²×C₄	—	134	2	no	—
467	C₂²×C₄	—	134	2	no	—
468	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
470	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
471	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
472	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
473	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
476	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
477	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
478	C₂²×C₄	#11761628	1	1	yes	characteristic
479	C₂²×C₄	#11761628	1	1	yes	characteristic
480	C₂²×C₄	#11761629	1	1	yes	characteristic
481	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
482	M₁₆	—	2	2	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	2	2	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	4	1	no	—
491	M₁₆	—	4	1	no	—
492	M₁₆	—	134	16	no	—
493	M₁₆	—	134	2	no	—
494	M₁₆	—	134	2	no	—
495	M₁₆	—	134	2	no	—
496	M₁₆	—	134	2	no	—
497	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
498	M₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	2	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	4	1	no	—
507	C₂×C₈	—	4	1	no	—
508	C₂×C₈	—	67	4	no	—
509	C₂×C₈	—	134	16	no	—
510	C₂×C₈	—	134	2	no	—
511	C₂×C₈	—	134	2	no	—
512	C₄⋊C₄	—	268	1	no	—
513	C₄⋊C₄	—	134	2	no	—
514	C₄⋊C₄	—	134	2	no	—
515	C₄⋊C₄	—	134	2	no	—
516	C₂²⋊C₄	—	268	1	no	—
517	C₂²⋊C₄	—	134	2	no	—
518	C₂²⋊C₄	—	134	2	no	—
519	C₂²⋊C₄	—	134	2	no	—
520	C₄²	—	268	1	no	—
521	C₄²	—	67	4	no	—
522	C₄²	—	134	2	no	—
523	C₄²	—	134	2	no	—
524	C₂³	—	8	1	no	—
525	C₂³	—	536	2	no	—
526	C₂³	—	134	2	no	—
527	C₂³	—	134	2	no	—
528	C₂³	—	134	2	no	—
529	C₂³	#14524005	1	1	yes	characteristic
530	C₂³	#14524005	1	1	yes	characteristic
531	C₂³	#14524019	1	1	yes	characteristic
532	C₂³	#14524019	1	1	yes	characteristic
533	C₂³	#14524028	1	1	yes	characteristic
534	C₂³	#14524028	1	1	yes	characteristic
535	C₂³	C₂×#11761629	1	1	yes	characteristic
536	Q₈	—	67	4	no	—
537	Q₈	—	134	2	no	—
538	Q₈	—	134	2	no	—
539	Q₈	—	134	2	no	—
540	D₄	—	67	4	no	—
541	D₄	—	134	4	no	—
542	D₄	—	134	4	no	—
543	D₄	—	134	4	no	—
544	D₄	—	134	4	no	—
545	D₄	—	134	2	no	—
546	D₄	—	134	2	no	—
547	D₄	—	134	2	no	—
548	C₂×C₄	—	268	2	no	—
549	C₂×C₄	—	536	2	no	—
550	C₂×C₄	—	536	2	no	—
551	C₂×C₄	—	134	4	no	—
552	C₂×C₄	—	134	2	no	—
553	C₂×C₄	—	134	2	no	—
554	C₂×C₄	—	134	2	no	—
555	C₂×C₄	—	134	4	no	—
556	C₂×C₄	—	134	4	no	—
557	C₂×C₄	—	134	4	no	—
558	C₂×C₄	—	134	4	no	—
559	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
560	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
561	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
562	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
563	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
564	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
565	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
566	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
567	C₂×C₄	#14524005	1	1	yes	characteristic
568	C₂×C₄	#14524005	1	1	yes	characteristic
569	C₂×C₄	#14524011	1	1	yes	characteristic
570	C₂×C₄	#14524011	1	1	yes	characteristic
571	C₂×C₄	#14524011	1	1	yes	characteristic
572	C₂×C₄	#14524011	1	1	yes	characteristic
573	C₂×C₄	#14524019	1	1	yes	characteristic
574	C₂×C₄	#14524019	1	1	yes	characteristic
575	C₂×C₄	#14524021	1	1	yes	characteristic
576	C₂×C₄	#14524021	1	1	yes	characteristic
577	C₂×C₄	#14524021	1	1	yes	characteristic
578	C₂×C₄	#14524021	1	1	yes	characteristic
579	C₂×C₄	#14524028	1	1	yes	characteristic
580	C₂×C₄	#14524028	1	1	yes	characteristic
581	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
582	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
585	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
587	C₈	—	2	4	no	—
588	C₈	—	2	4	no	—
589	C₈	—	4	2	no	—
590	C₈	—	134	4	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	268	4	no	—
599	C₂²	—	536	2	no	—
600	C₂²	—	536	2	no	—
601	C₂²	—	134	4	no	—
602	C₂²	#16863267	1	1	yes	characteristic
603	C₂²	#16863267	1	1	yes	characteristic
604	C₂²	#16863267	1	1	yes	characteristic
605	C₂²	#16863267	1	1	yes	characteristic
606	C₂²	C₂×#14524005	1	1	yes	characteristic
607	C₂²	C₂×#14524019	1	1	yes	characteristic
608	C₂²	C₂×#14524028	1	1	yes	characteristic
609	C₄	—	268	4	no	—
610	C₄	—	536	2	no	—
611	C₄	—	134	4	no	—
612	C₄	#16863267	1	1	yes	characteristic
613	C₄	#16863267	1	1	yes	characteristic
614	C₄	#16863267	1	1	yes	characteristic
615	C₄	#16863267	1	1	yes	characteristic
616	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
617	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
618	C₄	C₂×#14524011	1	1	yes	characteristic
619	C₄	C₂×#14524021	1	1	yes	characteristic
620	C₂	—	2	1	no	—
621	C₂	—	2	1	no	—
622	C₂	—	2	1	no	—
623	C₂	—	2	1	no	—
624	C₂	—	2	1	no	—
625	C₂	—	2	1	no	—
626	C₂	—	2	1	no	—
627	C₂	—	268	4	no	—
628	C₂	C₂×#16863267	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524011
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#430×C₆₇	C₂
socle	C₂×C₆₇	C₂×#385
prime radical	C₂²×#11761628	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#430	D₆₇
hypocenter	C₆₇	#20867
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
2. #22568772	C₂	semidirect
5. #22560446	C₂	semidirect
6. C₂×#16863168	C₂	semidirect
7. #22559920	C₂	semidirect
8. C₂×#430×C₆₇	C₂	semidirect
22. #16863419	C₂²	semidirect
24. #16863412	C₄	semidirect
25. #16863412	C₄	semidirect
29. #16863168	C₂²	semidirect
30. #16863157	C₂²	semidirect
35. #430×C₆₇	C₂²	semidirect
84. #14523969	C₂×C₄	semidirect
316. C₂×#430	D₆₇	semidirect
323. C₆₇	#20867	semidirect
339. #430	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864162	2
#22568772	2
C₂×#16863412	2
#22560446	2
C₂×#16863168	2
#22559920	2
C₂×#430×C₆₇	2
#20867	67