Subgroups — #30824939

Order 17152.

692 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568840	C₂	1	1	yes	maximal, characteristic
2	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863418	C₂	1	1	yes	maximal, characteristic
4	#22558999	C₂	1	2	yes	maximal, complemented
5	#22558996	C₂	1	2	yes	maximal, complemented
6	#22558982	C₂	1	2	yes	maximal, complemented
7	#22558978	C₂	1	2	yes	maximal, complemented
8	#22558962	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558959	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761611	—	2	1	no	—
13	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
14	#16863451	C₄	1	2	yes	—
15	#16863451	C₄	1	2	yes	—
16	#16863449	C₂²	1	1	yes	characteristic
17	#16863448	C₂²	1	1	yes	characteristic
18	#16863448	C₂²	1	1	yes	characteristic
19	#16863447	—	2	1	no	—
20	#16863447	C₂²	1	2	yes	—
21	#16863447	C₂²	1	2	yes	—
22	#16863446	—	2	1	no	—
23	#16863419	C₄	1	1	yes	characteristic
24	#16863419	C₄	1	1	yes	characteristic
25	#16863418	—	2	1	no	—
26	#16863418	—	2	1	no	—
27	#16863418	C₂²	1	2	yes	—
28	#16863418	C₂²	1	2	yes	—
29	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂×Dic₆₇⋊C₈	—	2	1	no	—
31	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
33	#16863359	C₄	1	1	yes	characteristic
34	#16863359	C₄	1	1	yes	characteristic
35	C₂²⋊C₈×D₆₇	—	2	1	no	—
36	C₄×#11761612	C₂²	1	1	yes	characteristic
37	C₄×C₈×D₆₇	—	2	1	no	—
38	#16863149	C₂²	1	2	yes	complemented
39	#16863148	C₂²	1	2	yes	complemented
40	#16863148	C₂²	1	2	yes	complemented
41	#16863147	C₂²	1	2	yes	complemented
42	#16863146	C₂²	1	2	yes	complemented
43	#16863146	C₂²	1	2	yes	complemented
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	#335×C₆₇	C₂²	1	2	yes	complemented
48	#334×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
52	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761611	—	2	2	no	—
55	C₂×#11761611	—	2	2	no	—
56	C₂×#11761611	—	4	4	no	—
57	C₂²×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₈×D₆₇	—	2	1	no	—
59	C₂×C₈×D₆₇	—	2	1	no	—
60	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
63	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₆₇	D₄	1	1	yes	characteristic
66	C₄²×D₆₇	D₄	1	1	yes	characteristic
67	C₄²×D₆₇	Q₈	1	1	yes	characteristic
68	C₄²×D₆₇	Q₈	1	1	yes	characteristic
69	C₄²×D₆₇	C₂³	1	1	yes	characteristic
70	C₄²×D₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
72	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
74	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
75	Dic₆₇⋊C₈	—	2	1	no	—
76	Dic₆₇⋊C₈	—	2	1	no	—
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
80	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
81	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
82	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
83	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
84	(C₂×D₆₇)⋊C₈	—	2	1	no	—
85	(C₂×D₆₇)⋊C₈	—	2	1	no	—
86	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
87	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
88	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
89	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
90	#14523954	C₂×C₄	1	2	yes	—
91	C₄×C₆₇⋊C₈	—	2	1	no	—
92	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
93	C₈×Dic₆₇	—	2	1	no	—
94	C₈×Dic₆₇	—	2	1	no	—
95	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₆₇	—	2	1	no	—
100	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
102	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²⋊C₈×C₆₇	—	2	1	no	—
104	#97×C₆₇	C₂×C₄	1	2	yes	—
105	C₄×C₈×C₆₇	—	2	1	no	—
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×D₆₇	—	2	2	no	—
112	C₂×C₄×D₆₇	—	2	2	no	—
113	C₂×C₄×D₆₇	—	2	2	no	—
114	C₂×C₄×D₆₇	—	2	2	no	—
115	C₂×C₄×D₆₇	—	4	2	no	—
116	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
127	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
128	#11761612	—	2	2	no	—
129	#11761612	—	2	2	no	—
130	#11761611	—	4	4	no	—
131	#11761611	—	4	4	no	—
132	C₂×C₆₇⋊C₈	—	2	1	no	—
133	C₂×C₆₇⋊C₈	—	2	1	no	—
134	C₂×C₆₇⋊C₈	—	2	2	no	—
135	C₂×C₆₇⋊C₈	—	4	2	no	—
136	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
138	C₈×D₆₇	—	4	2	no	—
139	C₈×D₆₇	—	4	2	no	—
140	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
141	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
142	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
143	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
144	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
149	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	4	2	no	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
163	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
166	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×D₆₇	—	2	2	no	—
169	C₂²×D₆₇	—	2	2	no	—
170	C₂²×D₆₇	—	4	1	no	—
171	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	#131	1	2	yes	—
180	C₂×C₄×C₆₇	#131	1	2	yes	—
181	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
184	C₂×Dic₆₇	—	2	2	no	—
185	C₂×Dic₆₇	—	2	2	no	—
186	C₂×Dic₆₇	—	2	2	no	—
187	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄×D₆₇	—	2	2	no	—
194	C₄×D₆₇	—	2	1	no	—
195	C₄×D₆₇	—	2	1	no	—
196	C₄×D₆₇	—	2	1	no	—
197	C₄×D₆₇	—	2	1	no	—
198	C₄×D₆₇	—	2	1	no	—
199	C₄×D₆₇	—	2	1	no	—
200	C₄×D₆₇	—	2	2	no	—
201	C₄×D₆₇	—	2	2	no	—
202	C₄×D₆₇	—	2	2	no	—
203	C₄×D₆₇	—	2	2	no	—
204	C₄×D₆₇	—	2	2	no	—
205	C₄×D₆₇	—	4	1	no	—
206	C₄×D₆₇	—	4	1	no	—
207	C₄×D₆₇	AGL(1,ℤ/8)	1	2	yes	—
208	C₄×D₆₇	#137	1	2	yes	—
209	C₈×C₆₇	—	2	2	no	—
210	C₈×C₆₇	—	2	2	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₈×C₆₇	—	4	1	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₆₇⋊C₈	—	2	2	no	—
215	C₆₇⋊C₈	—	2	2	no	—
216	C₆₇⋊C₈	—	4	1	no	—
217	C₆₇⋊C₈	—	4	2	no	—
218	C₆₇⋊C₈	—	4	1	no	—
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	#424	1	2	yes	—
222	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₂×D₆₇	—	2	1	no	—
224	C₂×D₆₇	—	2	1	no	—
225	C₂×D₆₇	—	2	1	no	—
226	C₂×D₆₇	—	2	1	no	—
227	C₂×D₆₇	—	2	1	no	—
228	C₂×D₆₇	—	4	4	no	—
229	C₂×D₆₇	#428	1	2	yes	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	#424	1	2	yes	—
234	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
235	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
236	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
237	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
238	Dic₆₇	—	2	1	no	—
239	Dic₆₇	—	2	1	no	—
240	Dic₆₇	—	2	1	no	—
241	Dic₆₇	—	2	1	no	—
242	Dic₆₇	#428	1	2	yes	—
243	Dic₆₇	#430	1	2	yes	—
244	#17809	—	67	1	no	maximal, Sylow
245	C₂×C₆₇	—	2	2	no	—
246	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
247	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
248	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
249	D₆₇	—	2	2	no	—
250	D₆₇	—	4	1	no	—
251	#2920	—	67	1	no	—
252	C₂×#431	—	67	1	no	—
253	C₂×#423	—	67	1	no	—
254	#1574	—	67	2	no	—
255	#1574	—	67	2	no	—
256	#1571	—	67	2	no	—
257	#1570	—	67	2	no	—
258	#1565	—	67	1	no	—
259	#1565	—	67	1	no	—
260	#1565	—	67	1	no	—
261	#1565	D₆₇	1	1	yes	characteristic, complemented
262	C₆₇	#17809	1	1	yes	characteristic, complemented, Sylow
263	C₂²×M₁₆	—	134	1	no	—
264	C₂²×C₄²	—	67	1	no	—
265	#432	—	67	2	no	—
266	#432	—	67	2	no	—
267	#431	—	67	2	no	—
268	#431	—	67	2	no	—
269	#431	—	67	1	no	—
270	#431	—	67	1	no	—
271	#431	—	67	1	no	—
272	#431	—	134	1	no	—
273	#431	—	134	1	no	—
274	#431	C₂×D₆₇	1	1	yes	characteristic
275	#423	—	67	1	no	—
276	#423	—	67	1	no	—
277	#423	—	67	2	no	—
278	#423	—	67	2	no	—
279	#423	—	134	1	no	—
280	#423	—	134	1	no	—
281	C₂×C₄⋊C₈	—	67	1	no	—
282	C₂×C₄⋊C₈	—	67	1	no	—
283	C₂×C₄⋊C₈	—	67	1	no	—
284	C₂×C₄⋊C₈	—	134	1	no	—
285	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
286	#407	—	67	1	no	—
287	#407	—	67	1	no	—
288	C₂×C₂²⋊C₈	—	134	1	no	—
289	C₄×M₁₆	—	67	1	no	—
290	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
291	C₂×C₄×C₈	—	134	1	no	—
292	#335	—	67	2	no	—
293	#335	—	67	2	no	—
294	#335	—	67	2	no	—
295	#335	C₂×D₆₇	1	2	yes	complemented
296	#334	—	67	2	no	—
297	#334	—	67	2	no	—
298	#334	—	67	2	no	—
299	#334	C₂×D₆₇	1	2	yes	complemented
300	C₂³×C₄	—	67	1	no	—
301	C₂³×C₄	—	67	1	no	—
302	C₂³×C₄	—	67	1	no	—
303	C₂×M₁₆	—	268	4	no	—
304	C₂×M₁₆	—	67	1	no	—
305	C₂×M₁₆	—	67	1	no	—
306	C₂×M₁₆	—	134	2	no	—
307	C₂×M₁₆	—	134	2	no	—
308	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
309	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	134	1	no	—
312	C₂²×C₈	—	134	1	no	—
313	C₂²×C₈	—	134	1	no	—
314	C₂×C₄²	—	67	1	no	—
315	C₂×C₄²	—	67	1	no	—
316	C₂×C₄²	—	67	1	no	—
317	C₂×C₄²	—	67	1	no	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	—	67	1	no	—
321	C₂×C₄²	—	67	1	no	—
322	C₂×C₄²	—	67	1	no	—
323	C₂×C₄²	—	67	1	no	—
324	C₂×C₄²	—	67	1	no	—
325	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	67	2	no	—
328	C₄⋊C₈	—	67	2	no	—
329	C₄⋊C₈	—	67	2	no	—
330	C₄⋊C₈	—	67	2	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	2	no	—
333	C₄⋊C₈	—	67	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	1	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	C₄⋊C₈	—	134	1	no	—
339	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
340	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
341	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	67	2	no	—
344	C₂²⋊C₈	—	67	2	no	—
345	C₂²⋊C₈	—	67	2	no	—
346	C₂²⋊C₈	—	67	2	no	—
347	C₂²⋊C₈	—	134	1	no	—
348	C₂²⋊C₈	—	134	1	no	—
349	C₂²⋊C₈	—	134	1	no	—
350	#97	—	67	2	no	—
351	#97	C₄×D₆₇	1	2	yes	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	67	2	no	—
354	C₄×C₈	—	134	1	no	—
355	C₄×C₈	—	134	1	no	—
356	C₄×C₈	—	134	1	no	—
357	C₄×C₈	C₂²×D₆₇	1	2	yes	—
358	C₂⁴	—	67	1	no	—
359	C₂²×C₄	—	268	2	no	—
360	C₂²×C₄	—	67	1	no	—
361	C₂²×C₄	—	67	1	no	—
362	C₂²×C₄	—	67	1	no	—
363	C₂²×C₄	—	67	1	no	—
364	C₂²×C₄	—	67	1	no	—
365	C₂²×C₄	—	67	1	no	—
366	C₂²×C₄	—	67	1	no	—
367	C₂²×C₄	—	67	1	no	—
368	C₂²×C₄	—	67	1	no	—
369	C₂²×C₄	—	67	1	no	—
370	C₂²×C₄	—	67	1	no	—
371	C₂²×C₄	—	67	1	no	—
372	C₂²×C₄	—	67	1	no	—
373	C₂²×C₄	—	67	1	no	—
374	C₂²×C₄	—	67	1	no	—
375	C₂²×C₄	—	134	2	no	—
376	C₂²×C₄	—	134	2	no	—
377	C₂²×C₄	—	134	2	no	—
378	C₂²×C₄	—	134	2	no	—
379	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
380	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
381	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
382	M₁₆	—	2	2	no	—
383	M₁₆	—	2	2	no	—
384	M₁₆	—	268	4	no	—
385	M₁₆	—	268	4	no	—
386	M₁₆	—	134	2	no	—
387	M₁₆	—	134	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	4	2	no	—
392	C₂×C₈	—	268	2	no	—
393	C₂×C₈	—	268	2	no	—
394	C₂×C₈	—	268	2	no	—
395	C₂×C₈	—	67	2	no	—
396	C₂×C₈	—	67	2	no	—
397	C₂×C₈	—	134	1	no	—
398	C₂×C₈	—	134	1	no	—
399	C₂×C₈	—	134	2	no	—
400	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
401	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
402	C₄²	—	67	1	no	—
403	C₄²	—	67	1	no	—
404	C₄²	—	67	1	no	—
405	C₄²	—	67	1	no	—
406	C₄²	—	67	1	no	—
407	C₄²	—	67	1	no	—
408	C₄²	—	67	1	no	—
409	C₄²	—	67	1	no	—
410	C₄²	—	67	1	no	—
411	C₄²	—	67	1	no	—
412	C₄²	—	67	1	no	—
413	C₄²	—	67	1	no	—
414	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₄²	D₄×D₆₇	1	1	yes	characteristic
416	C₄²	Q₈×D₆₇	1	1	yes	characteristic
417	C₄²	C₂³×D₆₇	1	1	yes	characteristic
418	C₂³	—	268	1	no	—
419	C₂³	—	67	1	no	—
420	C₂³	—	67	1	no	—
421	C₂³	—	134	2	no	—
422	C₂³	—	134	2	no	—
423	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	268	1	no	—
428	C₂×C₄	—	268	1	no	—
429	C₂×C₄	—	67	1	no	—
430	C₂×C₄	—	67	1	no	—
431	C₂×C₄	—	67	2	no	—
432	C₂×C₄	—	67	2	no	—
433	C₂×C₄	—	67	1	no	—
434	C₂×C₄	—	67	1	no	—
435	C₂×C₄	—	67	1	no	—
436	C₂×C₄	—	67	1	no	—
437	C₂×C₄	—	134	2	no	—
438	C₂×C₄	—	134	2	no	—
439	C₂×C₄	—	134	2	no	—
440	C₂×C₄	—	134	2	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	—	134	2	no	—
444	C₂×C₄	—	134	1	no	—
445	C₂×C₄	—	134	1	no	—
446	C₂×C₄	—	134	1	no	—
447	C₂×C₄	—	134	1	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	—	134	1	no	—
450	C₂×C₄	—	134	2	no	—
451	C₂×C₄	—	134	2	no	—
452	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
453	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
454	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
455	C₂×C₄	#14524049	1	2	yes	—
456	C₂×C₄	#14524051	1	2	yes	—
457	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
460	C₈	—	2	2	no	—
461	C₈	—	2	2	no	—
462	C₈	—	4	2	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	1	no	—
465	C₈	—	268	2	no	—
466	C₈	—	268	1	no	—
467	C₈	—	268	1	no	—
468	C₈	—	134	2	no	—
469	C₈	—	134	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	268	4	no	—
473	C₂²	—	67	2	no	—
474	C₂²	—	134	1	no	—
475	C₂²	—	134	1	no	—
476	C₂²	—	134	1	no	—
477	C₂²	—	134	1	no	—
478	C₂²	—	134	1	no	—
479	C₂²	#16863421	1	2	yes	—
480	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	2	1	no	—
484	C₄	—	67	2	no	—
485	C₄	—	67	2	no	—
486	C₄	—	134	1	no	—
487	C₄	—	134	1	no	—
488	C₄	—	134	1	no	—
489	C₄	—	134	1	no	—
490	C₄	#16863421	1	2	yes	—
491	C₄	C₂×#14524049	1	1	yes	characteristic
492	C₄	C₂×#14524051	1	1	yes	characteristic
493	C₄	AGL(1,ℤ/8)×D₆₇	1	1	yes	characteristic
494	C₄	#137×D₆₇	1	1	yes	characteristic
495	C₂	—	2	2	no	—
496	C₂	—	268	1	no	—
497	C₂	—	134	2	no	—
498	C₂	C₂×#16863421	1	1	yes	characteristic
499	C₂	#428×D₆₇	1	1	yes	characteristic
500	C₂	#430×D₆₇	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×#1565	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1565	D₆₇
hypocenter	C₆₇	#17809
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #22558999	C₂	semidirect
5. #22558996	C₂	semidirect
6. #22558982	C₂	semidirect
7. #22558978	C₂	semidirect
8. #22558962	C₂	semidirect
9. #22558961	C₂	semidirect
10. #22558959	C₂	semidirect
11. C₆₇×#1565	C₂	semidirect
38. #16863149	C₂²	semidirect
39. #16863148	C₂²	semidirect
40. #16863148	C₂²	semidirect
41. #16863147	C₂²	semidirect
42. #16863146	C₂²	semidirect
43. #16863146	C₂²	semidirect
47. #335×C₆₇	C₂²	semidirect
48. #334×C₆₇	C₂²	semidirect
261. #1565	D₆₇	semidirect
262. C₆₇	#17809	semidirect
295. #335	C₂×D₆₇	semidirect
299. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568840	2
#431×D₆₇	2
C₂×#16863418	2
#22558999	2
#22558996	2
#22558982	2
#22558978	2
#22558962	2
#22558961	2
#22558959	2
C₆₇×#1565	2
#17809	67