Subgroups — #30824968

Order 17152.

602 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	#22569323	C₂	1	1	yes	maximal, characteristic
2	#22569317	C₂	1	1	yes	maximal, characteristic
3	#22568848	C₂	1	1	yes	maximal, characteristic
4	#22558996	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558994	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558993	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558987	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558986	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558985	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558982	C₂	1	1	yes	maximal, characteristic, complemented
11	#22558960	C₂	1	1	yes	maximal, characteristic, complemented
12	#22558958	C₂	1	1	yes	maximal, characteristic, complemented
13	#22558954	C₂	1	1	yes	maximal, characteristic, complemented
14	#335×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#11761628	C₂²	1	1	yes	characteristic
17	#16863525	—	2	1	no	—
18	C₈×Dic₁₃₄	—	2	1	no	—
19	#16863472	—	2	1	no	—
20	#16863466	—	2	1	no	—
21	#16863465	—	2	1	no	—
22	C₈×D₂₆₈	—	2	1	no	—
23	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
24	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
25	#16863456	C₄	1	2	yes	—
26	#16863454	C₄	1	2	yes	—
27	#16863453	C₂²	1	1	yes	characteristic
28	#16863452	C₂²	1	1	yes	characteristic
29	#16863449	C₂²	1	1	yes	characteristic
30	#16863448	C₂²	1	1	yes	characteristic
31	#16863448	C₂²	1	1	yes	characteristic
32	#16863448	C₂²	1	1	yes	characteristic
33	#16863447	C₂²	1	1	yes	characteristic
34	#16863447	C₂²	1	1	yes	characteristic
35	#16863421	C₄	1	2	yes	—
36	#16863418	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
41	#16863361	C₄	1	2	yes	—
42	C₄×#11761611	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
44	#16863149	C₂²	1	1	yes	characteristic
45	#16863149	C₂²	1	1	yes	characteristic
46	#16863148	C₂²	1	1	yes	characteristic
47	#16863148	C₂²	1	1	yes	characteristic
48	#16863148	C₂²	1	1	yes	characteristic
49	#16863148	C₂²	1	1	yes	characteristic
50	#16863148	C₂²	1	1	yes	characteristic
51	#16863148	C₂²	1	1	yes	characteristic
52	#16863147	C₂²	1	1	yes	characteristic
53	#16863147	C₂²	1	1	yes	characteristic
54	#16863146	C₂²	1	1	yes	characteristic
55	#16863146	C₂²	1	1	yes	characteristic
56	#431×C₆₇	C₂²	1	1	yes	characteristic
57	#431×C₆₇	C₂²	1	1	yes	characteristic
58	#423×C₆₇	C₂²	1	1	yes	characteristic
59	#335×C₆₇	C₂²	1	1	yes	characteristic
60	#335×C₆₇	C₂²	1	1	yes	characteristic
61	#334×C₆₇	C₂²	1	1	yes	characteristic
62	#334×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
64	C₂×#11761611	—	2	1	no	—
65	C₂×#11761611	C₂×C₄	1	2	yes	—
66	C₂²×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₈×D₆₇	—	2	1	no	—
68	C₂×C₈×D₆₇	—	2	1	no	—
69	C₂×C₈×D₆₇	—	2	1	no	—
70	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
72	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
73	C₄×D₂₆₈	D₄	1	1	yes	characteristic
74	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
75	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
76	C₄×C₂²⋊D₆₇	D₄	1	1	yes	characteristic
77	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
78	C₄×C₂²⋊D₆₇	Q₈	1	1	yes	characteristic
79	#14524002	C₂×C₄	1	1	yes	characteristic
80	#14524000	C₂×C₄	1	1	yes	characteristic
81	#14524000	C₂×C₄	1	1	yes	characteristic
82	C₄²×D₆₇	C₂³	1	1	yes	characteristic
83	C₄²×D₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
85	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
86	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
87	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
88	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
89	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
90	Dic₆₇⋊C₈	—	2	1	no	—
91	Dic₆₇⋊C₈	—	2	1	no	—
92	Dic₆₇⋊C₈	—	2	1	no	—
93	Dic₆₇⋊C₈	—	2	1	no	—
94	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
95	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
96	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
97	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
98	(C₂×D₆₇)⋊C₈	—	2	1	no	—
99	(C₂×D₆₇)⋊C₈	—	2	1	no	—
100	(C₂×D₆₇)⋊C₈	—	2	1	no	—
101	(C₂×D₆₇)⋊C₈	—	2	1	no	—
102	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
103	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
104	#14523954	—	2	1	no	—
105	#14523953	—	2	1	no	—
106	#14523953	C₂×C₄	1	2	yes	—
107	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
108	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
109	C₈×Dic₆₇	—	2	1	no	—
110	C₈×Dic₆₇	—	2	1	no	—
111	C₈×Dic₆₇	—	2	1	no	—
112	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
113	C₂×M₁₆×C₆₇	—	2	1	no	—
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₄⋊C₈×C₆₇	—	2	1	no	—
116	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
121	C₂²⋊C₈×C₆₇	—	2	1	no	—
122	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
123	C₄×C₈×C₆₇	—	2	1	no	—
124	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
126	#11761628	—	2	4	no	—
127	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
128	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
136	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
137	#11761611	—	2	4	no	—
138	#11761611	—	8	1	no	—
139	C₂×C₆₇⋊C₈	—	2	1	no	—
140	C₂×C₆₇⋊C₈	—	2	1	no	—
141	C₂×C₆₇⋊C₈	—	2	1	no	—
142	C₂×C₆₇⋊C₈	—	2	1	no	—
143	C₂×C₆₇⋊C₈	—	8	1	no	—
144	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
145	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
146	C₈×D₆₇	—	2	4	no	—
147	C₈×D₆₇	—	8	1	no	—
148	C₈×D₆₇	—	8	1	no	—
149	C₈×D₆₇	—	8	1	no	—
150	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
151	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
152	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
153	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
154	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
155	Dic₆₇⋊C₄	C₄⋊C₄	1	1	yes	characteristic
156	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
157	C₂²⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
158	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
159	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
160	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
161	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	1	yes	characteristic
162	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
165	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
166	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
167	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
171	M₁₆×C₆₇	—	8	1	no	—
172	C₂×C₈×C₆₇	—	2	1	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	1	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
177	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
178	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
181	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
195	C₂²⋊D₆₇	—	2	2	no	—
196	C₂²⋊D₆₇	—	2	2	no	—
197	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
201	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
203	D₂₆₈	—	2	2	no	—
204	C₄×D₆₇	—	2	2	no	—
205	C₄×D₆₇	—	2	1	no	—
206	C₄×D₆₇	—	2	1	no	—
207	C₄×D₆₇	—	2	1	no	—
208	C₄×D₆₇	—	2	1	no	—
209	C₄×D₆₇	—	4	1	no	—
210	C₄×D₆₇	—	4	1	no	—
211	C₄×D₆₇	#131	1	4	yes	—
212	Dic₁₃₄	—	2	2	no	—
213	C₈×C₆₇	—	2	2	no	—
214	C₈×C₆₇	—	2	2	no	—
215	C₈×C₆₇	—	4	1	no	—
216	C₈×C₆₇	—	4	1	no	—
217	C₈×C₆₇	—	4	1	no	—
218	C₈×C₆₇	—	4	1	no	—
219	C₆₇⋊C₈	—	2	2	no	—
220	C₆₇⋊C₈	—	2	2	no	—
221	C₆₇⋊C₈	—	4	1	no	—
222	C₆₇⋊C₈	—	4	1	no	—
223	C₆₇⋊C₈	—	4	1	no	—
224	C₆₇⋊C₈	—	4	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
229	C₂×D₆₇	—	2	1	no	—
230	C₂×D₆₇	—	2	1	no	—
231	C₂×D₆₇	—	2	1	no	—
232	C₂×D₆₇	—	2	1	no	—
233	C₂×D₆₇	—	2	1	no	—
234	C₂×D₆₇	#424	1	2	yes	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	C₂×#131	1	2	yes	—
240	C₄×C₆₇	#576	1	1	yes	characteristic
241	C₄×C₆₇	#578	1	1	yes	characteristic
242	Dic₆₇	—	2	1	no	—
243	Dic₆₇	—	2	1	no	—
244	Dic₆₇	—	2	1	no	—
245	Dic₆₇	—	2	1	no	—
246	Dic₆₇	—	2	1	no	—
247	Dic₆₇	#424	1	2	yes	—
248	#17812	—	67	1	no	maximal, Sylow
249	C₂×C₆₇	—	4	1	no	—
250	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
251	C₂×C₆₇	#2913	1	1	yes	characteristic
252	C₂×C₆₇	#2916	1	1	yes	characteristic
253	D₆₇	—	2	2	no	—
254	D₆₇	—	4	1	no	—
255	#2989	—	67	1	no	—
256	#2988	—	67	1	no	—
257	#2922	—	67	1	no	—
258	#1574	—	67	1	no	—
259	#1574	D₆₇	1	1	yes	characteristic, complemented
260	#1573	—	67	1	no	—
261	#1573	—	67	1	no	—
262	#1572	—	67	1	no	—
263	#1572	—	67	1	no	—
264	#1571	—	67	1	no	—
265	#1571	—	67	1	no	—
266	#1565	—	67	1	no	—
267	#1565	—	67	1	no	—
268	#1564	—	67	1	no	—
269	C₂×#335	—	67	1	no	—
270	C₆₇	#17812	1	1	yes	characteristic, complemented, Sylow
271	C₄×#41	—	67	1	no	—
272	#446	—	134	1	no	—
273	C₈×Q₈	—	134	1	no	—
274	#436	—	134	1	no	—
275	#435	—	134	1	no	—
276	#435	—	134	1	no	—
277	C₈×D₄	—	134	1	no	—
278	C₈×D₄	—	134	1	no	—
279	C₈×D₄	—	134	1	no	—
280	#433	—	67	2	no	—
281	#433	—	67	2	no	—
282	#432	—	67	1	no	—
283	#432	—	67	1	no	—
284	#431	—	67	1	no	—
285	#431	—	67	1	no	—
286	#431	—	67	1	no	—
287	#431	—	67	1	no	—
288	#431	—	67	1	no	—
289	#431	—	67	1	no	—
290	#431	C₂×D₆₇	1	1	yes	characteristic
291	#431	C₂×D₆₇	1	1	yes	characteristic
292	#424	—	67	2	no	—
293	#423	—	67	1	no	—
294	#423	C₂×D₆₇	1	1	yes	characteristic
295	C₂×C₄⋊C₈	—	67	1	no	—
296	C₂×C₄⋊C₈	—	67	1	no	—
297	C₂×C₄⋊C₈	—	67	1	no	—
298	C₂×C₄⋊C₈	—	67	1	no	—
299	#408	—	67	2	no	—
300	C₄×M₁₆	—	67	1	no	—
301	C₂×C₄×C₈	—	67	1	no	—
302	#335	—	67	1	no	—
303	#335	—	67	1	no	—
304	#335	—	67	1	no	—
305	#335	—	67	1	no	—
306	#335	—	67	1	no	—
307	#335	—	67	1	no	—
308	#335	—	67	1	no	—
309	#335	—	67	1	no	—
310	#335	C₂×D₆₇	1	1	yes	characteristic
311	#335	C₂×D₆₇	1	1	yes	characteristic
312	#334	—	67	1	no	—
313	#334	—	67	1	no	—
314	#334	—	67	1	no	—
315	#334	—	67	1	no	—
316	#334	C₂×D₆₇	1	1	yes	characteristic
317	#334	C₂×D₆₇	1	1	yes	characteristic
318	C₂×#41	—	67	1	no	—
319	C₂×M₁₆	—	2	1	no	—
320	C₂×M₁₆	—	67	2	no	—
321	C₂×M₁₆	—	134	1	no	—
322	C₂²×C₈	—	67	2	no	—
323	C₂²×C₈	—	134	1	no	—
324	C₂²×C₈	—	134	1	no	—
325	C₂²×C₈	—	134	1	no	—
326	C₂²×C₈	—	134	1	no	—
327	C₄×Q₈	—	67	1	no	—
328	C₄×Q₈	—	67	1	no	—
329	C₄×D₄	—	67	1	no	—
330	C₄×D₄	—	67	1	no	—
331	C₄×D₄	—	67	1	no	—
332	C₄×D₄	—	67	1	no	—
333	C₄×D₄	—	67	1	no	—
334	C₄×D₄	—	67	1	no	—
335	#117	—	67	1	no	—
336	#117	—	67	1	no	—
337	#117	—	67	1	no	—
338	C₂×C₄²	—	67	1	no	—
339	C₂×C₄²	—	67	1	no	—
340	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	67	2	no	—
343	C₄⋊C₈	—	67	2	no	—
344	C₄⋊C₈	—	67	2	no	—
345	C₄⋊C₈	—	67	1	no	—
346	C₄⋊C₈	—	67	1	no	—
347	C₄⋊C₈	—	67	1	no	—
348	C₄⋊C₈	—	67	1	no	—
349	C₄⋊C₈	—	134	1	no	—
350	C₄⋊C₈	—	134	1	no	—
351	C₄⋊C₈	—	134	1	no	—
352	C₄⋊C₈	—	134	1	no	—
353	C₄⋊C₈	—	134	1	no	—
354	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
355	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
356	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
357	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
358	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
359	C₂²⋊C₈	—	2	1	no	—
360	C₂²⋊C₈	—	67	2	no	—
361	C₂²⋊C₈	—	67	2	no	—
362	C₂²⋊C₈	—	67	2	no	—
363	C₂²⋊C₈	—	134	1	no	—
364	C₂²⋊C₈	—	134	1	no	—
365	C₂²⋊C₈	—	134	1	no	—
366	C₂²⋊C₈	—	134	1	no	—
367	C₂²⋊C₈	—	134	1	no	—
368	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
369	#97	—	67	2	no	—
370	#97	—	134	1	no	—
371	#97	—	134	1	no	—
372	C₄×C₈	—	2	1	no	—
373	C₄×C₈	—	67	2	no	—
374	C₄×C₈	—	67	1	no	—
375	C₄×C₈	—	67	1	no	—
376	C₄×C₈	—	134	1	no	—
377	C₄×C₈	—	134	1	no	—
378	C₄×C₈	—	134	1	no	—
379	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
380	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
381	#41	—	134	4	no	—
382	C₂×Q₈	—	67	1	no	—
383	C₂×D₄	—	67	1	no	—
384	C₂×D₄	—	67	1	no	—
385	C₂×D₄	—	67	1	no	—
386	C₂²×C₄	—	67	1	no	—
387	C₂²×C₄	—	67	1	no	—
388	C₂²×C₄	—	67	1	no	—
389	C₂²×C₄	—	67	1	no	—
390	C₂²×C₄	—	67	1	no	—
391	C₂²×C₄	—	67	1	no	—
392	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
393	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
394	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
395	M₁₆	—	8	1	no	—
396	M₁₆	—	536	1	no	—
397	M₁₆	—	134	4	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	536	1	no	—
403	C₂×C₈	—	536	1	no	—
404	C₂×C₈	—	536	1	no	—
405	C₂×C₈	—	536	1	no	—
406	C₂×C₈	—	67	2	no	—
407	C₂×C₈	—	67	2	no	—
408	C₂×C₈	—	134	4	no	—
409	C₂×C₈	—	134	1	no	—
410	C₂×C₈	—	134	1	no	—
411	C₂×C₈	—	134	1	no	—
412	C₂×C₈	—	134	1	no	—
413	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
414	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
415	C₄⋊C₄	—	67	1	no	—
416	C₄⋊C₄	—	67	1	no	—
417	C₄⋊C₄	—	67	1	no	—
418	C₄⋊C₄	—	67	1	no	—
419	C₄⋊C₄	—	67	1	no	—
420	C₄⋊C₄	—	67	1	no	—
421	C₂²⋊C₄	—	67	1	no	—
422	C₂²⋊C₄	—	67	1	no	—
423	C₂²⋊C₄	—	67	1	no	—
424	C₂²⋊C₄	—	67	1	no	—
425	C₂²⋊C₄	—	67	1	no	—
426	C₂²⋊C₄	—	67	1	no	—
427	C₄²	—	67	1	no	—
428	C₄²	—	67	1	no	—
429	C₄²	—	67	1	no	—
430	C₄²	—	67	1	no	—
431	C₄²	—	67	1	no	—
432	C₄²	—	67	1	no	—
433	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
434	C₄²	D₄×D₆₇	1	1	yes	characteristic
435	C₄²	Q₈×D₆₇	1	1	yes	characteristic
436	C₄²	C₂³×D₆₇	1	1	yes	characteristic
437	C₂³	—	67	1	no	—
438	C₂³	—	67	1	no	—
439	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
440	Q₈	—	134	2	no	—
441	D₄	—	134	2	no	—
442	D₄	—	134	2	no	—
443	D₄	—	134	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	268	1	no	—
449	C₂×C₄	—	268	1	no	—
450	C₂×C₄	—	67	1	no	—
451	C₂×C₄	—	67	4	no	—
452	C₂×C₄	—	67	1	no	—
453	C₂×C₄	—	67	1	no	—
454	C₂×C₄	—	67	1	no	—
455	C₂×C₄	—	67	1	no	—
456	C₂×C₄	—	67	1	no	—
457	C₂×C₄	—	134	2	no	—
458	C₂×C₄	—	134	1	no	—
459	C₂×C₄	—	134	1	no	—
460	C₂×C₄	—	134	1	no	—
461	C₂×C₄	—	134	1	no	—
462	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
463	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
464	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
465	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
466	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	4	1	no	—
471	C₈	—	4	1	no	—
472	C₈	—	4	1	no	—
473	C₈	—	4	1	no	—
474	C₈	—	268	1	no	—
475	C₈	—	268	1	no	—
476	C₈	—	268	1	no	—
477	C₈	—	268	1	no	—
478	C₈	—	134	2	no	—
479	C₈	—	134	2	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	67	2	no	—
484	C₂²	—	134	1	no	—
485	C₂²	—	134	1	no	—
486	C₂²	—	134	1	no	—
487	C₂²	—	134	1	no	—
488	C₂²	—	134	1	no	—
489	C₂²	C₂×C₄⋊C₄×D₆₇	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	—	67	2	no	—
495	C₄	—	134	1	no	—
496	C₄	—	134	1	no	—
497	C₄	—	134	1	no	—
498	C₄	—	134	1	no	—
499	C₄	—	134	1	no	—
500	C₄	#131×D₆₇	1	2	yes	—
501	C₄	#16864200	1	1	yes	characteristic
502	C₄	#16864212	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	268	1	no	—
505	C₂	—	134	2	no	—
506	C₂	#424×D₆₇	1	1	yes	characteristic
507	C₂	#22568808	1	1	yes	characteristic
508	C₂	#22568817	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×#1574	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1574	D₆₇
hypocenter	C₆₇	#17812
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
4. #22558996	C₂	semidirect
5. #22558994	C₂	semidirect
6. #22558993	C₂	semidirect
7. #22558987	C₂	semidirect
8. #22558986	C₂	semidirect
9. #22558985	C₂	semidirect
10. #22558982	C₂	semidirect
11. #22558960	C₂	semidirect
12. #22558958	C₂	semidirect
13. #22558954	C₂	semidirect
14. #335×D₆₇	C₂	semidirect
15. C₆₇×#1574	C₂	semidirect
259. #1574	D₆₇	semidirect
270. C₆₇	#17812	semidirect

Maximal subgroups

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

subgroup	index
#22569323	2
#22569317	2
#22568848	2
#22558996	2
#22558994	2
#22558993	2
#22558987	2
#22558986	2
#22558985	2
#22558982	2
#22558960	2
#22558958	2
#22558954	2
#335×D₆₇	2
C₆₇×#1574	2
#17812	67