Subgroups — #30863661

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569997	C₂	1	1	yes	maximal, characteristic
2	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561153	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560808	C₂	1	2	yes	maximal, complemented
6	#22560484	C₂	1	2	yes	maximal
7	#22560229	C₂	1	1	yes	maximal, characteristic
8	#22560089	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560044	C₂	1	2	yes	maximal
11	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863953	C₂²	1	1	yes	characteristic
13	#16863703	C₄	1	2	yes	complemented
14	#16863696	C₄	1	2	yes	complemented
15	#16863691	C₄	1	1	yes	characteristic, complemented
16	#16863686	C₄	1	1	yes	characteristic, complemented
17	#16863681	C₄	1	1	yes	characteristic, complemented
18	#16863676	C₄	1	1	yes	characteristic, complemented
19	#16863526	—	2	1	no	—
20	#16863526	—	2	1	no	—
21	C₈×Dic₁₃₄	—	2	1	no	—
22	C₈×Dic₁₃₄	—	2	1	no	—
23	C₂×#14523978	C₂²	1	2	yes	—
24	#16863432	C₂²	1	2	yes	—
25	#16863422	C₂²	1	2	yes	—
26	#16863420	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×#14523970	C₂²	1	2	yes	—
29	#16863392	C₂²	1	2	yes	—
30	#16863351	C₂²	1	2	yes	—
31	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863160	C₂²	1	2	yes	—
34	#16863157	C₂²	1	2	yes	—
35	#16863157	C₂²	1	2	yes	—
36	#16863155	C₂²	1	2	yes	—
37	#16863150	C₂²	1	2	yes	—
38	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
41	#405×C₆₇	—	2	1	no	—
42	#405×C₆₇	—	2	1	no	—
43	#405×C₆₇	C₂²	1	1	yes	characteristic
44	#405×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₆₇	—	2	1	no	—
46	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₆₇	—	2	1	no	—
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761612	C₂³	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
51	#14524043	C₂×C₄	1	1	yes	characteristic
52	#14524043	C₂×C₄	1	1	yes	characteristic
53	#14524035	C₂×C₄	1	2	yes	—
54	#14524022	—	2	1	no	—
55	#14524022	—	2	1	no	—
56	C₄×Dic₁₃₄	—	2	1	no	—
57	C₄×Dic₁₃₄	—	2	1	no	—
58	#14524002	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₆₇	—	2	1	no	—
60	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
61	#14523978	—	2	4	no	—
62	#14523976	C₂×C₄	1	2	yes	—
63	#14523976	C₂×C₄	1	2	yes	—
64	#14523974	C₂×C₄	1	2	yes	—
65	#14523974	C₂×C₄	1	2	yes	—
66	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
69	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
71	Dic₆₇⋊C₈	—	2	2	no	—
72	Dic₆₇⋊C₈	—	2	2	no	—
73	#14523970	—	2	4	no	—
74	#14523967	C₂×C₄	1	2	yes	—
75	#14523967	C₂×C₄	1	2	yes	—
76	#14523964	C₂×C₄	1	2	yes	—
77	#14523964	C₂×C₄	1	2	yes	—
78	#14523953	—	2	2	no	—
79	C₈×Dic₆₇	—	2	2	no	—
80	C₈×Dic₆₇	—	2	4	no	—
81	C₆₇⋊#95	—	2	1	no	—
82	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
83	#131×C₆₇	—	2	2	no	—
84	#131×C₆₇	—	2	2	no	—
85	#131×C₆₇	—	2	2	no	—
86	#131×C₆₇	Q₈	1	4	yes	complemented
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×M₁₆×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	—	2	1	no	—
92	C₂²×C₈×C₆₇	—	2	1	no	—
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	—	2	1	no	—
99	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	—	2	1	no	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	#97×C₆₇	—	2	1	no	—
106	#97×C₆₇	—	2	1	no	—
107	#97×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	—	2	1	no	—
111	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₂×Dic₁₃₄	—	2	1	no	—
116	C₂×Dic₁₃₄	—	2	1	no	—
117	C₂²×Dic₆₇	—	2	2	no	—
118	#11761612	—	2	2	no	—
119	#11761612	—	2	2	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₂×C₆₇⋊C₈	—	2	2	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₆₇	—	2	4	no	—
126	C₄⋊Dic₆₇	—	2	1	no	—
127	C₄⋊Dic₆₇	—	2	1	no	—
128	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
129	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
130	Dic₆₇⋊C₄	—	2	2	no	—
131	Dic₆₇⋊C₄	—	2	2	no	—
132	C₂²⋊Dic₆₇	—	2	2	no	—
133	C₄×Dic₆₇	—	2	2	no	—
134	C₄×Dic₆₇	—	2	2	no	—
135	C₄×Dic₆₇	—	2	2	no	—
136	#41×C₆₇	—	2	1	no	—
137	#41×C₆₇	—	2	1	no	—
138	#41×C₆₇	C₂×Q₈	1	2	yes	—
139	#41×C₆₇	#41	1	2	yes	—
140	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×D₄×C₆₇	—	2	1	no	—
142	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₆₇	—	2	4	no	—
150	M₁₆×C₆₇	—	2	1	no	—
151	M₁₆×C₆₇	—	2	1	no	—
152	M₁₆×C₆₇	—	2	4	no	—
153	M₁₆×C₆₇	C₂×Q₈	1	2	yes	—
154	M₁₆×C₆₇	#41	1	2	yes	—
155	C₂×C₈×C₆₇	—	2	4	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
166	C₂×C₈×C₆₇	#41	1	2	yes	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₄⋊C₄×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	C₄²×C₆₇	—	2	1	no	—
175	C₄²×C₆₇	—	2	1	no	—
176	C₄²×C₆₇	—	2	1	no	—
177	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
183	Q₈×C₆₇	—	2	1	no	—
184	Q₈×C₆₇	C₄×Q₈	1	2	yes	—
185	D₄×C₆₇	—	2	2	no	—
186	D₄×C₆₇	—	2	2	no	—
187	D₄×C₆₇	—	2	1	no	—
188	D₄×C₆₇	C₄×Q₈	1	2	yes	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
208	C₂×Dic₆₇	—	2	4	no	—
209	C₂×Dic₆₇	—	2	2	no	—
210	C₂×Dic₆₇	—	2	2	no	—
211	C₂×Dic₆₇	—	4	2	no	—
212	C₂×Dic₆₇	—	4	2	no	—
213	Dic₁₃₄	—	4	2	no	—
214	Dic₁₃₄	—	4	2	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₈×C₆₇	—	2	1	no	—
217	C₈×C₆₇	—	2	2	no	—
218	C₈×C₆₇	—	2	1	no	—
219	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
220	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
221	C₆₇⋊C₈	—	2	2	no	—
222	C₆₇⋊C₈	—	2	2	no	—
223	C₆₇⋊C₈	—	4	2	no	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	4	1	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂²×C₆₇	#529	1	1	yes	characteristic
231	C₂²×C₆₇	#533	1	1	yes	characteristic
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	#443	1	2	yes	—
239	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
240	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
241	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
242	C₄×C₆₇	#525	1	1	yes	characteristic
243	Dic₆₇	—	4	4	no	—
244	Dic₆₇	—	4	2	no	—
245	Dic₆₇	—	4	2	no	—
246	#21627	—	67	1	no	maximal, Sylow
247	C₂×C₆₇	—	2	1	no	—
248	C₂×C₆₇	—	2	1	no	—
249	C₂×C₆₇	—	4	2	no	—
250	C₂×C₆₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
252	C₂×C₆₇	#2955	1	1	yes	characteristic
253	#3076	—	67	1	no	—
254	#2960	—	67	1	no	—
255	C₄×#131	D₆₇	1	1	yes	characteristic
256	#1986	—	67	1	no	—
257	#1950	—	67	1	no	—
258	#1922	—	67	2	no	—
259	#1866	—	67	2	no	—
260	#1806	—	67	1	no	—
261	#1772	—	67	2	no	—
262	C₈×C₄⋊C₄	—	67	1	no	—
263	#1764	—	67	2	no	—
264	C₆₇	#21627	1	1	yes	characteristic, complemented, Sylow
265	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
266	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
267	#533	—	67	1	no	—
268	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
269	#479	—	67	2	no	—
270	#478	—	67	2	no	—
271	#477	—	67	1	no	—
272	#476	—	67	1	no	—
273	#475	—	67	1	no	—
274	#474	—	67	1	no	—
275	#446	—	134	1	no	—
276	#446	—	134	1	no	—
277	C₈×Q₈	—	134	1	no	—
278	C₈×Q₈	—	134	1	no	—
279	C₂×#108	—	67	2	no	—
280	#427	—	67	2	no	—
281	#424	—	67	2	no	—
282	#423	—	67	1	no	—
283	C₂×C₄⋊C₈	—	67	1	no	—
284	C₂×#104	—	67	2	no	—
285	#416	—	67	2	no	—
286	#405	—	2	1	no	—
287	#405	—	2	1	no	—
288	#405	—	67	2	no	—
289	#405	C₂×D₆₇	1	1	yes	characteristic
290	#405	C₂×D₆₇	1	1	yes	characteristic
291	C₄×M₁₆	—	2	1	no	—
292	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
293	C₂×C₄×C₈	—	2	1	no	—
294	C₂×C₄×C₈	—	67	2	no	—
295	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
296	C₄×C₄⋊C₄	—	67	1	no	—
297	#341	—	67	2	no	—
298	#339	—	67	2	no	—
299	#339	—	67	2	no	—
300	#339	—	67	2	no	—
301	#336	—	67	2	no	—
302	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
303	#131	—	2	2	no	—
304	#131	—	2	2	no	—
305	#131	—	2	2	no	—
306	#131	Dic₁₃₄	1	4	yes	complemented
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	—	2	1	no	—
309	C₂×M₁₆	—	67	2	no	—
310	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
311	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	—	67	2	no	—
315	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
316	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
317	C₄⋊Q₈	—	67	1	no	—
318	C₄⋊Q₈	—	67	1	no	—
319	#125	—	67	2	no	—
320	C₂²⋊Q₈	—	134	1	no	—
321	C₂²⋊Q₈	—	134	1	no	—
322	C₄×Q₈	—	134	1	no	—
323	C₄×Q₈	—	134	1	no	—
324	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
325	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
329	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	—	67	2	no	—
332	#117	C₂²×D₆₇	1	1	yes	characteristic
333	C₂×C₄⋊C₄	—	134	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	67	2	no	—
336	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
337	#108	—	134	4	no	—
338	#107	—	67	2	no	—
339	#107	—	67	2	no	—
340	#106	—	67	2	no	—
341	#106	—	67	2	no	—
342	C₄⋊C₈	—	67	1	no	—
343	C₄⋊C₈	—	67	1	no	—
344	C₄⋊C₈	—	67	2	no	—
345	C₄⋊C₈	—	134	2	no	—
346	C₄⋊C₈	—	134	2	no	—
347	C₄⋊C₈	—	134	1	no	—
348	C₄⋊C₈	—	134	1	no	—
349	#104	—	134	4	no	—
350	#103	—	67	2	no	—
351	#103	—	67	2	no	—
352	#102	—	67	2	no	—
353	#102	—	67	2	no	—
354	#97	—	2	1	no	—
355	#97	—	2	1	no	—
356	#97	—	2	1	no	—
357	#97	—	134	2	no	—
358	C₄×C₈	—	2	1	no	—
359	C₄×C₈	—	2	1	no	—
360	C₄×C₈	—	2	1	no	—
361	C₄×C₈	—	134	2	no	—
362	C₄×C₈	—	134	4	no	—
363	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
364	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
365	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
366	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
367	#95	—	134	1	no	—
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	C₂×Dic₁₃₄	1	2	yes	—
371	#41	#11761628	1	2	yes	—
372	C₂×Q₈	—	134	1	no	—
373	C₂×Q₈	—	134	1	no	—
374	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	134	2	no	—
381	C₂²×C₄	#11761628	1	1	yes	characteristic
382	C₂²×C₄	#11761628	1	1	yes	characteristic
383	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
384	M₁₆	—	2	4	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	1	no	—
387	M₁₆	—	2	4	no	—
388	M₁₆	—	134	2	no	—
389	M₁₆	—	134	2	no	—
390	M₁₆	C₂×Dic₁₃₄	1	2	yes	—
391	M₁₆	#11761628	1	2	yes	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	4	no	—
398	C₂×C₈	—	67	2	no	—
399	C₂×C₈	—	67	2	no	—
400	C₂×C₈	—	134	2	no	—
401	C₂×C₈	—	134	2	no	—
402	C₂×C₈	—	134	2	no	—
403	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
404	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
405	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
406	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
407	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
408	C₂×C₈	#11761628	1	2	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	67	2	no	—
412	C₄⋊C₄	—	67	2	no	—
413	C₄⋊C₄	—	134	4	no	—
414	C₄⋊C₄	—	134	2	no	—
415	C₄⋊C₄	—	134	2	no	—
416	C₄⋊C₄	—	134	1	no	—
417	C₄⋊C₄	—	134	1	no	—
418	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
419	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	134	2	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	134	2	no	—
428	C₄²	—	134	2	no	—
429	C₄²	—	134	2	no	—
430	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
431	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
432	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
433	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
434	C₂³	—	2	1	no	—
435	C₂³	C₂×#11761628	1	1	yes	characteristic
436	Q₈	—	2	1	no	—
437	Q₈	—	268	2	no	—
438	Q₈	—	268	2	no	—
439	Q₈	C₄×Dic₁₃₄	1	2	yes	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	1	no	—
443	D₄	C₄×Dic₁₃₄	1	2	yes	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	268	2	no	—
454	C₂×C₄	—	268	2	no	—
455	C₂×C₄	—	134	4	no	—
456	C₂×C₄	—	134	2	no	—
457	C₂×C₄	—	134	2	no	—
458	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
459	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
460	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
461	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
465	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
466	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
467	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	1	no	—
472	C₈	—	268	2	no	—
473	C₈	—	134	2	no	—
474	C₈	—	134	2	no	—
475	C₈	#14524022	1	2	yes	—
476	C₈	#14524022	1	2	yes	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
483	C₂²	#16863930	1	1	yes	characteristic
484	C₂²	#16863953	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	2	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	1	no	—
491	C₄	—	268	4	no	—
492	C₄	—	268	2	no	—
493	C₄	—	268	2	no	—
494	C₄	#16863511	1	2	yes	—
495	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
496	C₄	C₄×#11761628	1	1	yes	characteristic
497	C₄	C₂×#14524022	1	1	yes	characteristic
498	C₄	#16863907	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	4	2	no	—
502	C₂	C₄×#14524022	1	1	yes	characteristic
503	C₂	C₂×#16863511	1	1	yes	characteristic
504	C₂	#22569110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#21627
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22569143	C₂	semidirect
3. #22561153	C₂	semidirect
4. #22560984	C₂	semidirect
5. #22560808	C₂	semidirect
8. #22560089	C₂	semidirect
9. C₈×C₄⋊Dic₆₇	C₂	semidirect
13. #16863703	C₄	semidirect
14. #16863696	C₄	semidirect
15. #16863691	C₄	semidirect
16. #16863686	C₄	semidirect
17. #16863681	C₄	semidirect
18. #16863676	C₄	semidirect
86. #131×C₆₇	Q₈	semidirect
264. C₆₇	#21627	semidirect
306. #131	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22569997	2
#22569143	2
#22561153	2
#22560984	2
#22560808	2
#22560484	2
#22560229	2
#22560089	2
C₈×C₄⋊Dic₆₇	2
#22560044	2
C₄×#131×C₆₇	2
#21627	67