Subgroups — #30869050

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569818	C₂	1	1	yes	maximal, characteristic
2	#22568912	C₂	1	2	yes	maximal, complemented
3	#22560858	C₂	1	2	yes	maximal, complemented
4	#22560809	C₂	1	2	yes	maximal, complemented
5	#22560652	C₂	1	2	yes	maximal, complemented
6	#22560535	C₂	1	2	yes	maximal, complemented
7	#22560487	C₂	1	2	yes	maximal, complemented
8	#22560251	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524065	C₂²	1	1	yes	characteristic
11	C₂×#14524049	C₂²	1	2	yes	—
12	#16863907	C₂²	1	2	yes	—
13	#16863654	C₄	1	1	yes	characteristic, complemented
14	#16863654	C₄	1	1	yes	characteristic, complemented
15	#16863649	C₄	1	1	yes	characteristic, complemented
16	#16863649	C₄	1	1	yes	characteristic, complemented
17	#16863642	C₄	1	2	yes	complemented
18	#16863642	C₄	1	2	yes	complemented
19	#16863472	—	2	2	no	—
20	#16863472	—	2	2	no	—
21	#16863465	—	2	2	no	—
22	#16863465	—	2	2	no	—
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	C₂×#14523979	C₂²	1	1	yes	characteristic
25	#16863434	C₂²	1	1	yes	characteristic
26	#16863422	C₂²	1	2	yes	—
27	C₂×#14523971	C₂²	1	2	yes	—
28	C₂×#14523971	C₂²	1	2	yes	—
29	#16863393	C₂²	1	2	yes	—
30	#16863362	C₂²	1	2	yes	—
31	#16863360	C₂²	1	2	yes	—
32	#16863351	C₂²	1	2	yes	—
33	#16863161	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	#16863156	C₂²	1	2	yes	—
36	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	C₂³	1	2	yes	—
42	#14524065	—	4	1	no	—
43	#14524065	—	4	1	no	—
44	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×#11761617	C₂×C₄	1	2	yes	complemented
46	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic, complemented
47	#14524049	—	4	2	no	—
48	#14524049	—	4	2	no	—
49	C₂×#11761612	C₂³	1	2	yes	—
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₂×#11761611	—	2	2	no	—
52	C₂×C₈×D₆₇	—	2	2	no	—
53	#14524026	—	2	2	no	—
54	#14524022	C₂×C₄	1	2	yes	—
55	#14524022	C₂×C₄	1	2	yes	—
56	#14524017	C₂×C₄	1	2	yes	—
57	#14524017	C₂×C₄	1	2	yes	—
58	#14524012	—	2	2	no	—
59	C₄×C₂²⋊D₆₇	—	2	2	no	—
60	C₄×C₂²⋊D₆₇	—	2	2	no	—
61	#14524002	C₂³	1	2	yes	—
62	#14523979	—	4	1	no	—
63	#14523979	—	4	1	no	—
64	#14523977	C₂×C₄	1	2	yes	complemented
65	#14523975	C₂×C₄	1	1	yes	characteristic, complemented
66	#14523975	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₆₇⋊C₈	—	2	2	no	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	#14523971	—	4	2	no	—
70	#14523971	—	4	2	no	—
71	#14523968	C₂×C₄	1	2	yes	complemented
72	#14523968	C₂×C₄	1	2	yes	complemented
73	#14523965	C₂×C₄	1	2	yes	complemented
74	#14523965	C₂×C₄	1	2	yes	complemented
75	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
76	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	#14523953	—	2	2	no	—
80	C₈×Dic₆₇	—	2	2	no	—
81	C₂³×C₄×C₆₇	—	2	1	no	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₆₇	—	4	1	no	—
84	C₂×M₁₆×C₆₇	—	4	1	no	—
85	C₂×M₁₆×C₆₇	—	4	1	no	—
86	C₂×M₁₆×C₆₇	—	4	1	no	—
87	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
92	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
93	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄²×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	—	4	1	no	—
99	C₂×C₄²×C₆₇	—	4	1	no	—
100	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
101	#11761628	—	2	2	no	—
102	#11761628	—	2	2	no	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
106	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
107	C₂×C₂²⋊D₆₇	—	2	2	no	—
108	C₂×C₂²⋊D₆₇	—	2	2	no	—
109	C₂×C₂²⋊D₆₇	—	2	2	no	—
110	C₂×C₄×D₆₇	—	2	2	no	—
111	Dic₂₆₈	—	4	1	no	—
112	#11761617	—	4	2	no	—
113	D₅₃₆	—	4	1	no	—
114	#11761612	—	2	2	no	—
115	#11761612	—	2	2	no	—
116	#11761612	—	2	2	no	—
117	#11761612	—	2	2	no	—
118	#11761611	—	4	4	no	—
119	C₂×C₆₇⋊C₈	—	2	2	no	—
120	C₂×C₆₇⋊C₈	—	2	2	no	—
121	C₈×D₆₇	—	4	4	no	—
122	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
123	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
124	Dic₆₇⋊C₄	—	2	2	no	—
125	Dic₆₇⋊C₄	—	2	2	no	—
126	C₂²⋊Dic₆₇	—	2	2	no	—
127	C₂²⋊Dic₆₇	—	2	2	no	—
128	C₂²⋊Dic₆₇	—	2	2	no	—
129	(C₂×D₆₇)⋊C₄	—	2	2	no	—
130	(C₂×D₆₇)⋊C₄	—	2	2	no	—
131	C₄×Dic₆₇	—	2	2	no	—
132	C₂⁴×C₆₇	#41	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	4	4	no	—
139	C₂²×C₄×C₆₇	—	4	2	no	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
146	M₁₆×C₆₇	—	4	1	no	—
147	M₁₆×C₆₇	—	4	1	no	—
148	M₁₆×C₆₇	—	4	1	no	—
149	M₁₆×C₆₇	—	4	1	no	—
150	M₁₆×C₆₇	—	4	1	no	—
151	M₁₆×C₆₇	—	4	2	no	—
152	M₁₆×C₆₇	—	4	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	4	2	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	—	4	2	no	—
171	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
172	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
173	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂²×D₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	8	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
197	C₂²⋊D₆₇	—	4	2	no	—
198	C₂²⋊D₆₇	—	4	2	no	—
199	C₂²⋊D₆₇	—	8	2	no	—
200	C₂²⋊D₆₇	—	8	2	no	—
201	C₂×Dic₆₇	—	2	2	no	—
202	C₂×Dic₆₇	—	2	2	no	—
203	C₂×Dic₆₇	—	2	2	no	—
204	D₂₆₈	—	2	2	no	—
205	D₂₆₈	—	2	2	no	—
206	C₄×D₆₇	—	4	2	no	—
207	C₄×D₆₇	—	4	2	no	—
208	Dic₁₃₄	—	2	2	no	—
209	Dic₁₃₄	—	2	2	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	2	2	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₆₇⋊C₈	—	4	2	no	—
217	C₆₇⋊C₈	—	4	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₆₇	#538	1	1	yes	characteristic
227	C₂²×C₆₇	#539	1	1	yes	characteristic
228	C₂×D₆₇	—	4	2	no	—
229	C₂×D₆₇	—	4	2	no	—
230	C₂×D₆₇	—	4	2	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	4	4	no	—
234	C₄×C₆₇	#518	1	1	yes	characteristic
235	C₄×C₆₇	#520	1	1	yes	characteristic
236	C₄×C₆₇	#534	1	1	yes	characteristic
237	C₄×C₆₇	#537	1	1	yes	characteristic
238	Dic₆₇	—	4	2	no	—
239	Dic₆₇	—	4	2	no	—
240	Dic₆₇	—	4	2	no	—
241	#22207	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	1	no	—
243	C₂×C₆₇	—	2	1	no	—
244	C₂×C₆₇	—	4	1	no	—
245	C₂×C₆₇	—	4	1	no	—
246	C₂×C₆₇	#2325	1	1	yes	characteristic
247	C₂×C₆₇	#2956	1	2	yes	—
248	D₆₇	—	8	2	no	—
249	#3054	—	67	1	no	—
250	#2930	—	67	2	no	—
251	#1929	—	67	2	no	—
252	#1922	—	67	2	no	—
253	#1896	—	67	2	no	—
254	#1876	—	67	2	no	—
255	#1866	—	67	2	no	—
256	#1811	—	67	1	no	—
257	#1728	D₆₇	1	1	yes	characteristic, complemented
258	C₆₇	#22207	1	1	yes	characteristic, complemented, Sylow
259	C₂×#135	—	67	1	no	—
260	C₂×#131	—	67	2	no	—
261	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
262	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
263	#525	—	67	2	no	—
264	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
265	#470	—	67	1	no	—
266	#470	—	67	1	no	—
267	#469	—	67	1	no	—
268	#469	—	67	1	no	—
269	#468	—	67	2	no	—
270	#468	—	67	2	no	—
271	#436	—	134	2	no	—
272	#436	—	134	2	no	—
273	#435	—	134	2	no	—
274	#435	—	134	2	no	—
275	C₂×#108	—	67	1	no	—
276	C₂×#108	—	67	1	no	—
277	#427	—	67	1	no	—
278	#424	—	67	2	no	—
279	C₂×#104	—	67	2	no	—
280	C₂×#104	—	67	2	no	—
281	#416	—	67	2	no	—
282	#408	—	67	2	no	—
283	#407	—	67	2	no	—
284	#405	—	67	2	no	—
285	#341	—	67	2	no	—
286	#339	—	67	2	no	—
287	#339	—	67	2	no	—
288	#336	C₂×D₆₇	1	2	yes	—
289	#336	C₂×D₆₇	1	2	yes	—
290	C₂×#41	—	67	2	no	—
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
293	#135	—	268	1	no	—
294	#135	—	268	1	no	—
295	C₂×Q₁₆	—	67	1	no	—
296	C₂×SD₁₆	—	67	2	no	—
297	C₂×D₈	—	67	1	no	—
298	#131	—	268	2	no	—
299	#131	—	268	2	no	—
300	C₂×M₁₆	—	4	1	no	—
301	C₂×M₁₆	—	4	1	no	—
302	C₂×M₁₆	—	4	1	no	—
303	C₂×M₁₆	—	4	1	no	—
304	C₂×M₁₆	—	67	2	no	—
305	C₂×M₁₆	—	67	2	no	—
306	C₂×M₁₆	—	134	2	no	—
307	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
308	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
309	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
310	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
311	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
312	C₂×M₁₆	D₂₆₈	1	2	yes	—
313	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
314	C₂²×C₈	—	134	2	no	—
315	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
316	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
317	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
318	#123	—	134	2	no	—
319	C₂²⋊Q₈	—	67	2	no	—
320	C₂²⋊Q₈	—	67	2	no	—
321	#121	—	67	2	no	—
322	#121	—	67	2	no	—
323	#120	—	134	2	no	—
324	C₄×D₄	—	134	2	no	—
325	C₄×D₄	—	134	2	no	—
326	#117	—	67	2	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	—	4	1	no	—
329	C₂×C₄²	—	4	1	no	—
330	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
331	#108	—	268	1	no	—
332	#108	—	268	1	no	—
333	#107	—	67	2	no	—
334	#106	—	67	1	no	—
335	#106	—	67	1	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	#104	—	268	2	no	—
339	#104	—	268	2	no	—
340	#103	—	67	2	no	—
341	#103	—	67	2	no	—
342	#102	—	67	2	no	—
343	#102	—	67	2	no	—
344	C₂²⋊C₈	—	134	2	no	—
345	C₂²⋊C₈	—	134	2	no	—
346	C₂²⋊C₈	—	134	2	no	—
347	C₂²⋊C₈	—	134	2	no	—
348	#97	—	134	2	no	—
349	C₄×C₈	—	134	2	no	—
350	C₂⁴	#11761628	1	1	yes	characteristic
351	#41	—	134	2	no	—
352	#41	—	134	2	no	—
353	#41	—	134	2	no	—
354	#41	—	134	2	no	—
355	C₂×Q₈	—	67	2	no	—
356	C₂×D₄	—	67	2	no	—
357	C₂×D₄	—	134	2	no	—
358	C₂×D₄	—	134	2	no	—
359	C₂×D₄	—	134	2	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	134	2	no	—
368	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
372	C₂²×C₄	#11761628	1	1	yes	characteristic
373	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
374	Q₁₆	—	268	1	no	—
375	SD₁₆	—	268	2	no	—
376	D₈	—	268	1	no	—
377	M₁₆	—	4	1	no	—
378	M₁₆	—	4	1	no	—
379	M₁₆	—	4	1	no	—
380	M₁₆	—	4	1	no	—
381	M₁₆	—	4	1	no	—
382	M₁₆	—	4	2	no	—
383	M₁₆	—	4	1	no	—
384	M₁₆	—	268	4	no	—
385	M₁₆	—	134	2	no	—
386	M₁₆	—	134	2	no	—
387	M₁₆	—	134	2	no	—
388	M₁₆	—	134	2	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	268	4	no	—
396	C₂×C₈	—	134	2	no	—
397	C₂×C₈	—	134	2	no	—
398	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
399	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
400	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
401	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
402	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
403	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
404	C₄⋊C₄	—	67	2	no	—
405	C₄⋊C₄	—	67	2	no	—
406	C₄⋊C₄	—	134	2	no	—
407	C₄⋊C₄	—	134	2	no	—
408	C₂²⋊C₄	—	134	2	no	—
409	C₂²⋊C₄	—	134	2	no	—
410	C₂²⋊C₄	—	134	2	no	—
411	C₂²⋊C₄	—	134	2	no	—
412	C₂²⋊C₄	—	134	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	4	2	no	—
416	C₄²	—	134	2	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	134	2	no	—
421	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
422	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
423	C₂³	C₂×#11761628	1	1	yes	characteristic
424	Q₈	—	134	2	no	—
425	Q₈	—	134	2	no	—
426	D₄	—	268	2	no	—
427	D₄	—	268	2	no	—
428	D₄	—	536	2	no	—
429	D₄	—	536	2	no	—
430	D₄	—	134	2	no	—
431	D₄	—	134	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	268	2	no	—
447	C₂×C₄	—	268	2	no	—
448	C₂×C₄	—	134	2	no	—
449	C₂×C₄	—	134	2	no	—
450	C₂×C₄	—	134	2	no	—
451	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
452	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
453	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
455	C₂×C₄	#14524088	1	1	yes	characteristic
456	C₂×C₄	#14524088	1	1	yes	characteristic
457	C₂×C₄	#14524088	1	1	yes	characteristic
458	C₂×C₄	#14524091	1	1	yes	characteristic
459	C₈	—	2	1	no	—
460	C₈	—	2	1	no	—
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	2	no	—
465	C₈	—	268	2	no	—
466	C₈	—	268	2	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	268	2	no	—
475	C₂²	—	268	2	no	—
476	C₂²	—	268	2	no	—
477	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
478	C₂²	#16863984	1	1	yes	characteristic
479	C₂²	#16863993	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	4	no	—
483	C₄	—	268	2	no	—
484	C₄	—	268	2	no	—
485	C₄	—	268	2	no	—
486	C₄	#16863869	1	1	yes	characteristic
487	C₄	#16863877	1	1	yes	characteristic
488	C₄	#16863958	1	1	yes	characteristic
489	C₄	#16863977	1	1	yes	characteristic
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	1	no	—
493	C₂	—	4	1	no	—
494	C₂	—	536	2	no	—
495	C₂	#22562912	1	1	yes	characteristic
496	C₂	#22569120	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1728	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22569818	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1728	D₆₇
hypocenter	C₆₇	#22207
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ #1728

Complements

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

N	H ≅ G/N	type
2. #22568912	C₂	semidirect
3. #22560858	C₂	semidirect
4. #22560809	C₂	semidirect
5. #22560652	C₂	semidirect
6. #22560535	C₂	semidirect
7. #22560487	C₂	semidirect
8. #22560251	C₂	semidirect
9. C₆₇×#1728	C₂	semidirect
13. #16863654	C₄	semidirect
14. #16863654	C₄	semidirect
15. #16863649	C₄	semidirect
16. #16863649	C₄	semidirect
17. #16863642	C₄	semidirect
18. #16863642	C₄	semidirect
44. C₂×Dic₂₆₈	C₂×C₄	semidirect
45. C₂×#11761617	C₂×C₄	semidirect
46. C₂×D₅₃₆	C₂×C₄	semidirect
64. #14523977	C₂×C₄	semidirect
65. #14523975	C₂×C₄	semidirect
66. #14523975	C₂×C₄	semidirect
71. #14523968	C₂×C₄	semidirect
72. #14523968	C₂×C₄	semidirect
73. #14523965	C₂×C₄	semidirect
74. #14523965	C₂×C₄	semidirect
257. #1728	D₆₇	semidirect
258. C₆₇	#22207	semidirect

Maximal subgroups

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

subgroup	index
#22569818	2
#22568912	2
#22560858	2
#22560809	2
#22560652	2
#22560535	2
#22560487	2
#22560251	2
C₆₇×#1728	2
#22207	67