Subgroups — #30862782

Order 17152.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569112	C₂	1	2	yes	maximal, complemented
2	C₂×#16863518	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568635	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561108	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561000	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560880	C₂	1	2	yes	maximal, complemented
7	#22560712	C₂	1	2	yes	maximal, complemented
8	#22560263	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560096	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560012	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3067	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	2	yes	—
13	C₂×#14524050	C₂²	1	1	yes	characteristic
14	#41×Dic₆₇	C₂²	1	1	yes	characteristic
15	#16863518	—	2	4	no	—
16	#16863518	—	2	1	no	—
17	#16863518	—	2	1	no	—
18	#16863518	—	2	1	no	—
19	#16863518	—	2	1	no	—
20	#16863513	—	2	4	no	—
21	#16863445	C₂²	1	2	yes	—
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	#16863434	C₂²	1	1	yes	characteristic
24	#16863413	C₂²	1	2	yes	—
25	#16863413	C₂²	1	2	yes	—
26	C₂×#14523970	C₂²	1	2	yes	—
27	#16863370	C₂²	1	2	yes	—
28	#16863354	—	2	1	no	—
29	#16863354	C₂²	1	1	yes	characteristic
30	#16863354	C₂²	1	2	yes	—
31	#16863351	—	2	1	no	—
32	#16863351	C₂²	1	1	yes	characteristic
33	M₁₆×Dic₆₇	—	2	1	no	—
34	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×#14523953	—	2	1	no	—
36	C₂×#14523953	C₂²	1	1	yes	characteristic
37	#16863168	C₂²	1	2	yes	—
38	#16863168	C₂²	1	2	yes	—
39	#16863157	C₂²	1	2	yes	—
40	#575×C₆₇	C₂²	1	2	yes	—
41	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
43	#473×C₆₇	C₄	1	2	yes	complemented
44	#472×C₆₇	C₄	1	2	yes	complemented
45	#471×C₆₇	C₄	1	4	yes	complemented
46	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
47	#413×C₆₇	C₂²	1	2	yes	—
48	#14524050	—	2	2	no	—
49	#14524050	—	2	2	no	—
50	#14524050	—	2	2	no	—
51	#14524050	—	2	2	no	—
52	#14524050	—	2	2	no	—
53	#14524050	D₄	1	4	yes	—
54	C₂×#11761612	—	2	1	no	—
55	C₂×#11761612	—	2	2	no	—
56	C₂×#11761612	C₂³	1	2	yes	—
57	C₂×#11761612	C₂³	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	—	2	1	no	—
59	C₂²×C₆₇⋊C₈	—	2	2	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	Q₈×Dic₆₇	—	2	1	no	—
62	D₄×Dic₆₇	—	2	2	no	—
63	D₄×Dic₆₇	—	2	1	no	—
64	#14524002	—	2	1	no	—
65	#14524002	C₂³	1	2	yes	—
66	#14524002	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₆₇	—	2	1	no	—
68	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
69	#14523979	—	2	4	no	—
70	#14523979	—	2	1	no	—
71	#14523979	—	2	1	no	—
72	#14523977	—	2	1	no	—
73	#14523975	—	2	1	no	—
74	#14523970	—	2	4	no	—
75	#14523970	—	2	2	no	—
76	#14523970	—	2	2	no	—
77	#14523970	—	2	4	no	—
78	#14523958	—	2	4	no	—
79	#14523953	—	2	2	no	—
80	#14523953	—	2	1	no	—
81	#14523953	—	2	1	no	—
82	#14523953	—	2	1	no	—
83	#14523953	D₄	1	1	yes	characteristic
84	#14523953	D₄	1	1	yes	characteristic
85	#14523953	D₄	1	1	yes	characteristic
86	#14523953	D₄	1	1	yes	characteristic
87	C₈×Dic₆₇	—	2	2	no	—
88	C₈×Dic₆₇	—	2	1	no	—
89	C₈×Dic₆₇	—	2	1	no	—
90	C₈×Dic₆₇	—	2	1	no	—
91	C₂×#41×C₆₇	C₂³	1	2	yes	—
92	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
93	#137×C₆₇	—	2	2	no	—
94	#137×C₆₇	C₂×C₄	1	4	yes	—
95	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
96	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	4	yes	—
97	#135×C₆₇	—	2	4	no	—
98	#135×C₆₇	—	2	1	no	—
99	#135×C₆₇	—	2	1	no	—
100	#135×C₆₇	—	2	1	no	—
101	#135×C₆₇	—	2	1	no	—
102	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
103	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
104	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
105	#131×C₆₇	—	2	1	no	—
106	#131×C₆₇	—	2	1	no	—
107	#131×C₆₇	C₂×C₄	1	2	yes	—
108	#131×C₆₇	C₂×C₄	1	2	yes	—
109	C₂×M₁₆×C₆₇	—	2	1	no	—
110	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
112	C₂²×C₈×C₆₇	—	2	1	no	—
113	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	#108×C₆₇	C₂×C₄	1	2	yes	complemented
115	#108×C₆₇	C₂×C₄	1	2	yes	complemented
116	#101×C₆₇	C₂×C₄	1	4	yes	complemented
117	#100×C₆₇	C₂×C₄	1	4	yes	complemented
118	C₂²×Dic₆₇	—	2	2	no	—
119	C₂²×Dic₆₇	—	2	1	no	—
120	#11761612	—	2	1	no	—
121	#11761612	—	2	4	no	—
122	#11761612	—	2	2	no	—
123	#11761612	—	2	2	no	—
124	#11761612	—	4	2	no	—
125	#11761612	—	4	2	no	—
126	#11761612	C₂×D₄	1	2	yes	—
127	C₂×C₆₇⋊C₈	—	2	1	no	—
128	C₂×C₆₇⋊C₈	—	2	4	no	—
129	C₂×C₆₇⋊C₈	—	2	2	no	—
130	C₂×C₆₇⋊C₈	—	4	2	no	—
131	C₂×C₆₇⋊C₈	—	4	2	no	—
132	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
133	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	#41	1	2	yes	—
136	C₄⋊Dic₆₇	—	2	2	no	—
137	C₄⋊Dic₆₇	—	2	1	no	—
138	C₄⋊Dic₆₇	—	2	2	no	—
139	C₂²⋊Dic₆₇	—	2	2	no	—
140	C₂²⋊Dic₆₇	—	2	2	no	—
141	C₂²⋊Dic₆₇	#41	1	2	yes	—
142	C₄×Dic₆₇	—	2	2	no	—
143	C₄×Dic₆₇	—	2	1	no	—
144	C₄×Dic₆₇	—	2	1	no	—
145	C₄×Dic₆₇	—	2	2	no	—
146	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
147	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
148	#41×C₆₇	—	2	1	no	—
149	#41×C₆₇	—	2	1	no	—
150	#41×C₆₇	—	2	2	no	—
151	#41×C₆₇	—	2	2	no	—
152	#41×C₆₇	—	2	2	no	—
153	#41×C₆₇	—	2	2	no	—
154	#41×C₆₇	C₂²×C₄	1	2	yes	—
155	#41×C₆₇	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
157	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
158	C₂×D₄×C₆₇	—	2	1	no	—
159	C₂×D₄×C₆₇	—	2	2	no	—
160	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₆₇	—	2	4	no	—
166	Q₁₆×C₆₇	—	2	1	no	—
167	Q₁₆×C₆₇	—	2	1	no	—
168	SD₁₆×C₆₇	—	2	4	no	—
169	SD₁₆×C₆₇	—	2	4	no	—
170	SD₁₆×C₆₇	—	2	2	no	—
171	SD₁₆×C₆₇	—	2	2	no	—
172	D₈×C₆₇	—	2	4	no	—
173	D₈×C₆₇	—	2	1	no	—
174	D₈×C₆₇	—	2	1	no	—
175	M₁₆×C₆₇	—	2	1	no	—
176	M₁₆×C₆₇	—	2	2	no	—
177	M₁₆×C₆₇	—	2	2	no	—
178	M₁₆×C₆₇	—	2	2	no	—
179	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
180	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
181	C₂×C₈×C₆₇	—	2	1	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₂×C₈×C₆₇	—	2	2	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
190	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
191	C₂³×C₆₇	—	2	1	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	1	no	—
195	Q₈×C₆₇	—	2	2	no	—
196	Q₈×C₆₇	—	2	2	no	—
197	Q₈×C₆₇	C₄×D₄	1	2	yes	—
198	D₄×C₆₇	—	2	2	no	—
199	D₄×C₆₇	—	2	2	no	—
200	D₄×C₆₇	—	2	2	no	—
201	D₄×C₆₇	—	2	2	no	—
202	D₄×C₆₇	—	2	1	no	—
203	D₄×C₆₇	—	4	2	no	—
204	D₄×C₆₇	—	4	2	no	—
205	D₄×C₆₇	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
218	C₂×Dic₆₇	—	2	1	no	—
219	C₂×Dic₆₇	—	2	2	no	—
220	C₂×Dic₆₇	—	2	1	no	—
221	C₂×Dic₆₇	—	4	2	no	—
222	C₂×Dic₆₇	—	8	1	no	—
223	C₂×Dic₆₇	—	8	2	no	—
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
227	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
228	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
229	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
230	C₆₇⋊C₈	—	2	2	no	—
231	C₆₇⋊C₈	—	4	2	no	—
232	C₆₇⋊C₈	#121	1	2	yes	—
233	C₆₇⋊C₈	#121	1	2	yes	—
234	C₂²×C₆₇	—	2	2	no	—
235	C₂²×C₆₇	—	4	1	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	8	2	no	—
239	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
240	C₂²×C₆₇	#525	1	1	yes	characteristic
241	C₂²×C₆₇	#532	1	1	yes	characteristic
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
245	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
246	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
247	C₄×C₆₇	#529	1	1	yes	characteristic
248	Dic₆₇	—	4	2	no	—
249	Dic₆₇	—	4	1	no	—
250	Dic₆₇	—	4	1	no	—
251	Dic₆₇	—	8	2	no	—
252	#21559	—	67	1	no	maximal, Sylow
253	C₂×C₆₇	—	2	1	no	—
254	C₂×C₆₇	—	2	1	no	—
255	C₂×C₆₇	—	2	1	no	—
256	C₂×C₆₇	—	4	2	no	—
257	C₂×C₆₇	—	8	2	no	—
258	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
259	#3067	D₆₇	1	1	yes	characteristic
260	#2955	—	67	2	no	—
261	C₂×#444	—	67	1	no	—
262	#2875	—	67	1	no	—
263	#1976	—	67	1	no	—
264	#1953	—	67	1	no	—
265	#1932	—	67	2	no	—
266	#1906	—	67	2	no	—
267	#1814	—	67	1	no	—
268	#1773	—	67	1	no	—
269	#1757	—	67	2	no	—
270	C₆₇	#21559	1	1	yes	characteristic, complemented, Sylow
271	#575	C₂×D₆₇	1	2	yes	—
272	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
273	C₂×#131	—	67	1	no	—
274	C₂×#131	—	67	2	no	—
275	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
276	C₄×#41	—	67	1	no	—
277	#473	Dic₆₇	1	2	yes	complemented
278	#472	Dic₆₇	1	2	yes	complemented
279	#471	Dic₆₇	1	4	yes	complemented
280	#444	—	134	1	no	—
281	#444	—	134	1	no	—
282	#444	—	134	1	no	—
283	#444	—	134	1	no	—
284	#444	—	134	4	no	—
285	#443	—	134	4	no	—
286	#430	—	67	2	no	—
287	C₂×#108	—	67	1	no	—
288	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
289	#427	—	67	1	no	—
290	#421	—	67	2	no	—
291	#421	—	67	2	no	—
292	C₂×#104	—	67	2	no	—
293	#413	C₂×D₆₇	1	2	yes	—
294	#410	—	67	2	no	—
295	#405	—	67	1	no	—
296	#405	—	67	2	no	—
297	#405	—	67	1	no	—
298	#405	—	134	1	no	—
299	#405	—	134	1	no	—
300	C₄×M₁₆	—	67	1	no	—
301	C₄×M₁₆	—	134	1	no	—
302	C₂×#97	—	67	1	no	—
303	C₂×#97	—	134	1	no	—
304	#344	—	67	2	no	—
305	#344	—	67	2	no	—
306	#339	—	67	2	no	—
307	C₂×#41	C₂²×D₆₇	1	2	yes	—
308	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
309	#137	—	2	2	no	—
310	#137	C₂×Dic₆₇	1	4	yes	—
311	AGL(1,ℤ/8)	—	2	2	no	—
312	AGL(1,ℤ/8)	C₂×Dic₆₇	1	4	yes	—
313	#135	—	2	4	no	—
314	#135	—	2	1	no	—
315	#135	—	2	1	no	—
316	#135	—	2	1	no	—
317	#135	—	2	1	no	—
318	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic, complemented
319	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	complemented
320	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
321	#131	—	2	1	no	—
322	#131	—	2	1	no	—
323	#131	—	67	4	no	—
324	#131	—	134	2	no	—
325	#131	—	134	2	no	—
326	#131	—	134	2	no	—
327	#131	—	134	2	no	—
328	#131	—	134	2	no	—
329	#131	C₂×Dic₆₇	1	2	yes	—
330	#131	C₂×Dic₆₇	1	2	yes	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	67	1	no	—
333	C₂×M₁₆	—	67	2	no	—
334	C₂×M₁₆	—	134	2	no	—
335	C₂×M₁₆	—	134	1	no	—
336	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
337	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	—	67	1	no	—
340	C₂²×C₈	—	134	2	no	—
341	C₂²×C₈	—	134	1	no	—
342	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
343	C₄×Q₈	—	134	1	no	—
344	C₄×D₄	—	134	2	no	—
345	C₄×D₄	—	134	1	no	—
346	#117	—	67	1	no	—
347	#117	—	67	2	no	—
348	#117	—	134	1	no	—
349	C₂×C₄²	—	67	1	no	—
350	C₂×C₄²	—	134	1	no	—
351	#108	—	134	4	no	—
352	#108	—	134	1	no	—
353	#108	—	134	1	no	—
354	#108	C₂×Dic₆₇	1	2	yes	complemented
355	#108	C₂×Dic₆₇	1	2	yes	complemented
356	#107	—	134	1	no	—
357	#106	—	134	1	no	—
358	#104	—	134	2	no	—
359	#104	—	134	2	no	—
360	#104	—	134	4	no	—
361	#104	—	134	4	no	—
362	#101	C₂×Dic₆₇	1	4	yes	complemented
363	#100	C₂×Dic₆₇	1	4	yes	complemented
364	#99	—	134	4	no	—
365	#97	—	67	1	no	—
366	#97	—	67	1	no	—
367	#97	—	67	1	no	—
368	#97	—	67	1	no	—
369	#97	—	134	1	no	—
370	#97	—	134	1	no	—
371	#97	—	134	2	no	—
372	#97	—	134	1	no	—
373	C₄×C₈	—	134	1	no	—
374	C₄×C₈	—	134	1	no	—
375	C₄×C₈	—	134	2	no	—
376	C₄×C₈	—	134	1	no	—
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	C₂²×Dic₆₇	1	2	yes	—
384	#41	D₄×D₆₇	1	2	yes	—
385	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
386	C₂×Q₈	#11761627	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
390	C₂×D₄	#11761627	1	1	yes	characteristic
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	134	2	no	—
394	C₂²×C₄	—	134	1	no	—
395	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
396	Q₁₆	—	2	4	no	—
397	Q₁₆	—	2	1	no	—
398	Q₁₆	—	2	1	no	—
399	SD₁₆	—	2	4	no	—
400	SD₁₆	—	2	4	no	—
401	SD₁₆	—	2	2	no	—
402	SD₁₆	—	2	2	no	—
403	D₈	—	2	4	no	—
404	D₈	—	2	1	no	—
405	D₈	—	2	1	no	—
406	M₁₆	—	2	1	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	268	2	no	—
411	M₁₆	—	268	2	no	—
412	M₁₆	—	67	2	no	—
413	M₁₆	—	134	2	no	—
414	M₁₆	—	134	2	no	—
415	M₁₆	—	134	4	no	—
416	M₁₆	—	134	1	no	—
417	M₁₆	C₂²×Dic₆₇	1	2	yes	—
418	M₁₆	C₂²×Dic₆₇	1	4	yes	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	268	2	no	—
424	C₂×C₈	—	268	2	no	—
425	C₂×C₈	—	67	1	no	—
426	C₂×C₈	—	67	2	no	—
427	C₂×C₈	—	67	1	no	—
428	C₂×C₈	—	67	2	no	—
429	C₂×C₈	—	134	2	no	—
430	C₂×C₈	—	134	4	no	—
431	C₂×C₈	—	134	1	no	—
432	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
433	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
434	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
435	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
436	C₂×C₈	#11761627	1	1	yes	characteristic
437	C₂×C₈	#11761627	1	1	yes	characteristic
438	C₄⋊C₄	—	134	2	no	—
439	C₄⋊C₄	—	134	2	no	—
440	C₄⋊C₄	—	134	1	no	—
441	C₂²⋊C₄	—	67	2	no	—
442	C₂²⋊C₄	—	134	2	no	—
443	C₂²⋊C₄	—	134	2	no	—
444	C₄²	—	67	1	no	—
445	C₄²	—	67	1	no	—
446	C₄²	—	134	2	no	—
447	C₄²	—	134	2	no	—
448	C₄²	—	134	1	no	—
449	C₄²	—	134	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	#41×D₆₇	1	1	yes	characteristic
453	Q₈	—	2	1	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	D₄×Dic₆₇	1	2	yes	—
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	2	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	1	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	2	no	—
464	D₄	D₄×Dic₆₇	1	2	yes	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	268	2	no	—
472	C₂×C₄	—	536	1	no	—
473	C₂×C₄	—	536	2	no	—
474	C₂×C₄	—	134	1	no	—
475	C₂×C₄	—	134	1	no	—
476	C₂×C₄	—	134	2	no	—
477	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
480	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
481	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
482	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	2	4	no	—
485	C₈	—	268	2	no	—
486	C₈	—	67	2	no	—
487	C₈	—	67	2	no	—
488	C₈	—	134	2	no	—
489	C₈	D₄×Dic₆₇	1	1	yes	characteristic
490	C₈	D₄×Dic₆₇	1	1	yes	characteristic
491	C₈	D₄×Dic₆₇	1	1	yes	characteristic
492	C₈	D₄×Dic₆₇	1	1	yes	characteristic
493	C₂²	—	2	2	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	#41×Dic₆₇	1	1	yes	characteristic
499	C₂²	#16863906	1	1	yes	characteristic
500	C₂²	#16863941	1	1	yes	characteristic
501	C₄	—	2	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	268	1	no	—
504	C₄	—	268	1	no	—
505	C₄	—	268	2	no	—
506	C₄	—	536	2	no	—
507	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
508	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
509	C₄	#121×D₆₇	1	1	yes	characteristic
510	C₄	#16863927	1	1	yes	characteristic
511	C₂	—	2	1	no	—
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	4	2	no	—
515	C₂	—	8	2	no	—
516	C₂	#121×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3067	C₂
socle	C₂×C₆₇	C₄×#121
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3067	D₆₇
hypocenter	C₆₇	#21559
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3067

Complements

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

N	H ≅ G/N	type
1. #22569112	C₂	semidirect
2. C₂×#16863518	C₂	semidirect
3. #22568635	C₂	semidirect
4. #22561108	C₂	semidirect
5. #22561000	C₂	semidirect
6. #22560880	C₂	semidirect
7. #22560712	C₂	semidirect
8. #22560263	C₂	semidirect
9. #22560096	C₂	semidirect
10. #22560012	C₂	semidirect
43. #473×C₆₇	C₄	semidirect
44. #472×C₆₇	C₄	semidirect
45. #471×C₆₇	C₄	semidirect
102. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
103. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
104. C₂×D₈×C₆₇	C₂×C₄	semidirect
114. #108×C₆₇	C₂×C₄	semidirect
115. #108×C₆₇	C₂×C₄	semidirect
116. #101×C₆₇	C₂×C₄	semidirect
117. #100×C₆₇	C₂×C₄	semidirect
270. C₆₇	#21559	semidirect
277. #473	Dic₆₇	semidirect
278. #472	Dic₆₇	semidirect
279. #471	Dic₆₇	semidirect
318. C₂×Q₁₆	C₂×Dic₆₇	semidirect
319. C₂×SD₁₆	C₂×Dic₆₇	semidirect
320. C₂×D₈	C₂×Dic₆₇	semidirect
354. #108	C₂×Dic₆₇	semidirect
355. #108	C₂×Dic₆₇	semidirect
362. #101	C₂×Dic₆₇	semidirect
363. #100	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569112	2
C₂×#16863518	2
#22568635	2
#22561108	2
#22561000	2
#22560880	2
#22560712	2
#22560263	2
#22560096	2
#22560012	2
C₆₇×#3067	2
#21559	67