Subgroups — #30874350

Order 17152.

812 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569124	C₂	1	2	yes	maximal
2	C₂×#16863511	C₂	1	1	yes	maximal, characteristic
3	#22568817	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560996	C₂	1	2	yes	maximal, complemented
5	#22560449	C₂	1	2	yes	maximal, complemented
6	#22560446	C₂	1	2	yes	maximal, complemented
7	#22560379	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560264	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560263	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560091	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	#16864162	C₂²	1	2	yes	—
14	C₂×#14524049	C₂²	1	1	yes	characteristic
15	#16863516	—	2	4	no	—
16	#16863511	—	2	4	no	—
17	#16863511	C₄	1	2	yes	complemented
18	#16863511	C₄	1	2	yes	complemented
19	#16863511	C₄	1	2	yes	complemented
20	#16863511	C₄	1	2	yes	complemented
21	#16863444	C₂²	1	2	yes	—
22	#16863443	—	2	2	no	—
23	C₂×#14523979	C₂²	1	1	yes	characteristic
24	C₂×#14523978	C₂²	1	2	yes	—
25	C₂×#14523978	C₂²	1	1	yes	characteristic
26	#108×D₆₇	—	2	2	no	—
27	#16863434	C₂²	1	1	yes	characteristic
28	#16863419	C₂²	1	2	yes	—
29	#16863413	C₂²	1	2	yes	—
30	#16863412	C₂²	1	2	yes	—
31	C₂×#14523970	C₂²	1	1	yes	characteristic
32	C₂×#14523969	C₂²	1	1	yes	characteristic
33	#16863354	C₂²	1	2	yes	—
34	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
35	#16863168	C₂²	1	2	yes	—
36	#16863167	C₂²	1	2	yes	—
37	#16863157	C₂²	1	2	yes	—
38	#568×C₆₇	C₂²	1	2	yes	—
39	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
40	#430×C₆₇	—	2	2	no	—
41	#430×C₆₇	C₂²	1	2	yes	—
42	C₂×#108×C₆₇	—	2	2	no	—
43	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761628	C₂³	1	1	yes	characteristic
46	#14524063	C₂×C₄	1	2	yes	—
47	#14524063	C₂×C₄	1	2	yes	—
48	#14524063	D₄	1	2	yes	—
49	#14524063	D₄	1	2	yes	—
50	C₂×#11761618	Q₈	1	1	yes	characteristic
51	C₂×#11761616	Q₈	1	1	yes	characteristic
52	C₂×#11761615	Q₈	1	1	yes	characteristic
53	C₂×#11761613	Q₈	1	1	yes	characteristic
54	#14524051	—	4	2	no	—
55	#14524049	—	2	4	no	—
56	#14524049	—	2	1	no	—
57	#14524049	—	2	1	no	—
58	#14524049	C₂×C₄	1	2	yes	—
59	#14524049	C₂×C₄	1	2	yes	—
60	C₂×#11761612	C₂³	1	2	yes	—
61	C₂×#11761612	C₂³	1	1	yes	characteristic
62	C₂×#11761611	—	2	1	no	—
63	M₁₆×D₆₇	—	2	4	no	—
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₆₇	—	2	1	no	—
66	#14524002	C₂³	1	2	yes	—
67	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
68	#14523979	—	2	1	no	—
69	#14523979	—	2	1	no	—
70	#14523978	—	2	2	no	—
71	#14523978	—	2	2	no	—
72	#14523978	—	2	4	no	—
73	#14523978	C₂×C₄	1	2	yes	—
74	#14523978	C₂×C₄	1	2	yes	—
75	#14523977	—	2	1	no	—
76	#14523975	—	2	1	no	—
77	Dic₆₇⋊C₈	—	4	2	no	—
78	#14523970	—	2	4	no	—
79	#14523970	C₂×C₄	1	2	yes	—
80	#14523970	C₂×C₄	1	2	yes	—
81	#14523969	—	2	4	no	—
82	#14523969	C₂×C₄	1	2	yes	—
83	#14523969	C₂×C₄	1	2	yes	—
84	#14523953	—	2	2	no	—
85	C₈×Dic₆₇	—	2	2	no	—
86	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
88	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
89	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
91	#131×C₆₇	—	2	1	no	—
92	#131×C₆₇	—	2	1	no	—
93	#131×C₆₇	—	2	4	no	—
94	#131×C₆₇	—	4	2	no	—
95	#131×C₆₇	C₂×C₄	1	2	yes	—
96	#131×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×M₁₆×C₆₇	—	2	1	no	—
98	C₂×M₁₆×C₆₇	—	2	4	no	—
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
102	C₂²×C₈×C₆₇	—	2	1	no	—
103	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
104	#108×C₆₇	—	2	4	no	—
105	#108×C₆₇	—	2	1	no	—
106	#108×C₆₇	—	2	1	no	—
107	#108×C₆₇	D₄	1	2	yes	—
108	#108×C₆₇	D₄	1	2	yes	—
109	#11761628	—	2	1	no	—
110	#11761628	—	2	1	no	—
111	#11761628	C₂²×C₄	1	2	yes	—
112	#11761628	C₂×D₄	1	2	yes	—
113	C₂×Dic₁₃₄	C₂×Q₈	1	1	yes	characteristic
114	C₂×D₂₆₈	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₂²⋊D₆₇	—	2	1	no	—
116	C₂²×Dic₆₇	—	2	1	no	—
117	C₂×C₄×D₆₇	—	2	1	no	—
118	#11761618	C₄⋊C₄	1	2	yes	—
119	#11761618	C₄⋊C₄	1	2	yes	—
120	#11761616	C₄⋊C₄	1	2	yes	—
121	#11761616	C₄⋊C₄	1	2	yes	—
122	#11761615	C₄⋊C₄	1	2	yes	—
123	#11761615	C₄⋊C₄	1	2	yes	—
124	#11761613	C₄⋊C₄	1	2	yes	—
125	#11761613	C₄⋊C₄	1	2	yes	—
126	#11761612	—	2	1	no	—
127	#11761612	—	2	2	no	—
128	#11761612	—	2	2	no	—
129	#11761612	C₂²×C₄	1	2	yes	—
130	#11761611	—	2	2	no	—
131	#11761611	—	2	2	no	—
132	#11761611	—	4	4	no	—
133	C₂×C₆₇⋊C₈	—	2	1	no	—
134	C₂×C₆₇⋊C₈	—	2	2	no	—
135	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₆₇⋊C₈	C₂×Q₈	1	1	yes	characteristic
140	C₂×C₆₇⋊C₈	C₂×Q₈	1	1	yes	characteristic
141	C₈×D₆₇	—	2	2	no	—
142	C₈×D₆₇	—	2	2	no	—
143	C₈×D₆₇	—	4	4	no	—
144	C₄⋊Dic₆₇	—	2	2	no	—
145	C₂²⋊Dic₆₇	—	2	2	no	—
146	C₄×Dic₆₇	—	2	1	no	—
147	C₄×Dic₆₇	—	2	2	no	—
148	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
150	#41×C₆₇	—	2	1	no	—
151	#41×C₆₇	—	2	1	no	—
152	#41×C₆₇	C₂²×C₄	1	2	yes	—
153	#41×C₆₇	C₂×D₄	1	2	yes	—
154	C₂×Q₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	M₁₆×C₆₇	—	2	1	no	—
160	M₁₆×C₆₇	—	2	2	no	—
161	M₁₆×C₆₇	—	2	2	no	—
162	M₁₆×C₆₇	—	2	2	no	—
163	M₁₆×C₆₇	—	2	2	no	—
164	M₁₆×C₆₇	—	2	2	no	—
165	M₁₆×C₆₇	—	4	4	no	—
166	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
167	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
168	C₂×C₈×C₆₇	—	2	1	no	—
169	C₂×C₈×C₆₇	—	2	2	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	4	4	no	—
174	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
179	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
182	C₂²×D₆₇	—	2	1	no	—
183	Q₈×C₆₇	—	2	1	no	—
184	Q₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
185	D₄×C₆₇	—	2	2	no	—
186	D₄×C₆₇	—	2	2	no	—
187	D₄×C₆₇	—	2	1	no	—
188	D₄×C₆₇	C₂×C₄⋊C₄	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	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂²⋊D₆₇	—	2	2	no	—
199	C₂²⋊D₆₇	—	2	2	no	—
200	C₂×Dic₆₇	—	2	1	no	—
201	C₂×Dic₆₇	—	2	1	no	—
202	C₂×Dic₆₇	—	2	1	no	—
203	C₂×Dic₆₇	—	4	2	no	—
204	C₂×Dic₆₇	—	4	2	no	—
205	D₂₆₈	—	2	1	no	—
206	D₂₆₈	C₂×C₄⋊C₄	1	2	yes	—
207	C₄×D₆₇	—	2	2	no	—
208	C₄×D₆₇	—	2	2	no	—
209	Dic₁₃₄	—	2	1	no	—
210	Dic₁₃₄	C₂×C₄⋊C₄	1	2	yes	—
211	C₈×C₆₇	—	2	2	no	—
212	C₈×C₆₇	—	2	4	no	—
213	C₈×C₆₇	—	4	2	no	—
214	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
215	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
216	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
217	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
218	C₆₇⋊C₈	—	2	2	no	—
219	C₆₇⋊C₈	—	4	2	no	—
220	C₆₇⋊C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
221	C₆₇⋊C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₆₇⋊C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₆₇⋊C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	4	1	no	—
226	C₂²×C₆₇	—	8	1	no	—
227	C₂²×C₆₇	#520	1	1	yes	characteristic
228	C₂²×C₆₇	#547	1	1	yes	characteristic
229	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
230	C₂×D₆₇	—	2	2	no	—
231	C₂×D₆₇	—	8	1	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₆₇	D₄²	1	1	yes	characteristic
236	C₄×C₆₇	#554	1	1	yes	characteristic
237	Dic₆₇	—	2	2	no	—
238	Dic₆₇	—	2	2	no	—
239	Dic₆₇	—	4	1	no	—
240	Dic₆₇	—	8	2	no	—
241	#22694	—	67	1	no	maximal, Sylow
242	C₂×C₆₇	—	2	1	no	—
243	C₂×C₆₇	—	2	1	no	—
244	C₂×C₆₇	—	2	1	no	—
245	C₂×C₆₇	—	4	2	no	—
246	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
247	D₆₇	—	4	2	no	—
248	#2957	—	67	2	no	—
249	C₂×#443	—	67	1	no	—
250	#2916	—	67	1	no	—
251	#2916	D₆₇	1	1	yes	characteristic, complemented
252	#1953	—	67	2	no	—
253	#1861	—	67	2	no	—
254	#1861	—	67	2	no	—
255	#1848	—	67	1	no	—
256	#1814	—	67	1	no	—
257	#1814	—	67	1	no	—
258	#1772	—	67	1	no	—
259	C₆₇	#22694	1	1	yes	characteristic, complemented, Sylow
260	C₂×#135	—	67	1	no	—
261	#568	—	67	2	no	—
262	#568	C₂×D₆₇	1	2	yes	—
263	C₂×#131	—	67	1	no	—
264	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
265	#444	—	134	4	no	—
266	#443	—	67	2	no	—
267	#443	—	67	2	no	—
268	#443	—	67	2	no	—
269	#443	—	67	2	no	—
270	#443	—	134	4	no	—
271	#430	—	2	2	no	—
272	#430	—	67	2	no	—
273	#430	—	134	2	no	—
274	#430	C₂×D₆₇	1	2	yes	—
275	C₂×#108	—	2	2	no	—
276	C₂×#108	—	67	1	no	—
277	C₂×#108	—	67	2	no	—
278	C₂×#108	—	67	1	no	—
279	C₂×#108	—	134	2	no	—
280	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
281	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
282	#427	—	67	1	no	—
283	#423	—	67	2	no	—
284	#421	—	67	2	no	—
285	#421	—	67	2	no	—
286	C₂×#104	—	67	1	no	—
287	C₂×#104	—	67	1	no	—
288	#405	—	67	2	no	—
289	C₂×C₄×C₈	—	67	1	no	—
290	#344	—	67	2	no	—
291	#344	—	67	2	no	—
292	#339	—	67	2	no	—
293	C₂×#41	—	67	1	no	—
294	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
295	#135	—	67	2	no	—
296	#135	—	67	2	no	—
297	#135	—	67	2	no	—
298	#135	—	67	2	no	—
299	C₂×Q₁₆	—	67	1	no	—
300	C₂×SD₁₆	—	67	1	no	—
301	C₂×SD₁₆	—	67	1	no	—
302	C₂×D₈	—	67	1	no	—
303	#131	—	2	1	no	—
304	#131	—	2	1	no	—
305	#131	—	2	4	no	—
306	#131	—	4	2	no	—
307	#131	—	268	2	no	—
308	#131	—	67	2	no	—
309	#131	—	67	2	no	—
310	#131	—	134	4	no	—
311	#131	—	134	1	no	—
312	#131	—	134	1	no	—
313	#131	C₄×D₆₇	1	2	yes	—
314	#131	C₄×D₆₇	1	2	yes	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	4	no	—
317	C₂×M₁₆	—	67	1	no	—
318	C₂×M₁₆	—	67	2	no	—
319	C₂×M₁₆	—	134	4	no	—
320	C₂×M₁₆	—	134	1	no	—
321	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
322	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
323	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	67	1	no	—
326	C₂²×C₈	—	134	1	no	—
327	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
328	#117	—	67	2	no	—
329	C₂×C₄²	—	67	1	no	—
330	#108	—	2	4	no	—
331	#108	—	2	1	no	—
332	#108	—	2	1	no	—
333	#108	—	67	2	no	—
334	#108	—	67	2	no	—
335	#108	—	134	2	no	—
336	#108	—	134	2	no	—
337	#108	—	134	4	no	—
338	#108	—	134	1	no	—
339	#108	—	134	1	no	—
340	#108	C₂²⋊D₆₇	1	2	yes	—
341	#108	C₂²⋊D₆₇	1	2	yes	—
342	#107	—	134	1	no	—
343	#106	—	134	1	no	—
344	C₄⋊C₈	—	268	2	no	—
345	#104	—	67	2	no	—
346	#104	—	67	2	no	—
347	#104	—	67	2	no	—
348	#104	—	67	2	no	—
349	#104	—	134	4	no	—
350	#104	—	134	4	no	—
351	#97	—	134	2	no	—
352	C₄×C₈	—	67	1	no	—
353	C₄×C₈	—	67	1	no	—
354	C₄×C₈	—	67	1	no	—
355	C₄×C₈	—	67	1	no	—
356	C₄×C₈	—	134	2	no	—
357	#41	—	2	1	no	—
358	#41	—	2	1	no	—
359	#41	—	67	2	no	—
360	#41	—	67	2	no	—
361	#41	—	134	1	no	—
362	#41	—	134	1	no	—
363	#41	C₂×C₄×D₆₇	1	2	yes	—
364	#41	D₄×D₆₇	1	2	yes	—
365	C₂×Q₈	—	67	1	no	—
366	C₂×Q₈	Q₈×D₆₇	1	1	yes	characteristic
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	67	1	no	—
369	C₂×D₄	—	134	1	no	—
370	C₂×D₄	Q₈×D₆₇	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	134	1	no	—
373	C₂²×C₄	—	134	1	no	—
374	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
375	Q₁₆	—	67	2	no	—
376	Q₁₆	—	67	2	no	—
377	SD₁₆	—	67	2	no	—
378	SD₁₆	—	67	2	no	—
379	SD₁₆	—	67	2	no	—
380	SD₁₆	—	67	2	no	—
381	D₈	—	67	2	no	—
382	D₈	—	67	2	no	—
383	M₁₆	—	2	1	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	4	4	no	—
390	M₁₆	—	268	4	no	—
391	M₁₆	—	67	2	no	—
392	M₁₆	—	134	2	no	—
393	M₁₆	—	134	2	no	—
394	M₁₆	—	134	1	no	—
395	M₁₆	—	134	2	no	—
396	M₁₆	—	134	2	no	—
397	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
398	M₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	4	4	no	—
405	C₂×C₈	—	268	4	no	—
406	C₂×C₈	—	67	1	no	—
407	C₂×C₈	—	67	1	no	—
408	C₂×C₈	—	67	1	no	—
409	C₂×C₈	—	67	1	no	—
410	C₂×C₈	—	67	1	no	—
411	C₂×C₈	—	67	1	no	—
412	C₂×C₈	—	134	2	no	—
413	C₂×C₈	—	134	1	no	—
414	C₂×C₈	—	134	2	no	—
415	C₂×C₈	—	134	2	no	—
416	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
417	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
418	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
419	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
420	C₂×C₈	#11761628	1	1	yes	characteristic
421	C₂×C₈	#11761628	1	1	yes	characteristic
422	C₄⋊C₄	—	134	2	no	—
423	C₂²⋊C₄	—	134	2	no	—
424	C₄²	—	67	1	no	—
425	C₄²	—	67	1	no	—
426	C₄²	—	134	2	no	—
427	C₄²	—	134	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	134	1	no	—
430	C₂³	#14524091	1	1	yes	characteristic
431	Q₈	—	2	1	no	—
432	Q₈	—	67	2	no	—
433	Q₈	—	134	1	no	—
434	Q₈	C₄⋊C₄×D₆₇	1	2	yes	—
435	D₄	—	2	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	1	no	—
438	D₄	—	67	2	no	—
439	D₄	—	134	2	no	—
440	D₄	—	134	2	no	—
441	D₄	—	134	1	no	—
442	D₄	C₄⋊C₄×D₆₇	1	2	yes	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	268	2	no	—
447	C₂×C₄	—	268	2	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	—	134	1	no	—
450	C₂×C₄	—	134	1	no	—
451	C₂×C₄	—	134	2	no	—
452	C₂×C₄	—	134	2	no	—
453	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
455	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
456	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
457	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
458	C₂×C₄	#14524088	1	1	yes	characteristic
459	C₈	—	2	4	no	—
460	C₈	—	2	2	no	—
461	C₈	—	4	2	no	—
462	C₈	—	268	2	no	—
463	C₈	—	67	1	no	—
464	C₈	—	67	1	no	—
465	C₈	—	67	1	no	—
466	C₈	—	67	1	no	—
467	C₈	—	134	2	no	—
468	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
469	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
470	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
471	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	8	1	no	—
475	C₂²	—	536	1	no	—
476	C₂²	—	134	2	no	—
477	C₂²	#16863877	1	1	yes	characteristic
478	C₂²	#16864047	1	1	yes	characteristic
479	C₂²	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	268	1	no	—
482	C₄	—	536	2	no	—
483	C₄	—	134	2	no	—
484	C₄	—	134	2	no	—
485	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
486	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
487	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
488	C₄	#16864101	1	1	yes	characteristic
489	C₂	—	2	1	no	—
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	2	no	—
493	C₂	—	268	2	no	—
494	C₂	C₄⋊C₄×C₂²⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2916	C₂
socle	C₂×C₆₇	D₄×C₄⋊C₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2916	D₆₇
hypocenter	C₆₇	#22694
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2916

Complements

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

N	H ≅ G/N	type
3. #22568817	C₂	semidirect
4. #22560996	C₂	semidirect
5. #22560449	C₂	semidirect
6. #22560446	C₂	semidirect
7. #22560379	C₂	semidirect
8. #22560264	C₂	semidirect
9. #22560263	C₂	semidirect
10. #22560091	C₂	semidirect
11. C₆₇×#2916	C₂	semidirect
17. #16863511	C₄	semidirect
18. #16863511	C₄	semidirect
19. #16863511	C₄	semidirect
20. #16863511	C₄	semidirect
251. #2916	D₆₇	semidirect
259. C₆₇	#22694	semidirect

Maximal subgroups

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

subgroup	index
#22569124	2
C₂×#16863511	2
#22568817	2
#22560996	2
#22560449	2
#22560446	2
#22560379	2
#22560264	2
#22560263	2
#22560091	2
C₆₇×#2916	2
#22694	67