Subgroups — #30854409

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863516	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863510	C₂	1	1	yes	maximal, characteristic
3	#22569005	C₂	1	1	yes	maximal, characteristic
4	#22568636	C₂	1	1	yes	maximal, characteristic, complemented
5	#22568635	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560114	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560094	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560087	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560056	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560052	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560046	C₂	1	1	yes	maximal, characteristic, complemented
12	#104×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560013	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#14523953	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524063	C₂²	1	1	yes	characteristic
17	C₂×#14524051	C₂²	1	1	yes	characteristic
18	C₂×#14524050	C₂²	1	1	yes	characteristic
19	C₂×#14524049	C₂²	1	1	yes	characteristic
20	#41×Dic₆₇	C₂²	1	1	yes	characteristic
21	#16863864	C₂²	1	1	yes	characteristic
22	#16863516	C₄	1	8	yes	complemented
23	#16863510	C₄	1	8	yes	complemented
24	#16863505	C₄	1	2	yes	complemented
25	#16863500	C₄	1	2	yes	complemented
26	#16863499	C₄	1	2	yes	complemented
27	#16863493	C₄	1	2	yes	complemented
28	C₂×#14523978	C₂²	1	1	yes	characteristic
29	C₂×#14523978	C₂²	1	1	yes	characteristic
30	#16863432	C₂²	1	1	yes	characteristic
31	C₂×#14523971	C₂²	1	1	yes	characteristic
32	C₂×#14523970	C₂²	1	1	yes	characteristic
33	C₂×#14523969	C₂²	1	1	yes	characteristic
34	#16863393	C₂²	1	1	yes	characteristic
35	#16863354	—	2	1	no	—
36	#16863354	C₂²	1	1	yes	characteristic
37	#16863353	—	2	1	no	—
38	#16863352	C₂²	1	1	yes	characteristic
39	#16863351	—	2	1	no	—
40	#16863351	C₂²	1	1	yes	characteristic
41	#16863350	—	2	1	no	—
42	M₁₆×Dic₆₇	—	2	1	no	—
43	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
44	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
45	C₄×#11761611	—	2	1	no	—
46	C₂×#14523953	—	2	1	no	—
47	C₂×#14523953	C₂²	1	1	yes	characteristic
48	C₂×#14523953	C₂²	1	1	yes	characteristic
49	C₂×#14523953	C₂²	1	1	yes	characteristic
50	C₂×#14523953	C₂²	1	1	yes	characteristic
51	#97×D₆₇	—	2	1	no	—
52	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
53	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
54	#16863161	C₂²	1	1	yes	characteristic
55	#16863157	C₂²	1	1	yes	characteristic
56	#16863156	C₂²	1	1	yes	characteristic
57	#16863155	C₂²	1	1	yes	characteristic
58	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
60	#416×C₆₇	C₂²	1	1	yes	characteristic
61	#405×C₆₇	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
63	#341×C₆₇	C₂²	1	1	yes	characteristic
64	#339×C₆₇	C₂²	1	1	yes	characteristic
65	C₂×#11761628	C₂³	1	1	yes	characteristic
66	#14524063	C₂×C₄	1	4	yes	—
67	#14524063	C₂×C₄	1	4	yes	—
68	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
69	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
70	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
71	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
72	#14524051	—	2	2	no	—
73	#14524051	C₂×C₄	1	4	yes	complemented
74	#14524050	—	2	2	no	—
75	#14524050	C₂×C₄	1	4	yes	complemented
76	#14524049	—	2	2	no	—
77	#14524049	C₂×C₄	1	4	yes	complemented
78	C₂×#11761612	—	2	1	no	—
79	C₂×#11761612	C₂³	1	1	yes	characteristic
80	C₂×#11761612	C₂³	1	1	yes	characteristic
81	C₂×#11761611	—	2	1	no	—
82	C₂×#11761611	—	2	1	no	—
83	C₂²×C₆₇⋊C₈	—	2	1	no	—
84	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
85	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₈×D₆₇	—	2	1	no	—
87	C₂×C₈×D₆₇	—	2	1	no	—
88	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
89	#14524008	C₂×C₄	1	2	yes	—
90	D₄×Dic₆₇	—	2	2	no	—
91	D₄×Dic₆₇	C₂×C₄	1	2	yes	—
92	#14524005	C₂×C₄	1	2	yes	—
93	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
94	#14524002	—	2	1	no	—
95	#14524002	C₂³	1	1	yes	characteristic
96	#14524000	—	2	1	no	—
97	C₂×C₄×Dic₆₇	—	2	1	no	—
98	C₂×C₄×Dic₆₇	—	2	2	no	—
99	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
101	C₄²×D₆₇	—	2	1	no	—
102	#14523978	C₂×C₄	1	4	yes	complemented
103	#14523978	C₂×C₄	1	4	yes	complemented
104	#14523976	C₂×C₄	1	2	yes	—
105	#14523974	C₂×C₄	1	2	yes	—
106	#14523971	C₂×C₄	1	4	yes	complemented
107	#14523970	C₂×C₄	1	4	yes	complemented
108	#14523969	C₂×C₄	1	4	yes	complemented
109	#14523968	C₂×C₄	1	2	yes	—
110	#14523965	C₂×C₄	1	2	yes	—
111	#14523954	—	2	1	no	—
112	#14523953	—	2	1	no	—
113	#14523953	—	2	2	no	—
114	#14523953	—	2	1	no	—
115	#14523953	—	2	1	no	—
116	#14523953	—	2	1	no	—
117	#14523953	C₂×C₄	1	2	yes	—
118	#14523953	C₂×C₄	1	2	yes	—
119	#14523953	C₂×C₄	1	4	yes	complemented
120	#14523953	D₄	1	1	yes	characteristic
121	#14523953	D₄	1	1	yes	characteristic
122	#14523953	D₄	1	1	yes	characteristic
123	#14523953	D₄	1	1	yes	characteristic
124	C₄×C₆₇⋊C₈	—	2	1	no	—
125	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
126	C₈×Dic₆₇	—	2	1	no	—
127	C₈×Dic₆₇	—	2	1	no	—
128	C₈×Dic₆₇	—	2	1	no	—
129	C₈×Dic₆₇	—	2	1	no	—
130	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
131	#131×C₆₇	—	2	2	no	—
132	#131×C₆₇	C₂×C₄	1	4	yes	complemented
133	C₂×M₁₆×C₆₇	—	2	1	no	—
134	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
136	C₂²×C₈×C₆₇	—	2	1	no	—
137	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
138	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
139	#117×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	#104×C₆₇	C₂×C₄	1	4	yes	complemented
142	#103×C₆₇	C₂×C₄	1	2	yes	—
143	#102×C₆₇	C₂×C₄	1	2	yes	—
144	#97×C₆₇	—	2	1	no	—
145	C₄×C₈×C₆₇	—	2	1	no	—
146	C₄×C₈×C₆₇	—	2	2	no	—
147	#11761628	—	2	1	no	—
148	#11761628	—	2	1	no	—
149	#11761628	C₂²×C₄	1	2	yes	—
150	#11761628	C₂²×C₄	1	2	yes	—
151	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₂²⋊D₆₇	—	2	1	no	—
154	C₂²×Dic₆₇	—	2	1	no	—
155	C₂²×Dic₆₇	—	2	2	no	—
156	C₂²×Dic₆₇	#41	1	2	yes	—
157	C₂×C₄×D₆₇	—	2	1	no	—
158	C₂×C₄×D₆₇	—	2	2	no	—
159	#11761618	C₄²	1	4	yes	complemented
160	#11761616	C₄²	1	4	yes	complemented
161	#11761615	C₄²	1	4	yes	complemented
162	#11761613	C₄²	1	4	yes	complemented
163	#11761612	—	2	1	no	—
164	#11761612	—	2	1	no	—
165	#11761612	—	2	4	no	—
166	#11761612	C₂²×C₄	1	2	yes	—
167	#11761612	C₂²×C₄	1	2	yes	—
168	#11761611	—	2	4	no	—
169	#11761611	—	2	4	no	—
170	C₂×C₆₇⋊C₈	—	2	1	no	—
171	C₂×C₆₇⋊C₈	—	2	1	no	—
172	C₂×C₆₇⋊C₈	—	2	4	no	—
173	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
174	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
177	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
178	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
180	C₂×C₆₇⋊C₈	#41	1	2	yes	—
181	C₈×D₆₇	—	2	4	no	—
182	C₈×D₆₇	—	2	4	no	—
183	C₄⋊Dic₆₇	—	2	2	no	—
184	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
185	Dic₆₇⋊C₄	—	2	2	no	—
186	C₂²⋊Dic₆₇	—	2	1	no	—
187	C₂²⋊Dic₆₇	—	2	2	no	—
188	(C₂×D₆₇)⋊C₄	—	2	2	no	—
189	C₄×Dic₆₇	—	2	1	no	—
190	C₄×Dic₆₇	—	2	1	no	—
191	C₄×Dic₆₇	—	2	2	no	—
192	C₄×Dic₆₇	—	2	2	no	—
193	C₄×Dic₆₇	—	4	4	no	—
194	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
195	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
196	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
197	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
198	#41×C₆₇	—	2	1	no	—
199	#41×C₆₇	—	2	1	no	—
200	#41×C₆₇	C₂²×C₄	1	2	yes	—
201	#41×C₆₇	C₂²×C₄	1	2	yes	—
202	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
203	C₂×D₄×C₆₇	—	2	1	no	—
204	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	—	2	1	no	—
206	C₂²×C₄×C₆₇	—	2	1	no	—
207	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
208	M₁₆×C₆₇	—	2	1	no	—
209	M₁₆×C₆₇	—	2	1	no	—
210	M₁₆×C₆₇	—	2	4	no	—
211	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
212	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
213	C₂×C₈×C₆₇	—	2	1	no	—
214	C₂×C₈×C₆₇	—	2	1	no	—
215	C₂×C₈×C₆₇	—	2	1	no	—
216	C₂×C₈×C₆₇	—	2	1	no	—
217	C₂×C₈×C₆₇	—	2	4	no	—
218	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
219	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
220	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
221	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
222	C₂×C₈×C₆₇	#41	1	2	yes	—
223	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
224	C₂²⋊C₄×C₆₇	—	2	1	no	—
225	C₄²×C₆₇	—	2	1	no	—
226	C₄²×C₆₇	—	2	1	no	—
227	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
228	C₂³×C₆₇	—	2	1	no	—
229	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
230	C₂²×D₆₇	—	2	1	no	—
231	Q₈×C₆₇	—	2	1	no	—
232	Q₈×C₆₇	C₂×C₄²	1	2	yes	—
233	D₄×C₆₇	—	2	2	no	—
234	D₄×C₆₇	—	2	2	no	—
235	D₄×C₆₇	—	2	1	no	—
236	D₄×C₆₇	C₂×C₄²	1	2	yes	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
249	C₂²⋊D₆₇	—	2	2	no	—
250	C₂²⋊D₆₇	—	2	2	no	—
251	C₂×Dic₆₇	—	2	1	no	—
252	C₂×Dic₆₇	—	2	2	no	—
253	C₂×Dic₆₇	—	2	2	no	—
254	C₂×Dic₆₇	—	4	2	no	—
255	C₂×Dic₆₇	—	4	2	no	—
256	C₂×Dic₆₇	—	8	2	no	—
257	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
258	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
259	D₂₆₈	—	2	1	no	—
260	D₂₆₈	C₂×C₄²	1	2	yes	—
261	C₄×D₆₇	—	2	2	no	—
262	C₄×D₆₇	—	2	2	no	—
263	C₄×D₆₇	—	8	2	no	—
264	Dic₁₃₄	—	2	1	no	—
265	Dic₁₃₄	C₂×C₄²	1	2	yes	—
266	C₈×C₆₇	—	2	2	no	—
267	C₈×C₆₇	—	2	2	no	—
268	C₈×C₆₇	—	2	2	no	—
269	C₈×C₆₇	C₄×D₄	1	2	yes	—
270	C₈×C₆₇	C₄×D₄	1	2	yes	—
271	C₆₇⋊C₈	—	2	2	no	—
272	C₆₇⋊C₈	—	2	2	no	—
273	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
274	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
275	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
276	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	2	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
282	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
283	C₂×D₆₇	—	2	2	no	—
284	C₂×D₆₇	—	4	2	no	—
285	C₄×C₆₇	—	2	4	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	2	no	—
288	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
289	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
290	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
291	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
292	Dic₆₇	—	2	2	no	—
293	Dic₆₇	—	2	2	no	—
294	Dic₆₇	—	2	2	no	—
295	Dic₆₇	—	4	2	no	—
296	Dic₆₇	—	4	2	no	—
297	#20512	—	67	1	no	maximal, Sylow
298	C₂×C₆₇	—	2	1	no	—
299	C₂×C₆₇	—	2	1	no	—
300	C₂×C₆₇	—	4	2	no	—
301	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
302	C₂×C₆₇	#2956	1	2	yes	—
303	D₆₇	—	4	2	no	—
304	C₂×#444	—	67	1	no	—
305	C₂×#443	—	67	1	no	—
306	#2943	—	67	1	no	—
307	#2875	—	67	1	no	—
308	#2875	—	67	1	no	—
309	#1778	—	67	1	no	—
310	#1773	—	67	1	no	—
311	#1772	—	67	1	no	—
312	#1765	—	67	1	no	—
313	#1765	—	67	1	no	—
314	#1764	—	67	1	no	—
315	#1764	D₆₇	1	1	yes	characteristic, complemented
316	C₄×#104	—	67	1	no	—
317	C₄×#104	—	67	1	no	—
318	C₄×#97	—	67	1	no	—
319	C₆₇	#20512	1	1	yes	characteristic, complemented, Sylow
320	C₂×#135	—	67	1	no	—
321	C₂×#131	—	67	1	no	—
322	C₂×#131	—	67	1	no	—
323	C₂×#131	—	67	1	no	—
324	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
325	C₄×#41	—	67	1	no	—
326	C₄×#41	—	67	1	no	—
327	#444	—	67	8	no	—
328	#443	—	67	8	no	—
329	#442	—	67	2	no	—
330	#441	—	67	2	no	—
331	#440	—	67	2	no	—
332	#440	—	67	2	no	—
333	C₂×#108	—	67	1	no	—
334	C₂×#108	—	67	1	no	—
335	#427	—	67	1	no	—
336	C₂×#104	—	67	1	no	—
337	C₂×#104	—	67	1	no	—
338	C₂×#104	—	67	1	no	—
339	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
340	#416	—	67	1	no	—
341	#416	C₂×D₆₇	1	1	yes	characteristic
342	#405	—	67	1	no	—
343	#405	—	67	1	no	—
344	#405	—	67	1	no	—
345	#405	—	134	1	no	—
346	#405	—	134	1	no	—
347	#405	—	134	1	no	—
348	#405	—	134	1	no	—
349	#405	C₂×D₆₇	1	1	yes	characteristic
350	C₄×M₁₆	—	67	1	no	—
351	C₄×M₁₆	—	67	1	no	—
352	C₄×M₁₆	—	134	1	no	—
353	C₄×M₁₆	—	134	1	no	—
354	C₂×#97	—	67	1	no	—
355	C₂×#97	—	67	1	no	—
356	C₂×#97	—	67	1	no	—
357	C₂×#97	—	67	1	no	—
358	C₂×#97	—	134	1	no	—
359	C₂×#97	—	134	1	no	—
360	C₂×C₄×C₈	—	67	1	no	—
361	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
362	C₄³	—	67	1	no	—
363	#341	—	67	1	no	—
364	#341	C₂×D₆₇	1	1	yes	characteristic
365	#339	—	67	1	no	—
366	#339	—	67	1	no	—
367	#339	—	67	1	no	—
368	#339	C₂×D₆₇	1	1	yes	characteristic
369	C₂×#41	—	67	1	no	—
370	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
371	#135	—	67	4	no	—
372	#135	—	67	4	no	—
373	C₂×Q₁₆	—	67	1	no	—
374	C₂×SD₁₆	—	67	1	no	—
375	C₂×SD₁₆	—	67	1	no	—
376	C₂×D₈	—	67	1	no	—
377	#131	—	2	2	no	—
378	#131	—	67	4	no	—
379	#131	—	67	4	no	—
380	#131	—	67	4	no	—
381	#131	—	134	2	no	—
382	#131	—	134	2	no	—
383	#131	—	134	2	no	—
384	#131	C₄×D₆₇	1	4	yes	complemented
385	C₂×M₁₆	—	2	1	no	—
386	C₂×M₁₆	—	67	1	no	—
387	C₂×M₁₆	—	67	1	no	—
388	C₂×M₁₆	—	134	1	no	—
389	C₂×M₁₆	—	134	1	no	—
390	C₂×M₁₆	—	134	1	no	—
391	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
392	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
393	C₂²×C₈	—	2	1	no	—
394	C₂²×C₈	—	67	1	no	—
395	C₂²×C₈	—	67	1	no	—
396	C₂²×C₈	—	134	1	no	—
397	C₂²×C₈	—	134	1	no	—
398	C₂²×C₈	—	134	1	no	—
399	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
400	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
401	C₄×Q₈	—	67	2	no	—
402	C₄×Q₈	—	67	2	no	—
403	C₄×D₄	—	67	2	no	—
404	C₄×D₄	—	67	2	no	—
405	C₄×D₄	—	134	2	no	—
406	C₄×D₄	—	134	2	no	—
407	#117	—	67	1	no	—
408	#117	—	134	1	no	—
409	#117	—	134	1	no	—
410	#117	C₂²×D₆₇	1	1	yes	characteristic
411	C₂×C₄²	—	67	1	no	—
412	C₂×C₄²	—	67	1	no	—
413	C₂×C₄²	—	134	1	no	—
414	C₂×C₄²	—	134	2	no	—
415	C₂×C₄²	—	134	1	no	—
416	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
417	#108	—	67	4	no	—
418	#108	—	67	4	no	—
419	#107	—	67	2	no	—
420	#106	—	67	2	no	—
421	#104	—	67	4	no	—
422	#104	—	67	4	no	—
423	#104	—	67	4	no	—
424	#104	C₄×D₆₇	1	4	yes	complemented
425	#103	—	67	2	no	—
426	#103	C₄×D₆₇	1	2	yes	—
427	#102	—	67	2	no	—
428	#102	C₄×D₆₇	1	2	yes	—
429	#97	—	2	1	no	—
430	#97	—	67	2	no	—
431	#97	—	67	2	no	—
432	#97	—	67	4	no	—
433	#97	—	67	1	no	—
434	#97	—	67	1	no	—
435	#97	—	67	1	no	—
436	#97	—	67	1	no	—
437	#97	—	134	1	no	—
438	#97	—	134	1	no	—
439	#97	—	134	1	no	—
440	#97	—	134	2	no	—
441	#97	—	134	1	no	—
442	#97	—	134	1	no	—
443	C₄×C₈	—	2	1	no	—
444	C₄×C₈	—	2	2	no	—
445	C₄×C₈	—	67	4	no	—
446	C₄×C₈	—	134	1	no	—
447	C₄×C₈	—	134	1	no	—
448	C₄×C₈	—	134	1	no	—
449	C₄×C₈	—	134	1	no	—
450	C₄×C₈	—	134	1	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	—	67	2	no	—
454	#41	—	67	2	no	—
455	#41	—	134	1	no	—
456	#41	—	134	1	no	—
457	#41	C₂×C₄×D₆₇	1	2	yes	—
458	#41	C₂×C₄×D₆₇	1	2	yes	—
459	C₂×Q₈	—	67	1	no	—
460	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	67	1	no	—
463	C₂×D₄	—	134	1	no	—
464	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	67	2	no	—
468	C₂²×C₄	—	134	2	no	—
469	C₂²×C₄	—	134	1	no	—
470	C₂²×C₄	—	134	2	no	—
471	C₂²×C₄	—	134	1	no	—
472	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
473	Q₁₆	—	67	4	no	—
474	SD₁₆	—	67	4	no	—
475	SD₁₆	—	67	4	no	—
476	D₈	—	67	4	no	—
477	M₁₆	—	2	1	no	—
478	M₁₆	—	2	1	no	—
479	M₁₆	—	2	4	no	—
480	M₁₆	—	67	2	no	—
481	M₁₆	—	67	2	no	—
482	M₁₆	—	134	4	no	—
483	M₁₆	—	134	4	no	—
484	M₁₆	—	134	1	no	—
485	M₁₆	—	134	1	no	—
486	M₁₆	—	134	4	no	—
487	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
488	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	4	no	—
494	C₂×C₈	—	67	2	no	—
495	C₂×C₈	—	67	2	no	—
496	C₂×C₈	—	67	1	no	—
497	C₂×C₈	—	67	1	no	—
498	C₂×C₈	—	67	2	no	—
499	C₂×C₈	—	67	1	no	—
500	C₂×C₈	—	67	1	no	—
501	C₂×C₈	—	67	2	no	—
502	C₂×C₈	—	134	4	no	—
503	C₂×C₈	—	134	4	no	—
504	C₂×C₈	—	134	1	no	—
505	C₂×C₈	—	134	1	no	—
506	C₂×C₈	—	134	4	no	—
507	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
508	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
509	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
510	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
511	C₂×C₈	#11761627	1	2	yes	—
512	C₄⋊C₄	—	67	2	no	—
513	C₄⋊C₄	—	134	2	no	—
514	C₄⋊C₄	—	134	2	no	—
515	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	134	2	no	—
518	C₂²⋊C₄	—	134	2	no	—
519	C₂²⋊C₄	—	134	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	268	4	no	—
523	C₄²	—	67	2	no	—
524	C₄²	—	67	1	no	—
525	C₄²	—	67	1	no	—
526	C₄²	—	67	2	no	—
527	C₄²	—	134	2	no	—
528	C₄²	—	134	2	no	—
529	C₄²	—	134	1	no	—
530	C₄²	—	134	1	no	—
531	C₄²	C₂×C₄×D₆₇	1	2	yes	—
532	C₂³	—	2	1	no	—
533	C₂³	—	134	1	no	—
534	C₂³	#41×D₆₇	1	1	yes	characteristic
535	Q₈	—	2	1	no	—
536	Q₈	—	67	2	no	—
537	Q₈	—	134	1	no	—
538	Q₈	C₄²×D₆₇	1	2	yes	—
539	D₄	—	2	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	1	no	—
542	D₄	—	67	2	no	—
543	D₄	—	134	2	no	—
544	D₄	—	134	2	no	—
545	D₄	—	134	1	no	—
546	D₄	C₄²×D₆₇	1	2	yes	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	268	2	no	—
554	C₂×C₄	—	268	2	no	—
555	C₂×C₄	—	536	2	no	—
556	C₂×C₄	—	536	2	no	—
557	C₂×C₄	—	67	2	no	—
558	C₂×C₄	—	67	2	no	—
559	C₂×C₄	—	134	2	no	—
560	C₂×C₄	—	134	2	no	—
561	C₂×C₄	—	134	1	no	—
562	C₂×C₄	—	134	2	no	—
563	C₂×C₄	—	134	2	no	—
564	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
566	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
567	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
568	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
569	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₈	—	2	2	no	—
573	C₈	—	67	2	no	—
574	C₈	—	67	2	no	—
575	C₈	—	67	2	no	—
576	C₈	—	67	2	no	—
577	C₈	—	134	2	no	—
578	C₈	—	134	2	no	—
579	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
580	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	268	2	no	—
585	C₂²	—	134	2	no	—
586	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
587	C₂²	C₄×#11761627	1	1	yes	characteristic
588	C₂²	#16863864	1	1	yes	characteristic
589	C₄	—	2	4	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	268	2	no	—
593	C₄	—	268	2	no	—
594	C₄	—	134	2	no	—
595	C₄	—	134	2	no	—
596	C₄	—	134	2	no	—
597	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
598	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
599	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
600	C₄	C₄×#11761627	1	1	yes	characteristic
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	4	2	no	—
604	C₂	—	268	2	no	—
605	C₂	C₄×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
606	C₂	#22569114	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1764	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1764	D₆₇
hypocenter	C₆₇	#20512
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1764

Complements

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

N	H ≅ G/N	type
4. #22568636	C₂	semidirect
5. #22568635	C₂	semidirect
6. #22560114	C₂	semidirect
7. #22560094	C₂	semidirect
8. #22560087	C₂	semidirect
9. #22560056	C₂	semidirect
10. #22560052	C₂	semidirect
11. #22560046	C₂	semidirect
12. #104×Dic₆₇	C₂	semidirect
13. #22560013	C₂	semidirect
14. C₄×#14523953	C₂	semidirect
15. C₆₇×#1764	C₂	semidirect
22. #16863516	C₄	semidirect
23. #16863510	C₄	semidirect
24. #16863505	C₄	semidirect
25. #16863500	C₄	semidirect
26. #16863499	C₄	semidirect
27. #16863493	C₄	semidirect
73. #14524051	C₂×C₄	semidirect
75. #14524050	C₂×C₄	semidirect
77. #14524049	C₂×C₄	semidirect
102. #14523978	C₂×C₄	semidirect
103. #14523978	C₂×C₄	semidirect
106. #14523971	C₂×C₄	semidirect
107. #14523970	C₂×C₄	semidirect
108. #14523969	C₂×C₄	semidirect
119. #14523953	C₂×C₄	semidirect
125. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
132. #131×C₆₇	C₂×C₄	semidirect
141. #104×C₆₇	C₂×C₄	semidirect
159. #11761618	C₄²	semidirect
160. #11761616	C₄²	semidirect
161. #11761615	C₄²	semidirect
162. #11761613	C₄²	semidirect
315. #1764	D₆₇	semidirect
319. C₆₇	#20512	semidirect
384. #131	C₄×D₆₇	semidirect
424. #104	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863516	2
C₂×#16863510	2
#22569005	2
#22568636	2
#22568635	2
#22560114	2
#22560094	2
#22560087	2
#22560056	2
#22560052	2
#22560046	2
#104×Dic₆₇	2
#22560013	2
C₄×#14523953	2
C₆₇×#1764	2
#20512	67