Subgroups — #1383×D₆₇ #30770730

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#106×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558309	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558308	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558307	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₆₇	—	2	1	no	—
9	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×D₆₇	—	2	1	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₂×#14523982	C₄	1	2	yes	complemented
13	M₃₂×D₆₇	—	2	2	no	—
14	M₃₂×D₆₇	—	2	2	no	—
15	C₂×C₁₆×D₆₇	C₄	1	2	yes	complemented
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	C₂×#14523978	—	2	1	no	—
18	#108×D₆₇	—	2	2	no	—
19	#108×D₆₇	C₄	1	4	yes	complemented
20	C₂×#14523975	C₂²	1	1	yes	characteristic
21	C₂×#14523974	—	2	1	no	—
22	#106×D₆₇	—	2	2	no	—
23	#106×D₆₇	C₄	1	4	yes	—
24	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₆₇	—	4	1	no	—
28	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
29	C₂×#11761612	—	2	1	no	—
30	C₂×#11761611	—	4	1	no	—
31	M₁₆×D₆₇	—	2	4	no	—
32	M₁₆×D₆₇	—	4	2	no	—
33	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₈×D₆₇	—	4	1	no	—
35	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₆₇	D₄	1	2	yes	—
40	C₂×C₈×D₆₇	D₄	1	2	yes	—
41	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
43	C₂×C₄⋊Dic₆₇	—	2	1	no	—
44	C₂×Dic₆₇⋊C₄	—	4	1	no	—
45	C₄⋊C₄×D₆₇	—	2	4	no	—
46	C₄⋊C₄×D₆₇	—	4	2	no	—
47	#14523983	—	2	1	no	—
48	#14523983	—	2	1	no	—
49	#14523982	—	2	4	no	—
50	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
51	C₁₆×D₆₇	—	2	4	no	—
52	#14523979	—	2	1	no	—
53	#14523979	C₂×C₄	1	2	yes	complemented
54	#14523978	—	2	4	no	—
55	#14523975	—	2	1	no	—
56	#14523975	C₂×C₄	1	2	yes	—
57	#14523974	—	2	4	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	M₃₂×C₆₇	—	2	1	no	—
62	M₃₂×C₆₇	—	2	1	no	—
63	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
64	#108×C₆₇	—	2	1	no	—
65	#108×C₆₇	C₂×C₄	1	2	yes	complemented
66	#106×C₆₇	—	2	1	no	—
67	#106×C₆₇	C₂×C₄	1	2	yes	—
68	C₂³×D₆₇	SD₁₆	1	1	yes	characteristic
69	C₂²×Dic₆₇	—	4	1	no	—
70	C₂²×Dic₆₇	SD₁₆	1	1	yes	characteristic
71	C₂×C₄×D₆₇	—	4	4	no	—
72	C₂×C₄×D₆₇	—	8	4	no	—
73	C₂×C₄×D₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×D₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₆₇	D₈	1	2	yes	—
78	C₂×C₄×D₆₇	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₆₇	SD₁₆	1	1	yes	characteristic
80	C₂×C₄×D₆₇	Q₁₆	1	2	yes	—
81	#11761612	—	2	2	no	—
82	#11761612	—	4	1	no	—
83	#11761611	—	4	4	no	—
84	C₂×C₆₇⋊C₈	—	4	1	no	—
85	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₈×D₆₇	—	4	4	no	—
90	C₈×D₆₇	C₄²	1	2	yes	—
91	C₈×D₆₇	C₄²	1	2	yes	—
92	C₈×D₆₇	C₂²⋊C₄	1	2	yes	—
93	C₈×D₆₇	C₂²⋊C₄	1	2	yes	—
94	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
95	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
96	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
97	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
98	C₄⋊Dic₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	4	1	no	—
100	Dic₆₇⋊C₄	—	4	4	no	—
101	C₆₇⋊C₁₆	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	M₁₆×C₆₇	—	2	2	no	—
105	M₁₆×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₁₆×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂²×D₆₇	—	2	4	no	—
118	C₂²×D₆₇	#102	1	2	yes	—
119	C₂²×D₆₇	#103	1	2	yes	—
120	C₂²×D₆₇	#106	1	1	yes	characteristic
121	C₂²×D₆₇	#106	1	1	yes	characteristic
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	8	2	no	—
132	C₂×Dic₆₇	#102	1	2	yes	—
133	C₂×Dic₆₇	#103	1	2	yes	—
134	C₂×Dic₆₇	#106	1	1	yes	characteristic
135	C₂×Dic₆₇	#106	1	1	yes	characteristic
136	C₄×D₆₇	—	8	8	no	—
137	C₄×D₆₇	#95	1	2	yes	—
138	C₄×D₆₇	#95	1	2	yes	—
139	C₄×D₆₇	#102	1	2	yes	—
140	C₄×D₆₇	#102	1	2	yes	—
141	C₄×D₆₇	#103	1	2	yes	—
142	C₄×D₆₇	#103	1	2	yes	—
143	C₄×D₆₇	#107	1	2	yes	—
144	C₄×D₆₇	#107	1	2	yes	—
145	C₈×C₆₇	—	4	2	no	—
146	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
147	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₆₇⋊C₈	—	4	2	no	—
151	C₆₇⋊C₈	#95	1	2	yes	—
152	C₆₇⋊C₈	#95	1	2	yes	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
157	C₂×D₆₇	—	2	2	no	—
158	C₂×D₆₇	—	2	4	no	—
159	C₂×D₆₇	—	2	4	no	—
160	C₂×D₆₇	#340	1	2	yes	—
161	C₂×D₆₇	#340	1	2	yes	—
162	C₂×D₆₇	#365	1	4	yes	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
165	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
166	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
167	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
168	Dic₆₇	—	8	2	no	—
169	Dic₆₇	#340	1	2	yes	—
170	Dic₆₇	#340	1	2	yes	—
171	C₂×#1383	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	2	1	no	—
173	C₂×C₆₇	—	2	1	no	—
174	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#365	1	2	yes	—
176	D₆₇	—	2	2	no	—
177	D₆₇	#1383	1	4	yes	direct product factor
178	C₂²×M₃₂	—	67	1	no	—
179	C₂²×#108	—	67	1	no	—
180	C₂²×#106	—	67	1	no	—
181	#1383	—	67	1	no	—
182	#1383	—	67	1	no	—
183	#1383	—	67	1	no	—
184	#1383	D₆₇	1	1	yes	characteristic, direct product factor
185	C₆₇	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
186	C₂²×M₁₆	—	134	1	no	—
187	C₂³×C₈	—	67	1	no	—
188	C₂²×C₄⋊C₄	—	134	1	no	—
189	C₂×M₃₂	—	67	1	no	—
190	C₂×M₃₂	—	67	2	no	—
191	C₂×M₃₂	—	134	2	no	—
192	C₂×M₃₂	—	134	2	no	—
193	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
194	C₂²×C₁₆	—	67	2	no	—
195	C₂×#108	—	67	1	no	—
196	C₂×#108	—	67	4	no	—
197	C₂×#108	—	134	2	no	—
198	C₂×#108	—	134	1	no	—
199	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
200	C₂×#106	—	67	1	no	—
201	C₂×#106	—	67	4	no	—
202	C₂×#106	—	134	2	no	—
203	C₂×#106	—	134	1	no	—
204	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
205	C₂³×C₄	—	268	1	no	—
206	C₂³×C₄	—	67	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	268	1	no	—
209	C₂×M₁₆	—	268	2	no	—
210	C₂×M₁₆	—	134	4	no	—
211	C₂×M₁₆	—	134	1	no	—
212	C₂²×C₈	—	268	1	no	—
213	C₂²×C₈	—	67	1	no	—
214	C₂²×C₈	—	67	2	no	—
215	C₂²×C₈	—	67	2	no	—
216	C₂²×C₈	—	67	1	no	—
217	C₂²×C₈	—	67	1	no	—
218	C₂²×C₈	—	67	2	no	—
219	C₂²×C₈	—	67	2	no	—
220	C₂²×C₈	—	67	1	no	—
221	C₂²×C₈	—	67	1	no	—
222	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	268	2	no	—
225	C₂×C₄⋊C₄	—	268	1	no	—
226	C₂×C₄⋊C₄	—	134	4	no	—
227	C₂×C₄⋊C₄	—	134	1	no	—
228	M₃₂	—	2	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	134	4	no	—
231	M₃₂	—	134	1	no	—
232	M₃₂	—	134	1	no	—
233	C₂×C₁₆	—	67	2	no	—
234	C₂×C₁₆	—	134	4	no	—
235	C₂×C₁₆	C₄×D₆₇	1	2	yes	complemented
236	#108	—	2	1	no	—
237	#108	—	67	2	no	—
238	#108	—	134	4	no	—
239	#108	—	134	1	no	—
240	#108	C₄×D₆₇	1	2	yes	complemented
241	#106	—	2	1	no	—
242	#106	—	67	2	no	—
243	#106	—	134	4	no	—
244	#106	—	134	1	no	—
245	#106	C₄×D₆₇	1	2	yes	—
246	C₂⁴	—	67	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	268	4	no	—
249	C₂²×C₄	—	268	1	no	—
250	C₂²×C₄	—	536	4	no	—
251	C₂²×C₄	—	67	1	no	—
252	C₂²×C₄	—	67	2	no	—
253	C₂²×C₄	—	67	2	no	—
254	C₂²×C₄	—	67	1	no	—
255	C₂²×C₄	—	67	1	no	—
256	C₂²×C₄	—	67	2	no	—
257	C₂²×C₄	—	67	1	no	—
258	C₂²×C₄	—	67	1	no	—
259	C₂²×C₄	—	67	2	no	—
260	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	268	4	no	—
264	M₁₆	—	268	1	no	—
265	M₁₆	—	134	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	268	4	no	—
268	C₂×C₈	—	268	1	no	—
269	C₂×C₈	—	67	2	no	—
270	C₂×C₈	—	67	2	no	—
271	C₂×C₈	—	67	1	no	—
272	C₂×C₈	—	67	1	no	—
273	C₂×C₈	—	67	2	no	—
274	C₂×C₈	—	67	2	no	—
275	C₂×C₈	—	67	2	no	—
276	C₂×C₈	—	67	2	no	—
277	C₂×C₈	—	67	2	no	—
278	C₂×C₈	—	67	2	no	—
279	C₂×C₈	—	67	2	no	—
280	C₂×C₈	—	67	2	no	—
281	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
282	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
283	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
284	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
285	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
286	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	268	4	no	—
290	C₄⋊C₄	—	268	1	no	—
291	C₄⋊C₄	—	134	2	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	—	134	2	no	—
294	C₂³	—	67	1	no	—
295	C₂³	—	67	2	no	—
296	C₂³	—	67	2	no	—
297	C₂³	—	67	1	no	—
298	C₂³	—	134	4	no	—
299	C₂³	SD₁₆×D₆₇	1	1	yes	characteristic
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	536	8	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	—	67	2	no	—
306	C₂×C₄	—	67	2	no	—
307	C₂×C₄	—	67	1	no	—
308	C₂×C₄	—	67	1	no	—
309	C₂×C₄	—	67	2	no	—
310	C₂×C₄	—	67	2	no	—
311	C₂×C₄	—	67	2	no	—
312	C₂×C₄	—	67	2	no	—
313	C₂×C₄	—	67	2	no	—
314	C₂×C₄	—	67	2	no	—
315	C₂×C₄	—	67	2	no	—
316	C₂×C₄	—	67	2	no	—
317	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
318	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
319	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
320	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
321	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
322	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
323	C₈	—	4	2	no	—
324	C₈	—	268	2	no	—
325	C₈	—	67	2	no	—
326	C₈	—	67	2	no	—
327	C₈	C₄²×D₆₇	1	1	yes	characteristic
328	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
329	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
330	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
331	C₂²	—	2	2	no	—
332	C₂²	—	67	2	no	—
333	C₂²	—	67	4	no	—
334	C₂²	—	67	2	no	—
335	C₂²	—	134	4	no	—
336	C₂²	—	134	4	no	—
337	C₂²	—	134	2	no	—
338	C₂²	#102×D₆₇	1	1	yes	characteristic
339	C₂²	#103×D₆₇	1	1	yes	characteristic
340	C₂²	#106×D₆₇	1	1	yes	characteristic
341	C₄	—	8	2	no	—
342	C₄	—	536	2	no	—
343	C₄	—	67	2	no	—
344	C₄	—	67	2	no	—
345	C₄	#95×D₆₇	1	1	yes	characteristic
346	C₄	#102×D₆₇	1	1	yes	characteristic
347	C₄	#103×D₆₇	1	1	yes	characteristic
348	C₄	#107×D₆₇	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	67	4	no	—
352	C₂	—	134	2	no	—
353	C₂	#340×D₆₇	1	1	yes	characteristic
354	C₂	#365×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²	#106×D₆₇
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1383	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×D₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1383	D₆₇
hypocenter	C₆₇	C₂×#1383
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
4. #22558309	C₂	semidirect
5. #22558308	C₂	semidirect
6. #22558307	C₂	semidirect
7. C₆₇×#1383	C₂	semidirect
12. C₂×#14523982	C₄	semidirect
15. C₂×C₁₆×D₆₇	C₄	semidirect
19. #108×D₆₇	C₄	semidirect
50. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
53. #14523979	C₂×C₄	semidirect
63. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
65. #108×C₆₇	C₂×C₄	semidirect
177. D₆₇	#1383	direct (N × H)
184. #1383	D₆₇	direct (N × H)
185. C₆₇	C₂×#1383	semidirect
235. C₂×C₁₆	C₄×D₆₇	semidirect
240. #108	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×M₃₂×D₆₇	2
C₂×#108×D₆₇	2
C₂×#106×D₆₇	2
#22558309	2
#22558308	2
#22558307	2
C₆₇×#1383	2
C₂×#1383	67

Subgroups — #1383×D67 #30770730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1383×D₆₇ #30770730