Subgroups — #1382×D₆₇ #30770722

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#427×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558306	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558305	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558304	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₆₇	—	2	1	no	—
9	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
10	#117×D₆₇	—	2	1	no	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₆₇	—	2	1	no	—
13	C₂×C₁₆×D₆₇	—	2	1	no	—
14	C₂×C₁₆×D₆₇	—	2	2	no	—
15	C₂×C₁₆×D₆₇	C₄	1	4	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	#16863434	C₂²	1	1	yes	characteristic
21	#16863433	C₄	1	2	yes	—
22	#16863432	—	2	1	no	—
23	#16863431	—	2	1	no	—
24	#107×D₆₇	C₄	1	2	yes	—
25	#106×D₆₇	—	2	1	no	—
26	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
28	#427×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
30	C₂×#11761612	—	2	1	no	—
31	C₂×#11761611	—	4	1	no	—
32	M₁₆×D₆₇	—	2	4	no	—
33	M₁₆×D₆₇	—	4	2	no	—
34	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₆₇	—	4	1	no	—
36	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
41	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
42	C₂×C₈×D₆₇	D₄	1	2	yes	—
43	C₂×C₈×D₆₇	Q₈	1	2	yes	—
44	#14524002	—	2	1	no	—
45	#14524001	—	2	2	no	—
46	#14524000	—	4	1	no	—
47	#14523999	—	4	1	no	—
48	C₄⋊C₄×D₆₇	—	2	2	no	—
49	C₂²⋊C₄×D₆₇	—	4	1	no	—
50	C₄²×D₆₇	—	4	1	no	—
51	C₂×C₆₇⋊C₁₆	—	2	1	no	—
52	C₂×C₆₇⋊C₁₆	—	2	1	no	—
53	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
54	C₁₆×D₆₇	—	2	4	no	—
55	C₁₆×D₆₇	—	2	4	no	—
56	#14523979	—	2	1	no	—
57	#14523979	C₂×C₄	1	2	yes	complemented
58	#14523978	—	2	4	no	—
59	#14523977	C₂×C₄	1	2	yes	—
60	#14523976	—	2	2	no	—
61	#14523975	—	2	1	no	—
62	#14523974	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#117×C₆₇	—	2	1	no	—
66	C₂×C₁₆×C₆₇	—	2	1	no	—
67	C₂×C₁₆×C₆₇	—	2	1	no	—
68	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
69	#108×C₆₇	—	2	1	no	—
70	#108×C₆₇	C₂×C₄	1	2	yes	complemented
71	#107×C₆₇	C₂×C₄	1	2	yes	—
72	#106×C₆₇	—	2	1	no	—
73	C₂³×D₆₇	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₆₇	SD₁₆	1	1	yes	characteristic
75	C₂×C₄×D₆₇	—	4	2	no	—
76	C₂×C₄×D₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
81	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
82	C₂×C₄×D₆₇	SD₁₆	1	2	yes	—
83	C₂×C₄×D₆₇	Q₁₆	1	2	yes	—
84	#11761612	—	2	2	no	—
85	#11761612	—	4	1	no	—
86	#11761611	—	4	4	no	—
87	C₂×C₆₇⋊C₈	—	4	1	no	—
88	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
92	C₈×D₆₇	—	4	4	no	—
93	C₈×D₆₇	C₄²	1	2	yes	—
94	C₈×D₆₇	C₄²	1	2	yes	—
95	C₈×D₆₇	C₂²⋊C₄	1	2	yes	—
96	C₈×D₆₇	C₂²⋊C₄	1	2	yes	—
97	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
98	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
99	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
100	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
101	C₄⋊Dic₆₇	—	2	2	no	—
102	Dic₆₇⋊C₄	—	4	2	no	—
103	C₂²⋊Dic₆₇	—	4	1	no	—
104	(C₂×D₆₇)⋊C₄	—	4	2	no	—
105	C₄×Dic₆₇	—	4	1	no	—
106	C₄×Dic₆₇	—	4	2	no	—
107	C₆₇⋊C₁₆	—	2	2	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	M₁₆×C₆₇	—	2	2	no	—
110	M₁₆×C₆₇	—	4	1	no	—
111	C₂×C₈×C₆₇	—	4	1	no	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₂²⋊C₄×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	—	4	1	no	—
121	C₁₆×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×D₆₇	—	2	4	no	—
124	C₂²×D₆₇	#102	1	1	yes	characteristic
125	C₂²×D₆₇	#102	1	1	yes	characteristic
126	C₂²×D₆₇	#103	1	2	yes	—
127	C₂²×D₆₇	#106	1	2	yes	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×Dic₆₇	—	4	2	no	—
136	C₂×Dic₆₇	#102	1	1	yes	characteristic
137	C₂×Dic₆₇	#102	1	1	yes	characteristic
138	C₂×Dic₆₇	#103	1	2	yes	—
139	C₂×Dic₆₇	#106	1	2	yes	—
140	C₄×D₆₇	—	8	4	no	—
141	C₄×D₆₇	#95	1	2	yes	—
142	C₄×D₆₇	#95	1	2	yes	—
143	C₄×D₆₇	#102	1	2	yes	—
144	C₄×D₆₇	#102	1	2	yes	—
145	C₄×D₆₇	#103	1	2	yes	—
146	C₄×D₆₇	#103	1	2	yes	—
147	C₄×D₆₇	#107	1	2	yes	—
148	C₄×D₆₇	#107	1	2	yes	—
149	C₈×C₆₇	—	4	2	no	—
150	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
151	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₆₇⋊C₈	—	4	2	no	—
155	C₆₇⋊C₈	#95	1	2	yes	—
156	C₆₇⋊C₈	#95	1	2	yes	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
161	C₂×D₆₇	—	2	2	no	—
162	C₂×D₆₇	—	2	4	no	—
163	C₂×D₆₇	—	2	4	no	—
164	C₂×D₆₇	#340	1	2	yes	—
165	C₂×D₆₇	#340	1	2	yes	—
166	C₂×D₆₇	#359	1	4	yes	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
169	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
170	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
171	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
172	Dic₆₇	—	8	2	no	—
173	Dic₆₇	#340	1	2	yes	—
174	Dic₆₇	#340	1	2	yes	—
175	C₂×#1382	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#359	1	2	yes	—
180	D₆₇	—	2	2	no	—
181	D₆₇	#1382	1	4	yes	direct product factor
182	C₂³×C₁₆	—	67	1	no	—
183	C₂²×#108	—	67	1	no	—
184	C₂×#427	—	67	1	no	—
185	#1382	—	67	1	no	—
186	#1382	—	67	1	no	—
187	#1382	—	67	1	no	—
188	#1382	D₆₇	1	1	yes	characteristic, direct product factor
189	C₆₇	C₂×#1382	1	1	yes	characteristic, complemented, Sylow
190	C₂²×M₁₆	—	134	1	no	—
191	C₂³×C₈	—	67	1	no	—
192	C₂×#117	—	134	1	no	—
193	C₂²×C₁₆	—	67	1	no	—
194	C₂²×C₁₆	—	67	4	no	—
195	C₂²×C₁₆	—	134	1	no	—
196	C₂²×C₁₆	—	134	1	no	—
197	C₂²×C₁₆	—	134	2	no	—
198	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#108	—	67	1	no	—
200	C₂×#108	—	67	4	no	—
201	C₂×#108	—	134	2	no	—
202	C₂×#108	—	134	1	no	—
203	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
204	#427	—	67	1	no	—
205	#427	—	67	2	no	—
206	#427	—	134	1	no	—
207	#427	—	134	1	no	—
208	#427	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#107	—	67	2	no	—
210	C₂×#106	—	134	1	no	—
211	C₂³×C₄	—	67	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	268	1	no	—
214	C₂×M₁₆	—	268	2	no	—
215	C₂×M₁₆	—	134	4	no	—
216	C₂×M₁₆	—	134	1	no	—
217	C₂²×C₈	—	268	1	no	—
218	C₂²×C₈	—	67	1	no	—
219	C₂²×C₈	—	67	1	no	—
220	C₂²×C₈	—	67	1	no	—
221	C₂²×C₈	—	67	2	no	—
222	C₂²×C₈	—	67	2	no	—
223	C₂²×C₈	—	67	1	no	—
224	C₂²×C₈	—	67	1	no	—
225	C₂²×C₈	—	67	2	no	—
226	C₂²×C₈	—	67	2	no	—
227	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
228	#117	—	2	1	no	—
229	#117	—	268	1	no	—
230	#117	—	268	1	no	—
231	#117	—	134	2	no	—
232	#117	—	134	1	no	—
233	C₂×C₄⋊C₄	—	134	2	no	—
234	C₂×C₂²⋊C₄	—	268	1	no	—
235	C₂×C₄²	—	268	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	67	2	no	—
239	C₂×C₁₆	—	134	4	no	—
240	C₂×C₁₆	—	134	4	no	—
241	C₂×C₁₆	—	134	1	no	—
242	C₂×C₁₆	—	134	1	no	—
243	C₂×C₁₆	C₄×D₆₇	1	2	yes	complemented
244	#108	—	2	1	no	—
245	#108	—	67	2	no	—
246	#108	—	134	4	no	—
247	#108	—	134	1	no	—
248	#108	C₄×D₆₇	1	2	yes	complemented
249	#107	—	67	2	no	—
250	#107	—	134	2	no	—
251	#107	C₄×D₆₇	1	2	yes	—
252	#106	—	2	1	no	—
253	#106	—	134	2	no	—
254	#106	—	134	1	no	—
255	C₂⁴	—	67	1	no	—
256	C₂²×C₄	—	268	2	no	—
257	C₂²×C₄	—	67	1	no	—
258	C₂²×C₄	—	67	1	no	—
259	C₂²×C₄	—	67	1	no	—
260	C₂²×C₄	—	67	2	no	—
261	C₂²×C₄	—	67	2	no	—
262	C₂²×C₄	—	67	1	no	—
263	C₂²×C₄	—	67	1	no	—
264	C₂²×C₄	—	67	2	no	—
265	C₂²×C₄	—	67	2	no	—
266	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	268	4	no	—
270	M₁₆	—	268	1	no	—
271	M₁₆	—	134	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	268	4	no	—
274	C₂×C₈	—	268	1	no	—
275	C₂×C₈	—	67	1	no	—
276	C₂×C₈	—	67	1	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	—	67	2	no	—
282	C₂×C₈	—	67	2	no	—
283	C₂×C₈	—	67	2	no	—
284	C₂×C₈	—	67	2	no	—
285	C₂×C₈	—	67	2	no	—
286	C₂×C₈	—	67	2	no	—
287	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	268	2	no	—
295	C₄⋊C₄	—	134	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	268	2	no	—
298	C₂²⋊C₄	—	268	1	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	268	2	no	—
301	C₄²	—	268	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	134	2	no	—
304	C₂³	—	67	1	no	—
305	C₂³	—	67	1	no	—
306	C₂³	—	67	2	no	—
307	C₂³	—	67	2	no	—
308	C₂³	—	134	4	no	—
309	C₂³	SD₁₆×D₆₇	1	1	yes	characteristic
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	268	2	no	—
312	C₂×C₄	—	536	4	no	—
313	C₂×C₄	—	67	1	no	—
314	C₂×C₄	—	67	1	no	—
315	C₂×C₄	—	67	2	no	—
316	C₂×C₄	—	67	2	no	—
317	C₂×C₄	—	67	2	no	—
318	C₂×C₄	—	67	2	no	—
319	C₂×C₄	—	67	2	no	—
320	C₂×C₄	—	67	2	no	—
321	C₂×C₄	—	67	2	no	—
322	C₂×C₄	—	67	2	no	—
323	C₂×C₄	—	67	2	no	—
324	C₂×C₄	—	67	2	no	—
325	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
328	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
330	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	268	2	no	—
333	C₈	—	67	2	no	—
334	C₈	—	67	2	no	—
335	C₈	C₄²×D₆₇	1	1	yes	characteristic
336	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
337	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
338	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	67	4	no	—
341	C₂²	—	67	2	no	—
342	C₂²	—	67	2	no	—
343	C₂²	—	134	4	no	—
344	C₂²	—	134	4	no	—
345	C₂²	—	134	2	no	—
346	C₂²	#102×D₆₇	1	1	yes	characteristic
347	C₂²	#103×D₆₇	1	1	yes	characteristic
348	C₂²	#106×D₆₇	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	536	2	no	—
351	C₄	—	67	2	no	—
352	C₄	—	67	2	no	—
353	C₄	#95×D₆₇	1	1	yes	characteristic
354	C₄	#102×D₆₇	1	1	yes	characteristic
355	C₄	#103×D₆₇	1	1	yes	characteristic
356	C₄	#107×D₆₇	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	67	4	no	—
360	C₂	—	134	2	no	—
361	C₂	#340×D₆₇	1	1	yes	characteristic
362	C₂	#359×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1382	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×D₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1382	D₆₇
hypocenter	C₆₇	C₂×#1382
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
4. #22558306	C₂	semidirect
5. #22558305	C₂	semidirect
6. #22558304	C₂	semidirect
7. C₆₇×#1382	C₂	semidirect
15. C₂×C₁₆×D₆₇	C₄	semidirect
19. #108×D₆₇	C₄	semidirect
53. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
57. #14523979	C₂×C₄	semidirect
68. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
70. #108×C₆₇	C₂×C₄	semidirect
181. D₆₇	#1382	direct (N × H)
188. #1382	D₆₇	direct (N × H)
189. C₆₇	C₂×#1382	semidirect
243. C₂×C₁₆	C₄×D₆₇	semidirect
248. #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
C₂²×C₁₆×D₆₇	2
C₂×#108×D₆₇	2
#427×D₆₇	2
#22558306	2
#22558305	2
#22558304	2
C₆₇×#1382	2
C₂×#1382	67

Subgroups — #1382×D67 #30770722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1382×D₆₇ #30770722