Subgroups — C₄×Q₃₂×D₆₇ #31009102

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₆₇	C₂	1	2	yes	maximal
3	C₄×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523989	C₂	1	2	yes	maximal, complemented
7	#22562065	C₂	1	2	yes	maximal, complemented
8	#366×D₆₇	C₂	1	1	yes	maximal, characteristic
9	#358×D₆₇	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
13	C₄×Q₈×D₆₇	—	2	2	no	—
14	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
15	C₂×#14523989	C₂²	1	2	yes	—
16	Q₃₂×D₆₇	C₄	1	8	yes	direct product factor
17	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₆₈	C₂²	1	2	yes	—
20	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
21	#16863490	C₂²	1	2	yes	—
22	C₄×#11761618	—	2	2	no	—
23	#16863488	—	2	2	no	—
24	#107×D₆₇	C₂²	1	2	yes	—
25	#103×D₆₇	—	2	2	no	—
26	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
27	#16863222	C₂²	1	1	yes	characteristic
28	#16863221	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863200	C₂²	1	2	yes	—
31	#16863199	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₆₇	—	2	2	no	—
41	C₂×Dic₂₆₈	C₂³	1	2	yes	—
42	C₂×#11761618	—	2	2	no	—
43	Q₁₆×D₆₇	—	2	4	no	—
44	Q₁₆×D₆₇	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	Q₈×Dic₆₇	—	2	2	no	—
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×Dic₁₃₄	—	4	2	no	—
50	#14524008	—	2	2	no	—
51	#14524008	—	4	2	no	—
52	C₄⋊C₄×D₆₇	—	2	2	no	—
53	C₄⋊C₄×D₆₇	—	4	2	no	—
54	C₄²×D₆₇	—	4	2	no	—
55	C₄²×D₆₇	D₄	1	1	yes	characteristic
56	Dic₅₃₆	C₂×C₄	1	4	yes	complemented
57	#14523989	C₂×C₄	1	8	yes	complemented
58	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₆₇	—	2	2	no	—
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	—
62	#14523977	C₂³	1	2	yes	—
63	#14523976	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523967	—	2	2	no	—
66	#14523966	—	2	2	no	—
67	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₆₇	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	Q₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₃₄	—	2	2	no	—
79	C₂×Dic₁₃₄	—	4	2	no	—
80	Q₈×D₆₇	—	2	8	no	—
81	Q₈×D₆₇	—	4	4	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	C₂×C₄×D₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
85	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
86	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
87	Dic₂₆₈	—	2	2	no	—
88	Dic₂₆₈	C₂²×C₄	1	4	yes	—
89	#11761618	—	2	8	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₈×D₆₇	—	2	2	no	—
93	C₈×D₆₇	C₂²×C₄	1	2	yes	—
94	C₈×D₆₇	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	C₄×Dic₆₇	—	4	2	no	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	D₈	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₇	—	2	4	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
132	C₄×D₆₇	—	2	2	no	—
133	C₄×D₆₇	—	4	8	no	—
134	C₄×D₆₇	—	8	4	no	—
135	C₄×D₆₇	Q₃₂	1	4	yes	direct product factor
136	C₄×D₆₇	C₄×D₄	1	2	yes	—
137	C₄×D₆₇	#135	1	2	yes	—
138	Dic₁₃₄	—	2	4	no	—
139	Dic₁₃₄	—	4	2	no	—
140	Dic₁₃₄	—	4	8	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₆₇	C₄×D₈	1	2	yes	—
148	C₂×D₆₇	C₂×Q₃₂	1	2	yes	—
149	C₂×D₆₇	#508	1	2	yes	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	4	4	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	C₂×Q₃₂	1	2	yes	complemented
154	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
156	Dic₆₇	—	2	1	no	—
157	Dic₆₇	—	4	4	no	—
158	Dic₆₇	—	8	2	no	—
159	Dic₆₇	C₄×D₈	1	2	yes	—
160	Dic₆₇	C₂×Q₃₂	1	2	yes	complemented
161	C₂×C₄×Q₃₂	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
165	D₆₇	C₄×Q₃₂	1	4	yes	direct product factor
166	C₂²×Q₃₂	—	67	1	no	—
167	C₂×C₄×Q₁₆	—	67	2	no	—
168	C₄×Q₃₂	—	67	1	no	—
169	C₄×Q₃₂	—	67	2	no	—
170	C₄×Q₃₂	—	67	1	no	—
171	C₄×Q₃₂	—	67	2	no	—
172	C₄×Q₃₂	—	67	1	no	—
173	C₄×Q₃₂	D₆₇	1	1	yes	characteristic, direct product factor
174	C₂×#366	—	67	1	no	—
175	C₂×#358	—	67	2	no	—
176	C₂×C₄×C₁₆	—	67	1	no	—
177	C₆₇	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	67	2	no	—
179	C₂×C₄×Q₈	—	134	2	no	—
180	C₂×Q₃₂	—	67	2	no	—
181	C₂×Q₃₂	—	67	8	no	—
182	C₂×Q₃₂	—	67	1	no	—
183	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
184	C₂²×C₁₆	—	67	1	no	—
185	C₂²×C₁₆	—	67	1	no	—
186	C₄×Q₁₆	—	67	2	no	—
187	C₄×Q₁₆	—	67	2	no	—
188	C₄×Q₁₆	—	67	2	no	—
189	C₄×Q₁₆	—	134	2	no	—
190	C₄×Q₁₆	—	134	2	no	—
191	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
192	C₂×#107	—	67	2	no	—
193	C₂×#103	—	134	2	no	—
194	C₂×C₄×C₈	—	67	1	no	—
195	#366	—	67	1	no	—
196	#366	—	67	2	no	—
197	#366	C₂×D₆₇	1	1	yes	characteristic
198	#358	—	67	2	no	—
199	#358	—	67	2	no	—
200	#358	—	67	2	no	—
201	#358	C₂×D₆₇	1	2	yes	—
202	C₄×C₁₆	—	67	1	no	—
203	C₄×C₁₆	—	67	1	no	—
204	C₄×C₁₆	—	67	1	no	—
205	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
206	C₂²×Q₈	—	134	2	no	—
207	C₂×Q₁₆	—	67	8	no	—
208	C₂×Q₁₆	—	67	2	no	—
209	C₂×Q₁₆	—	134	4	no	—
210	C₂×Q₁₆	—	134	2	no	—
211	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
212	C₂²×C₈	—	67	1	no	—
213	C₂²×C₈	—	67	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	268	2	no	—
216	C₄×Q₈	—	268	2	no	—
217	C₄×Q₈	—	134	2	no	—
218	C₄×Q₈	—	134	2	no	—
219	C₄×Q₈	—	134	2	no	—
220	C₂×C₄⋊C₄	—	268	2	no	—
221	C₂×C₄⋊C₄	—	134	2	no	—
222	C₂×C₄²	—	268	2	no	—
223	C₂×C₄²	—	67	1	no	—
224	Q₃₂	—	67	8	no	—
225	Q₃₂	—	67	4	no	—
226	Q₃₂	C₄×D₆₇	1	4	yes	direct product factor
227	C₂×C₁₆	—	67	1	no	—
228	C₂×C₁₆	—	67	4	no	—
229	C₂×C₁₆	—	67	1	no	—
230	C₂×C₁₆	—	134	2	no	—
231	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
233	#107	—	67	2	no	—
234	#107	—	134	2	no	—
235	#107	C₂²×D₆₇	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	134	2	no	—
238	#103	—	134	2	no	—
239	#103	—	134	2	no	—
240	C₄×C₈	—	67	1	no	—
241	C₄×C₈	—	67	1	no	—
242	C₄×C₈	—	67	1	no	—
243	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	268	2	no	—
246	C₂×Q₈	—	268	4	no	—
247	C₂×Q₈	—	134	8	no	—
248	C₂×Q₈	—	134	2	no	—
249	C₂²×C₄	—	268	2	no	—
250	C₂²×C₄	—	268	2	no	—
251	C₂²×C₄	—	67	1	no	—
252	C₂²×C₄	—	67	1	no	—
253	C₂²×C₄	—	67	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	67	4	no	—
256	Q₁₆	—	134	8	no	—
257	Q₁₆	—	134	2	no	—
258	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
259	C₂×C₈	—	67	1	no	—
260	C₂×C₈	—	67	2	no	—
261	C₂×C₈	—	67	2	no	—
262	C₂×C₈	—	67	1	no	—
263	C₂×C₈	—	134	2	no	—
264	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	268	2	no	—
269	C₄⋊C₄	—	268	2	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	134	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	268	2	no	—
275	C₄²	—	268	2	no	—
276	C₄²	—	268	2	no	—
277	C₄²	—	67	1	no	—
278	C₄²	—	67	1	no	—
279	C₄²	—	67	1	no	—
280	C₄²	D₄×D₆₇	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	67	2	no	—
283	C₁₆	—	134	1	no	—
284	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
285	C₂³	—	67	1	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	268	8	no	—
289	Q₈	—	268	2	no	—
290	Q₈	—	134	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	268	8	no	—
294	C₂×C₄	—	268	2	no	—
295	C₂×C₄	—	268	2	no	—
296	C₂×C₄	—	536	4	no	—
297	C₂×C₄	—	67	1	no	—
298	C₂×C₄	—	67	1	no	—
299	C₂×C₄	—	67	4	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	67	2	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	134	2	no	—
304	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
305	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	67	2	no	—
309	C₈	—	134	1	no	—
310	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
311	C₈	#41×D₆₇	1	1	yes	characteristic
312	C₂²	—	67	2	no	—
313	C₂²	—	67	2	no	—
314	C₂²	—	67	2	no	—
315	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	268	4	no	—
320	C₄	—	536	2	no	—
321	C₄	—	67	2	no	—
322	C₄	—	67	2	no	—
323	C₄	—	134	1	no	—
324	C₄	Q₃₂×D₆₇	1	2	yes	direct product factor
325	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
326	C₄	#135×D₆₇	1	1	yes	characteristic
327	C₂	—	67	4	no	—
328	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
329	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic
330	C₂	#508×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×Q₃₂	D₆₇
hypocenter	C₆₇	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₅₃₆	C₂	semidirect
4. Q₃₂×Dic₆₇	C₂	semidirect
5. #22562067	C₂	semidirect
6. C₄×#14523989	C₂	semidirect
7. #22562065	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
16. Q₃₂×D₆₇	C₄	direct (N × H)
56. Dic₅₃₆	C₂×C₄	semidirect
57. #14523989	C₂×C₄	semidirect
72. Q₃₂×C₆₇	C₂×C₄	semidirect
135. C₄×D₆₇	Q₃₂	direct (N × H)
153. C₄×C₆₇	C₂×Q₃₂	semidirect
160. Dic₆₇	C₂×Q₃₂	semidirect
165. D₆₇	C₄×Q₃₂	direct (N × H)
173. C₄×Q₃₂	D₆₇	direct (N × H)
177. C₆₇	C₂×C₄×Q₃₂	semidirect
226. Q₃₂	C₄×D₆₇	direct (N × H)
324. C₄	Q₃₂×D₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₆₇	2
C₄×Q₁₆×D₆₇	2
C₄×Dic₅₃₆	2
Q₃₂×Dic₆₇	2
#22562067	2
C₄×#14523989	2
#22562065	2
#366×D₆₇	2
#358×D₆₇	2
C₄×C₁₆×D₆₇	2
C₄×Q₃₂×C₆₇	2
C₂×C₄×Q₃₂	67

Subgroups — C4×Q32×D67 #31009102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₃₂×D₆₇ #31009102