Subgroups — C₆₇×#33689 #30712940

Order 17152.

520 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2956	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2239	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2230	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2214	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2191	C₂	1	2	yes	maximal
7	C₆₇×#2180	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	#575×C₆₇	C₂²	1	1	yes	characteristic
10	#568×C₆₇	—	2	1	no	—
11	#533×C₆₇	—	2	1	no	—
12	#510×C₆₇	C₂²	1	4	yes	—
13	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#487×C₆₇	—	2	1	no	—
16	#470×C₆₇	—	2	1	no	—
17	#470×C₆₇	C₂²	1	2	yes	—
18	#444×C₆₇	—	2	1	no	—
19	#444×C₆₇	C₂²	1	2	yes	—
20	#443×C₆₇	—	2	1	no	—
21	#443×C₆₇	C₂²	1	2	yes	—
22	#441×C₆₇	—	2	1	no	—
23	#441×C₆₇	C₂²	1	2	yes	complemented
24	#430×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#104×C₆₇	—	2	1	no	—
26	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	#365×C₆₇	C₂²	1	4	yes	complemented
28	#364×C₆₇	C₂²	1	2	yes	complemented
29	#362×C₆₇	C₂²	1	4	yes	complemented
30	#360×C₆₇	C₂²	1	4	yes	—
31	#349×C₆₇	C₂²	1	2	yes	—
32	#347×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#41×C₆₇	—	2	1	no	—
34	#137×C₆₇	—	2	2	no	—
35	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
36	#135×C₆₇	—	2	1	no	—
37	#135×C₆₇	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₆₇	—	2	1	no	—
39	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
40	#131×C₆₇	—	2	2	no	—
41	#131×C₆₇	—	2	2	no	—
42	#131×C₆₇	—	4	2	no	—
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	—	4	2	no	—
45	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	#125×C₆₇	—	4	1	no	—
50	C₂²⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	4	1	no	—
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	—	4	1	no	—
54	#117×C₆₇	—	4	1	no	—
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	Q₃₂×C₆₇	—	2	4	no	—
57	SD₃₂×C₆₇	—	2	4	no	—
58	M₃₂×C₆₇	C₂³	1	4	yes	—
59	#108×C₆₇	—	2	1	no	—
60	#108×C₆₇	C₂³	1	2	yes	—
61	#106×C₆₇	—	2	1	no	—
62	#106×C₆₇	C₂³	1	2	yes	—
63	#104×C₆₇	—	2	2	no	—
64	#104×C₆₇	—	2	2	no	—
65	#103×C₆₇	—	2	2	no	—
66	#103×C₆₇	—	2	2	no	—
67	#97×C₆₇	D₄	1	1	yes	characteristic
68	#97×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	—	2	2	no	—
73	#41×C₆₇	—	4	1	no	—
74	#41×C₆₇	—	4	1	no	—
75	C₂×Q₈×C₆₇	—	2	1	no	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×D₄×C₆₇	—	2	1	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₆₇	—	2	1	no	—
84	Q₁₆×C₆₇	—	2	4	no	—
85	Q₁₆×C₆₇	—	4	2	no	—
86	Q₁₆×C₆₇	#41	1	2	yes	—
87	SD₁₆×C₆₇	—	2	2	no	—
88	SD₁₆×C₆₇	—	2	2	no	—
89	D₈×C₆₇	—	2	1	no	—
90	D₈×C₆₇	#41	1	2	yes	—
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	—	2	2	no	—
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	—	4	2	no	—
95	M₁₆×C₆₇	—	4	1	no	—
96	M₁₆×C₆₇	—	4	2	no	—
97	C₂×C₈×C₆₇	—	4	2	no	—
98	C₂×C₈×C₆₇	—	4	1	no	—
99	C₂×C₈×C₆₇	—	4	2	no	—
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₆₇	—	2	4	no	—
116	C₂³×C₆₇	—	4	1	no	—
117	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	4	1	no	—
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	8	2	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	4	1	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
143	C₈×C₆₇	C₂×#41	1	2	yes	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	4	1	no	—
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	—	16	1	no	—
148	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	4	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	#525	1	1	yes	characteristic
156	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	#33689	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	4	1	no	—
160	C₂×C₆₇	—	8	2	no	—
161	C₂×C₆₇	#3109	1	1	yes	characteristic
162	#3185	C₂×C₆₇	1	1	yes	characteristic, complemented
163	#2956	C₂×C₆₇	1	1	yes	characteristic
164	#2239	C₂×C₆₇	1	2	yes	complemented
165	#2230	C₂×C₆₇	1	2	yes	complemented
166	#2214	C₂×C₆₇	1	2	yes	complemented
167	#2191	C₂×C₆₇	1	2	yes	—
168	#2180	C₂×C₆₇	1	4	yes	complemented
169	#2165	C₂×C₆₇	1	1	yes	characteristic, complemented
170	C₆₇	#33689	1	1	yes	characteristic, direct product factor, Sylow
171	#575	C₂²×C₆₇	1	1	yes	characteristic
172	#568	—	2	1	no	—
173	#533	—	2	1	no	—
174	#510	C₂²×C₆₇	1	4	yes	—
175	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
176	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
177	#487	—	2	1	no	—
178	#470	—	2	1	no	—
179	#470	C₂²×C₆₇	1	2	yes	—
180	#444	—	2	1	no	—
181	#444	C₂²×C₆₇	1	2	yes	—
182	#443	—	2	1	no	—
183	#443	C₂²×C₆₇	1	2	yes	—
184	#441	—	2	1	no	—
185	#441	C₂²×C₆₇	1	2	yes	complemented
186	#430	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×#104	—	2	1	no	—
188	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
189	#365	C₂²×C₆₇	1	4	yes	complemented
190	#364	C₂²×C₆₇	1	2	yes	complemented
191	#362	C₂²×C₆₇	1	4	yes	complemented
192	#360	C₂²×C₆₇	1	4	yes	—
193	#349	C₂²×C₆₇	1	2	yes	—
194	#347	C₂²×C₆₇	1	2	yes	complemented
195	C₂×#41	—	2	1	no	—
196	#137	—	2	2	no	—
197	AGL(1,ℤ/8)	—	2	2	no	—
198	#135	—	2	1	no	—
199	#135	C₂³×C₆₇	1	2	yes	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	4	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	2	no	—
207	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
208	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	4	1	no	—
216	#117	—	4	1	no	—
217	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
218	Q₃₂	—	2	4	no	—
219	SD₃₂	—	2	4	no	—
220	M₃₂	C₂³×C₆₇	1	4	yes	—
221	#108	—	2	1	no	—
222	#108	C₂³×C₆₇	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	C₂³×C₆₇	1	2	yes	—
225	#104	—	2	2	no	—
226	#104	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#97	D₄×C₆₇	1	1	yes	characteristic
230	#97	C₂³×C₆₇	1	1	yes	characteristic
231	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
245	Q₁₆	—	2	1	no	—
246	Q₁₆	—	2	4	no	—
247	Q₁₆	—	4	2	no	—
248	Q₁₆	#41×C₆₇	1	2	yes	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	D₈	—	2	1	no	—
252	D₈	#41×C₆₇	1	2	yes	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₁₆	—	2	4	no	—
278	C₂³	—	4	1	no	—
279	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	4	1	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	4	2	no	—
304	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
305	C₈	C₂×#41×C₆₇	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	16	1	no	—
310	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	#525×C₆₇	1	1	yes	characteristic
318	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	4	1	no	—
321	C₂	—	8	2	no	—
322	C₂	C₆₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	#575×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3185	C₂	semidirect
3. C₆₇×#2239	C₂	semidirect
4. C₆₇×#2230	C₂	semidirect
5. C₆₇×#2214	C₂	semidirect
7. C₆₇×#2180	C₂	semidirect
8. C₆₇×#2165	C₂	semidirect
13. #498×C₆₇	C₂²	semidirect
14. #494×C₆₇	C₂²	semidirect
23. #441×C₆₇	C₂²	semidirect
27. #365×C₆₇	C₂²	semidirect
28. #364×C₆₇	C₂²	semidirect
29. #362×C₆₇	C₂²	semidirect
32. #347×C₆₇	C₂²	semidirect
157. #33689	C₆₇	direct (N × H)
162. #3185	C₂×C₆₇	semidirect
164. #2239	C₂×C₆₇	semidirect
165. #2230	C₂×C₆₇	semidirect
166. #2214	C₂×C₆₇	semidirect
168. #2180	C₂×C₆₇	semidirect
169. #2165	C₂×C₆₇	semidirect
170. C₆₇	#33689	direct (N × H)
175. #498	C₂²×C₆₇	semidirect
176. #494	C₂²×C₆₇	semidirect
185. #441	C₂²×C₆₇	semidirect
189. #365	C₂²×C₆₇	semidirect
190. #364	C₂²×C₆₇	semidirect
191. #362	C₂²×C₆₇	semidirect
194. #347	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3185	2
C₆₇×#2956	2
C₆₇×#2239	2
C₆₇×#2230	2
C₆₇×#2214	2
C₆₇×#2191	2
C₆₇×#2180	2
C₆₇×#2165	2
#33689	67

Subgroups — C67×#33689 #30712940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33689 #30712940