Subgroups — C₆₇×#33688 #30712939

Order 17152.

564 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2238	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2230	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2213	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2190	C₂	1	2	yes	maximal, complemented
7	C₆₇×AGL(1,ℤ/16)	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	#532×C₆₇	—	2	1	no	—
12	#509×C₆₇	C₂²	1	4	yes	—
13	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#486×C₆₇	—	2	1	no	—
16	#469×C₆₇	—	2	1	no	—
17	#469×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	#442×C₆₇	—	2	1	no	—
23	#442×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	#361×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	C₂×#41×C₆₇	—	4	1	no	—
35	#137×C₆₇	—	2	2	no	—
36	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
37	#135×C₆₇	—	2	1	no	—
38	#135×C₆₇	C₂³	1	2	yes	—
39	C₂×D₈×C₆₇	—	2	1	no	—
40	C₂×D₈×C₆₇	C₂³	1	2	yes	—
41	#131×C₆₇	—	2	2	no	—
42	#131×C₆₇	—	2	2	no	—
43	#131×C₆₇	—	4	2	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	4	2	no	—
46	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
47	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	#127×C₆₇	—	2	1	no	—
50	#124×C₆₇	—	4	1	no	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	4	1	no	—
53	C₄×D₄×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	—	4	1	no	—
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	SD₃₂×C₆₇	—	2	4	no	—
57	D₁₆×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	#102×C₆₇	—	2	2	no	—
66	#102×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	#41×C₆₇	—	8	4	no	—
76	C₂×Q₈×C₆₇	—	2	1	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×D₄×C₆₇	—	2	1	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂×D₄×C₆₇	—	4	1	no	—
82	C₂×D₄×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₆₇	—	2	1	no	—
89	Q₁₆×C₆₇	#41	1	2	yes	complemented
90	SD₁₆×C₆₇	—	2	2	no	—
91	SD₁₆×C₆₇	—	2	2	no	—
92	D₈×C₆₇	—	2	1	no	—
93	D₈×C₆₇	—	2	4	no	—
94	D₈×C₆₇	—	4	2	no	—
95	D₈×C₆₇	#41	1	2	yes	complemented
96	M₁₆×C₆₇	—	2	2	no	—
97	M₁₆×C₆₇	—	2	2	no	—
98	M₁₆×C₆₇	—	2	2	no	—
99	M₁₆×C₆₇	—	4	2	no	—
100	M₁₆×C₆₇	—	4	1	no	—
101	M₁₆×C₆₇	—	4	2	no	—
102	C₂×C₈×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	4	1	no	—
104	C₂×C₈×C₆₇	—	4	2	no	—
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	—	2	1	no	—
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₆₇	—	2	4	no	—
117	C₂³×C₆₇	—	4	1	no	—
118	C₂³×C₆₇	—	4	2	no	—
119	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	4	1	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	D₄×C₆₇	—	4	2	no	—
128	D₄×C₆₇	—	4	2	no	—
129	D₄×C₆₇	—	8	2	no	—
130	D₄×C₆₇	—	8	2	no	—
131	D₄×C₆₇	—	8	4	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	8	4	no	—
142	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	4	2	no	—
147	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₆₇	C₂×#41	1	2	yes	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	16	1	no	—
156	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	4	1	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	#525	1	1	yes	characteristic
163	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	#33688	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	—	4	1	no	—
167	C₂×C₆₇	—	8	2	no	—
168	C₂×C₆₇	—	16	2	no	—
169	C₂×C₆₇	#3109	1	1	yes	characteristic
170	#3184	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#2956	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#2238	C₂×C₆₇	1	2	yes	complemented
173	#2230	C₂×C₆₇	1	2	yes	complemented
174	#2213	C₂×C₆₇	1	2	yes	complemented
175	#2190	C₂×C₆₇	1	2	yes	complemented
176	AGL(1,ℤ/16)	C₂×C₆₇	1	4	yes	complemented
177	#2165	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₆₇	#33688	1	1	yes	characteristic, direct product factor, Sylow
179	#575	C₂²×C₆₇	1	1	yes	characteristic
180	#568	—	2	1	no	—
181	#532	—	2	1	no	—
182	#509	C₂²×C₆₇	1	4	yes	—
183	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
184	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
185	#486	—	2	1	no	—
186	#469	—	2	1	no	—
187	#469	C₂²×C₆₇	1	2	yes	—
188	#444	—	2	1	no	—
189	#444	C₂²×C₆₇	1	2	yes	—
190	#443	—	2	1	no	—
191	#443	C₂²×C₆₇	1	2	yes	—
192	#442	—	2	1	no	—
193	#442	C₂²×C₆₇	1	2	yes	complemented
194	#430	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×#104	—	2	1	no	—
196	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
197	#365	C₂²×C₆₇	1	4	yes	complemented
198	#364	C₂²×C₆₇	1	2	yes	complemented
199	#361	C₂²×C₆₇	1	4	yes	complemented
200	#360	C₂²×C₆₇	1	4	yes	—
201	#349	C₂²×C₆₇	1	2	yes	—
202	#347	C₂²×C₆₇	1	2	yes	complemented
203	C₂×#41	—	2	1	no	—
204	C₂×#41	—	4	1	no	—
205	#137	—	2	2	no	—
206	AGL(1,ℤ/8)	—	2	2	no	—
207	#135	—	2	1	no	—
208	#135	C₂³×C₆₇	1	2	yes	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂³×C₆₇	1	2	yes	—
211	#131	—	2	2	no	—
212	#131	—	2	2	no	—
213	#131	—	4	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
217	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	#127	—	2	1	no	—
220	#124	—	4	1	no	—
221	#121	—	2	2	no	—
222	#121	—	4	1	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	4	1	no	—
225	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
226	SD₃₂	—	2	4	no	—
227	D₁₆	—	2	4	no	—
228	M₃₂	C₂³×C₆₇	1	4	yes	—
229	#108	—	2	1	no	—
230	#108	C₂³×C₆₇	1	2	yes	—
231	#106	—	2	1	no	—
232	#106	C₂³×C₆₇	1	2	yes	—
233	#104	—	2	2	no	—
234	#104	—	2	2	no	—
235	#102	—	2	2	no	—
236	#102	—	2	2	no	—
237	#97	D₄×C₆₇	1	1	yes	characteristic
238	#97	C₂³×C₆₇	1	1	yes	characteristic
239	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	#41×C₆₇	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	D₈	#41×C₆₇	1	2	yes	complemented
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₁₆	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	2	no	—
317	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
318	C₈	C₂×#41×C₆₇	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	16	1	no	—
326	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	8	2	no	—
332	C₄	#525×C₆₇	1	1	yes	characteristic
333	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	4	1	no	—
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₆₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
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×#3184	C₂	semidirect
2. C₆₇×#2956	C₂	semidirect
3. C₆₇×#2238	C₂	semidirect
4. C₆₇×#2230	C₂	semidirect
5. C₆₇×#2213	C₂	semidirect
6. C₆₇×#2190	C₂	semidirect
7. C₆₇×AGL(1,ℤ/16)	C₂	semidirect
8. C₆₇×#2165	C₂	semidirect
13. #498×C₆₇	C₂²	semidirect
14. #493×C₆₇	C₂²	semidirect
23. #442×C₆₇	C₂²	semidirect
27. #365×C₆₇	C₂²	semidirect
28. #364×C₆₇	C₂²	semidirect
29. #361×C₆₇	C₂²	semidirect
32. #347×C₆₇	C₂²	semidirect
89. Q₁₆×C₆₇	#41	semidirect
95. D₈×C₆₇	#41	semidirect
164. #33688	C₆₇	direct (N × H)
170. #3184	C₂×C₆₇	semidirect
171. #2956	C₂×C₆₇	semidirect
172. #2238	C₂×C₆₇	semidirect
173. #2230	C₂×C₆₇	semidirect
174. #2213	C₂×C₆₇	semidirect
175. #2190	C₂×C₆₇	semidirect
176. AGL(1,ℤ/16)	C₂×C₆₇	semidirect
177. #2165	C₂×C₆₇	semidirect
178. C₆₇	#33688	direct (N × H)
183. #498	C₂²×C₆₇	semidirect
184. #493	C₂²×C₆₇	semidirect
193. #442	C₂²×C₆₇	semidirect
197. #365	C₂²×C₆₇	semidirect
198. #364	C₂²×C₆₇	semidirect
199. #361	C₂²×C₆₇	semidirect
202. #347	C₂²×C₆₇	semidirect
259. Q₁₆	#41×C₆₇	semidirect
265. D₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3184	2
C₆₇×#2956	2
C₆₇×#2238	2
C₆₇×#2230	2
C₆₇×#2213	2
C₆₇×#2190	2
C₆₇×AGL(1,ℤ/16)	2
C₆₇×#2165	2
#33688	67

Subgroups — C67×#33688 #30712939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33688 #30712939