Subgroups — C₆₇×#33637 #30712888

Order 17152.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2231	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2230	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2155	C₂	1	2	yes	maximal, complemented
6	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2145	C₂	1	2	yes	maximal
8	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#131×C₆₇	—	2	1	no	—
10	#533×C₆₇	—	2	1	no	—
11	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	—	2	1	no	—
13	#499×C₆₇	—	2	1	no	—
14	#499×C₆₇	C₂²	1	2	yes	complemented
15	#498×C₆₇	C₂²	1	2	yes	complemented
16	#443×C₆₇	—	2	1	no	—
17	#443×C₆₇	—	2	1	no	—
18	#443×C₆₇	C₂²	1	2	yes	—
19	#443×C₆₇	C₂²	1	2	yes	—
20	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
21	#427×C₆₇	—	2	1	no	—
22	#427×C₆₇	C₂²	1	2	yes	—
23	C₂×#104×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#365×C₆₇	C₂²	1	8	yes	complemented
26	#364×C₆₇	C₂²	1	2	yes	complemented
27	#364×C₆₇	C₂²	1	2	yes	complemented
28	#360×C₆₇	C₂²	1	8	yes	—
29	C₂×#41×C₆₇	—	2	1	no	—
30	#135×C₆₇	—	2	1	no	—
31	#135×C₆₇	—	2	1	no	—
32	#135×C₆₇	Q₈	1	2	yes	complemented
33	#135×C₆₇	C₂³	1	2	yes	—
34	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₆₇	—	2	1	no	—
36	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
37	#131×C₆₇	—	2	2	no	—
38	#131×C₆₇	—	2	2	no	—
39	#131×C₆₇	—	4	1	no	—
40	#131×C₆₇	—	4	1	no	—
41	C₂×M₁₆×C₆₇	—	2	1	no	—
42	C₂×M₁₆×C₆₇	—	4	1	no	—
43	C₂²×C₈×C₆₇	—	4	1	no	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	#125×C₆₇	—	2	2	no	—
48	C₂²⋊Q₈×C₆₇	—	4	2	no	—
49	C₄×Q₈×C₆₇	—	4	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	M₃₂×C₆₇	—	2	2	no	—
53	M₃₂×C₆₇	C₂³	1	4	yes	—
54	C₂×C₁₆×C₆₇	—	2	2	no	—
55	#108×C₆₇	—	2	1	no	—
56	#108×C₆₇	C₂³	1	2	yes	—
57	#107×C₆₇	—	2	2	no	—
58	#107×C₆₇	—	2	2	no	—
59	#106×C₆₇	—	2	2	no	—
60	#106×C₆₇	C₂³	1	4	yes	—
61	#104×C₆₇	—	2	2	no	—
62	#104×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	#41×C₆₇	—	2	2	no	—
68	#41×C₆₇	—	2	2	no	—
69	#41×C₆₇	—	4	1	no	—
70	#41×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×Q₈×C₆₇	—	4	2	no	—
73	C₂×D₄×C₆₇	—	2	1	no	—
74	C₂×D₄×C₆₇	—	4	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	4	1	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₆₇	—	2	1	no	—
79	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
80	SD₁₆×C₆₇	—	2	2	no	—
81	SD₁₆×C₆₇	—	2	2	no	—
82	D₈×C₆₇	—	2	1	no	—
83	D₈×C₆₇	C₂×Q₈	1	2	yes	—
84	M₁₆×C₆₇	—	2	2	no	—
85	M₁₆×C₆₇	—	4	1	no	—
86	M₁₆×C₆₇	—	4	2	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	4	1	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	4	2	no	—
91	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
92	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
101	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	4	4	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₂²⋊C₄×C₆₇	—	4	2	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₁₆×C₆₇	—	2	4	no	—
112	C₂³×C₆₇	—	4	1	no	—
113	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₈×C₆₇	—	2	2	no	—
115	Q₈×C₆₇	—	4	1	no	—
116	Q₈×C₆₇	—	8	4	no	—
117	D₄×C₆₇	—	2	2	no	—
118	D₄×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₆₇	—	4	2	no	—
135	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
140	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
141	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂²×C₆₇	—	4	1	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	8	2	no	—
146	C₂²×C₆₇	#476	1	1	yes	characteristic
147	C₂²×C₆₇	#476	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	4	1	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	8	4	no	—
153	C₄×C₆₇	#476	1	1	yes	characteristic
154	C₄×C₆₇	#476	1	1	yes	characteristic
155	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
157	#33637	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	1	no	—
161	C₂×C₆₇	—	8	2	no	—
162	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
163	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
164	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
165	#2231	C₂×C₆₇	1	4	yes	complemented
166	#2230	C₂×C₆₇	1	4	yes	complemented
167	#2155	C₂×C₆₇	1	2	yes	complemented
168	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2145	C₂×C₆₇	1	2	yes	—
170	C₆₇	#33637	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×#131	—	2	1	no	—
173	#533	—	2	1	no	—
174	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
175	#500	—	2	1	no	—
176	#499	—	2	1	no	—
177	#499	C₂²×C₆₇	1	2	yes	complemented
178	#498	C₂²×C₆₇	1	2	yes	complemented
179	#443	—	2	1	no	—
180	#443	—	2	1	no	—
181	#443	C₂²×C₆₇	1	2	yes	—
182	#443	C₂²×C₆₇	1	2	yes	—
183	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
184	#427	—	2	1	no	—
185	#427	C₂²×C₆₇	1	2	yes	—
186	C₂×#104	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
188	#365	C₂²×C₆₇	1	8	yes	complemented
189	#364	C₂²×C₆₇	1	2	yes	complemented
190	#364	C₂²×C₆₇	1	2	yes	complemented
191	#360	C₂²×C₆₇	1	8	yes	—
192	C₂×#41	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Q₈×C₆₇	1	2	yes	complemented
196	#135	C₂³×C₆₇	1	2	yes	—
197	C₂×Q₁₆	Q₈×C₆₇	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Q₈×C₆₇	1	1	yes	characteristic, complemented
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	4	1	no	—
203	#131	—	4	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	4	1	no	—
206	C₂²×C₈	—	4	1	no	—
207	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	2	no	—
210	#125	—	2	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	C₄×Q₈	—	4	2	no	—
213	#117	—	2	2	no	—
214	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
215	M₃₂	—	2	2	no	—
216	M₃₂	C₂³×C₆₇	1	4	yes	—
217	C₂×C₁₆	—	2	2	no	—
218	#108	—	2	1	no	—
219	#108	C₂³×C₆₇	1	2	yes	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	C₂³×C₆₇	1	4	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
227	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	#41	—	4	1	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	C₂×Q₈×C₆₇	1	2	yes	—
243	SD₁₆	—	2	2	no	—
244	SD₁₆	—	2	2	no	—
245	D₈	—	2	1	no	—
246	D₈	C₂×Q₈×C₆₇	1	2	yes	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
255	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
256	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
257	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
263	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
264	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₁₆	—	2	4	no	—
275	C₂³	—	4	1	no	—
276	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	8	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
295	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
299	C₈	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
300	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
301	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
302	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
303	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
304	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
305	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#476×C₆₇	1	1	yes	characteristic
310	C₂²	#476×C₆₇	1	1	yes	characteristic
311	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	8	4	no	—
316	C₄	#476×C₆₇	1	1	yes	characteristic
317	C₄	#476×C₆₇	1	1	yes	characteristic
318	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
319	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#476×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×#476
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3160	C₂	semidirect
3. C₆₇×#2231	C₂	semidirect
4. C₆₇×#2230	C₂	semidirect
5. C₆₇×#2155	C₂	semidirect
6. C₂×#364×C₆₇	C₂	semidirect
14. #499×C₆₇	C₂²	semidirect
15. #498×C₆₇	C₂²	semidirect
25. #365×C₆₇	C₂²	semidirect
26. #364×C₆₇	C₂²	semidirect
27. #364×C₆₇	C₂²	semidirect
32. #135×C₆₇	Q₈	semidirect
34. C₂×Q₁₆×C₆₇	Q₈	semidirect
36. C₂×D₈×C₆₇	Q₈	semidirect
157. #33637	C₆₇	direct (N × H)
163. #3160	C₂×C₆₇	semidirect
165. #2231	C₂×C₆₇	semidirect
166. #2230	C₂×C₆₇	semidirect
167. #2155	C₂×C₆₇	semidirect
168. C₂×#364	C₂×C₆₇	semidirect
170. C₆₇	#33637	direct (N × H)
177. #499	C₂²×C₆₇	semidirect
178. #498	C₂²×C₆₇	semidirect
188. #365	C₂²×C₆₇	semidirect
189. #364	C₂²×C₆₇	semidirect
190. #364	C₂²×C₆₇	semidirect
195. #135	Q₈×C₆₇	semidirect
197. C₂×Q₁₆	Q₈×C₆₇	semidirect
199. C₂×D₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3160	2
C₂×#443×C₆₇	2
C₆₇×#2231	2
C₆₇×#2230	2
C₆₇×#2155	2
C₂×#364×C₆₇	2
C₆₇×#2145	2
#33637	67

Subgroups — C67×#33637 #30712888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33637 #30712888