Subgroups — C₆₇×#33064 #30712315

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3038	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1965	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1931	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1907	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
11	C₂×#131×C₆₇	C₂²	1	2	yes	—
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#490×C₆₇	—	2	1	no	—
16	#490×C₆₇	—	2	1	no	—
17	#490×C₆₇	—	2	2	no	—
18	#479×C₆₇	—	2	2	no	—
19	#478×C₆₇	—	2	2	no	—
20	#460×C₆₇	—	2	2	no	—
21	#459×C₆₇	—	2	2	no	—
22	#428×C₆₇	C₂²	1	2	yes	—
23	#423×C₆₇	C₂²	1	1	yes	characteristic
24	#423×C₆₇	C₂²	1	1	yes	characteristic
25	#418×C₆₇	C₂²	1	2	yes	—
26	#417×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	4	yes	—
28	C₂×#100×C₆₇	C₂²	1	4	yes	—
29	#405×C₆₇	—	2	2	no	—
30	#405×C₆₇	C₂²	1	2	yes	—
31	C₄×M₁₆×C₆₇	—	2	1	no	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₆₇	—	2	1	no	—
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	#343×C₆₇	C₂²	1	4	yes	—
36	#341×C₆₇	C₂²	1	4	yes	—
37	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
40	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	—	2	2	no	—
43	#131×C₆₇	—	2	4	no	—
44	#131×C₆₇	—	2	4	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
48	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	#127×C₆₇	—	2	2	no	—
52	#127×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#125×C₆₇	—	2	2	no	—
55	#125×C₆₇	—	2	1	no	—
56	#125×C₆₇	—	2	1	no	—
57	C₄×Q₈×C₆₇	D₄	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	D₄	1	2	yes	—
60	#117×C₆₇	—	2	1	no	—
61	#117×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#107×C₆₇	—	2	2	no	—
67	#106×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#102×C₆₇	—	2	2	no	—
72	#102×C₆₇	—	2	4	no	—
73	#102×C₆₇	—	2	4	no	—
74	#102×C₆₇	—	2	4	no	—
75	#100×C₆₇	—	2	8	no	—
76	#97×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	1	no	—
78	#97×C₆₇	D₄	1	1	yes	characteristic
79	#97×C₆₇	D₄	1	1	yes	characteristic
80	#97×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	1	no	—
83	C₂⁴×C₆₇	—	2	2	no	—
84	#41×C₆₇	—	2	2	no	—
85	#41×C₆₇	#41	1	4	yes	—
86	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	2	no	—
94	C₂×D₄×C₆₇	—	8	4	no	—
95	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	#41	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	SD₁₆×C₆₇	—	4	4	no	—
102	D₈×C₆₇	—	4	4	no	—
103	M₁₆×C₆₇	—	2	8	no	—
104	M₁₆×C₆₇	—	2	2	no	—
105	M₁₆×C₆₇	—	2	4	no	—
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	#41	1	4	yes	—
108	C₂×C₈×C₆₇	—	2	8	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₆₇	—	2	1	no	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	2	yes	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂³×C₆₇	—	2	1	no	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	4	no	—
132	C₂³×C₆₇	—	8	4	no	—
133	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	1	no	—
135	Q₈×C₆₇	#124	1	2	yes	—
136	D₄×C₆₇	—	2	4	no	—
137	D₄×C₆₇	—	2	1	no	—
138	D₄×C₆₇	—	4	4	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	8	4	no	—
142	D₄×C₆₇	—	8	4	no	—
143	D₄×C₆₇	#124	1	2	yes	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	8	2	no	—
150	C₂×C₄×C₆₇	—	8	2	no	—
151	C₂×C₄×C₆₇	#121	1	2	yes	—
152	C₂×C₄×C₆₇	#121	1	2	yes	—
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
156	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
157	C₈×C₆₇	—	2	2	no	—
158	C₈×C₆₇	—	2	4	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₈×C₆₇	—	4	2	no	—
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	8	2	no	—
166	C₂²×C₆₇	—	8	4	no	—
167	C₂²×C₆₇	—	16	4	no	—
168	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
169	C₂²×C₆₇	#529	1	2	yes	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	—	8	2	no	—
175	C₄×C₆₇	#525	1	2	yes	—
176	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
177	C₄×C₆₇	#532	1	1	yes	characteristic
178	#33064	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂×C₆₇	—	2	2	no	—
180	C₂×C₆₇	—	4	2	no	—
181	C₂×C₆₇	—	16	2	no	—
182	C₂×C₆₇	#2850	1	1	yes	characteristic
183	C₂×C₆₇	#3068	1	2	yes	—
184	C₂×#490	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#3080	C₂×C₆₇	1	1	yes	characteristic, complemented
186	#3038	C₂×C₆₇	1	1	yes	characteristic
187	#2875	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
189	#1965	C₂×C₆₇	1	1	yes	characteristic, complemented
190	#1931	C₂×C₆₇	1	4	yes	complemented
191	#1907	C₂×C₆₇	1	4	yes	complemented
192	#1838	C₂×C₆₇	1	1	yes	characteristic, complemented
193	C₆₇	#33064	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
195	C₂×#131	C₂²×C₆₇	1	2	yes	—
196	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
198	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
199	#490	—	2	1	no	—
200	#490	—	2	1	no	—
201	#490	—	2	2	no	—
202	#479	—	2	2	no	—
203	#478	—	2	2	no	—
204	#460	—	2	2	no	—
205	#459	—	2	2	no	—
206	#428	C₂²×C₆₇	1	2	yes	—
207	#423	C₂²×C₆₇	1	1	yes	characteristic
208	#423	C₂²×C₆₇	1	1	yes	characteristic
209	#418	C₂²×C₆₇	1	2	yes	—
210	#417	C₂²×C₆₇	1	2	yes	—
211	C₂×#102	C₂²×C₆₇	1	4	yes	—
212	C₂×#100	C₂²×C₆₇	1	4	yes	—
213	#405	—	2	2	no	—
214	#405	C₂²×C₆₇	1	2	yes	—
215	C₄×M₁₆	—	2	1	no	—
216	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×#97	—	2	1	no	—
218	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
219	#343	C₂²×C₆₇	1	4	yes	—
220	#341	C₂²×C₆₇	1	4	yes	—
221	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×D₄	—	2	2	no	—
223	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
224	AGL(1,ℤ/8)	—	2	8	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	—	2	2	no	—
227	#131	—	2	4	no	—
228	#131	—	2	4	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
231	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
232	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
235	#127	—	2	2	no	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#125	—	2	2	no	—
239	#125	—	2	1	no	—
240	#125	—	2	1	no	—
241	C₄×Q₈	D₄×C₆₇	1	2	yes	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	D₄×C₆₇	1	2	yes	—
244	#117	—	2	1	no	—
245	#117	C₂³×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊C₄	—	2	2	no	—
247	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
250	#107	—	2	2	no	—
251	#106	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	2	2	no	—
254	#103	—	2	2	no	—
255	#102	—	2	2	no	—
256	#102	—	2	4	no	—
257	#102	—	2	4	no	—
258	#102	—	2	4	no	—
259	#100	—	2	8	no	—
260	#97	—	2	2	no	—
261	#97	—	2	1	no	—
262	#97	D₄×C₆₇	1	1	yes	characteristic
263	#97	D₄×C₆₇	1	1	yes	characteristic
264	#97	D₄×C₆₇	1	2	yes	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	2	1	no	—
267	C₂⁴	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	#41×C₆₇	1	4	yes	—
270	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	4	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	8	4	no	—
279	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	#41×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
285	SD₁₆	—	4	4	no	—
286	D₈	—	4	4	no	—
287	M₁₆	—	2	8	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	2	4	no	—
291	M₁₆	#41×C₆₇	1	4	yes	—
292	C₂×C₈	—	2	8	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
299	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
306	C₂²⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₆₇	1	2	yes	—
313	C₂³	—	2	1	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	4	no	—
316	C₂³	—	8	4	no	—
317	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
318	Q₈	—	2	1	no	—
319	Q₈	#124×C₆₇	1	2	yes	—
320	D₄	—	2	4	no	—
321	D₄	—	2	1	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	4	no	—
326	D₄	—	8	4	no	—
327	D₄	#124×C₆₇	1	2	yes	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	#121×C₆₇	1	2	yes	—
336	C₂×C₄	#121×C₆₇	1	2	yes	—
337	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
340	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
341	C₈	—	2	2	no	—
342	C₈	—	2	4	no	—
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	8	4	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	4	no	—
352	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
353	C₂²	#529×C₆₇	1	2	yes	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	#525×C₆₇	1	2	yes	—
360	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
361	C₄	#532×C₆₇	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	2	no	—
364	C₂	—	16	2	no	—
365	C₂	C₆₇×#2850	1	1	yes	characteristic
366	C₂	C₆₇×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#490×C₆₇	C₂	semidirect
2. C₆₇×#3080	C₂	semidirect
4. C₆₇×#2875	C₂	semidirect
5. C₆₇×#1998	C₂	semidirect
6. C₆₇×#1965	C₂	semidirect
7. C₆₇×#1931	C₂	semidirect
8. C₆₇×#1907	C₂	semidirect
9. C₆₇×#1838	C₂	semidirect
178. #33064	C₆₇	direct (N × H)
184. C₂×#490	C₂×C₆₇	semidirect
185. #3080	C₂×C₆₇	semidirect
187. #2875	C₂×C₆₇	semidirect
188. #1998	C₂×C₆₇	semidirect
189. #1965	C₂×C₆₇	semidirect
190. #1931	C₂×C₆₇	semidirect
191. #1907	C₂×C₆₇	semidirect
192. #1838	C₂×C₆₇	semidirect
193. C₆₇	#33064	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×#490×C₆₇	2
C₆₇×#3080	2
C₆₇×#3038	2
C₆₇×#2875	2
C₆₇×#1998	2
C₆₇×#1965	2
C₆₇×#1931	2
C₆₇×#1907	2
C₆₇×#1838	2
#33064	67

Subgroups — C67×#33064 #30712315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33064 #30712315