Subgroups — C₆₇×#33443 #30712694

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3407	C₂	1	2	yes	maximal, complemented
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2171	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2170	C₂	1	4	yes	maximal, complemented
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2117	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
9	C₂×#131×C₆₇	C₂²	1	2	yes	—
10	#504×C₆₇	—	2	4	no	—
11	#504×C₆₇	C₂²	1	8	yes	complemented
12	C₂×M₃₂×C₆₇	—	2	4	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
16	#443×C₆₇	C₄	1	2	yes	—
17	#443×C₆₇	C₄	1	2	yes	—
18	#443×C₆₇	C₂²	1	2	yes	—
19	#443×C₆₇	C₂²	1	2	yes	—
20	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#104×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#364×C₆₇	C₂²	1	2	yes	complemented
24	#364×C₆₇	C₂²	1	2	yes	complemented
25	#350×C₆₇	C₂²	1	8	yes	complemented
26	#346×C₆₇	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
29	#135×C₆₇	C₈	1	2	yes	—
30	#135×C₆₇	C₈	1	2	yes	—
31	#135×C₆₇	C₂×C₄	1	2	yes	—
32	#135×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₆₇	C₈	1	1	yes	characteristic
34	C₂×SD₁₆×C₆₇	C₈	1	2	yes	—
35	C₂×D₈×C₆₇	C₈	1	1	yes	characteristic
36	#131×C₆₇	—	2	2	no	—
37	#131×C₆₇	—	2	2	no	—
38	#131×C₆₇	C₂×C₄	1	4	yes	—
39	#131×C₆₇	C₂³	1	4	yes	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₆₇	—	2	2	no	—
47	M₃₂×C₆₇	—	2	1	no	—
48	M₃₂×C₆₇	—	2	1	no	—
49	M₃₂×C₆₇	—	2	8	no	—
50	M₃₂×C₆₇	D₄	1	2	yes	—
51	M₃₂×C₆₇	D₄	1	2	yes	—
52	M₃₂×C₆₇	C₂³	1	4	yes	—
53	C₂×C₁₆×C₆₇	—	2	2	no	—
54	C₂×C₁₆×C₆₇	—	2	8	no	—
55	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
56	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
57	#108×C₆₇	C₂×C₄	1	2	yes	—
58	#108×C₆₇	C₂×C₄	1	2	yes	—
59	#104×C₆₇	C₂×C₄	1	4	yes	—
60	#104×C₆₇	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#41×C₆₇	—	2	2	no	—
66	#41×C₆₇	—	2	2	no	—
67	#41×C₆₇	C₂×C₈	1	4	yes	—
68	#41×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₆₇	—	2	2	no	—
71	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
76	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
77	SD₁₆×C₆₇	C₂×C₈	1	4	yes	—
78	SD₁₆×C₆₇	C₂×C₈	1	4	yes	—
79	D₈×C₆₇	C₂×C₈	1	2	yes	—
80	D₈×C₆₇	C₂×C₈	1	2	yes	—
81	M₁₆×C₆₇	—	2	2	no	—
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	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	#41	1	1	yes	characteristic
98	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₁₆×C₆₇	—	2	4	no	—
104	C₁₆×C₆₇	—	2	2	no	—
105	C₁₆×C₆₇	C₂×D₄	1	4	yes	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	#131	1	1	yes	characteristic
108	Q₈×C₆₇	—	2	2	no	—
109	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
110	D₄×C₆₇	—	2	4	no	—
111	D₄×C₆₇	—	2	4	no	—
112	D₄×C₆₇	—	2	2	no	—
113	D₄×C₆₇	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
128	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
129	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
132	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
133	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	—	8	2	no	—
138	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	4	1	no	—
144	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
145	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
146	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
148	#33443	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	4	4	no	—
153	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
154	#3407	C₂×C₆₇	1	2	yes	complemented
155	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
156	#2171	C₂×C₆₇	1	4	yes	complemented
157	#2170	C₂×C₆₇	1	4	yes	complemented
158	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
159	#2117	C₂×C₆₇	1	2	yes	complemented
160	C₄×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
161	C₆₇	#33443	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#135	C₄×C₆₇	1	1	yes	characteristic
163	C₂×#131	C₂²×C₆₇	1	2	yes	—
164	#504	—	2	4	no	—
165	#504	C₂²×C₆₇	1	8	yes	complemented
166	C₂×M₃₂	—	2	4	no	—
167	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
168	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
169	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
170	#443	C₄×C₆₇	1	2	yes	—
171	#443	C₄×C₆₇	1	2	yes	—
172	#443	C₂²×C₆₇	1	2	yes	—
173	#443	C₂²×C₆₇	1	2	yes	—
174	C₂×#108	C₄×C₆₇	1	1	yes	characteristic
175	C₂×#104	C₄×C₆₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
177	#364	C₂²×C₆₇	1	2	yes	complemented
178	#364	C₂²×C₆₇	1	2	yes	complemented
179	#350	C₂²×C₆₇	1	8	yes	complemented
180	#346	C₂²×C₆₇	1	2	yes	complemented
181	C₄×C₁₆	C₂²×C₆₇	1	2	yes	complemented
182	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
183	#135	C₈×C₆₇	1	2	yes	—
184	#135	C₈×C₆₇	1	2	yes	—
185	#135	C₂×C₄×C₆₇	1	2	yes	—
186	#135	C₂×C₄×C₆₇	1	2	yes	—
187	C₂×Q₁₆	C₈×C₆₇	1	1	yes	characteristic
188	C₂×SD₁₆	C₈×C₆₇	1	2	yes	—
189	C₂×D₈	C₈×C₆₇	1	1	yes	characteristic
190	#131	—	2	2	no	—
191	#131	—	2	2	no	—
192	#131	C₂×C₄×C₆₇	1	4	yes	—
193	#131	C₂³×C₆₇	1	4	yes	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
200	M₃₂	—	2	2	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	8	no	—
204	M₃₂	D₄×C₆₇	1	2	yes	—
205	M₃₂	D₄×C₆₇	1	2	yes	—
206	M₃₂	C₂³×C₆₇	1	4	yes	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	2	8	no	—
209	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
210	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
211	#108	C₂×C₄×C₆₇	1	2	yes	—
212	#108	C₂×C₄×C₆₇	1	2	yes	—
213	#104	C₂×C₄×C₆₇	1	4	yes	—
214	#104	C₂×C₄×C₆₇	1	4	yes	—
215	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	C₂×C₈×C₆₇	1	4	yes	—
222	#41	C₂²×C₄×C₆₇	1	4	yes	—
223	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
229	Q₁₆	C₂×C₈×C₆₇	1	2	yes	—
230	Q₁₆	C₂×C₈×C₆₇	1	2	yes	—
231	SD₁₆	C₂×C₈×C₆₇	1	4	yes	—
232	SD₁₆	C₂×C₈×C₆₇	1	4	yes	—
233	D₈	C₂×C₈×C₆₇	1	2	yes	—
234	D₈	C₂×C₈×C₆₇	1	2	yes	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
252	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
256	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
257	C₁₆	—	2	4	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	C₂×D₄×C₆₇	1	4	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	#131×C₆₇	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂²×C₈×C₆₇	1	4	yes	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
278	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
282	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
283	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
284	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
285	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
286	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
287	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
288	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
293	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
294	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	C₈×D₄×C₆₇	1	1	yes	characteristic
299	C₄	C₈×D₄×C₆₇	1	1	yes	characteristic
300	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
301	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	C₂×#135×C₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂×M₃₂×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3407	C₂	semidirect
3. C₆₇×#2171	C₂	semidirect
4. C₆₇×#2170	C₂	semidirect
5. C₂×#364×C₆₇	C₂	semidirect
6. C₆₇×#2117	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
11. #504×C₆₇	C₂²	semidirect
23. #364×C₆₇	C₂²	semidirect
24. #364×C₆₇	C₂²	semidirect
25. #350×C₆₇	C₂²	semidirect
26. #346×C₆₇	C₂²	semidirect
27. C₄×C₁₆×C₆₇	C₂²	semidirect
148. #33443	C₆₇	direct (N × H)
154. #3407	C₂×C₆₇	semidirect
156. #2171	C₂×C₆₇	semidirect
157. #2170	C₂×C₆₇	semidirect
158. C₂×#364	C₂×C₆₇	semidirect
159. #2117	C₂×C₆₇	semidirect
160. C₄×M₃₂	C₂×C₆₇	semidirect
161. C₆₇	#33443	direct (N × H)
165. #504	C₂²×C₆₇	semidirect
177. #364	C₂²×C₆₇	semidirect
178. #364	C₂²×C₆₇	semidirect
179. #350	C₂²×C₆₇	semidirect
180. #346	C₂²×C₆₇	semidirect
181. C₄×C₁₆	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×#3407	2
C₂×#443×C₆₇	2
C₆₇×#2171	2
C₆₇×#2170	2
C₂×#364×C₆₇	2
C₆₇×#2117	2
C₄×M₃₂×C₆₇	2
#33443	67

Subgroups — C67×#33443 #30712694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33443 #30712694