Subgroups — C₆₇×#20779 #30700030

Order 17152.

802 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	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1840	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1832	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1817	C₂	1	2	yes	maximal
5	C₆₇×#1797	C₂	1	2	yes	maximal
6	C₆₇×#1768	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1746	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#376×C₆₇	C₄	1	2	yes	—
19	#376×C₆₇	C₄	1	2	yes	—
20	#376×C₆₇	C₂²	1	1	yes	characteristic
21	#376×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	C₂²	1	1	yes	characteristic
23	#376×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#95×C₆₇	C₄	1	2	yes	—
25	C₂×#95×C₆₇	C₄	1	2	yes	—
26	#336×C₆₇	C₂²	1	4	yes	—
27	#336×C₆₇	C₂²	1	4	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	#336×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₆₇	—	2	4	no	—
39	C₂×C₄²×C₆₇	—	2	4	no	—
40	C₂×C₄²×C₆₇	—	2	4	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	—	2	8	no	—
52	C₂²⋊C₈×C₆₇	—	2	4	no	—
53	C₂²⋊C₈×C₆₇	—	2	4	no	—
54	#97×C₆₇	—	2	4	no	—
55	C₄×C₈×C₆₇	—	2	4	no	—
56	#95×C₆₇	C₂×C₄	1	2	yes	—
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	#95×C₆₇	C₂×C₄	1	2	yes	—
60	#95×C₆₇	C₂×C₄	1	2	yes	—
61	#95×C₆₇	C₂×C₄	1	2	yes	—
62	#95×C₆₇	C₂×C₄	1	2	yes	—
63	#95×C₆₇	C₂×C₄	1	2	yes	—
64	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	#41	1	4	yes	—
103	C₄²×C₆₇	#41	1	4	yes	—
104	C₂³×C₆₇	—	2	1	no	—
105	C₂³×C₆₇	—	2	1	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	#117	1	2	yes	—
108	C₂³×C₆₇	#131	1	4	yes	—
109	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	4	8	no	—
132	C₂×C₄×C₆₇	—	4	8	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	#117	1	2	yes	—
136	C₂×C₄×C₆₇	#117	1	4	yes	—
137	C₂×C₄×C₆₇	#117	1	4	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₈×C₆₇	—	4	4	no	—
142	C₈×C₆₇	—	4	4	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	4	no	—
145	C₂²×C₆₇	—	2	4	no	—
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	#433	1	4	yes	—
148	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#131	1	2	yes	—
152	C₂²×C₆₇	#568	1	2	yes	—
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	2	4	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	4	1	no	—
157	C₄×C₆₇	—	4	1	no	—
158	C₄×C₆₇	—	4	1	no	—
159	C₄×C₆₇	—	4	1	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	#529	1	4	yes	—
163	#20779	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	2	4	no	—
165	C₂×C₆₇	#2283	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
167	C₂×C₆₇	#2921	1	1	yes	characteristic
168	C₂×C₆₇	#2924	1	2	yes	—
169	C₂×C₆₇	#2924	1	2	yes	—
170	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
171	#1840	C₂×C₆₇	1	2	yes	complemented
172	#1832	C₂×C₆₇	1	2	yes	complemented
173	#1817	C₂×C₆₇	1	2	yes	—
174	#1797	C₂×C₆₇	1	2	yes	—
175	#1768	C₂×C₆₇	1	2	yes	complemented
176	#1749	C₂×C₆₇	1	1	yes	characteristic
177	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
178	#1746	C₂×C₆₇	1	2	yes	complemented
179	C₆₇	#20779	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
181	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
182	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
184	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
185	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
186	C₂×#97	C₂²×C₆₇	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
188	#376	C₄×C₆₇	1	2	yes	—
189	#376	C₄×C₆₇	1	2	yes	—
190	#376	C₂²×C₆₇	1	1	yes	characteristic
191	#376	C₂²×C₆₇	1	1	yes	characteristic
192	#376	C₂²×C₆₇	1	1	yes	characteristic
193	#376	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×#95	C₄×C₆₇	1	2	yes	—
195	C₂×#95	C₄×C₆₇	1	2	yes	—
196	#336	C₂²×C₆₇	1	4	yes	—
197	#336	C₂²×C₆₇	1	4	yes	—
198	#336	C₂²×C₆₇	1	4	yes	—
199	#336	C₂²×C₆₇	1	2	yes	—
200	#336	C₂²×C₆₇	1	2	yes	—
201	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
202	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
203	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
204	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
205	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
207	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
221	C₄⋊C₈	—	2	8	no	—
222	C₂²⋊C₈	—	2	4	no	—
223	C₂²⋊C₈	—	2	4	no	—
224	#97	—	2	4	no	—
225	C₄×C₈	—	2	4	no	—
226	#95	C₂×C₄×C₆₇	1	2	yes	—
227	#95	C₂×C₄×C₆₇	1	2	yes	—
228	#95	C₂×C₄×C₆₇	1	2	yes	—
229	#95	C₂×C₄×C₆₇	1	2	yes	—
230	#95	C₂×C₄×C₆₇	1	2	yes	—
231	#95	C₂×C₄×C₆₇	1	2	yes	—
232	#95	C₂×C₄×C₆₇	1	2	yes	—
233	#95	C₂×C₄×C₆₇	1	2	yes	—
234	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	#41×C₆₇	1	4	yes	—
257	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	#41×C₆₇	1	4	yes	—
273	C₄²	#41×C₆₇	1	4	yes	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	2	no	—
277	C₂³	#117×C₆₇	1	2	yes	—
278	C₂³	#131×C₆₇	1	4	yes	—
279	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	#117×C₆₇	1	2	yes	—
306	C₂×C₄	#117×C₆₇	1	4	yes	—
307	C₂×C₄	#117×C₆₇	1	4	yes	—
308	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
309	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
310	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₈	—	4	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	#433×C₆₇	1	4	yes	—
318	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
319	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
320	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
321	C₂²	C₂×#131×C₆₇	1	2	yes	—
322	C₂²	#568×C₆₇	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	#529×C₆₇	1	4	yes	—
333	C₂	—	2	4	no	—
334	C₂	C₆₇×#2283	1	1	yes	characteristic
335	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
336	C₂	C₆₇×#2921	1	1	yes	characteristic
337	C₂	C₆₇×#2924	1	2	yes	—
338	C₂	C₆₇×#2924	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1840	C₂	semidirect
3. C₆₇×#1832	C₂	semidirect
6. C₆₇×#1768	C₂	semidirect
9. C₆₇×#1746	C₂	semidirect
163. #20779	C₆₇	direct (N × H)
171. #1840	C₂×C₆₇	semidirect
172. #1832	C₂×C₆₇	semidirect
175. #1768	C₂×C₆₇	semidirect
178. #1746	C₂×C₆₇	semidirect
179. C₆₇	#20779	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×#376×C₆₇	2
C₆₇×#1840	2
C₆₇×#1832	2
C₆₇×#1817	2
C₆₇×#1797	2
C₆₇×#1768	2
C₆₇×#1749	2
C₄×C₂²⋊C₈×C₆₇	2
C₆₇×#1746	2
#20779	67

Subgroups — C67×#20779 #30700030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20779 #30700030