Subgroups — C₆₇×#20469 #30699720

Order 17152.

1002 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1768	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1749	C₂	1	2	yes	maximal
6	C₄×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
7	C₆₇×#1746	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#432×C₆₇	C₄	1	8	yes	complemented
12	#431×C₆₇	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₆₇	C₂²	1	4	yes	—
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#376×C₆₇	C₄	1	2	yes	—
23	#376×C₆₇	C₄	1	2	yes	—
24	#376×C₆₇	C₂²	1	1	yes	characteristic
25	#376×C₆₇	C₂²	1	2	yes	—
26	#376×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#95×C₆₇	C₄	1	2	yes	—
28	C₂×#95×C₆₇	C₄	1	2	yes	—
29	#336×C₆₇	C₂²	1	4	yes	—
30	#336×C₆₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₇	—	2	8	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	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	2	yes	—
48	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
49	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
50	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
53	#97×C₆₇	—	2	8	no	—
54	#97×C₆₇	C₂×C₄	1	8	yes	complemented
55	C₄×C₈×C₆₇	—	2	8	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
57	#95×C₆₇	C₂×C₄	1	2	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	#95×C₆₇	C₂×C₄	1	4	yes	—
60	#95×C₆₇	C₂×C₄	1	4	yes	—
61	#95×C₆₇	C₂×C₄	1	2	yes	—
62	#95×C₆₇	C₂×C₄	1	2	yes	—
63	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	8	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
89	C₄²×C₆₇	—	2	8	no	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	C₄²	1	4	yes	—
92	C₄²×C₆₇	C₄²	1	4	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	#41	1	8	yes	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
101	C₂³×C₆₇	#131	1	4	yes	—
102	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	8	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	16	no	—
118	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₆₇	#117	1	8	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
127	C₈×C₆₇	—	2	4	no	—
128	C₈×C₆₇	—	2	8	no	—
129	C₈×C₆₇	—	2	4	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	8	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	#405	1	4	yes	—
134	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#117	1	2	yes	—
136	C₂²×C₆₇	C₂×#131	1	2	yes	—
137	C₂²×C₆₇	#568	1	2	yes	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	8	no	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	4	4	no	—
145	C₄×C₆₇	C₄×#41	1	2	yes	—
146	C₄×C₆₇	C₄×#41	1	2	yes	—
147	#20469	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₆₇	—	2	4	no	—
149	C₂×C₆₇	C₄×#117	1	1	yes	characteristic
150	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
151	C₂×C₆₇	#2873	1	1	yes	characteristic
152	C₂×C₆₇	#2924	1	4	yes	—
153	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
154	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
155	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
156	#1768	C₂×C₆₇	1	4	yes	complemented
157	#1749	C₂×C₆₇	1	2	yes	—
158	C₄×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
159	#1746	C₂×C₆₇	1	2	yes	complemented
160	#1726	C₂×C₆₇	1	2	yes	complemented
161	C₆₇	#20469	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
163	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
164	#432	C₄×C₆₇	1	8	yes	complemented
165	#431	C₄×C₆₇	1	8	yes	complemented
166	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
171	C₂×#97	C₂²×C₆₇	1	4	yes	—
172	C₂×#97	C₂²×C₆₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
174	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
175	#376	C₄×C₆₇	1	2	yes	—
176	#376	C₄×C₆₇	1	2	yes	—
177	#376	C₂²×C₆₇	1	1	yes	characteristic
178	#376	C₂²×C₆₇	1	1	yes	characteristic
179	#376	C₂²×C₆₇	1	2	yes	—
180	C₂×#95	C₄×C₆₇	1	2	yes	—
181	C₂×#95	C₄×C₆₇	1	2	yes	—
182	#336	C₂²×C₆₇	1	4	yes	—
183	#336	C₂²×C₆₇	1	4	yes	—
184	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
185	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
189	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
190	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
191	C₂×C₄²	—	2	8	no	—
192	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
197	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
198	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
202	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
203	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
204	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
205	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
206	#97	—	2	8	no	—
207	#97	C₂×C₄×C₆₇	1	8	yes	complemented
208	C₄×C₈	—	2	8	no	—
209	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
210	#95	C₂×C₄×C₆₇	1	2	yes	—
211	#95	C₂×C₄×C₆₇	1	2	yes	—
212	#95	C₂×C₄×C₆₇	1	4	yes	—
213	#95	C₂×C₄×C₆₇	1	4	yes	—
214	#95	C₂×C₄×C₆₇	1	2	yes	—
215	#95	C₂×C₄×C₆₇	1	2	yes	—
216	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
224	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₄²×C₆₇	1	4	yes	—
245	C₄²	C₄²×C₆₇	1	4	yes	—
246	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
247	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
248	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
249	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
250	C₄²	#41×C₆₇	1	8	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
254	C₂³	#131×C₆₇	1	4	yes	—
255	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	16	no	—
271	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
274	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
276	C₂×C₄	#117×C₆₇	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	8	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	8	no	—
286	C₂²	#405×C₆₇	1	4	yes	—
287	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
288	C₂²	C₂×#117×C₆₇	1	2	yes	—
289	C₂²	C₂×#131×C₆₇	1	2	yes	—
290	C₂²	#568×C₆₇	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	4	no	—
298	C₄	C₄×#41×C₆₇	1	2	yes	—
299	C₄	C₄×#41×C₆₇	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₄×#117×C₆₇	1	1	yes	characteristic
302	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
303	C₂	C₆₇×#2873	1	1	yes	characteristic
304	C₂	C₆₇×#2924	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1768	C₂	semidirect
7. C₆₇×#1746	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
11. #432×C₆₇	C₄	semidirect
12. #431×C₆₇	C₄	semidirect
48. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
49. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
54. #97×C₆₇	C₂×C₄	semidirect
56. C₄×C₈×C₆₇	C₂×C₄	semidirect
147. #20469	C₆₇	direct (N × H)
156. #1768	C₂×C₆₇	semidirect
159. #1746	C₂×C₆₇	semidirect
160. #1726	C₂×C₆₇	semidirect
161. C₆₇	#20469	direct (N × H)
164. #432	C₄×C₆₇	semidirect
165. #431	C₄×C₆₇	semidirect
201. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
202. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
207. #97	C₂×C₄×C₆₇	semidirect
209. C₄×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₂×#376×C₆₇	2
C₆₇×#1768	2
C₆₇×#1749	2
C₄×C₂²⋊C₈×C₆₇	2
C₆₇×#1746	2
C₆₇×#1726	2
#20469	67

Subgroups — C67×#20469 #30699720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20469 #30699720