Subgroups — C₆₇×#20470 #30699721

Order 17152.

958 conjugacy classes of subgroups.

The subgroup lattice has 318 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	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1767	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1750	C₂	1	4	yes	maximal
7	C₆₇×#1746	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1726	C₂	1	2	yes	maximal, complemented
9	C₂×#117×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	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
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	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
26	#376×C₆₇	C₂²	1	1	yes	characteristic
27	#376×C₆₇	C₂²	1	1	yes	characteristic
28	C₄³×C₆₇	C₂²	1	2	yes	—
29	#336×C₆₇	C₂²	1	4	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	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	#117×C₆₇	—	2	4	no	—
38	#117×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
53	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
55	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
56	#97×C₆₇	—	2	8	no	—
57	#97×C₆₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₆₇	—	2	8	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	8	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	4	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₆₇	—	2	4	no	—
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	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
88	C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₄²×C₆₇	—	2	8	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	4	4	no	—
94	C₄²×C₆₇	C₄²	1	4	yes	—
95	C₄²×C₆₇	C₄²	1	4	yes	—
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	#41	1	4	yes	—
101	C₄²×C₆₇	#41	1	4	yes	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
106	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	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	4	no	—
116	C₂×C₄×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₆₇	#117	1	8	yes	—
127	C₂×C₄×C₆₇	#131	1	4	yes	—
128	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
132	C₈×C₆₇	—	2	4	no	—
133	C₈×C₆₇	—	2	8	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#117	1	2	yes	—
141	C₂²×C₆₇	C₂×#131	1	2	yes	—
142	C₂²×C₆₇	#568	1	2	yes	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	8	no	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	4	4	no	—
149	C₄×C₆₇	#405	1	4	yes	—
150	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
151	C₄×C₆₇	C₄×#41	1	2	yes	—
152	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
153	#20470	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₆₇	—	4	2	no	—
155	C₂×C₆₇	C₄×#117	1	1	yes	characteristic
156	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
157	C₂×C₆₇	#2873	1	1	yes	characteristic
158	C₂×C₆₇	#2925	1	4	yes	—
159	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
160	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
161	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
162	#1767	C₂×C₆₇	1	2	yes	complemented
163	C₄×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
164	#1750	C₂×C₆₇	1	4	yes	—
165	#1746	C₂×C₆₇	1	2	yes	complemented
166	#1726	C₂×C₆₇	1	2	yes	complemented
167	C₆₇	#20470	1	1	yes	characteristic, direct product factor, Sylow
168	C₂×#117	C₂²×C₆₇	1	2	yes	—
169	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
170	#432	C₄×C₆₇	1	8	yes	complemented
171	#431	C₄×C₆₇	1	8	yes	complemented
172	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
175	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
176	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
177	C₂×#97	C₂²×C₆₇	1	4	yes	—
178	C₂×#97	C₂²×C₆₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
180	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
181	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
182	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
183	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
184	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
185	#376	C₂²×C₆₇	1	1	yes	characteristic
186	#376	C₂²×C₆₇	1	1	yes	characteristic
187	C₄³	C₂²×C₆₇	1	2	yes	—
188	#336	C₂²×C₆₇	1	4	yes	—
189	#336	C₂²×C₆₇	1	4	yes	—
190	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
191	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
195	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
200	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
202	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
205	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
206	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
207	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
210	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
211	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
212	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
213	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
214	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
215	#97	—	2	8	no	—
216	#97	C₂×C₄×C₆₇	1	8	yes	—
217	C₄×C₈	—	2	8	no	—
218	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
219	#95	C₂×C₄×C₆₇	1	2	yes	—
220	#95	C₂×C₄×C₆₇	1	2	yes	—
221	#95	C₂×C₄×C₆₇	1	2	yes	—
222	#95	C₂×C₄×C₆₇	1	2	yes	—
223	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
229	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
245	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
246	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₄²×C₆₇	1	4	yes	—
254	C₄²	C₄²×C₆₇	1	4	yes	—
255	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
256	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
257	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
258	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
259	C₄²	#41×C₆₇	1	4	yes	—
260	C₄²	#41×C₆₇	1	4	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
265	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
281	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
282	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
283	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
284	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
285	C₂×C₄	#117×C₆₇	1	8	yes	—
286	C₂×C₄	#131×C₆₇	1	4	yes	—
287	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
290	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
299	C₂²	C₂×#117×C₆₇	1	2	yes	—
300	C₂²	C₂×#131×C₆₇	1	2	yes	—
301	C₂²	#568×C₆₇	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	8	no	—
307	C₄	—	4	4	no	—
308	C₄	#405×C₆₇	1	4	yes	—
309	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
310	C₄	C₄×#41×C₆₇	1	2	yes	—
311	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
312	C₂	—	4	2	no	—
313	C₂	C₄×#117×C₆₇	1	1	yes	characteristic
314	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
315	C₂	C₆₇×#2873	1	1	yes	characteristic
316	C₂	C₆₇×#2925	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×#1767	C₂	semidirect
5. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
7. C₆₇×#1746	C₂	semidirect
8. C₆₇×#1726	C₂	semidirect
11. #432×C₆₇	C₄	semidirect
12. #431×C₆₇	C₄	semidirect
153. #20470	C₆₇	direct (N × H)
162. #1767	C₂×C₆₇	semidirect
163. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
165. #1746	C₂×C₆₇	semidirect
166. #1726	C₂×C₆₇	semidirect
167. C₆₇	#20470	direct (N × H)
170. #432	C₄×C₆₇	semidirect
171. #431	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
C₄×#117×C₆₇	2
C₆₇×#1767	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1750	2
C₆₇×#1746	2
C₆₇×#1726	2
#20470	67

Subgroups — C67×#20470 #30699721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20470 #30699721