Subgroups — C₆₇×#20558 #30699809

Order 17152.

1146 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal
2	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1770	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1752	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1749	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#336×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
13	C₈×D₄×C₆₇	C₄	1	16	yes	complemented
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	4	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	1	yes	characteristic
27	#336×C₆₇	C₂²	1	2	yes	—
28	#336×C₆₇	C₂²	1	4	yes	—
29	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	—	4	4	no	—
33	C₂²×C₈×C₆₇	—	4	4	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
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	C₄×D₄×C₆₇	—	2	8	no	—
39	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	—	4	4	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂²⋊C₈×C₆₇	C₂×C₄	1	16	yes	complemented
54	#97×C₆₇	—	4	4	no	—
55	#97×C₆₇	—	4	2	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	4	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₆₇	C₄²	1	4	yes	—
61	C₂×D₄×C₆₇	M₁₆	1	8	yes	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	8	no	—
66	C₂²×C₄×C₆₇	—	2	8	no	—
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	4	16	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	C₄²	1	4	yes	—
88	C₂²⋊C₄×C₆₇	—	2	8	no	—
89	C₂²⋊C₄×C₆₇	C₄²	1	8	yes	—
90	C₄²×C₆₇	—	2	4	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	4	4	no	—
97	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	4	no	—
101	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
102	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
103	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	D₄×C₆₇	#97	1	16	yes	complemented
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	8	no	—
111	C₂×C₄×C₆₇	—	2	8	no	—
112	C₂×C₄×C₆₇	—	2	4	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	2	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	4	16	no	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₆₇	#131	1	4	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	—	2	8	no	—
130	C₈×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	—	2	8	no	—
133	C₂²×C₆₇	—	2	8	no	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	C₂×#97	1	8	yes	—
136	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
137	C₂²×C₆₇	#518	1	1	yes	characteristic
138	C₂²×C₆₇	#520	1	1	yes	characteristic
139	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
140	C₂²×C₆₇	C₂×#131	1	2	yes	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	2	4	no	—
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	4	4	no	—
145	C₄×C₆₇	—	4	2	no	—
146	C₄×C₆₇	C₂×#97	1	4	yes	—
147	C₄×C₆₇	#518	1	2	yes	—
148	C₄×C₆₇	#520	1	2	yes	—
149	#20558	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₆₇	—	2	8	no	—
151	C₂×C₆₇	#2275	1	1	yes	characteristic
152	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
153	C₂×C₆₇	C₄×#131	1	1	yes	characteristic
154	C₂×C₆₇	#2974	1	2	yes	—
155	C₂×C₆₇	#2990	1	2	yes	—
156	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	—
157	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
158	#1770	C₂×C₆₇	1	2	yes	complemented
159	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
160	#1752	C₂×C₆₇	1	4	yes	complemented
161	#1749	C₂×C₆₇	1	2	yes	complemented
162	#1744	C₂×C₆₇	1	1	yes	characteristic, complemented
163	C₂×#336	C₂×C₆₇	1	2	yes	complemented
164	C₆₇	#20558	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
166	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
167	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
169	C₈×D₄	C₄×C₆₇	1	16	yes	complemented
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
172	C₂×#97	C₂²×C₆₇	1	4	yes	—
173	C₂×#97	C₂²×C₆₇	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
177	C₄×C₂²⋊C₄	C₄×C₆₇	1	4	yes	—
178	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₄³	C₂²×C₆₇	1	1	yes	characteristic
180	#336	C₂²×C₆₇	1	1	yes	characteristic
181	#336	C₂²×C₆₇	1	2	yes	—
182	#336	C₂²×C₆₇	1	4	yes	—
183	#336	C₂²×C₆₇	1	1	yes	characteristic
184	#336	C₂²×C₆₇	1	2	yes	—
185	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
187	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
188	C₂²×C₈	—	4	4	no	—
189	C₂²×C₈	—	4	4	no	—
190	C₂²×C₈	C₂×C₄×C₆₇	1	16	yes	complemented
191	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
193	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
194	C₄×D₄	—	2	8	no	—
195	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
200	C₂×C₄²	—	4	4	no	—
201	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
204	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
207	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
208	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
209	C₂²⋊C₈	C₂×C₄×C₆₇	1	16	yes	complemented
210	#97	—	4	2	no	—
211	#97	—	4	4	no	—
212	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
213	#95	C₂×C₄×C₆₇	1	2	yes	—
214	#95	C₂×C₄×C₆₇	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
216	C₂×D₄	C₄²×C₆₇	1	4	yes	—
217	C₂×D₄	M₁₆×C₆₇	1	8	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂²×C₄×C₆₇	1	2	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	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
229	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	16	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	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄²×C₆₇	1	4	yes	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	C₄²×C₆₇	1	8	yes	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	C₂×C₄²×C₆₇	1	4	yes	—
258	C₂³	C₂×M₁₆×C₆₇	1	8	yes	—
259	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
260	D₄	#97×C₆₇	1	16	yes	complemented
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
277	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
279	C₂×C₄	#131×C₆₇	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
281	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
282	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
283	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	8	no	—
286	C₈	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	8	no	—
291	C₂²	C₂×#97×C₆₇	1	8	yes	—
292	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
293	C₂²	#518×C₆₇	1	1	yes	characteristic
294	C₂²	#520×C₆₇	1	1	yes	characteristic
295	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
296	C₂²	C₂×#131×C₆₇	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	2	no	—
302	C₄	C₂×#97×C₆₇	1	4	yes	—
303	C₄	#518×C₆₇	1	2	yes	—
304	C₄	#520×C₆₇	1	2	yes	—
305	C₂	—	2	8	no	—
306	C₂	C₆₇×#2275	1	1	yes	characteristic
307	C₂	C₂²×#97×C₆₇	1	1	yes	characteristic
308	C₂	C₄×#131×C₆₇	1	1	yes	characteristic
309	C₂	C₆₇×#2974	1	2	yes	—
310	C₂	C₆₇×#2990	1	2	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	C₂²×D₄×C₆₇	C₂×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
3. C₆₇×#1770	C₂	semidirect
4. C₆₇×#1767	C₂	semidirect
5. C₆₇×#1752	C₂	semidirect
6. C₆₇×#1749	C₂	semidirect
7. C₆₇×#1744	C₂	semidirect
8. C₂×#336×C₆₇	C₂	semidirect
13. C₈×D₄×C₆₇	C₄	semidirect
34. C₂²×C₈×C₆₇	C₂×C₄	semidirect
52. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
53. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
56. C₄×C₈×C₆₇	C₂×C₄	semidirect
104. D₄×C₆₇	#97	semidirect
149. #20558	C₆₇	direct (N × H)
158. #1770	C₂×C₆₇	semidirect
159. #1767	C₂×C₆₇	semidirect
160. #1752	C₂×C₆₇	semidirect
161. #1749	C₂×C₆₇	semidirect
162. #1744	C₂×C₆₇	semidirect
163. C₂×#336	C₂×C₆₇	semidirect
164. C₆₇	#20558	direct (N × H)
169. C₈×D₄	C₄×C₆₇	semidirect
190. C₂²×C₈	C₂×C₄×C₆₇	semidirect
208. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
209. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
212. C₄×C₈	C₂×C₄×C₆₇	semidirect
260. D₄	#97×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₇	2
C₄²×D₄×C₆₇	2
C₆₇×#1770	2
C₆₇×#1767	2
C₆₇×#1752	2
C₆₇×#1749	2
C₆₇×#1744	2
C₂×#336×C₆₇	2
#20558	67

Subgroups — C67×#20558 #30699809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20558 #30699809