Subgroups — C₂×C₆₇×#1560 #30696986

Order 17152.

1322 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1560	C₂	1	8	yes	maximal, direct product factor
4	C₂×#333×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₄	1	2	yes	—
7	C₂²×M₁₆×C₆₇	—	2	2	no	—
8	#532×C₆₇	—	2	4	no	—
9	#532×C₆₇	C₄	1	4	yes	—
10	#532×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	C₄	1	2	yes	—
14	C₂×#121×C₆₇	—	2	2	no	—
15	C₂×C₄×D₄×C₆₇	—	2	2	no	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	—	2	4	no	—
18	#423×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
21	#333×C₆₇	C₂²	1	8	yes	complemented
22	#331×C₆₇	C₂²	1	8	yes	complemented
23	C₂×#41×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	—	4	8	no	—
25	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	2	no	—
29	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	2	yes	—
33	C₂×M₁₆×C₆₇	—	2	8	no	—
34	C₂×M₁₆×C₆₇	—	4	8	no	—
35	C₂²×C₈×C₆₇	—	2	4	no	—
36	C₄⋊Q₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
39	#127×C₆₇	—	2	2	no	—
40	#127×C₆₇	C₂×C₄	1	2	yes	—
41	#127×C₆₇	C₂³	1	2	yes	—
42	#124×C₆₇	—	2	4	no	—
43	#124×C₆₇	C₂×C₄	1	8	yes	—
44	#121×C₆₇	—	2	8	no	—
45	#121×C₆₇	—	2	8	no	—
46	C₄×D₄×C₆₇	—	2	8	no	—
47	C₄×D₄×C₆₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
56	C₄⋊C₈×C₆₇	—	2	8	no	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
59	C₂⁴×C₆₇	—	2	2	no	—
60	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	#41×C₆₇	—	4	16	no	—
62	#41×C₆₇	—	4	16	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	—	4	4	no	—
65	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂×D₄×C₆₇	—	2	8	no	—
67	C₂×D₄×C₆₇	—	2	4	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	—	2	4	no	—
70	C₂×D₄×C₆₇	—	4	4	no	—
71	C₂×D₄×C₆₇	—	4	8	no	—
72	C₂×D₄×C₆₇	—	4	4	no	—
73	C₂×D₄×C₆₇	—	4	4	no	—
74	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	8	no	—
80	C₂²×C₄×C₆₇	—	4	8	no	—
81	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	M₁₆×C₆₇	—	4	16	no	—
90	C₂×C₈×C₆₇	—	2	8	no	—
91	C₂×C₈×C₆₇	—	4	8	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₂²⋊C₄×C₆₇	—	2	16	no	—
95	C₄²×C₆₇	C₂²⋊C₄	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	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	4	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	4	no	—
104	C₂³×C₆₇	—	2	4	no	—
105	C₂³×C₆₇	—	4	4	no	—
106	C₂³×C₆₇	—	4	8	no	—
107	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	4	4	no	—
110	Q₈×C₆₇	—	4	4	no	—
111	D₄×C₆₇	—	4	4	no	—
112	D₄×C₆₇	—	4	4	no	—
113	D₄×C₆₇	—	4	16	no	—
114	D₄×C₆₇	—	4	4	no	—
115	D₄×C₆₇	—	4	4	no	—
116	D₄×C₆₇	—	4	8	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	4	no	—
123	C₂×C₄×C₆₇	—	2	8	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	16	no	—
127	C₂×C₄×C₆₇	#100	1	4	yes	—
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
133	C₂×C₄×C₆₇	#137	1	4	yes	—
134	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
136	C₈×C₆₇	—	4	8	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	4	4	no	—
140	C₂²×C₆₇	—	4	4	no	—
141	C₂²×C₆₇	—	4	8	no	—
142	C₂²×C₆₇	—	8	8	no	—
143	C₂²×C₆₇	C₂×#100	1	2	yes	—
144	C₂²×C₆₇	#417	1	4	yes	—
145	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	4	4	no	—
150	C₄×C₆₇	C₂×#100	1	4	yes	—
151	C₄×C₆₇	#417	1	4	yes	—
152	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
153	C₄×C₆₇	C₂×#137	1	2	yes	—
154	C₂×#1560	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₇	—	4	2	no	—
156	C₂×C₆₇	—	8	4	no	—
157	C₂×C₆₇	#1560	1	4	yes	direct product factor
158	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#417	1	2	yes	—
160	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
161	C₂×#423	C₂×C₆₇	1	2	yes	complemented
162	#1560	C₂×C₆₇	1	8	yes	direct product factor
163	C₂×#333	C₂×C₆₇	1	2	yes	complemented
164	C₂×#331	C₂×C₆₇	1	2	yes	complemented
165	C₆₇	C₂×#1560	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×#41	C₄×C₆₇	1	2	yes	—
167	C₂²×M₁₆	—	2	2	no	—
168	#532	—	2	4	no	—
169	#532	C₄×C₆₇	1	4	yes	—
170	#532	C₂²×C₆₇	1	4	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×#124	C₄×C₆₇	1	2	yes	—
174	C₂×#121	—	2	2	no	—
175	C₂×C₄×D₄	—	2	2	no	—
176	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
177	#423	—	2	4	no	—
178	#423	C₂²×C₆₇	1	8	yes	complemented
179	C₂×C₄⋊C₈	—	2	2	no	—
180	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
181	#333	C₂²×C₆₇	1	8	yes	complemented
182	#331	C₂²×C₆₇	1	8	yes	complemented
183	C₂×#41	—	2	4	no	—
184	C₂×#41	—	4	8	no	—
185	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
186	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
192	C₂³×C₄	D₄×C₆₇	1	2	yes	—
193	C₂×M₁₆	—	2	8	no	—
194	C₂×M₁₆	—	4	8	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
199	#127	—	2	2	no	—
200	#127	C₂×C₄×C₆₇	1	2	yes	—
201	#127	C₂³×C₆₇	1	2	yes	—
202	#124	—	2	4	no	—
203	#124	C₂×C₄×C₆₇	1	8	yes	—
204	#121	—	2	8	no	—
205	#121	—	2	8	no	—
206	C₄×D₄	—	2	8	no	—
207	C₄×D₄	—	2	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
212	C₂×C₄²	D₄×C₆₇	1	2	yes	—
213	C₂×C₄²	D₄×C₆₇	1	4	yes	—
214	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
216	C₄⋊C₈	—	2	8	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
221	#41	—	4	16	no	—
222	#41	—	4	16	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
226	C₂×D₄	—	2	8	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	4	8	no	—
241	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
244	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
249	M₁₆	—	4	16	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	—	4	8	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	16	no	—
255	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
256	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
257	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
258	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₆₇	1	4	yes	—
260	C₄²	C₂×D₄×C₆₇	1	2	yes	—
261	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	8	no	—
267	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
268	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	D₄	—	4	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	16	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	#100×C₆₇	1	4	yes	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
292	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
293	C₂×C₄	#137×C₆₇	1	4	yes	—
294	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
296	C₈	—	4	8	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	8	8	no	—
303	C₂²	C₂×#100×C₆₇	1	2	yes	—
304	C₂²	#417×C₆₇	1	4	yes	—
305	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	4	no	—
309	C₄	—	4	4	no	—
310	C₄	C₂×#100×C₆₇	1	4	yes	—
311	C₄	#417×C₆₇	1	4	yes	—
312	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
313	C₄	C₂×#137×C₆₇	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	—	8	4	no	—
316	C₂	C₆₇×#1560	1	4	yes	direct product factor
317	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
318	C₂	C₂×#417×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#423×C₆₇	C₂	semidirect
3. C₆₇×#1560	C₂	direct (N × H)
4. C₂×#333×C₆₇	C₂	semidirect
5. C₂×#331×C₆₇	C₂	semidirect
18. #423×C₆₇	C₂²	semidirect
21. #333×C₆₇	C₂²	semidirect
22. #331×C₆₇	C₂²	semidirect
154. C₂×#1560	C₆₇	direct (N × H)
157. C₂×C₆₇	#1560	direct (N × H)
161. C₂×#423	C₂×C₆₇	semidirect
162. #1560	C₂×C₆₇	direct (N × H)
163. C₂×#333	C₂×C₆₇	semidirect
164. C₂×#331	C₂×C₆₇	semidirect
165. C₆₇	C₂×#1560	direct (N × H)
178. #423	C₂²×C₆₇	semidirect
181. #333	C₂²×C₆₇	semidirect
182. #331	C₂²×C₆₇	semidirect
316. C₂	C₆₇×#1560	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×#532×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#1560	2
C₂×#333×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1560	67

Subgroups — C2×C67×#1560 #30696986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1560 #30696986