Subgroups — C₂×C₆₇×#2221 #30712865

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#472×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2221	C₂	1	8	yes	maximal, direct product factor
5	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#361×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
8	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×D₁₆×C₆₇	—	2	1	no	—
11	C₂×D₁₆×C₆₇	—	2	1	no	—
12	C₂×D₁₆×C₆₇	—	2	2	no	—
13	C₂×D₁₆×C₆₇	C₂²	1	4	yes	complemented
14	#504×C₆₇	—	2	2	no	—
15	#504×C₆₇	C₂²	1	4	yes	complemented
16	C₂×M₃₂×C₆₇	—	2	1	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₆₇	—	2	1	no	—
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#472×C₆₇	—	2	4	no	—
21	#472×C₆₇	C₂²	1	8	yes	complemented
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	C₂×#100×C₆₇	—	2	2	no	—
24	#368×C₆₇	C₂²	1	4	yes	complemented
25	#361×C₆₇	C₂²	1	8	yes	complemented
26	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
29	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
30	C₂×SD₁₆×C₆₇	—	4	2	no	—
31	C₂×D₈×C₆₇	—	2	2	no	—
32	C₂×D₈×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	—	2	2	no	—
34	C₂×D₈×C₆₇	—	2	2	no	—
35	C₂×D₈×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	—	4	2	no	—
37	C₂×D₈×C₆₇	C₂³	1	4	yes	—
38	#131×C₆₇	—	2	1	no	—
39	#131×C₆₇	—	2	1	no	—
40	#131×C₆₇	D₄	1	2	yes	—
41	#131×C₆₇	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₇	—	2	1	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	D₁₆×C₆₇	—	2	4	no	—
48	D₁₆×C₆₇	—	2	4	no	—
49	M₃₂×C₆₇	—	2	1	no	—
50	M₃₂×C₆₇	—	2	4	no	—
51	M₃₂×C₆₇	C₂³	1	2	yes	—
52	C₂×C₁₆×C₆₇	—	2	1	no	—
53	C₂×C₁₆×C₆₇	—	2	4	no	—
54	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
57	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
58	#108×C₆₇	—	2	2	no	—
59	#108×C₆₇	C₂³	1	4	yes	—
60	#100×C₆₇	—	2	8	no	—
61	C₂⁴×C₆₇	—	4	2	no	—
62	#41×C₆₇	—	2	1	no	—
63	#41×C₆₇	—	2	1	no	—
64	#41×C₆₇	D₈	1	2	yes	—
65	#41×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
67	C₂×D₄×C₆₇	—	2	1	no	—
68	C₂×D₄×C₆₇	—	2	4	no	—
69	C₂×D₄×C₆₇	—	4	2	no	—
70	C₂×D₄×C₆₇	—	4	2	no	—
71	C₂×D₄×C₆₇	—	4	2	no	—
72	C₂×D₄×C₆₇	—	4	2	no	—
73	C₂×D₄×C₆₇	—	4	2	no	—
74	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	SD₁₆×C₆₇	—	4	8	no	—
78	D₈×C₆₇	—	2	4	no	—
79	D₈×C₆₇	—	2	4	no	—
80	D₈×C₆₇	—	4	8	no	—
81	D₈×C₆₇	—	4	2	no	—
82	D₈×C₆₇	—	4	2	no	—
83	M₁₆×C₆₇	—	2	1	no	—
84	M₁₆×C₆₇	—	2	2	no	—
85	M₁₆×C₆₇	—	2	2	no	—
86	M₁₆×C₆₇	—	2	4	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₁₆×C₆₇	—	2	2	no	—
100	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
101	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂³×C₆₇	—	4	4	no	—
104	C₂³×C₆₇	—	8	2	no	—
105	C₂³×C₆₇	—	8	2	no	—
106	C₂³×C₆₇	—	16	4	no	—
107	C₂³×C₆₇	#135	1	1	yes	characteristic
108	Q₈×C₆₇	—	2	1	no	—
109	Q₈×C₆₇	C₂×D₈	1	2	yes	—
110	D₄×C₆₇	—	2	2	no	—
111	D₄×C₆₇	—	2	2	no	—
112	D₄×C₆₇	—	2	1	no	—
113	D₄×C₆₇	—	4	4	no	—
114	D₄×C₆₇	—	4	4	no	—
115	D₄×C₆₇	—	8	2	no	—
116	D₄×C₆₇	—	8	2	no	—
117	D₄×C₆₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	—	4	4	no	—
129	C₈×C₆₇	#121	1	1	yes	characteristic
130	C₈×C₆₇	#121	1	1	yes	characteristic
131	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	8	4	no	—
137	C₂²×C₆₇	—	16	4	no	—
138	C₂²×C₆₇	—	16	4	no	—
139	C₂²×C₆₇	—	16	4	no	—
140	C₂²×C₆₇	#466	1	1	yes	characteristic
141	C₂²×C₆₇	#466	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	#466	1	1	yes	characteristic
145	C₄×C₆₇	#466	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
147	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
148	C₂×#2221	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	4	2	no	—
152	C₂×C₆₇	—	16	4	no	—
153	C₂×C₆₇	#2221	1	2	yes	direct product factor
154	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
155	C₂²×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
156	C₂×#504	C₂×C₆₇	1	1	yes	characteristic, complemented
157	C₂×#472	C₂×C₆₇	1	2	yes	complemented
158	#2221	C₂×C₆₇	1	8	yes	direct product factor
159	C₂×#368	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₂×#361	C₂×C₆₇	1	2	yes	complemented
161	C₆₇	C₂×#2221	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×AGL(1,ℤ/8)	—	2	2	no	—
163	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
164	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×D₁₆	—	2	1	no	—
166	C₂×D₁₆	—	2	1	no	—
167	C₂×D₁₆	—	2	2	no	—
168	C₂×D₁₆	C₂²×C₆₇	1	4	yes	complemented
169	#504	—	2	2	no	—
170	#504	C₂²×C₆₇	1	4	yes	complemented
171	C₂×M₃₂	—	2	1	no	—
172	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
173	C₂²×C₁₆	—	2	1	no	—
174	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
175	#472	—	2	4	no	—
176	#472	C₂²×C₆₇	1	8	yes	complemented
177	C₂×#108	C₂²×C₆₇	1	2	yes	—
178	C₂×#100	—	2	2	no	—
179	#368	C₂²×C₆₇	1	4	yes	complemented
180	#361	C₂²×C₆₇	1	8	yes	complemented
181	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
182	C₂²×D₄	—	2	2	no	—
183	AGL(1,ℤ/8)	—	2	8	no	—
184	AGL(1,ℤ/8)	—	4	4	no	—
185	C₂×SD₁₆	—	4	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	4	2	no	—
192	C₂×D₈	C₂³×C₆₇	1	4	yes	—
193	#131	—	2	1	no	—
194	#131	—	2	1	no	—
195	#131	D₄×C₆₇	1	2	yes	—
196	#131	C₂³×C₆₇	1	2	yes	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
202	D₁₆	—	2	4	no	—
203	D₁₆	—	2	4	no	—
204	M₃₂	—	2	1	no	—
205	M₃₂	—	2	4	no	—
206	M₃₂	C₂³×C₆₇	1	2	yes	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	4	no	—
209	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
210	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
211	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
212	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
213	#108	—	2	2	no	—
214	#108	C₂³×C₆₇	1	4	yes	—
215	#100	—	2	8	no	—
216	C₂⁴	—	4	2	no	—
217	#41	—	2	1	no	—
218	#41	—	2	1	no	—
219	#41	D₈×C₆₇	1	2	yes	—
220	#41	C₂×D₄×C₆₇	1	2	yes	—
221	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
232	SD₁₆	—	4	8	no	—
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	8	no	—
236	D₈	—	4	2	no	—
237	D₈	—	4	2	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
252	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
254	C₁₆	—	2	2	no	—
255	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
256	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	16	4	no	—
262	C₂³	#135×C₆₇	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	C₂×D₈×C₆₇	1	2	yes	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	2	1	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	C₂×D₈×C₆₇	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
277	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
280	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	4	no	—
284	C₈	#121×C₆₇	1	1	yes	characteristic
285	C₈	#121×C₆₇	1	1	yes	characteristic
286	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	16	4	no	—
294	C₂²	—	16	4	no	—
295	C₂²	#466×C₆₇	1	1	yes	characteristic
296	C₂²	#466×C₆₇	1	1	yes	characteristic
297	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	#466×C₆₇	1	1	yes	characteristic
300	C₄	#466×C₆₇	1	1	yes	characteristic
301	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
302	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	2	no	—
306	C₂	—	16	4	no	—
307	C₂	C₆₇×#2221	1	2	yes	direct product factor
308	C₂	C₂×#466×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₆₇	C₂	semidirect
2. C₂×#504×C₆₇	C₂	semidirect
3. C₂×#472×C₆₇	C₂	semidirect
4. C₆₇×#2221	C₂	direct (N × H)
5. C₂×#368×C₆₇	C₂	semidirect
6. C₂×#361×C₆₇	C₂	semidirect
13. C₂×D₁₆×C₆₇	C₂²	semidirect
15. #504×C₆₇	C₂²	semidirect
21. #472×C₆₇	C₂²	semidirect
24. #368×C₆₇	C₂²	semidirect
25. #361×C₆₇	C₂²	semidirect
148. C₂×#2221	C₆₇	direct (N × H)
153. C₂×C₆₇	#2221	direct (N × H)
155. C₂²×D₁₆	C₂×C₆₇	semidirect
156. C₂×#504	C₂×C₆₇	semidirect
157. C₂×#472	C₂×C₆₇	semidirect
158. #2221	C₂×C₆₇	direct (N × H)
159. C₂×#368	C₂×C₆₇	semidirect
160. C₂×#361	C₂×C₆₇	semidirect
161. C₆₇	C₂×#2221	direct (N × H)
168. C₂×D₁₆	C₂²×C₆₇	semidirect
170. #504	C₂²×C₆₇	semidirect
176. #472	C₂²×C₆₇	semidirect
179. #368	C₂²×C₆₇	semidirect
180. #361	C₂²×C₆₇	semidirect
307. C₂	C₆₇×#2221	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₆₇	2
C₂×#504×C₆₇	2
C₂×#472×C₆₇	2
C₆₇×#2221	2
C₂×#368×C₆₇	2
C₂×#361×C₆₇	2
C₂×#2221	67

Subgroups — C2×C67×#2221 #30712865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2221 #30712865