Subgroups — C₂×C₆₇×#2170 #30712690

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2170	C₂	1	8	yes	maximal, direct product factor
4	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#350×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
8	C₂×#131×C₆₇	C₂²	1	2	yes	—
9	#504×C₆₇	—	2	4	no	—
10	#504×C₆₇	C₂²	1	8	yes	complemented
11	C₂×M₃₂×C₆₇	—	2	2	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₆₇	—	2	2	no	—
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#443×C₆₇	C₄	1	4	yes	—
17	#443×C₆₇	C₂²	1	4	yes	—
18	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#104×C₆₇	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#364×C₆₇	C₂²	1	4	yes	complemented
22	#350×C₆₇	C₂²	1	8	yes	complemented
23	C₄×C₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
25	#135×C₆₇	C₈	1	4	yes	—
26	#135×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×Q₁₆×C₆₇	C₈	1	1	yes	characteristic
28	C₂×SD₁₆×C₆₇	C₈	1	2	yes	—
29	C₂×D₈×C₆₇	C₈	1	1	yes	characteristic
30	#131×C₆₇	—	2	2	no	—
31	#131×C₆₇	—	2	2	no	—
32	#131×C₆₇	C₂×C₄	1	4	yes	—
33	#131×C₆₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	2	2	no	—
41	M₃₂×C₆₇	—	2	8	no	—
42	M₃₂×C₆₇	C₂³	1	4	yes	—
43	C₂×C₁₆×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	—	2	1	no	—
45	C₂×C₁₆×C₆₇	—	2	1	no	—
46	C₂×C₁₆×C₆₇	—	2	8	no	—
47	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
49	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
52	#108×C₆₇	C₂×C₄	1	4	yes	—
53	#104×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	#41×C₆₇	—	2	2	no	—
57	#41×C₆₇	—	2	2	no	—
58	#41×C₆₇	C₂×C₈	1	4	yes	—
59	#41×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	Q₁₆×C₆₇	C₂×C₈	1	4	yes	—
67	SD₁₆×C₆₇	C₂×C₈	1	8	yes	—
68	D₈×C₆₇	C₂×C₈	1	4	yes	—
69	M₁₆×C₆₇	—	2	2	no	—
70	M₁₆×C₆₇	—	2	4	no	—
71	M₁₆×C₆₇	—	2	4	no	—
72	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₁₆×C₆₇	—	2	4	no	—
89	C₁₆×C₆₇	—	2	2	no	—
90	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
91	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
92	C₂³×C₆₇	—	2	2	no	—
93	C₂³×C₆₇	#131	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	2	no	—
95	Q₈×C₆₇	C₂²×C₈	1	4	yes	—
96	D₄×C₆₇	—	2	4	no	—
97	D₄×C₆₇	—	2	4	no	—
98	D₄×C₆₇	—	2	2	no	—
99	D₄×C₆₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₆₇	—	2	4	no	—
112	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₈×C₆₇	C₂²×C₈	1	2	yes	—
115	C₈×C₆₇	C₂²×C₈	1	2	yes	—
116	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	—	4	4	no	—
121	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
124	C₄×C₆₇	—	2	4	no	—
125	C₄×C₆₇	—	2	4	no	—
126	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
127	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
128	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
130	C₂×#2170	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	—	4	4	no	—
134	C₂×C₆₇	#2170	1	2	yes	direct product factor
135	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
136	C₂×#504	C₂×C₆₇	1	2	yes	complemented
137	C₂×#443	C₂×C₆₇	1	1	yes	characteristic
138	#2170	C₂×C₆₇	1	8	yes	direct product factor
139	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₂×#350	C₂×C₆₇	1	2	yes	complemented
141	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
142	C₆₇	C₂×#2170	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#135	C₄×C₆₇	1	1	yes	characteristic
144	C₂×#131	C₂²×C₆₇	1	2	yes	—
145	#504	—	2	4	no	—
146	#504	C₂²×C₆₇	1	8	yes	complemented
147	C₂×M₃₂	—	2	2	no	—
148	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
149	C₂²×C₁₆	—	2	2	no	—
150	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
152	#443	C₄×C₆₇	1	4	yes	—
153	#443	C₂²×C₆₇	1	4	yes	—
154	C₂×#108	C₄×C₆₇	1	1	yes	characteristic
155	C₂×#104	C₄×C₆₇	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	#364	C₂²×C₆₇	1	4	yes	complemented
158	#350	C₂²×C₆₇	1	8	yes	complemented
159	C₄×C₁₆	C₂²×C₆₇	1	4	yes	complemented
160	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
161	#135	C₈×C₆₇	1	4	yes	—
162	#135	C₂×C₄×C₆₇	1	4	yes	—
163	C₂×Q₁₆	C₈×C₆₇	1	1	yes	characteristic
164	C₂×SD₁₆	C₈×C₆₇	1	2	yes	—
165	C₂×D₈	C₈×C₆₇	1	1	yes	characteristic
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	C₂×C₄×C₆₇	1	4	yes	—
169	#131	C₂³×C₆₇	1	4	yes	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
176	M₃₂	—	2	2	no	—
177	M₃₂	—	2	8	no	—
178	M₃₂	C₂³×C₆₇	1	4	yes	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	8	no	—
183	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
184	C₂×C₁₆	D₄×C₆₇	1	2	yes	—
185	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
186	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
187	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
188	#108	C₂×C₄×C₆₇	1	4	yes	—
189	#104	C₂×C₄×C₆₇	1	8	yes	—
190	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
191	C₄×C₈	C₂³×C₆₇	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	C₂×C₈×C₆₇	1	4	yes	—
195	#41	C₂²×C₄×C₆₇	1	4	yes	—
196	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	Q₁₆	C₂×C₈×C₆₇	1	4	yes	—
203	SD₁₆	C₂×C₈×C₆₇	1	8	yes	—
204	D₈	C₂×C₈×C₆₇	1	4	yes	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
213	C₂×C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
214	C₂×C₈	C₂×C₈×C₆₇	1	2	yes	—
215	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
219	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
220	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	2	2	no	—
226	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
227	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	#131×C₆₇	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	C₂²×C₈×C₆₇	1	4	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	D₄	C₂²×C₈×C₆₇	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
245	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
249	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
250	C₈	C₂²×C₈×C₆₇	1	2	yes	—
251	C₈	C₂²×C₈×C₆₇	1	2	yes	—
252	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
253	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
258	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
259	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	C₈×D₄×C₆₇	1	1	yes	characteristic
263	C₄	C₈×D₄×C₆₇	1	1	yes	characteristic
264	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
265	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	C₆₇×#2170	1	2	yes	direct product factor
270	C₂	C₂×C₈×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₆₇	D₄
commutator	C₄	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₈×D₄
prime radical	C₂×#135×C₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₆₇ ◁ C₂²×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#504×C₆₇	C₂	semidirect
3. C₆₇×#2170	C₂	direct (N × H)
4. C₂×#364×C₆₇	C₂	semidirect
5. C₂×#350×C₆₇	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
10. #504×C₆₇	C₂²	semidirect
21. #364×C₆₇	C₂²	semidirect
22. #350×C₆₇	C₂²	semidirect
23. C₄×C₁₆×C₆₇	C₂²	semidirect
130. C₂×#2170	C₆₇	direct (N × H)
134. C₂×C₆₇	#2170	direct (N × H)
136. C₂×#504	C₂×C₆₇	semidirect
138. #2170	C₂×C₆₇	direct (N × H)
139. C₂×#364	C₂×C₆₇	semidirect
140. C₂×#350	C₂×C₆₇	semidirect
141. C₂×C₄×C₁₆	C₂×C₆₇	semidirect
142. C₆₇	C₂×#2170	direct (N × H)
146. #504	C₂²×C₆₇	semidirect
157. #364	C₂²×C₆₇	semidirect
158. #350	C₂²×C₆₇	semidirect
159. C₄×C₁₆	C₂²×C₆₇	semidirect
269. C₂	C₆₇×#2170	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×#504×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2170	2
C₂×#364×C₆₇	2
C₂×#350×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×#2170	67

Subgroups — C2×C67×#2170 #30712690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2170 #30712690