Subgroups — C₂×C₇₁×#2069 #31844492

Order 18176.

1078 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₇₁	C₂	1	3	yes	maximal, complemented
2	C₂²×#100×C₇₁	C₂	1	3	yes	maximal
3	C₇₁×#2069	C₂	1	8	yes	maximal, direct product factor
4	C₂×#338×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₇₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₇₁	C₂²	1	3	yes	—
7	C₂²×M₁₆×C₇₁	C₂²	1	3	yes	—
8	#430×C₇₁	—	2	6	no	—
9	#430×C₇₁	C₂²	1	12	yes	complemented
10	C₂×#108×C₇₁	—	2	6	no	—
11	C₂×#100×C₇₁	—	2	6	no	—
12	C₂×#100×C₇₁	—	2	6	no	—
13	C₂×#100×C₇₁	C₂²	1	12	yes	—
14	#338×C₇₁	C₂²	1	4	yes	complemented
15	C₂⁵×C₇₁	Q₈	1	2	yes	—
16	C₂²×D₄×C₇₁	—	2	3	no	—
17	C₂²×D₄×C₇₁	—	2	3	no	—
18	C₂²×D₄×C₇₁	—	2	3	no	—
19	C₂²×D₄×C₇₁	—	2	3	no	—
20	C₂²×D₄×C₇₁	Q₈	1	2	yes	—
21	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
22	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₇₁	—	2	3	no	—
24	C₂×M₁₆×C₇₁	—	2	3	no	—
25	C₂×M₁₆×C₇₁	—	2	3	no	—
26	C₂×M₁₆×C₇₁	—	2	3	no	—
27	C₂×M₁₆×C₇₁	—	2	3	no	—
28	C₂×M₁₆×C₇₁	—	2	3	no	—
29	C₂×M₁₆×C₇₁	—	2	3	no	—
30	C₂×M₁₆×C₇₁	—	2	3	no	—
31	C₂×M₁₆×C₇₁	D₄	1	6	yes	—
32	C₂×M₁₆×C₇₁	D₄	1	6	yes	—
33	C₂×M₁₆×C₇₁	C₂³	1	6	yes	—
34	C₂×M₁₆×C₇₁	C₂³	1	6	yes	—
35	C₂²×C₈×C₇₁	—	2	3	no	—
36	C₂²×C₈×C₇₁	—	2	3	no	—
37	#108×C₇₁	—	2	24	no	—
38	#100×C₇₁	—	2	24	no	—
39	C₂⁴×C₇₁	—	2	6	no	—
40	C₂⁴×C₇₁	—	2	6	no	—
41	C₂⁴×C₇₁	—	8	4	no	—
42	C₂⁴×C₇₁	C₂×Q₈	1	4	yes	—
43	C₂⁴×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×D₄×C₇₁	—	2	6	no	—
45	C₂×D₄×C₇₁	—	2	6	no	—
46	C₂×D₄×C₇₁	—	2	6	no	—
47	C₂×D₄×C₇₁	—	2	6	no	—
48	C₂×D₄×C₇₁	—	4	3	no	—
49	C₂×D₄×C₇₁	—	4	3	no	—
50	C₂×D₄×C₇₁	—	4	3	no	—
51	C₂×D₄×C₇₁	—	4	3	no	—
52	C₂×D₄×C₇₁	—	4	3	no	—
53	C₂×D₄×C₇₁	—	4	1	no	—
54	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	#41	1	3	yes	—
58	C₂²×C₄×C₇₁	#41	1	3	yes	—
59	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₇₁	—	2	6	no	—
61	M₁₆×C₇₁	—	2	6	no	—
62	M₁₆×C₇₁	—	2	6	no	—
63	M₁₆×C₇₁	—	2	6	no	—
64	M₁₆×C₇₁	—	2	6	no	—
65	M₁₆×C₇₁	—	4	3	no	—
66	M₁₆×C₇₁	—	4	3	no	—
67	M₁₆×C₇₁	C₂×D₄	1	12	yes	—
68	C₂×C₈×C₇₁	—	2	6	no	—
69	C₂×C₈×C₇₁	—	2	6	no	—
70	C₂×C₈×C₇₁	—	2	6	no	—
71	C₂×C₈×C₇₁	—	2	6	no	—
72	C₂×C₈×C₇₁	—	4	3	no	—
73	C₂×C₈×C₇₁	—	4	3	no	—
74	C₂³×C₇₁	—	2	12	no	—
75	C₂³×C₇₁	—	4	2	no	—
76	C₂³×C₇₁	—	4	6	no	—
77	C₂³×C₇₁	—	4	2	no	—
78	C₂³×C₇₁	—	8	12	no	—
79	C₂³×C₇₁	—	8	4	no	—
80	C₂³×C₇₁	—	8	12	no	—
81	C₂³×C₇₁	C₂²⋊Q₈	1	3	yes	—
82	C₂³×C₇₁	C₂²⋊Q₈	1	3	yes	—
83	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
84	D₄×C₇₁	—	4	6	no	—
85	D₄×C₇₁	—	4	2	no	—
86	D₄×C₇₁	—	4	6	no	—
87	D₄×C₇₁	—	4	2	no	—
88	C₂×C₄×C₇₁	#121	1	3	yes	—
89	C₂×C₄×C₇₁	#121	1	3	yes	—
90	C₂×C₄×C₇₁	#121	1	3	yes	—
91	C₂×C₄×C₇₁	#121	1	3	yes	—
92	C₂×C₄×C₇₁	C₂²⋊Q₈	1	3	yes	—
93	C₂×C₄×C₇₁	C₂²⋊Q₈	1	3	yes	—
94	C₂×C₄×C₇₁	#126	1	3	yes	—
95	C₂×C₄×C₇₁	#126	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
97	C₂×C₄×C₇₁	C₂×#41	1	3	yes	—
98	C₈×C₇₁	—	2	12	no	—
99	C₈×C₇₁	—	4	6	no	—
100	C₂²×C₇₁	—	2	6	no	—
101	C₂²×C₇₁	—	4	3	no	—
102	C₂²×C₇₁	—	4	1	no	—
103	C₂²×C₇₁	—	4	4	no	—
104	C₂²×C₇₁	—	8	12	no	—
105	C₂²×C₇₁	—	8	12	no	—
106	C₂²×C₇₁	—	8	4	no	—
107	C₂²×C₇₁	#396	1	3	yes	—
108	C₂²×C₇₁	#396	1	1	yes	characteristic
109	C₂²×C₇₁	C₂×C₂²⋊Q₈	1	3	yes	—
110	C₄×C₇₁	#396	1	3	yes	—
111	C₄×C₇₁	#396	1	1	yes	characteristic
112	C₄×C₇₁	C₂×#121	1	3	yes	—
113	C₄×C₇₁	C₂×#126	1	1	yes	characteristic
114	C₂×#2069	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₇₁	—	2	3	no	—
116	C₂×C₇₁	—	2	3	no	—
117	C₂×C₇₁	—	8	4	no	—
118	C₂×C₇₁	#2069	1	2	yes	direct product factor
119	C₂×C₇₁	C₂×#396	1	1	yes	characteristic
120	C₂×#430	C₂×C₇₁	1	3	yes	complemented
121	C₂²×#100	C₂×C₇₁	1	3	yes	—
122	#2069	C₂×C₇₁	1	8	yes	direct product factor
123	C₂×#338	C₂×C₇₁	1	1	yes	characteristic, complemented
124	C₇₁	C₂×#2069	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×D₄	C₂²×C₇₁	1	1	yes	characteristic
126	C₂²×M₁₆	C₂²×C₇₁	1	3	yes	—
127	C₂²×M₁₆	C₂²×C₇₁	1	3	yes	—
128	#430	—	2	6	no	—
129	#430	C₂²×C₇₁	1	12	yes	complemented
130	C₂×#108	—	2	6	no	—
131	C₂×#100	—	2	6	no	—
132	C₂×#100	—	2	6	no	—
133	C₂×#100	C₂²×C₇₁	1	12	yes	—
134	#338	C₂²×C₇₁	1	4	yes	complemented
135	C₂⁵	Q₈×C₇₁	1	2	yes	—
136	C₂²×D₄	—	2	3	no	—
137	C₂²×D₄	—	2	3	no	—
138	C₂²×D₄	—	2	3	no	—
139	C₂²×D₄	—	2	3	no	—
140	C₂²×D₄	Q₈×C₇₁	1	2	yes	—
141	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
142	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
143	C₂×M₁₆	—	2	3	no	—
144	C₂×M₁₆	—	2	3	no	—
145	C₂×M₁₆	—	2	3	no	—
146	C₂×M₁₆	—	2	3	no	—
147	C₂×M₁₆	—	2	3	no	—
148	C₂×M₁₆	—	2	3	no	—
149	C₂×M₁₆	—	2	3	no	—
150	C₂×M₁₆	—	2	3	no	—
151	C₂×M₁₆	D₄×C₇₁	1	6	yes	—
152	C₂×M₁₆	D₄×C₇₁	1	6	yes	—
153	C₂×M₁₆	C₂³×C₇₁	1	6	yes	—
154	C₂×M₁₆	C₂³×C₇₁	1	6	yes	—
155	C₂²×C₈	—	2	3	no	—
156	C₂²×C₈	—	2	3	no	—
157	#108	—	2	24	no	—
158	#100	—	2	24	no	—
159	C₂⁴	—	2	6	no	—
160	C₂⁴	—	2	6	no	—
161	C₂⁴	—	8	4	no	—
162	C₂⁴	C₂×Q₈×C₇₁	1	4	yes	—
163	C₂⁴	C₂×Q₈×C₇₁	1	1	yes	characteristic
164	C₂×D₄	—	2	6	no	—
165	C₂×D₄	—	2	6	no	—
166	C₂×D₄	—	2	6	no	—
167	C₂×D₄	—	2	6	no	—
168	C₂×D₄	—	4	3	no	—
169	C₂×D₄	—	4	3	no	—
170	C₂×D₄	—	4	3	no	—
171	C₂×D₄	—	4	3	no	—
172	C₂×D₄	—	4	3	no	—
173	C₂×D₄	—	4	1	no	—
174	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
175	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
176	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
177	C₂²×C₄	#41×C₇₁	1	3	yes	—
178	C₂²×C₄	#41×C₇₁	1	3	yes	—
179	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
180	M₁₆	—	2	6	no	—
181	M₁₆	—	2	6	no	—
182	M₁₆	—	2	6	no	—
183	M₁₆	—	2	6	no	—
184	M₁₆	—	2	6	no	—
185	M₁₆	—	4	3	no	—
186	M₁₆	—	4	3	no	—
187	M₁₆	C₂×D₄×C₇₁	1	12	yes	—
188	C₂×C₈	—	2	6	no	—
189	C₂×C₈	—	2	6	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	—	2	6	no	—
192	C₂×C₈	—	4	3	no	—
193	C₂×C₈	—	4	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	6	no	—
197	C₂³	—	4	2	no	—
198	C₂³	—	8	12	no	—
199	C₂³	—	8	4	no	—
200	C₂³	—	8	12	no	—
201	C₂³	C₂²⋊Q₈×C₇₁	1	3	yes	—
202	C₂³	C₂²⋊Q₈×C₇₁	1	3	yes	—
203	C₂³	C₂²×Q₈×C₇₁	1	1	yes	characteristic
204	D₄	—	4	6	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	6	no	—
207	D₄	—	4	2	no	—
208	C₂×C₄	#121×C₇₁	1	3	yes	—
209	C₂×C₄	#121×C₇₁	1	3	yes	—
210	C₂×C₄	#121×C₇₁	1	3	yes	—
211	C₂×C₄	#121×C₇₁	1	3	yes	—
212	C₂×C₄	C₂²⋊Q₈×C₇₁	1	3	yes	—
213	C₂×C₄	C₂²⋊Q₈×C₇₁	1	3	yes	—
214	C₂×C₄	#126×C₇₁	1	3	yes	—
215	C₂×C₄	#126×C₇₁	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
217	C₂×C₄	C₂×#41×C₇₁	1	3	yes	—
218	C₈	—	2	12	no	—
219	C₈	—	4	6	no	—
220	C₂²	—	2	6	no	—
221	C₂²	—	4	3	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	4	no	—
224	C₂²	—	8	12	no	—
225	C₂²	—	8	12	no	—
226	C₂²	—	8	4	no	—
227	C₂²	#396×C₇₁	1	3	yes	—
228	C₂²	#396×C₇₁	1	1	yes	characteristic
229	C₂²	C₂×C₂²⋊Q₈×C₇₁	1	3	yes	—
230	C₄	#396×C₇₁	1	3	yes	—
231	C₄	#396×C₇₁	1	1	yes	characteristic
232	C₄	C₂×#121×C₇₁	1	3	yes	—
233	C₄	C₂×#126×C₇₁	1	1	yes	characteristic
234	C₂	—	2	3	no	—
235	C₂	—	2	3	no	—
236	C₂	—	8	4	no	—
237	C₂	C₇₁×#2069	1	2	yes	direct product factor
238	C₂	C₂×#396×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂³×D₄×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#430×C₇₁	C₂	semidirect
3. C₇₁×#2069	C₂	direct (N × H)
4. C₂×#338×C₇₁	C₂	semidirect
9. #430×C₇₁	C₂²	semidirect
14. #338×C₇₁	C₂²	semidirect
114. C₂×#2069	C₇₁	direct (N × H)
118. C₂×C₇₁	#2069	direct (N × H)
120. C₂×#430	C₂×C₇₁	semidirect
122. #2069	C₂×C₇₁	direct (N × H)
123. C₂×#338	C₂×C₇₁	semidirect
124. C₇₁	C₂×#2069	direct (N × H)
129. #430	C₂²×C₇₁	semidirect
134. #338	C₂²×C₇₁	semidirect
237. C₂	C₇₁×#2069	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×#430×C₇₁	2
C₂²×#100×C₇₁	2
C₇₁×#2069	2
C₂×#338×C₇₁	2
C₂×#2069	71

Subgroups — C2×C71×#2069 #31844492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#2069 #31844492