Subgroups — C₂×C₇₁×#2823 #31857554

Order 18176.

2502 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₇₁	C₂	1	2	yes	maximal, complemented
2	C₇₁×#2823	C₂	1	16	yes	maximal, direct product factor
3	C₂×#398×C₇₁	C₂	1	3	yes	maximal, complemented
4	C₂×#396×C₇₁	C₂	1	6	yes	maximal, complemented
5	C₂×#394×C₇₁	C₂	1	3	yes	maximal
6	C₂×#384×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×#126×C₇₁	—	2	16	no	—
9	C₂×#126×C₇₁	C₂²	1	16	yes	complemented
10	C₂²×C₄⋊C₄×C₇₁	C₂²	1	6	yes	—
11	C₂²×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
12	C₂²×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
13	C₂²×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
14	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	6	yes	—
15	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	6	yes	—
16	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	#398×C₇₁	C₂²	1	24	yes	complemented
18	#396×C₇₁	C₂²	1	48	yes	complemented
19	#394×C₇₁	C₂²	1	24	yes	—
20	#384×C₇₁	C₂²	1	8	yes	complemented
21	C₂×#95×C₇₁	C₂²	1	3	yes	—
22	C₂×#95×C₇₁	C₂²	1	3	yes	—
23	C₂⁵×C₇₁	C₂³	1	2	yes	—
24	C₂²×D₄×C₇₁	—	4	6	no	—
25	C₂²×D₄×C₇₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
27	C₂³×C₄×C₇₁	C₂³	1	6	yes	—
28	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
30	#126×C₇₁	—	2	64	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	24	no	—
32	C₂×C₄⋊C₄×C₇₁	—	4	6	no	—
33	C₂×C₄⋊C₄×C₇₁	—	4	6	no	—
34	C₂×C₄⋊C₄×C₇₁	—	4	6	no	—
35	C₂×C₄⋊C₄×C₇₁	C₂³	1	24	yes	—
36	C₂×C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
38	C₂×C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
39	C₂×C₂²⋊C₄×C₇₁	—	2	24	no	—
40	C₂×C₂²⋊C₄×C₇₁	—	4	12	no	—
41	C₂×C₂²⋊C₄×C₇₁	C₂³	1	24	yes	—
42	C₂×C₂²⋊C₄×C₇₁	C₂³	1	24	yes	—
43	C₂×C₄²×C₇₁	D₄	1	8	yes	complemented
44	C₂×C₄²×C₇₁	C₂³	1	4	yes	—
45	#95×C₇₁	C₂³	1	12	yes	—
46	#95×C₇₁	C₂³	1	12	yes	—
47	C₂⁴×C₇₁	—	2	4	no	—
48	C₂⁴×C₇₁	—	2	12	no	—
49	C₂⁴×C₇₁	—	2	12	no	—
50	C₂⁴×C₇₁	C₂⁴	1	4	yes	—
51	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₄×C₇₁	—	4	12	no	—
53	C₂×D₄×C₇₁	—	4	16	no	—
54	C₂²×C₄×C₇₁	—	2	6	no	—
55	C₂²×C₄×C₇₁	—	2	6	no	—
56	C₂²×C₄×C₇₁	—	2	12	no	—
57	C₂²×C₄×C₇₁	—	2	12	no	—
58	C₂²×C₄×C₇₁	—	2	12	no	—
59	C₂²×C₄×C₇₁	—	2	6	no	—
60	C₂²×C₄×C₇₁	—	2	6	no	—
61	C₂²×C₄×C₇₁	—	2	12	no	—
62	C₂²×C₄×C₇₁	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
64	C₂²×C₄×C₇₁	C₂⁴	1	12	yes	—
65	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
67	C₄⋊C₄×C₇₁	—	2	48	no	—
68	C₄⋊C₄×C₇₁	—	4	12	no	—
69	C₄⋊C₄×C₇₁	—	4	12	no	—
70	C₄⋊C₄×C₇₁	—	4	12	no	—
71	C₂²⋊C₄×C₇₁	—	2	48	no	—
72	C₂²⋊C₄×C₇₁	—	4	24	no	—
73	C₄²×C₇₁	C₂×D₄	1	16	yes	complemented
74	C₂³×C₇₁	—	2	4	no	—
75	C₂³×C₇₁	—	2	12	no	—
76	C₂³×C₇₁	—	2	12	no	—
77	C₂³×C₇₁	—	4	24	no	—
78	C₂³×C₇₁	—	4	8	no	—
79	C₂³×C₇₁	—	4	24	no	—
80	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
81	C₂³×C₇₁	H(5,𝔽₂)	1	6	yes	—
82	C₂³×C₇₁	2₋¹⁺⁴	1	6	yes	—
83	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
84	D₄×C₇₁	—	4	16	no	—
85	C₂×C₄×C₇₁	—	2	24	no	—
86	C₂×C₄×C₇₁	—	2	6	no	—
87	C₂×C₄×C₇₁	—	2	6	no	—
88	C₂×C₄×C₇₁	—	2	12	no	—
89	C₂×C₄×C₇₁	—	2	12	no	—
90	C₂×C₄×C₇₁	—	2	12	no	—
91	C₂×C₄×C₇₁	—	2	6	no	—
92	C₂×C₄×C₇₁	—	2	6	no	—
93	C₂×C₄×C₇₁	—	2	12	no	—
94	C₂×C₄×C₇₁	—	4	24	no	—
95	C₂×C₄×C₇₁	—	4	12	no	—
96	C₂×C₄×C₇₁	—	4	4	no	—
97	C₂×C₄×C₇₁	C₂²×D₄	1	12	yes	—
98	C₂²×C₇₁	—	4	12	no	—
99	C₂²×C₇₁	—	4	4	no	—
100	C₂²×C₇₁	—	4	12	no	—
101	C₂²×C₇₁	—	8	16	no	—
102	C₂²×C₇₁	#537	1	12	yes	—
103	C₂²×C₇₁	#560	1	4	yes	—
104	C₂²×C₇₁	#563	1	12	yes	—
105	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
106	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	3	yes	—
107	C₂²×C₇₁	C₂×2₋¹⁺⁴	1	3	yes	—
108	C₄×C₇₁	—	2	12	no	—
109	C₄×C₇₁	—	4	12	no	—
110	C₄×C₇₁	—	4	6	no	—
111	C₄×C₇₁	—	4	2	no	—
112	C₂×#2823	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₇₁	—	8	4	no	—
114	C₂×C₇₁	#2823	1	8	yes	direct product factor
115	C₂×C₇₁	C₂×#537	1	3	yes	—
116	C₂×C₇₁	C₂×#560	1	1	yes	characteristic
117	C₂×C₇₁	C₂×#563	1	3	yes	—
118	C₂²×#126	C₂×C₇₁	1	2	yes	complemented
119	#2823	C₂×C₇₁	1	16	yes	direct product factor
120	C₂×#398	C₂×C₇₁	1	3	yes	complemented
121	C₂×#396	C₂×C₇₁	1	6	yes	complemented
122	C₂×#394	C₂×C₇₁	1	3	yes	—
123	C₂×#384	C₂×C₇₁	1	1	yes	characteristic, complemented
124	C₇₁	C₂×#2823	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×D₄	C₂²×C₇₁	1	1	yes	characteristic
126	C₂×#126	—	2	16	no	—
127	C₂×#126	C₂²×C₇₁	1	16	yes	complemented
128	C₂²×C₄⋊C₄	C₂²×C₇₁	1	6	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
132	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	6	yes	—
133	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	6	yes	—
134	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
135	#398	C₂²×C₇₁	1	24	yes	complemented
136	#396	C₂²×C₇₁	1	48	yes	complemented
137	#394	C₂²×C₇₁	1	24	yes	—
138	#384	C₂²×C₇₁	1	8	yes	complemented
139	C₂×#95	C₂²×C₇₁	1	3	yes	—
140	C₂×#95	C₂²×C₇₁	1	3	yes	—
141	C₂⁵	C₂³×C₇₁	1	2	yes	—
142	C₂²×D₄	—	4	6	no	—
143	C₂²×D₄	C₂³×C₇₁	1	4	yes	—
144	C₂³×C₄	C₂³×C₇₁	1	3	yes	—
145	C₂³×C₄	C₂³×C₇₁	1	6	yes	—
146	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
147	C₂³×C₄	C₂³×C₇₁	1	3	yes	—
148	#126	—	2	64	no	—
149	C₂×C₄⋊C₄	—	2	24	no	—
150	C₂×C₄⋊C₄	—	4	6	no	—
151	C₂×C₄⋊C₄	—	4	6	no	—
152	C₂×C₄⋊C₄	—	4	6	no	—
153	C₂×C₄⋊C₄	C₂³×C₇₁	1	24	yes	—
154	C₂×C₄⋊C₄	C₂³×C₇₁	1	12	yes	—
155	C₂×C₄⋊C₄	C₂³×C₇₁	1	12	yes	—
156	C₂×C₄⋊C₄	C₂³×C₇₁	1	12	yes	—
157	C₂×C₂²⋊C₄	—	2	24	no	—
158	C₂×C₂²⋊C₄	—	4	12	no	—
159	C₂×C₂²⋊C₄	C₂³×C₇₁	1	24	yes	—
160	C₂×C₂²⋊C₄	C₂³×C₇₁	1	24	yes	—
161	C₂×C₄²	D₄×C₇₁	1	8	yes	complemented
162	C₂×C₄²	C₂³×C₇₁	1	4	yes	—
163	#95	C₂³×C₇₁	1	12	yes	—
164	#95	C₂³×C₇₁	1	12	yes	—
165	C₂⁴	—	2	4	no	—
166	C₂⁴	—	2	12	no	—
167	C₂⁴	—	2	12	no	—
168	C₂⁴	C₂⁴×C₇₁	1	4	yes	—
169	C₂⁴	C₂⁴×C₇₁	1	1	yes	characteristic
170	C₂×D₄	—	4	12	no	—
171	C₂×D₄	—	4	16	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	C₂×D₄×C₇₁	1	12	yes	—
181	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
182	C₂²×C₄	C₂⁴×C₇₁	1	12	yes	—
183	C₂²×C₄	C₂⁴×C₇₁	1	6	yes	—
184	C₂²×C₄	C₂⁴×C₇₁	1	6	yes	—
185	C₄⋊C₄	—	2	48	no	—
186	C₄⋊C₄	—	4	12	no	—
187	C₄⋊C₄	—	4	12	no	—
188	C₄⋊C₄	—	4	12	no	—
189	C₂²⋊C₄	—	2	48	no	—
190	C₂²⋊C₄	—	4	24	no	—
191	C₄²	C₂×D₄×C₇₁	1	16	yes	complemented
192	C₂³	—	2	4	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	4	8	no	—
197	C₂³	—	4	24	no	—
198	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
199	C₂³	H(5,𝔽₂)×C₇₁	1	6	yes	—
200	C₂³	2₋¹⁺⁴×C₇₁	1	6	yes	—
201	C₂³	C₂⁵×C₇₁	1	1	yes	characteristic
202	D₄	—	4	16	no	—
203	C₂×C₄	—	2	24	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	12	no	—
207	C₂×C₄	—	2	12	no	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	4	24	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	12	no	—
215	C₂×C₄	C₂²×D₄×C₇₁	1	12	yes	—
216	C₂²	—	4	12	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	12	no	—
219	C₂²	—	8	16	no	—
220	C₂²	#537×C₇₁	1	12	yes	—
221	C₂²	#560×C₇₁	1	4	yes	—
222	C₂²	#563×C₇₁	1	12	yes	—
223	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
224	C₂²	C₂×H(5,𝔽₂)×C₇₁	1	3	yes	—
225	C₂²	C₂×2₋¹⁺⁴×C₇₁	1	3	yes	—
226	C₄	—	2	12	no	—
227	C₄	—	4	12	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	6	no	—
230	C₂	—	8	4	no	—
231	C₂	C₇₁×#2823	1	8	yes	direct product factor
232	C₂	C₂×#537×C₇₁	1	3	yes	—
233	C₂	C₂×#560×C₇₁	1	1	yes	characteristic
234	C₂	C₂×#563×C₇₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇₁
prime radical	C₂³×D₄×C₇₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#126×C₇₁	C₂	semidirect
2. C₇₁×#2823	C₂	direct (N × H)
3. C₂×#398×C₇₁	C₂	semidirect
4. C₂×#396×C₇₁	C₂	semidirect
6. C₂×#384×C₇₁	C₂	semidirect
9. C₂×#126×C₇₁	C₂²	semidirect
17. #398×C₇₁	C₂²	semidirect
18. #396×C₇₁	C₂²	semidirect
20. #384×C₇₁	C₂²	semidirect
43. C₂×C₄²×C₇₁	D₄	semidirect
73. C₄²×C₇₁	C₂×D₄	semidirect
112. C₂×#2823	C₇₁	direct (N × H)
114. C₂×C₇₁	#2823	direct (N × H)
118. C₂²×#126	C₂×C₇₁	semidirect
119. #2823	C₂×C₇₁	direct (N × H)
120. C₂×#398	C₂×C₇₁	semidirect
121. C₂×#396	C₂×C₇₁	semidirect
123. C₂×#384	C₂×C₇₁	semidirect
124. C₇₁	C₂×#2823	direct (N × H)
127. C₂×#126	C₂²×C₇₁	semidirect
135. #398	C₂²×C₇₁	semidirect
136. #396	C₂²×C₇₁	semidirect
138. #384	C₂²×C₇₁	semidirect
161. C₂×C₄²	D₄×C₇₁	semidirect
191. C₄²	C₂×D₄×C₇₁	semidirect
231. C₂	C₇₁×#2823	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×#126×C₇₁	2
C₇₁×#2823	2
C₂×#398×C₇₁	2
C₂×#396×C₇₁	2
C₂×#394×C₇₁	2
C₂×#384×C₇₁	2
C₂×#2823	71

Subgroups — C2×C71×#2823 #31857554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#2823 #31857554