Subgroups — C₂×C₃₁×#1778 #21477216

Order 7936.

1046 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	C₂²×C₃₁×#108	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1778	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#341	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#108	C₄	1	16	yes	complemented
11	C₂×C₃₁×#108	C₂²	1	8	yes	—
12	C₄×M₁₆×C₃₁	—	2	4	no	—
13	C₄×M₁₆×C₃₁	C₂²	1	8	yes	—
14	C₂×C₃₁×#97	—	2	2	no	—
15	C₂×C₃₁×#97	C₄	1	4	yes	—
16	C₂×C₃₁×#97	C₄	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	—	2	2	no	—
19	C₃₁×#341	C₂²	1	8	yes	—
20	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	4	no	—
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	16	yes	—
25	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	D₄	1	2	yes	—
39	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#108	C₂×C₄	1	32	yes	complemented
42	C₃₁×#97	—	2	8	no	—
43	C₃₁×#97	C₂×C₄	1	16	yes	—
44	C₄×C₈×C₃₁	—	2	8	no	—
45	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	4	4	no	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	#41	1	4	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₃₁	—	2	16	no	—
59	M₁₆×C₃₁	—	2	8	no	—
60	M₁₆×C₃₁	C₂²×C₄	1	16	yes	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	2	4	no	—
63	C₂×C₈×C₃₁	—	2	16	no	—
64	C₂×C₈×C₃₁	—	2	8	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	C₄²	1	8	yes	—
67	C₂×C₈×C₃₁	C₄²	1	8	yes	—
68	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
71	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
72	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
74	C₄²×C₃₁	C₂×D₄	1	4	yes	—
75	C₂×#1778	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	2	4	no	—
77	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
80	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	—	2	8	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	#117	1	2	yes	—
90	C₂×C₄×C₃₁	#117	1	2	yes	—
91	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
94	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
98	C₈×C₃₁	—	2	4	no	—
99	C₈×C₃₁	—	2	16	no	—
100	C₈×C₃₁	C₂×C₄²	1	8	yes	—
101	C₂²×#108	C₂×C₃₁	1	2	yes	—
102	C₂×C₄×M₁₆	C₂×C₃₁	1	2	yes	—
103	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
104	#1778	C₂×C₃₁	1	8	yes	direct product factor
105	C₂×#341	C₂×C₃₁	1	2	yes	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	8	no	—
109	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
110	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₂²×C₃₁	#430	1	4	yes	—
112	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₃₁	—	2	8	no	—
115	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
116	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
117	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
119	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
120	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
121	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×#108	C₄×C₃₁	1	16	yes	complemented
125	C₂×#108	C₂²×C₃₁	1	8	yes	—
126	C₄×M₁₆	—	2	4	no	—
127	C₄×M₁₆	C₂²×C₃₁	1	8	yes	—
128	C₂×#97	—	2	2	no	—
129	C₂×#97	C₄×C₃₁	1	4	yes	—
130	C₂×#97	C₄×C₃₁	1	4	yes	—
131	C₂×#97	C₂²×C₃₁	1	4	yes	—
132	C₂×C₄×C₈	—	2	2	no	—
133	#341	C₂²×C₃₁	1	8	yes	—
134	C₂×C₃₁	—	2	2	no	—
135	C₂×C₃₁	—	2	2	no	—
136	C₂×C₃₁	#1778	1	4	yes	direct product factor
137	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₃₁	C₂×#430	1	2	yes	—
139	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
140	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	8	no	—
143	C₂×M₁₆	C₂×C₄×C₃₁	1	16	yes	—
144	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
148	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
149	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
152	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
153	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×C₄²	D₄×C₃₁	1	2	yes	—
157	C₂×C₄²	D₄×C₃₁	1	2	yes	—
158	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
160	#108	C₂×C₄×C₃₁	1	32	yes	complemented
161	#97	—	2	8	no	—
162	#97	C₂×C₄×C₃₁	1	16	yes	—
163	C₄×C₈	—	2	8	no	—
164	C₃₁	C₂×#1778	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
168	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
172	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
175	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
176	C₂²×C₄	#41×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
178	M₁₆	—	2	16	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	C₂²×C₄×C₃₁	1	16	yes	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₄²×C₃₁	1	8	yes	—
187	C₂×C₈	C₄²×C₃₁	1	8	yes	—
188	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
191	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
192	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
193	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
194	C₄²	C₂×D₄×C₃₁	1	4	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
199	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂×C₄	C₃₁×#117	1	2	yes	—
209	C₂×C₄	C₃₁×#117	1	2	yes	—
210	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
211	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
212	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
213	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
215	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	2	16	no	—
219	C₈	C₂×C₄²×C₃₁	1	8	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
224	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
225	C₂²	C₃₁×#430	1	4	yes	—
226	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
227	C₂²	C₂×C₄×Q₈×C₃₁	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
230	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
231	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
232	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
233	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	C₃₁×#1778	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂	C₂×C₃₁×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1778	C₂	direct (N × H)
10. C₂×C₃₁×#108	C₄	semidirect
41. C₃₁×#108	C₂×C₄	semidirect
75. C₂×#1778	C₃₁	direct (N × H)
104. #1778	C₂×C₃₁	direct (N × H)
124. C₂×#108	C₄×C₃₁	semidirect
136. C₂×C₃₁	#1778	direct (N × H)
160. #108	C₂×C₄×C₃₁	semidirect
164. C₃₁	C₂×#1778	direct (N × H)
236. C₂	C₃₁×#1778	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×C₃₁×#108	2
C₂×C₄×M₁₆×C₃₁	2
C₂²×C₃₁×#97	2
C₃₁×#1778	2
C₂×C₃₁×#341	2
C₂×#1778	31

Subgroups — C2×C31×#1778 #21477216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1778 #21477216