Subgroups — C₃₁×#17977 #21474736

Order 7936.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1584	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1493	C₂	1	2	yes	maximal
6	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂²	1	2	yes	—
8	C₃₁×#500	C₄	1	1	yes	characteristic
9	C₃₁×#494	C₄	1	1	yes	characteristic
10	C₃₁×#487	C₄	1	2	yes	—
11	C₃₁×#467	C₄	1	4	yes	—
12	C₃₁×#446	—	2	2	no	—
13	C₃₁×#446	C₂²	1	4	yes	—
14	C₈×Q₈×C₃₁	—	2	2	no	—
15	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#424	—	2	2	no	—
18	C₃₁×#423	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	2	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#335	C₂²	1	4	yes	—
24	C₃₁×Q₈⋊C₈	C₂²	1	8	yes	—
25	C₃₁×#322	C₂²	1	4	yes	—
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₃₁×#125	—	2	2	no	—
35	C₃₁×C₂²⋊Q₈	—	2	2	no	—
36	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₃₁	—	2	2	no	—
38	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
39	C₃₁×#117	—	2	2	no	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	C₂×C₄	1	4	yes	—
42	C₃₁×C₄⋊C₈	—	2	4	no	—
43	C₃₁×C₄⋊C₈	—	2	4	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
46	C₃₁×#103	C₂×C₄	1	8	yes	—
47	C₃₁×#97	—	2	4	no	—
48	C₄×C₈×C₃₁	—	2	4	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	D₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	C₄×C₈×C₃₁	C₂³	1	2	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
61	Q₁₆×C₃₁	—	4	4	no	—
62	M₁₆×C₃₁	—	4	4	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	Q₁₆	1	4	yes	complemented
69	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₁	#41	1	1	yes	characteristic
84	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
85	#17977	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
87	Q₈×C₃₁	—	2	8	no	—
88	Q₈×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₃₁	#135	1	2	yes	—
98	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
101	C₈×C₃₁	—	2	2	no	—
102	C₈×C₃₁	—	2	2	no	—
103	C₈×C₃₁	—	2	2	no	—
104	C₈×C₃₁	—	4	2	no	—
105	C₈×C₃₁	—	4	4	no	—
106	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
107	#3170	C₂×C₃₁	1	1	yes	characteristic
108	#2960	C₂×C₃₁	1	2	yes	—
109	#1584	C₂×C₃₁	1	8	yes	complemented
110	#1567	C₂×C₃₁	1	1	yes	characteristic
111	#1493	C₂×C₃₁	1	2	yes	—
112	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	C₄×Q₁₆	1	2	yes	—
116	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	2	1	no	—
119	C₄×C₃₁	—	4	4	no	—
120	C₄×C₃₁	—	4	4	no	—
121	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
122	C₄×C₃₁	#444	1	2	yes	—
123	C₄×C₃₁	#568	1	2	yes	—
124	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
125	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
126	#533	C₂²×C₃₁	1	2	yes	—
127	#500	C₄×C₃₁	1	1	yes	characteristic
128	#494	C₄×C₃₁	1	1	yes	characteristic
129	#487	C₄×C₃₁	1	2	yes	—
130	#467	C₄×C₃₁	1	4	yes	—
131	#446	—	2	2	no	—
132	#446	C₂²×C₃₁	1	4	yes	—
133	C₈×Q₈	—	2	2	no	—
134	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
135	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
136	#424	—	2	2	no	—
137	#423	C₂²×C₃₁	1	2	yes	—
138	#405	—	2	2	no	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
142	#335	C₂²×C₃₁	1	4	yes	—
143	Q₈⋊C₈	C₂²×C₃₁	1	8	yes	—
144	#322	C₂²×C₃₁	1	4	yes	—
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	#2932	1	1	yes	characteristic
147	C₂×C₃₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
148	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
149	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
153	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
156	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
157	#125	—	2	2	no	—
158	C₂²⋊Q₈	—	2	2	no	—
159	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
162	#117	—	2	2	no	—
163	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
164	#107	C₂×C₄×C₃₁	1	4	yes	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
169	#103	C₂×C₄×C₃₁	1	8	yes	—
170	#97	—	2	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₄×C₈	D₄×C₃₁	1	2	yes	—
176	C₄×C₈	C₂³×C₃₁	1	2	yes	—
177	C₄×C₈	C₂³×C₃₁	1	2	yes	—
178	C₄×C₈	C₂³×C₃₁	1	2	yes	—
179	C₃₁	#17977	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
185	Q₁₆	—	4	4	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	Q₁₆×C₃₁	1	4	yes	complemented
193	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₄²	#41×C₃₁	1	1	yes	characteristic
208	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
209	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
220	C₂×C₄	C₃₁×#135	1	2	yes	—
221	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	4	4	no	—
229	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Q₁₆×C₃₁	1	2	yes	—
233	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	4	4	no	—
238	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
239	C₄	C₃₁×#444	1	2	yes	—
240	C₄	C₃₁×#568	1	2	yes	—
241	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
242	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	C₃₁×#2932	1	1	yes	characteristic
245	C₂	C₂×C₄×Q₁₆×C₃₁	1	1	yes	characteristic
246	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1584	C₂	semidirect
68. C₂×C₈×C₃₁	Q₁₆	semidirect
85. #17977	C₃₁	direct (N × H)
109. #1584	C₂×C₃₁	semidirect
179. C₃₁	#17977	direct (N × H)
192. C₂×C₈	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3170	2
C₃₁×#2960	2
C₃₁×#1584	2
C₃₁×#1567	2
C₃₁×#1493	2
C₂×C₃₁×#322	2
#17977	31

Subgroups — C31×#17977 #21474736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17977 #21474736