Subgroups — C₃₁×#21300 #21478059

Order 7936.

1198 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1984	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1729	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#533	—	2	4	no	—
9	C₃₁×#533	—	2	4	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
11	C₂×C₃₁×#125	C₄	1	2	yes	—
12	C₂×C₃₁×C₂²⋊Q₈	C₄	1	4	yes	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#423	—	2	4	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	—	2	4	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#336	C₂²	1	16	yes	—
21	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	—	2	8	no	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	Q₈	1	8	yes	complemented
29	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
31	C₃₁×C₄⋊Q₈	—	2	8	no	—
32	C₃₁×#125	—	2	8	no	—
33	C₃₁×C₂²⋊Q₈	—	2	16	no	—
34	C₄×Q₈×C₃₁	—	2	8	no	—
35	C₄×Q₈×C₃₁	—	2	8	no	—
36	C₃₁×#117	—	2	4	no	—
37	C₃₁×#117	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	4	yes	—
45	C₂×C₄²×C₃₁	D₄	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
48	C₃₁×C₄⋊C₈	—	2	8	no	—
49	C₄×C₈×C₃₁	—	2	8	no	—
50	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	2	8	no	—
52	C₂×Q₈×C₃₁	—	2	16	no	—
53	C₂²×C₄×C₃₁	—	2	8	no	—
54	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₃₁	#41	1	4	yes	—
63	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₃₁	—	4	8	no	—
65	C₂×C₈×C₃₁	—	2	16	no	—
66	C₂×C₈×C₃₁	—	2	4	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	8	no	—
70	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
71	C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₄⋊C₄×C₃₁	—	2	16	no	—
73	C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₂²⋊C₄×C₃₁	—	2	8	no	—
75	C₄²×C₃₁	—	2	4	no	—
76	C₄²×C₃₁	—	2	4	no	—
77	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
78	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
79	C₄²×C₃₁	C₂×D₄	1	4	yes	—
80	C₄²×C₃₁	C₂×D₄	1	4	yes	—
81	#21300	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂³×C₃₁	C₄×Q₈	1	4	yes	—
85	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₃₁	—	4	16	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	16	no	—
89	C₂×C₄×C₃₁	—	2	8	no	—
90	C₂×C₄×C₃₁	—	2	16	no	—
91	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
92	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₃₁	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
96	C₂×C₄×C₃₁	#124	1	4	yes	—
97	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
98	C₂×C₄×C₃₁	#131	1	8	yes	—
99	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
103	C₈×C₃₁	—	2	8	no	—
104	C₈×C₃₁	—	4	4	no	—
105	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
106	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
107	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
108	#1984	C₂×C₃₁	1	8	yes	complemented
109	#1729	C₂×C₃₁	1	4	yes	—
110	C₂²×C₃₁	—	2	4	no	—
111	C₂²×C₃₁	—	2	8	no	—
112	C₂²×C₃₁	#391	1	4	yes	—
113	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₃₁	C₂×C₄×Q₈	1	2	yes	—
115	C₂²×C₃₁	C₂×#131	1	4	yes	—
116	C₄×C₃₁	—	2	4	no	—
117	C₄×C₃₁	—	4	8	no	—
118	C₄×C₃₁	#391	1	4	yes	—
119	C₄×C₃₁	#408	1	8	yes	—
120	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
121	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
122	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
124	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
125	#533	—	2	4	no	—
126	#533	—	2	4	no	—
127	C₂×C₄⋊Q₈	C₄×C₃₁	1	2	yes	—
128	C₂×#125	C₄×C₃₁	1	2	yes	—
129	C₂×C₂²⋊Q₈	C₄×C₃₁	1	4	yes	—
130	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
131	C₂×#117	C₂²×C₃₁	1	2	yes	—
132	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
133	#423	—	2	4	no	—
134	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
137	#336	C₂²×C₃₁	1	16	yes	—
138	C₂×C₃₁	—	2	4	no	—
139	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
140	C₂×C₃₁	C₂×#408	1	2	yes	—
141	C₂×C₃₁	#2960	1	4	yes	—
142	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
143	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
144	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	2	8	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	Q₈×C₃₁	1	8	yes	complemented
150	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
151	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
152	C₄⋊Q₈	—	2	8	no	—
153	#125	—	2	8	no	—
154	C₂²⋊Q₈	—	2	16	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	8	no	—
157	#117	—	2	4	no	—
158	#117	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
160	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
162	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₃₁	1	4	yes	—
166	C₂×C₄²	D₄×C₃₁	1	4	yes	—
167	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
168	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
169	C₄⋊C₈	—	2	8	no	—
170	C₄×C₈	—	2	8	no	—
171	C₃₁	#21300	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	—	2	16	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
178	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
184	C₂²×C₄	#41×C₃₁	1	4	yes	—
185	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
186	M₁₆	—	4	8	no	—
187	C₂×C₈	—	2	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	16	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
200	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
201	C₄²	C₂×D₄×C₃₁	1	4	yes	—
202	C₄²	C₂×D₄×C₃₁	1	4	yes	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
205	C₂³	C₄×Q₈×C₃₁	1	4	yes	—
206	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
207	Q₈	—	4	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
215	C₂×C₄	C₄×Q₈×C₃₁	1	4	yes	—
216	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
217	C₂×C₄	C₃₁×#124	1	4	yes	—
218	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
219	C₂×C₄	C₃₁×#131	1	8	yes	—
220	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
223	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
224	C₈	—	2	8	no	—
225	C₈	—	4	4	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	8	no	—
228	C₂²	C₃₁×#391	1	4	yes	—
229	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂²	C₂×C₄×Q₈×C₃₁	1	2	yes	—
231	C₂²	C₂×C₃₁×#131	1	4	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	C₃₁×#391	1	4	yes	—
235	C₄	C₃₁×#408	1	8	yes	—
236	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
237	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
238	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
239	C₂	—	2	4	no	—
240	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
241	C₂	C₂×C₃₁×#408	1	2	yes	—
242	C₂	C₃₁×#2960	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1984	C₂	semidirect
28. C₂²×C₈×C₃₁	Q₈	semidirect
81. #21300	C₃₁	direct (N × H)
108. #1984	C₂×C₃₁	semidirect
149. C₂²×C₈	Q₈×C₃₁	semidirect
171. C₃₁	#21300	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×C₃₁×#533	2
C₂×C₃₁×#423	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1984	2
C₃₁×#1729	2
#21300	31

Subgroups — C31×#21300 #21478059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21300 #21478059