Subgroups — C₃₁×#3187 #16843120

Order 3968.

468 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#483	C₂	1	2	yes	maximal, complemented
4	C₃₁×#481	C₂	1	2	yes	maximal, complemented
5	C₃₁×#465	C₂	1	4	yes	maximal, complemented
6	C₃₁×#450	C₂	1	2	yes	maximal, complemented
7	C₃₁×#448	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
10	C₂²×D₄×C₃₁	—	2	1	no	—
11	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
14	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
15	C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₃₁×C₂²⋊Q₈	—	2	1	no	—
17	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
19	C₃₁×#121	—	2	1	no	—
20	C₃₁×#121	—	2	1	no	—
21	C₃₁×#121	C₂²	1	2	yes	—
22	C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
28	C₃₁×#106	C₂²	1	4	yes	complemented
29	C₃₁×#103	C₂²	1	4	yes	complemented
30	C₃₁×#102	C₂²	1	4	yes	complemented
31	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₂⁴×C₃₁	—	2	1	no	—
34	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₃₁	—	2	1	no	—
36	C₂×Q₈×C₃₁	—	4	2	no	—
37	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
38	C₂×D₄×C₃₁	—	2	2	no	—
39	C₂×D₄×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	1	no	—
41	C₂×D₄×C₃₁	—	4	2	no	—
42	C₂×D₄×C₃₁	—	4	4	no	—
43	C₂×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₄×C₃₁	—	2	1	no	—
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	4	2	no	—
48	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	SD₁₆×C₃₁	—	4	4	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	C₂³	1	4	yes	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	—	2	2	no	—
68	C₂³×C₃₁	—	4	1	no	—
69	C₂³×C₃₁	—	4	1	no	—
70	C₂³×C₃₁	—	4	2	no	—
71	C₂³×C₃₁	—	4	2	no	—
72	C₂³×C₃₁	—	4	2	no	—
73	C₂³×C₃₁	SD₁₆	1	4	yes	complemented
74	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₃₁	—	2	4	no	—
78	Q₈×C₃₁	—	4	2	no	—
79	D₄×C₃₁	—	2	4	no	—
80	D₄×C₃₁	—	4	2	no	—
81	D₄×C₃₁	—	4	4	no	—
82	D₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	1	no	—
90	C₂×C₄×C₃₁	—	4	1	no	—
91	C₂×C₄×C₃₁	—	4	1	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₃₁	—	2	4	no	—
100	#3187	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	—	2	1	no	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	—	2	1	no	—
107	C₂²×C₃₁	—	4	4	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	—	4	2	no	—
110	C₂²×C₃₁	—	4	2	no	—
111	C₂²×C₃₁	—	8	4	no	—
112	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
113	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
114	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
115	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	2	1	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	4	2	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
123	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂×#121	C₂×C₃₁	1	1	yes	characteristic
125	#483	C₂×C₃₁	1	2	yes	complemented
126	#481	C₂×C₃₁	1	2	yes	complemented
127	#465	C₂×C₃₁	1	4	yes	complemented
128	#450	C₂×C₃₁	1	2	yes	complemented
129	#448	C₂×C₃₁	1	2	yes	complemented
130	C₂×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂×C₃₁	—	2	2	no	—
132	C₂×C₃₁	—	2	2	no	—
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	—	8	2	no	—
135	C₂×C₃₁	#534	1	1	yes	characteristic
136	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
137	C₂×C₃₁	#575	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
139	C₂²×D₄	—	2	1	no	—
140	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
143	C₂²×C₈	C₂²×C₃₁	1	2	yes	complemented
144	C₂²⋊Q₈	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
147	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
148	#121	—	2	1	no	—
149	#121	—	2	1	no	—
150	#121	C₂²×C₃₁	1	2	yes	—
151	#121	C₂²×C₃₁	1	2	yes	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
155	C₂×C₂²⋊C₄	—	2	1	no	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	#106	C₂²×C₃₁	1	4	yes	complemented
158	#103	C₂²×C₃₁	1	4	yes	complemented
159	#102	C₂²×C₃₁	1	4	yes	complemented
160	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
162	C₃₁	#3187	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	—	2	1	no	—
164	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	C₂³×C₃₁	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
184	SD₁₆	—	4	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₂³×C₃₁	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
192	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	—	4	2	no	—
201	C₂³	—	4	2	no	—
202	C₂³	—	4	2	no	—
203	C₂³	SD₁₆×C₃₁	1	4	yes	complemented
204	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
207	Q₈	—	2	4	no	—
208	Q₈	—	4	2	no	—
209	D₄	—	2	4	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	8	4	no	—
241	C₂²	C₂×SD₁₆×C₃₁	1	2	yes	—
242	C₂²	C₂×SD₁₆×C₃₁	1	2	yes	—
243	C₂²	C₂×SD₁₆×C₃₁	1	2	yes	—
244	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	8	2	no	—
256	C₂	C₃₁×#534	1	1	yes	characteristic
257	C₂	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
258	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#483	C₂	semidirect
4. C₃₁×#481	C₂	semidirect
5. C₃₁×#465	C₂	semidirect
6. C₃₁×#450	C₂	semidirect
7. C₃₁×#448	C₂	semidirect
8. C₂×C₃₁×C₂²⋊C₈	C₂	semidirect
9. C₂²×Q₈×C₃₁	C₂²	semidirect
13. C₂×SD₁₆×C₃₁	C₂²	semidirect
14. C₂²×C₈×C₃₁	C₂²	semidirect
18. C₃₁×C₂²⋊Q₈	C₂²	semidirect
25. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
28. C₃₁×#106	C₂²	semidirect
29. C₃₁×#103	C₂²	semidirect
30. C₃₁×#102	C₂²	semidirect
32. C₃₁×C₂²⋊C₈	C₂²	semidirect
73. C₂³×C₃₁	SD₁₆	semidirect
100. #3187	C₃₁	direct (N × H)
123. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
125. #483	C₂×C₃₁	semidirect
126. #481	C₂×C₃₁	semidirect
127. #465	C₂×C₃₁	semidirect
128. #450	C₂×C₃₁	semidirect
129. #448	C₂×C₃₁	semidirect
130. C₂×C₂²⋊C₈	C₂×C₃₁	semidirect
138. C₂²×Q₈	C₂²×C₃₁	semidirect
142. C₂×SD₁₆	C₂²×C₃₁	semidirect
143. C₂²×C₈	C₂²×C₃₁	semidirect
147. C₂²⋊Q₈	C₂²×C₃₁	semidirect
154. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
157. #106	C₂²×C₃₁	semidirect
158. #103	C₂²×C₃₁	semidirect
159. #102	C₂²×C₃₁	semidirect
161. C₂²⋊C₈	C₂²×C₃₁	semidirect
162. C₃₁	#3187	direct (N × H)
203. C₂³	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₃₁×#121	2
C₃₁×#483	2
C₃₁×#481	2
C₃₁×#465	2
C₃₁×#450	2
C₃₁×#448	2
C₂×C₃₁×C₂²⋊C₈	2
#3187	31

Subgroups — C31×#3187 #16843120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3187 #16843120