Subgroups — C₃₁×#9800 #21466559

Order 7936.

478 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	1	no	—
11	C₃₁×#486	—	2	1	no	—
12	C₃₁×#486	—	2	2	no	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₃₁×#423	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	C₄	1	4	yes	—
17	C₂×C₃₁×#100	—	2	2	no	—
18	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#343	—	2	2	no	—
21	C₃₁×#334	—	2	2	no	—
22	C₃₁×#332	—	2	2	no	—
23	C₃₁×#331	—	2	2	no	—
24	C₃₁×#322	—	2	1	no	—
25	C₈²×C₃₁	—	2	1	no	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	—	2	1	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×#127	—	2	1	no	—
39	C₃₁×#127	—	2	1	no	—
40	C₃₁×#127	—	2	1	no	—
41	C₃₁×#127	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₃₁×#102	—	2	4	no	—
51	C₃₁×#102	—	2	4	no	—
52	C₃₁×#100	—	4	4	no	—
53	C₃₁×C₂²⋊C₈	—	2	1	no	—
54	C₃₁×#97	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	2	yes	—
60	C₂⁴×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	—	4	1	no	—
63	C₂×Q₈×C₃₁	—	8	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	4	4	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	8	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
78	M₁₆×C₃₁	—	2	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	SD₁₆	1	4	yes	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	1	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	4	1	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	#41	1	1	yes	characteristic
97	#9800	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	4	2	no	—
99	C₂³×C₃₁	—	4	4	no	—
100	C₂³×C₃₁	—	8	2	no	—
101	C₂³×C₃₁	—	8	2	no	—
102	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₃₁	—	8	2	no	—
104	Q₈×C₃₁	—	8	2	no	—
105	D₄×C₃₁	—	4	4	no	—
106	D₄×C₃₁	—	8	2	no	—
107	D₄×C₃₁	—	8	2	no	—
108	D₄×C₃₁	—	8	2	no	—
109	D₄×C₃₁	—	8	2	no	—
110	D₄×C₃₁	—	8	1	no	—
111	D₄×C₃₁	—	8	1	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₃₁	#135	1	2	yes	—
124	C₈×C₃₁	—	2	2	no	—
125	C₈×C₃₁	—	2	2	no	—
126	C₈×C₃₁	—	2	4	no	—
127	C₈×C₃₁	—	4	2	no	—
128	C₈×C₃₁	—	8	1	no	—
129	C₈×C₃₁	—	8	1	no	—
130	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
131	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1965	C₂×C₃₁	1	1	yes	characteristic
133	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#1545	C₂×C₃₁	1	1	yes	characteristic
136	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	16	4	no	—
143	C₂²×C₃₁	#390	1	1	yes	characteristic
144	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	8	2	no	—
148	C₄×C₃₁	#413	1	2	yes	—
149	C₄×C₃₁	#465	1	2	yes	—
150	C₄×C₃₁	#472	1	2	yes	—
151	C₄×C₃₁	#492	1	1	yes	characteristic
152	C₄×C₃₁	#495	1	1	yes	characteristic
153	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
155	#486	—	2	1	no	—
156	#486	—	2	1	no	—
157	#486	—	2	2	no	—
158	#431	C₂²×C₃₁	1	1	yes	characteristic
159	#423	C₂²×C₃₁	1	1	yes	characteristic
160	#423	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×#102	C₄×C₃₁	1	4	yes	—
162	C₂×#100	—	2	2	no	—
163	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#343	—	2	2	no	—
166	#334	—	2	2	no	—
167	#332	—	2	2	no	—
168	#331	—	2	2	no	—
169	#322	—	2	1	no	—
170	C₈²	—	2	1	no	—
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	—	16	2	no	—
173	C₂×C₃₁	#1967	1	1	yes	characteristic
174	C₂×C₃₁	#1968	1	1	yes	characteristic
175	C₂×C₃₁	#1978	1	1	yes	characteristic
176	C₂²×Q₈	—	2	1	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
183	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	#127	—	2	1	no	—
189	#127	—	2	1	no	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
200	#102	—	2	4	no	—
201	#102	—	2	4	no	—
202	#100	—	4	4	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₃₁	1	2	yes	—
210	C₃₁	#9800	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	—	2	2	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	SD₁₆×C₃₁	1	4	yes	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
239	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₄²	#41×C₃₁	1	1	yes	characteristic
248	C₂³	—	4	2	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
253	Q₈	—	8	2	no	—
254	Q₈	—	8	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	8	2	no	—
257	D₄	—	8	2	no	—
258	D₄	—	8	2	no	—
259	D₄	—	8	2	no	—
260	D₄	—	8	1	no	—
261	D₄	—	8	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
269	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
272	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
273	C₂×C₄	C₃₁×#135	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	8	1	no	—
279	C₈	—	8	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	16	4	no	—
286	C₂²	C₃₁×#390	1	1	yes	characteristic
287	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	C₃₁×#413	1	2	yes	—
292	C₄	C₃₁×#465	1	2	yes	—
293	C₄	C₃₁×#472	1	2	yes	—
294	C₄	C₃₁×#492	1	1	yes	characteristic
295	C₄	C₃₁×#495	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₃₁×#1967	1	1	yes	characteristic
299	C₂	C₃₁×#1968	1	1	yes	characteristic
300	C₂	C₃₁×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1997	C₂	semidirect
4. C₃₁×#1566	C₂	semidirect
5. C₃₁×#1549	C₂	semidirect
7. C₈×M₁₆×C₃₁	C₂	semidirect
97. #9800	C₃₁	direct (N × H)
131. #1997	C₂×C₃₁	semidirect
133. #1566	C₂×C₃₁	semidirect
134. #1549	C₂×C₃₁	semidirect
136. C₈×M₁₆	C₂×C₃₁	semidirect
210. C₃₁	#9800	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×#486	2
C₃₁×#1997	2
C₃₁×#1965	2
C₃₁×#1566	2
C₃₁×#1549	2
C₃₁×#1545	2
C₈×M₁₆×C₃₁	2
#9800	31

Subgroups — C31×#9800 #21466559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9800 #21466559