Subgroups — C₃₁×#12744 #21469503

Order 7936.

874 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#362	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2143	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
7	C₃₁×#578	—	2	4	no	—
8	C₃₁×#578	—	2	4	no	—
9	C₂×C₃₁×#137	—	2	1	no	—
10	C₂×C₃₁×#137	—	2	2	no	—
11	C₂×C₃₁×#135	C₄	1	2	yes	—
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₃₁	C₄	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×#108	C₂²	1	2	yes	—
18	C₃₁×#413	—	2	1	no	—
19	C₃₁×#362	—	2	4	no	—
20	C₃₁×#359	—	2	4	no	—
21	C₃₁×#349	—	2	2	no	—
22	C₃₁×#338	—	2	1	no	—
23	C₃₁×2₋¹⁺⁴	—	2	8	no	—
24	C₃₁×2₋¹⁺⁴	—	4	4	no	—
25	C₂×#41×C₃₁	—	2	2	no	—
26	C₂×#41×C₃₁	—	2	1	no	—
27	C₂×#41×C₃₁	—	4	1	no	—
28	C₂×#41×C₃₁	—	4	2	no	—
29	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₃₁	—	2	2	no	—
31	C₂²×Q₈×C₃₁	—	2	1	no	—
32	C₂²×Q₈×C₃₁	—	4	1	no	—
33	C₃₁×#137	—	2	8	no	—
34	C₃₁×#137	—	2	16	no	—
35	C₃₁×#135	—	2	4	no	—
36	C₃₁×#135	—	2	4	no	—
37	C₃₁×#135	—	2	2	no	—
38	C₃₁×#135	D₄	1	4	yes	—
39	C₂×Q₁₆×C₃₁	—	2	4	no	—
40	C₂×Q₁₆×C₃₁	—	2	4	no	—
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×Q₁₆×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	1	no	—
49	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₃₁×#131	—	2	2	no	—
51	C₃₁×#131	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×M₃₂	—	2	2	no	—
58	C₃₁×M₃₂	—	2	2	no	—
59	C₂×C₁₆×C₃₁	—	2	2	no	—
60	C₃₁×#108	—	2	2	no	—
61	C₃₁×#108	—	2	2	no	—
62	C₃₁×#101	—	4	1	no	—
63	C₃₁×#100	—	4	1	no	—
64	#41×C₃₁	—	2	2	no	—
65	#41×C₃₁	—	2	2	no	—
66	#41×C₃₁	—	2	8	no	—
67	#41×C₃₁	—	2	4	no	—
68	#41×C₃₁	—	4	4	no	—
69	#41×C₃₁	—	4	16	no	—
70	#41×C₃₁	—	4	8	no	—
71	#41×C₃₁	—	4	2	no	—
72	C₂×Q₈×C₃₁	—	2	8	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	4	4	no	—
77	C₂×Q₈×C₃₁	—	4	8	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	4	2	no	—
81	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
86	C₂×D₄×C₃₁	SD₁₆	1	2	yes	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	—	4	1	no	—
91	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₃₁	—	2	8	no	—
94	Q₁₆×C₃₁	—	2	1	no	—
95	Q₁₆×C₃₁	—	2	2	no	—
96	Q₁₆×C₃₁	—	2	4	no	—
97	Q₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
98	Q₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
99	SD₁₆×C₃₁	—	2	8	no	—
100	SD₁₆×C₃₁	—	2	4	no	—
101	D₈×C₃₁	—	2	1	no	—
102	D₈×C₃₁	C₂²⋊C₄	1	2	yes	—
103	M₁₆×C₃₁	—	2	2	no	—
104	M₁₆×C₃₁	—	2	4	no	—
105	M₁₆×C₃₁	—	4	2	no	—
106	M₁₆×C₃₁	—	4	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	4	2	no	—
110	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₃₁	—	4	2	no	—
116	#12744	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	1	no	—
118	C₂³×C₃₁	#102	1	2	yes	—
119	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
120	Q₈×C₃₁	—	2	2	no	—
121	Q₈×C₃₁	—	2	4	no	—
122	Q₈×C₃₁	—	2	2	no	—
123	Q₈×C₃₁	—	4	8	no	—
124	Q₈×C₃₁	—	4	1	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	2	4	no	—
128	D₄×C₃₁	—	2	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	4	8	no	—
131	D₄×C₃₁	—	4	4	no	—
132	D₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	4	8	no	—
136	C₂×C₄×C₃₁	—	4	8	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	4	no	—
140	C₂×C₄×C₃₁	—	4	4	no	—
141	C₂×C₄×C₃₁	#102	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#120	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
146	C₈×C₃₁	—	2	2	no	—
147	C₈×C₃₁	—	8	2	no	—
148	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₈×C₃₁	#120	1	2	yes	—
151	C₂×#578	C₂×C₃₁	1	1	yes	characteristic
152	C₂×#362	C₂×C₃₁	1	2	yes	complemented
153	#2143	C₂×C₃₁	1	2	yes	complemented
154	#2115	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	—	4	2	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
162	C₂²×C₃₁	#450	1	2	yes	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	4	4	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	#379	1	1	yes	characteristic
167	C₄×C₃₁	#418	1	1	yes	characteristic
168	C₄×C₃₁	#447	1	2	yes	—
169	C₂×2₋¹⁺⁴	—	2	1	no	—
170	#578	—	2	4	no	—
171	#578	—	2	4	no	—
172	C₂×#137	—	2	1	no	—
173	C₂×#137	—	2	2	no	—
174	C₂×#135	C₄×C₃₁	1	2	yes	—
175	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
176	C₂²×Q₁₆	C₄×C₃₁	1	2	yes	—
177	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
180	C₂×#108	C₂²×C₃₁	1	2	yes	—
181	#413	—	2	1	no	—
182	#362	—	2	4	no	—
183	#359	—	2	4	no	—
184	#349	—	2	2	no	—
185	#338	—	2	1	no	—
186	C₂×C₃₁	—	2	1	no	—
187	C₂×C₃₁	—	2	2	no	—
188	C₂×C₃₁	—	4	2	no	—
189	C₂×C₃₁	—	8	2	no	—
190	C₂×C₃₁	#1786	1	1	yes	characteristic
191	2₋¹⁺⁴	—	2	8	no	—
192	2₋¹⁺⁴	—	4	4	no	—
193	C₂×#41	—	2	2	no	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	—	4	1	no	—
196	C₂×#41	—	4	2	no	—
197	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	4	1	no	—
201	#137	—	2	8	no	—
202	#137	—	2	16	no	—
203	#135	—	2	4	no	—
204	#135	—	2	4	no	—
205	#135	—	2	2	no	—
206	#135	D₄×C₃₁	1	4	yes	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×Q₁₆	—	2	4	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
213	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
222	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
223	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	M₃₂	—	2	2	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	#108	—	2	2	no	—
229	#108	—	2	2	no	—
230	#101	—	4	1	no	—
231	#100	—	4	1	no	—
232	C₃₁	#12744	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	2	8	no	—
236	#41	—	2	4	no	—
237	#41	—	4	4	no	—
238	#41	—	4	16	no	—
239	#41	—	4	8	no	—
240	#41	—	4	2	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	4	4	no	—
246	C₂×Q₈	—	4	8	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
255	C₂×D₄	SD₁₆×C₃₁	1	2	yes	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
262	Q₁₆	—	2	8	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	4	no	—
266	Q₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
267	Q₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
268	SD₁₆	—	2	8	no	—
269	SD₁₆	—	2	4	no	—
270	D₈	—	2	1	no	—
271	D₈	C₂²⋊C₄×C₃₁	1	2	yes	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	4	2	no	—
275	M₁₆	—	4	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
280	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₁₆	—	4	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₃₁×#102	1	2	yes	—
287	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	2	4	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	4	8	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	4	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	8	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	C₃₁×#102	1	2	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄	C₃₁×#120	1	2	yes	—
312	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
313	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	—	8	2	no	—
316	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
317	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
318	C₈	C₃₁×#120	1	2	yes	—
319	C₂²	—	2	2	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
325	C₂²	C₃₁×#450	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	4	2	no	—
329	C₄	C₃₁×#379	1	1	yes	characteristic
330	C₄	C₃₁×#418	1	1	yes	characteristic
331	C₄	C₃₁×#447	1	2	yes	—
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	4	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₃₁×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1786
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#362	C₂	semidirect
3. C₃₁×#2143	C₂	semidirect
4. C₃₁×#2115	C₂	semidirect
5. C₃₁×#1813	C₂	semidirect
116. #12744	C₃₁	direct (N × H)
152. C₂×#362	C₂×C₃₁	semidirect
153. #2143	C₂×C₃₁	semidirect
154. #2115	C₂×C₃₁	semidirect
155. #1813	C₂×C₃₁	semidirect
232. C₃₁	#12744	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×#578	2
C₂×C₃₁×#362	2
C₃₁×#2143	2
C₃₁×#2115	2
C₃₁×#1813	2
#12744	31

Subgroups — C31×#12744 #21469503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12744 #21469503