Subgroups — C₃₁×#19268 #21476027

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	2	yes	maximal
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1670	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1669	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1549	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₃₁	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#494	—	2	1	no	—
12	C₃₁×#494	—	2	1	no	—
13	C₃₁×#494	C₂²	1	4	yes	—
14	C₃₁×#477	—	2	2	no	—
15	C₃₁×#476	—	2	2	no	—
16	C₃₁×#475	C₂²	1	4	yes	—
17	C₃₁×#474	C₂²	1	4	yes	—
18	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₈×D₄×C₃₁	—	2	2	no	—
20	C₈×D₄×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#431	—	2	1	no	—
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₃₁×#428	—	2	2	no	—
24	C₃₁×#423	C₂²	1	2	yes	—
25	C₃₁×#417	—	2	2	no	—
26	C₂×C₄×C₈×C₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#335	C₂²	1	4	yes	—
29	C₃₁×#332	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×Q₁₆×C₃₁	—	4	8	no	—
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#117	—	2	1	no	—
49	C₃₁×#117	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	—	2	4	no	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
59	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
60	C₃₁×#103	—	2	4	no	—
61	C₃₁×#102	—	2	4	no	—
62	C₃₁×C₂²⋊C₈	—	2	1	no	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	2	yes	—
69	#41×C₃₁	—	2	2	no	—
70	#41×C₃₁	Q₁₆	1	4	yes	complemented
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	4	8	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	—	2	1	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
84	Q₁₆×C₃₁	—	4	8	no	—
85	Q₁₆×C₃₁	—	4	8	no	—
86	M₁₆×C₃₁	—	4	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	4	4	no	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊C₄×C₃₁	—	2	1	no	—
102	C₂²⋊C₄×C₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	#41	1	1	yes	characteristic
110	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
111	#19268	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	2	1	no	—
113	C₂³×C₃₁	#121	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	1	no	—
115	Q₈×C₃₁	—	4	8	no	—
116	Q₈×C₃₁	—	8	4	no	—
117	Q₈×C₃₁	C₂×Q₁₆	1	2	yes	complemented
118	D₄×C₃₁	—	2	4	no	—
119	D₄×C₃₁	—	2	1	no	—
120	D₄×C₃₁	C₂×Q₁₆	1	2	yes	complemented
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	4	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₃₁	#135	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₃₁	—	2	2	no	—
140	C₈×C₃₁	—	4	4	no	—
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
143	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#3079	C₂×C₃₁	1	2	yes	—
145	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
146	#1670	C₂×C₃₁	1	4	yes	complemented
147	#1669	C₂×C₃₁	1	4	yes	complemented
148	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#1549	C₂×C₃₁	1	2	yes	complemented
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	#467	1	2	yes	—
156	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	2	no	—
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	4	1	no	—
161	C₄×C₃₁	—	4	1	no	—
162	C₄×C₃₁	—	8	4	no	—
163	C₄×C₃₁	#467	1	2	yes	—
164	C₄×C₃₁	#473	1	2	yes	—
165	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
166	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
167	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
168	C₂²×Q₁₆	—	2	2	no	—
169	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
170	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
171	#494	—	2	1	no	—
172	#494	—	2	1	no	—
173	#494	C₂²×C₃₁	1	4	yes	—
174	#477	—	2	2	no	—
175	#476	—	2	2	no	—
176	#475	C₂²×C₃₁	1	4	yes	—
177	#474	C₂²×C₃₁	1	4	yes	—
178	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
179	C₈×D₄	—	2	2	no	—
180	C₈×D₄	C₂²×C₃₁	1	2	yes	—
181	#431	—	2	1	no	—
182	#431	C₂²×C₃₁	1	1	yes	characteristic
183	#428	—	2	2	no	—
184	#423	C₂²×C₃₁	1	2	yes	—
185	#417	—	2	2	no	—
186	C₂×C₄×C₈	—	2	1	no	—
187	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
188	#335	C₂²×C₃₁	1	4	yes	—
189	#332	C₂²×C₃₁	1	8	yes	—
190	C₂×C₃₁	—	2	2	no	—
191	C₂×C₃₁	—	4	2	no	—
192	C₂×C₃₁	#3038	1	1	yes	characteristic
193	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
194	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
195	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×Q₈	—	2	2	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	—	4	8	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	D₄×C₃₁	1	2	yes	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
207	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
208	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
212	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
213	#117	—	2	1	no	—
214	#117	D₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	#107	—	2	4	no	—
220	#106	—	2	4	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
224	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
225	#103	—	2	4	no	—
226	#102	—	2	4	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
232	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₃₁	1	2	yes	—
234	C₃₁	#19268	1	1	yes	characteristic, direct product factor, Sylow
235	#41	—	2	2	no	—
236	#41	Q₁₆×C₃₁	1	4	yes	complemented
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	8	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
250	Q₁₆	—	4	8	no	—
251	Q₁₆	—	4	8	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	4	no	—
257	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₄²	#41×C₃₁	1	1	yes	characteristic
276	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	C₃₁×#121	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	8	4	no	—
282	Q₈	C₂×Q₁₆×C₃₁	1	2	yes	complemented
283	D₄	—	2	4	no	—
284	D₄	—	2	1	no	—
285	D₄	C₂×Q₁₆×C₃₁	1	2	yes	complemented
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
298	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
300	C₂×C₄	C₃₁×#135	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	2	no	—
307	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₃₁×#467	1	2	yes	—
314	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	8	4	no	—
321	C₄	C₃₁×#467	1	2	yes	—
322	C₄	C₃₁×#473	1	2	yes	—
323	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
324	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
325	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	C₃₁×#3038	1	1	yes	characteristic
329	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
330	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₄² ▷ 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
1. C₂×C₃₁×#494	C₂	semidirect
4. C₃₁×#1670	C₂	semidirect
5. C₃₁×#1669	C₂	semidirect
6. C₃₁×#1567	C₂	semidirect
7. C₃₁×#1549	C₂	semidirect
70. #41×C₃₁	Q₁₆	semidirect
111. #19268	C₃₁	direct (N × H)
117. Q₈×C₃₁	C₂×Q₁₆	semidirect
120. D₄×C₃₁	C₂×Q₁₆	semidirect
143. C₂×#494	C₂×C₃₁	semidirect
146. #1670	C₂×C₃₁	semidirect
147. #1669	C₂×C₃₁	semidirect
148. #1567	C₂×C₃₁	semidirect
149. #1549	C₂×C₃₁	semidirect
234. C₃₁	#19268	direct (N × H)
236. #41	Q₁₆×C₃₁	semidirect
282. Q₈	C₂×Q₁₆×C₃₁	semidirect
285. D₄	C₂×Q₁₆×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×#494	2
C₃₁×#3079	2
C₈×#41×C₃₁	2
C₃₁×#1670	2
C₃₁×#1669	2
C₃₁×#1567	2
C₃₁×#1549	2
#19268	31

Subgroups — C31×#19268 #21476027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19268 #21476027