Subgroups — C₃₁×#21210 #21477969

Order 7936.

1570 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#421	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#104	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1963	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1962	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#537	—	2	2	no	—
11	C₃₁×#536	C₄	1	4	yes	complemented
12	C₃₁×#535	C₄	1	4	yes	complemented
13	C₃₁×#534	—	2	2	no	—
14	C₃₁×#532	C₂²	1	4	yes	—
15	C₃₁×#525	—	2	4	no	—
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#424	—	2	4	no	—
20	C₃₁×#423	C₂²	1	4	yes	complemented
21	C₃₁×#421	—	2	8	no	—
22	C₂×C₃₁×#104	—	2	8	no	—
23	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
24	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#104	C₂²	1	8	yes	complemented
26	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
28	C₃₁×#376	C₂²	1	2	yes	complemented
29	C₄³×C₃₁	C₂²	1	2	yes	complemented
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	4	no	—
33	C₂×#41×C₃₁	—	2	4	no	—
34	C₂×#41×C₃₁	—	2	4	no	—
35	C₂×#41×C₃₁	—	2	4	no	—
36	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×#41×C₃₁	C₂³	1	4	yes	—
38	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂³×C₄×C₃₁	—	2	2	no	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
50	C₃₁×#127	C₂×C₄	1	4	yes	complemented
51	C₃₁×#124	C₂×C₄	1	8	yes	complemented
52	C₃₁×#123	—	2	8	no	—
53	C₃₁×C₂²⋊Q₈	—	2	8	no	—
54	C₃₁×#121	—	2	8	no	—
55	C₃₁×#121	—	2	8	no	—
56	C₃₁×#120	—	2	8	no	—
57	C₄×D₄×C₃₁	—	2	8	no	—
58	C₃₁×#117	—	2	2	no	—
59	C₃₁×#117	C₂×C₄	1	4	yes	—
60	C₃₁×#117	D₄	1	4	yes	—
61	C₃₁×#117	D₄	1	4	yes	complemented
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄²×C₃₁	—	2	1	no	—
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	4	no	—
69	C₂×C₄²×C₃₁	D₄	1	4	yes	complemented
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	—	2	8	no	—
75	C₃₁×#104	—	2	8	no	—
76	C₃₁×#104	—	2	8	no	—
77	C₃₁×#104	—	2	16	no	—
78	C₃₁×#95	—	2	4	no	—
79	C₂⁴×C₃₁	—	2	2	no	—
80	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
81	#41×C₃₁	—	2	8	no	—
82	#41×C₃₁	—	2	8	no	—
83	#41×C₃₁	—	2	4	no	—
84	#41×C₃₁	—	2	4	no	—
85	#41×C₃₁	—	2	4	no	—
86	#41×C₃₁	—	2	4	no	—
87	#41×C₃₁	—	4	4	no	—
88	#41×C₃₁	—	4	4	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	4	4	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	4	no	—
93	C₂×Q₈×C₃₁	—	2	4	no	—
94	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	4	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	8	no	—
99	C₂×D₄×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	4	4	no	—
102	C₂×D₄×C₃₁	—	4	4	no	—
103	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	4	4	no	—
112	C₂²×C₄×C₃₁	—	4	4	no	—
113	C₂²×C₄×C₃₁	—	4	4	no	—
114	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	#41	1	2	yes	—
121	C₂²×C₄×C₃₁	#41	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₃₁	—	2	4	no	—
124	M₁₆×C₃₁	—	2	4	no	—
125	M₁₆×C₃₁	—	4	4	no	—
126	C₂×C₈×C₃₁	—	2	4	no	—
127	C₂×C₈×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	—	2	8	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
131	C₂²⋊C₄×C₃₁	—	2	4	no	—
132	C₂²⋊C₄×C₃₁	—	2	16	no	—
133	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
134	C₄²×C₃₁	—	2	4	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	8	no	—
139	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
140	C₄²×C₃₁	C₂×D₄	1	2	yes	—
141	C₄²×C₃₁	C₂×D₄	1	2	yes	—
142	C₄²×C₃₁	C₂×D₄	1	4	yes	—
143	C₄²×C₃₁	C₂×D₄	1	4	yes	—
144	#21210	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	4	no	—
148	C₂³×C₃₁	—	8	4	no	—
149	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂³×C₃₁	#121	1	2	yes	—
151	C₂³×C₃₁	#121	1	2	yes	—
152	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	4	no	—
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	4	4	no	—
156	D₄×C₃₁	—	2	4	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	4	no	—
160	D₄×C₃₁	—	4	8	no	—
161	D₄×C₃₁	—	4	4	no	—
162	D₄×C₃₁	—	8	8	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	—	4	8	no	—
170	C₂×C₄×C₃₁	—	4	4	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	4	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	4	4	no	—
175	C₂×C₄×C₃₁	—	4	8	no	—
176	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
177	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
179	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₃₁	#121	1	2	yes	—
181	C₂×C₄×C₃₁	#121	1	2	yes	—
182	C₂×C₄×C₃₁	#124	1	4	yes	—
183	C₂×C₄×C₃₁	#127	1	4	yes	—
184	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	4	4	no	—
191	#3449	C₂×C₃₁	1	1	yes	characteristic
192	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂×#421	C₂×C₃₁	1	2	yes	complemented
194	C₂²×#104	C₂×C₃₁	1	2	yes	complemented
195	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1963	C₂×C₃₁	1	4	yes	complemented
197	#1962	C₂×C₃₁	1	4	yes	complemented
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	4	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	8	no	—
203	C₂²×C₃₁	—	4	4	no	—
204	C₂²×C₃₁	—	8	8	no	—
205	C₂²×C₃₁	#390	1	4	yes	—
206	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
208	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	2	8	no	—
211	C₄×C₃₁	—	4	4	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	4	no	—
214	C₄×C₃₁	#390	1	4	yes	—
215	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
218	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
219	C₂²×#41	C₂²×C₃₁	1	2	yes	—
220	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
221	#537	—	2	2	no	—
222	#536	C₄×C₃₁	1	4	yes	complemented
223	#535	C₄×C₃₁	1	4	yes	complemented
224	#534	—	2	2	no	—
225	#532	C₂²×C₃₁	1	4	yes	—
226	#525	—	2	4	no	—
227	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
229	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
230	#424	—	2	4	no	—
231	#423	C₂²×C₃₁	1	4	yes	complemented
232	#421	—	2	8	no	—
233	C₂×#104	—	2	8	no	—
234	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
235	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
236	C₂×#104	C₂²×C₃₁	1	8	yes	complemented
237	C₄×C₄⋊C₄	—	2	2	no	—
238	C₄×C₂²⋊C₄	—	2	2	no	—
239	#376	C₂²×C₃₁	1	2	yes	complemented
240	C₄³	C₂²×C₃₁	1	2	yes	complemented
241	C₂×C₃₁	—	2	1	no	—
242	C₂×C₃₁	—	2	1	no	—
243	C₂×C₃₁	—	2	4	no	—
244	C₂×C₃₁	—	8	4	no	—
245	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
246	C₂×C₃₁	#2901	1	2	yes	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	—	2	2	no	—
249	C₂×#41	—	2	4	no	—
250	C₂×#41	—	2	4	no	—
251	C₂×#41	—	2	4	no	—
252	C₂×#41	—	2	4	no	—
253	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
254	C₂×#41	C₂³×C₃₁	1	4	yes	—
255	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
256	C₂²×D₄	—	2	2	no	—
257	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	—	2	2	no	—
260	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	4	no	—
263	C₂×M₁₆	—	2	4	no	—
264	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
265	C₂²×C₈	—	2	2	no	—
266	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
267	#127	C₂×C₄×C₃₁	1	4	yes	complemented
268	#124	C₂×C₄×C₃₁	1	8	yes	complemented
269	#123	—	2	8	no	—
270	C₂²⋊Q₈	—	2	8	no	—
271	#121	—	2	8	no	—
272	#121	—	2	8	no	—
273	#120	—	2	8	no	—
274	C₄×D₄	—	2	8	no	—
275	#117	—	2	2	no	—
276	#117	C₂×C₄×C₃₁	1	4	yes	—
277	#117	D₄×C₃₁	1	4	yes	—
278	#117	D₄×C₃₁	1	4	yes	complemented
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
281	C₂×C₂²⋊C₄	—	2	1	no	—
282	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	—	2	4	no	—
286	C₂×C₄²	D₄×C₃₁	1	4	yes	complemented
287	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
291	C₄⋊C₈	—	2	8	no	—
292	#104	—	2	8	no	—
293	#104	—	2	8	no	—
294	#104	—	2	16	no	—
295	#95	—	2	4	no	—
296	C₃₁	#21210	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	—	2	2	no	—
298	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
299	#41	—	2	8	no	—
300	#41	—	2	8	no	—
301	#41	—	2	4	no	—
302	#41	—	2	4	no	—
303	#41	—	2	4	no	—
304	#41	—	2	4	no	—
305	#41	—	4	4	no	—
306	#41	—	4	4	no	—
307	#41	—	4	4	no	—
308	#41	—	4	4	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	4	4	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	4	4	no	—
332	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
338	C₂²×C₄	#41×C₃₁	1	2	yes	—
339	C₂²×C₄	#41×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
341	M₁₆	—	2	4	no	—
342	M₁₆	—	2	4	no	—
343	M₁₆	—	4	4	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	4	4	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
349	C₂²⋊C₄	—	2	4	no	—
350	C₂²⋊C₄	—	2	16	no	—
351	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
352	C₄²	—	2	4	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	4	8	no	—
357	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
358	C₄²	C₂×D₄×C₃₁	1	2	yes	—
359	C₄²	C₂×D₄×C₃₁	1	2	yes	—
360	C₄²	C₂×D₄×C₃₁	1	4	yes	—
361	C₄²	C₂×D₄×C₃₁	1	4	yes	—
362	C₂³	—	2	4	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	4	no	—
365	C₂³	—	8	4	no	—
366	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
367	C₂³	C₃₁×#121	1	2	yes	—
368	C₂³	C₃₁×#121	1	2	yes	—
369	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	4	4	no	—
373	D₄	—	2	4	no	—
374	D₄	—	2	4	no	—
375	D₄	—	4	4	no	—
376	D₄	—	4	4	no	—
377	D₄	—	4	8	no	—
378	D₄	—	4	4	no	—
379	D₄	—	8	8	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	4	no	—
392	C₂×C₄	—	4	8	no	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
394	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
396	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
397	C₂×C₄	C₃₁×#121	1	2	yes	—
398	C₂×C₄	C₃₁×#121	1	2	yes	—
399	C₂×C₄	C₃₁×#124	1	4	yes	—
400	C₂×C₄	C₃₁×#127	1	4	yes	—
401	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
407	C₈	—	4	4	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	8	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	8	8	no	—
415	C₂²	C₃₁×#390	1	4	yes	—
416	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
417	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
418	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
419	C₄	—	2	4	no	—
420	C₄	—	2	8	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	4	no	—
424	C₄	C₃₁×#390	1	4	yes	—
425	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
426	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
427	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
428	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
429	C₂	—	2	1	no	—
430	C₂	—	2	1	no	—
431	C₂	—	2	4	no	—
432	C₂	—	8	4	no	—
433	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
434	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2905	C₂	semidirect
3. C₂×C₃₁×#421	C₂	semidirect
4. C₂²×C₃₁×#104	C₂	semidirect
5. C₄×C₃₁×#117	C₂	semidirect
6. C₃₁×#1963	C₂	semidirect
7. C₃₁×#1962	C₂	semidirect
11. C₃₁×#536	C₄	semidirect
12. C₃₁×#535	C₄	semidirect
20. C₃₁×#423	C₂²	semidirect
23. C₂×C₃₁×#104	C₂²	semidirect
24. C₂×C₃₁×#104	C₂²	semidirect
25. C₂×C₃₁×#104	C₂²	semidirect
28. C₃₁×#376	C₂²	semidirect
29. C₄³×C₃₁	C₂²	semidirect
49. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
50. C₃₁×#127	C₂×C₄	semidirect
51. C₃₁×#124	C₂×C₄	semidirect
61. C₃₁×#117	D₄	semidirect
69. C₂×C₄²×C₃₁	D₄	semidirect
144. #21210	C₃₁	direct (N × H)
192. #2905	C₂×C₃₁	semidirect
193. C₂×#421	C₂×C₃₁	semidirect
194. C₂²×#104	C₂×C₃₁	semidirect
195. C₄×#117	C₂×C₃₁	semidirect
196. #1963	C₂×C₃₁	semidirect
197. #1962	C₂×C₃₁	semidirect
222. #536	C₄×C₃₁	semidirect
223. #535	C₄×C₃₁	semidirect
231. #423	C₂²×C₃₁	semidirect
234. C₂×#104	C₂²×C₃₁	semidirect
235. C₂×#104	C₂²×C₃₁	semidirect
236. C₂×#104	C₂²×C₃₁	semidirect
239. #376	C₂²×C₃₁	semidirect
240. C₄³	C₂²×C₃₁	semidirect
266. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
267. #127	C₂×C₄×C₃₁	semidirect
268. #124	C₂×C₄×C₃₁	semidirect
278. #117	D₄×C₃₁	semidirect
286. C₂×C₄²	D₄×C₃₁	semidirect
296. C₃₁	#21210	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3449	2
C₃₁×#2905	2
C₂×C₃₁×#421	2
C₂²×C₃₁×#104	2
C₄×C₃₁×#117	2
C₃₁×#1963	2
C₃₁×#1962	2
#21210	31

Subgroups — C31×#21210 #21477969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21210 #21477969