Subgroups — C₃₁×#33361 #21490120

Order 7936.

1252 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2883	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2129	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2127	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2126	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#354	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×2₋¹⁺⁴	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
9	C₃₁×#548	—	2	2	no	—
10	C₃₁×#546	—	2	4	no	—
11	C₃₁×#546	—	2	2	no	—
12	C₃₁×#536	C₄	1	2	yes	complemented
13	C₃₁×#536	C₂²	1	1	yes	characteristic
14	C₃₁×#535	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#535	C₂²	1	2	yes	—
16	C₃₁×#532	C₄	1	2	yes	complemented
17	C₃₁×#532	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#124	C₄	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#124	C₂²	1	2	yes	—
20	C₃₁×#518	—	2	2	no	—
21	C₄×#41×C₃₁	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#410	—	2	2	no	—
23	C₃₁×#410	C₂²	1	4	yes	—
24	C₂×C₃₁×#99	—	2	4	no	—
25	C₂×C₃₁×#99	C₂²	1	8	yes	—
26	C₃₁×#354	C₂²	1	4	yes	complemented
27	C₃₁×#354	C₂²	1	8	yes	complemented
28	C₃₁×#353	C₂²	1	4	yes	complemented
29	C₃₁×2₋¹⁺⁴	—	4	4	no	—
30	C₃₁×H(5,𝔽₂)	—	4	4	no	—
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
36	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×#41×C₃₁	D₄	1	2	yes	—
38	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
42	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	D₄	1	4	yes	—
44	C₂²×D₄×C₃₁	D₄	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
47	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
48	C₃₁×#127	C₂×C₄	1	2	yes	complemented
49	C₃₁×#127	C₂³	1	1	yes	characteristic
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×#124	—	2	4	no	—
52	C₃₁×#124	C₂×C₄	1	2	yes	complemented
53	C₃₁×#124	C₂×C₄	1	2	yes	complemented
54	C₃₁×#124	C₂×C₄	1	2	yes	complemented
55	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
56	C₃₁×#124	C₂³	1	1	yes	characteristic
57	C₃₁×#124	C₂³	1	2	yes	—
58	C₃₁×#123	—	2	4	no	—
59	C₃₁×#123	—	2	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×#121	—	2	4	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	2	4	no	—
66	C₃₁×#120	—	2	2	no	—
67	C₃₁×#120	—	2	4	no	—
68	C₄×Q₈×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	4	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	—	4	8	no	—
74	C₃₁×#117	—	2	4	no	—
75	C₃₁×#117	C₂×C₄	1	1	yes	characteristic, complemented
76	C₃₁×#117	C₂×C₄	1	2	yes	complemented
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
80	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
81	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	complemented
82	C₃₁×#99	—	2	4	no	—
83	C₃₁×#99	—	2	8	no	—
84	C₃₁×#99	C₂³	1	8	yes	—
85	C₂⁴×C₃₁	C₂²⋊C₄	1	2	yes	—
86	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
87	#41×C₃₁	—	2	4	no	—
88	#41×C₃₁	—	4	2	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	4	4	no	—
91	#41×C₃₁	—	4	4	no	—
92	#41×C₃₁	—	4	2	no	—
93	#41×C₃₁	—	4	8	no	—
94	#41×C₃₁	—	4	4	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	—	4	4	no	—
99	C₂×Q₈×C₃₁	—	4	4	no	—
100	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₄×C₃₁	—	2	8	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	8	no	—
112	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
113	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
114	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
115	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
120	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	4	8	no	—
128	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	—	2	4	no	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	2	no	—
142	C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂²⋊C₄×C₃₁	—	2	4	no	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	8	no	—
146	C₂²⋊C₄×C₃₁	—	4	8	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	4	4	no	—
150	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
153	#33361	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	8	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	8	4	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
160	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
163	Q₈×C₃₁	—	2	2	no	—
164	Q₈×C₃₁	—	4	1	no	—
165	Q₈×C₃₁	—	4	2	no	—
166	Q₈×C₃₁	—	4	1	no	—
167	Q₈×C₃₁	—	4	4	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	2	8	no	—
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	4	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	4	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	1	no	—
176	D₄×C₃₁	—	4	1	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	4	no	—
179	D₄×C₃₁	—	8	4	no	—
180	D₄×C₃₁	—	8	2	no	—
181	D₄×C₃₁	#99	1	4	yes	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	4	4	no	—
191	C₂×C₄×C₃₁	—	4	4	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	8	2	no	—
197	C₂×C₄×C₃₁	—	8	1	no	—
198	C₂×C₄×C₃₁	—	16	8	no	—
199	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
201	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
203	#3500	C₂×C₃₁	1	1	yes	characteristic
204	#2883	C₂×C₃₁	1	2	yes	complemented
205	#2129	C₂×C₃₁	1	2	yes	complemented
206	#2127	C₂×C₃₁	1	4	yes	complemented
207	#2126	C₂×C₃₁	1	2	yes	complemented
208	C₂×#354	C₂×C₃₁	1	4	yes	complemented
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	1	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	4	2	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	4	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	8	8	no	—
219	C₂²×C₃₁	—	16	2	no	—
220	C₂²×C₃₁	C₂×#99	1	4	yes	—
221	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	4	1	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	—	8	8	no	—
231	C₄×C₃₁	C₂×#99	1	2	yes	—
232	C₂×2₋¹⁺⁴	C₄×C₃₁	1	1	yes	characteristic, complemented
233	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
234	#548	—	2	2	no	—
235	#546	—	2	4	no	—
236	#546	—	2	2	no	—
237	#536	C₄×C₃₁	1	2	yes	complemented
238	#536	C₂²×C₃₁	1	1	yes	characteristic
239	#535	C₄×C₃₁	1	1	yes	characteristic, complemented
240	#535	C₂²×C₃₁	1	2	yes	—
241	#532	C₄×C₃₁	1	2	yes	complemented
242	#532	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#124	C₄×C₃₁	1	1	yes	characteristic, complemented
244	C₂×#124	C₂²×C₃₁	1	2	yes	—
245	#518	—	2	2	no	—
246	C₄×#41	C₄×C₃₁	1	1	yes	characteristic, complemented
247	#410	—	2	2	no	—
248	#410	C₂²×C₃₁	1	4	yes	—
249	C₂×#99	—	2	4	no	—
250	C₂×#99	C₂²×C₃₁	1	8	yes	—
251	#354	C₂²×C₃₁	1	4	yes	complemented
252	#354	C₂²×C₃₁	1	8	yes	complemented
253	#353	C₂²×C₃₁	1	4	yes	complemented
254	C₂×C₃₁	—	2	1	no	—
255	C₂×C₃₁	—	2	4	no	—
256	C₂×C₃₁	—	4	2	no	—
257	C₂×C₃₁	—	4	4	no	—
258	C₂×C₃₁	—	4	1	no	—
259	C₂×C₃₁	—	8	1	no	—
260	C₂×C₃₁	C₂²×#99	1	1	yes	characteristic
261	2₋¹⁺⁴	—	4	4	no	—
262	H(5,𝔽₂)	—	4	4	no	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	2	1	no	—
265	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
266	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
268	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
269	C₂×#41	D₄×C₃₁	1	2	yes	—
270	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
273	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
274	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×D₄	D₄×C₃₁	1	4	yes	—
276	C₂²×D₄	D₄×C₃₁	1	2	yes	—
277	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
278	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
279	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
280	#127	C₂×C₄×C₃₁	1	2	yes	complemented
281	#127	C₂³×C₃₁	1	1	yes	characteristic
282	#124	—	2	2	no	—
283	#124	—	2	4	no	—
284	#124	C₂×C₄×C₃₁	1	2	yes	complemented
285	#124	C₂×C₄×C₃₁	1	2	yes	complemented
286	#124	C₂×C₄×C₃₁	1	2	yes	complemented
287	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
288	#124	C₂³×C₃₁	1	1	yes	characteristic
289	#124	C₂³×C₃₁	1	2	yes	—
290	#123	—	2	4	no	—
291	#123	—	2	2	no	—
292	C₂²⋊Q₈	—	2	4	no	—
293	C₂²⋊Q₈	—	2	2	no	—
294	#121	—	2	4	no	—
295	#121	—	2	2	no	—
296	#121	—	2	2	no	—
297	#121	—	2	4	no	—
298	#120	—	2	2	no	—
299	#120	—	2	4	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	—	2	1	no	—
303	C₄×D₄	—	2	4	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	4	8	no	—
306	#117	—	2	4	no	—
307	#117	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
308	#117	C₂×C₄×C₃₁	1	2	yes	complemented
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	4	no	—
311	C₂×C₂²⋊C₄	—	2	8	no	—
312	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
313	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	complemented
314	#99	—	2	4	no	—
315	#99	—	2	8	no	—
316	#99	C₂³×C₃₁	1	8	yes	—
317	C₃₁	#33361	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	C₂²⋊C₄×C₃₁	1	2	yes	—
319	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
320	#41	—	2	4	no	—
321	#41	—	4	2	no	—
322	#41	—	4	4	no	—
323	#41	—	4	4	no	—
324	#41	—	4	4	no	—
325	#41	—	4	2	no	—
326	#41	—	4	8	no	—
327	#41	—	4	4	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	4	no	—
332	C₂×Q₈	—	4	4	no	—
333	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
334	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
336	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	4	no	—
344	C₂×D₄	—	4	8	no	—
345	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
346	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
347	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
348	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
349	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
353	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	8	no	—
361	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	1	no	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	2	8	no	—
379	C₂²⋊C₄	—	4	8	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	4	no	—
383	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
384	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
385	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂³	—	2	8	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	8	4	no	—
390	C₂³	—	8	2	no	—
391	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
392	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
393	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
394	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
395	Q₈	—	2	2	no	—
396	Q₈	—	4	1	no	—
397	Q₈	—	4	2	no	—
398	Q₈	—	4	1	no	—
399	Q₈	—	4	4	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	8	no	—
402	D₄	—	2	4	no	—
403	D₄	—	2	4	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	4	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	1	no	—
408	D₄	—	4	1	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	4	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	2	no	—
413	D₄	C₃₁×#99	1	4	yes	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	8	1	no	—
430	C₂×C₄	—	16	8	no	—
431	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
432	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
435	C₂²	—	2	2	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	8	8	no	—
445	C₂²	—	16	2	no	—
446	C₂²	C₂×C₃₁×#99	1	4	yes	—
447	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	C₄	—	2	1	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	1	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	8	8	no	—
457	C₄	C₂×C₃₁×#99	1	2	yes	—
458	C₂	—	2	1	no	—
459	C₂	—	2	4	no	—
460	C₂	—	4	2	no	—
461	C₂	—	4	4	no	—
462	C₂	—	4	1	no	—
463	C₂	—	8	1	no	—
464	C₂	C₂²×C₃₁×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₃₁
prime radical	C₃₁×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×D₄×C₃₁ ◁ C₂×C₃₁×H(5,𝔽₂) ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2883	C₂	semidirect
3. C₃₁×#2129	C₂	semidirect
4. C₃₁×#2127	C₂	semidirect
5. C₃₁×#2126	C₂	semidirect
6. C₂×C₃₁×#354	C₂	semidirect
7. C₂×C₃₁×2₋¹⁺⁴	C₄	semidirect
12. C₃₁×#536	C₄	semidirect
14. C₃₁×#535	C₄	semidirect
16. C₃₁×#532	C₄	semidirect
18. C₂×C₃₁×#124	C₄	semidirect
21. C₄×#41×C₃₁	C₄	semidirect
26. C₃₁×#354	C₂²	semidirect
27. C₃₁×#354	C₂²	semidirect
28. C₃₁×#353	C₂²	semidirect
35. C₂×#41×C₃₁	C₂×C₄	semidirect
36. C₂×#41×C₃₁	C₂×C₄	semidirect
40. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
41. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
46. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
47. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
48. C₃₁×#127	C₂×C₄	semidirect
52. C₃₁×#124	C₂×C₄	semidirect
53. C₃₁×#124	C₂×C₄	semidirect
54. C₃₁×#124	C₂×C₄	semidirect
55. C₃₁×#124	C₂×C₄	semidirect
75. C₃₁×#117	C₂×C₄	semidirect
76. C₃₁×#117	C₂×C₄	semidirect
80. C₂×C₄²×C₃₁	C₂×C₄	semidirect
81. C₂×C₄²×C₃₁	C₂×C₄	semidirect
153. #33361	C₃₁	direct (N × H)
204. #2883	C₂×C₃₁	semidirect
205. #2129	C₂×C₃₁	semidirect
206. #2127	C₂×C₃₁	semidirect
207. #2126	C₂×C₃₁	semidirect
208. C₂×#354	C₂×C₃₁	semidirect
232. C₂×2₋¹⁺⁴	C₄×C₃₁	semidirect
237. #536	C₄×C₃₁	semidirect
239. #535	C₄×C₃₁	semidirect
241. #532	C₄×C₃₁	semidirect
243. C₂×#124	C₄×C₃₁	semidirect
246. C₄×#41	C₄×C₃₁	semidirect
251. #354	C₂²×C₃₁	semidirect
252. #354	C₂²×C₃₁	semidirect
253. #353	C₂²×C₃₁	semidirect
267. C₂×#41	C₂×C₄×C₃₁	semidirect
268. C₂×#41	C₂×C₄×C₃₁	semidirect
272. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
273. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
278. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
279. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
280. #127	C₂×C₄×C₃₁	semidirect
284. #124	C₂×C₄×C₃₁	semidirect
285. #124	C₂×C₄×C₃₁	semidirect
286. #124	C₂×C₄×C₃₁	semidirect
287. #124	C₂×C₄×C₃₁	semidirect
307. #117	C₂×C₄×C₃₁	semidirect
308. #117	C₂×C₄×C₃₁	semidirect
312. C₂×C₄²	C₂×C₄×C₃₁	semidirect
313. C₂×C₄²	C₂×C₄×C₃₁	semidirect
317. C₃₁	#33361	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×#3500	2
C₃₁×#2883	2
C₃₁×#2129	2
C₃₁×#2127	2
C₃₁×#2126	2
C₂×C₃₁×#354	2
#33361	31

Subgroups — C31×#33361 #21490120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33361 #21490120