Subgroups — C₃₁×#33359 #21490118

Order 7936.

1356 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2883	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2128	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×#353	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×H(5,𝔽₂)	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
9	C₃₁×#550	—	2	2	no	—
10	D₄²×C₃₁	—	2	4	no	—
11	D₄²×C₃₁	—	2	2	no	—
12	C₃₁×#535	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#535	C₄	1	2	yes	complemented
14	C₃₁×#535	C₂²	1	1	yes	characteristic
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×#127	C₄	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#127	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	complemented
24	C₂×C₃₁×#99	—	2	4	no	—
25	C₂×C₃₁×#99	C₂²	1	8	yes	complemented
26	C₃₁×#354	C₂²	1	4	yes	complemented
27	C₃₁×#353	C₂²	1	4	yes	complemented
28	C₃₁×#353	C₂²	1	8	yes	complemented
29	C₃₁×H(5,𝔽₂)	—	4	4	no	—
30	C₃₁×H(5,𝔽₂)	—	4	4	no	—
31	C₂×#41×C₃₁	—	2	1	no	—
32	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
35	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×#41×C₃₁	D₄	1	2	yes	—
37	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	4	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
44	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
45	C₂²×D₄×C₃₁	D₄	1	4	yes	—
46	C₂²×D₄×C₃₁	D₄	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
49	C₃₁×#127	—	2	2	no	—
50	C₃₁×#127	—	2	4	no	—
51	C₃₁×#127	C₂×C₄	1	2	yes	complemented
52	C₃₁×#127	C₂×C₄	1	2	yes	complemented
53	C₃₁×#127	C₂×C₄	1	2	yes	complemented
54	C₃₁×#127	C₂³	1	1	yes	characteristic
55	C₃₁×#127	C₂³	1	2	yes	—
56	C₃₁×#124	C₂×C₄	1	2	yes	complemented
57	C₃₁×#124	C₂×C₄	1	2	yes	complemented
58	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
59	C₃₁×#124	C₂³	1	1	yes	characteristic
60	C₃₁×#121	—	2	4	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	2	4	no	—
64	C₃₁×#121	—	2	4	no	—
65	C₃₁×#121	—	2	2	no	—
66	C₃₁×#120	—	2	2	no	—
67	C₃₁×#120	—	2	4	no	—
68	C₃₁×#120	—	2	4	no	—
69	C₃₁×#120	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	4	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	4	8	no	—
76	C₃₁×#117	—	2	4	no	—
77	C₃₁×#117	C₂×C₄	1	1	yes	characteristic, complemented
78	C₃₁×#117	C₂×C₄	1	2	yes	complemented
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	—	2	1	no	—
86	C₂⁴×C₃₁	—	2	2	no	—
87	C₂⁴×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	4	2	no	—
91	#41×C₃₁	—	4	4	no	—
92	#41×C₃₁	—	4	4	no	—
93	#41×C₃₁	—	4	4	no	—
94	#41×C₃₁	—	4	2	no	—
95	#41×C₃₁	—	4	4	no	—
96	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×D₄×C₃₁	—	2	8	no	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	—	2	4	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	—	4	8	no	—
114	C₂×D₄×C₃₁	—	4	8	no	—
115	C₂×D₄×C₃₁	—	4	4	no	—
116	C₂×D₄×C₃₁	—	8	8	no	—
117	C₂×D₄×C₃₁	—	8	4	no	—
118	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
119	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
120	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
121	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
122	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	4	8	no	—
133	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	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	4	no	—
146	C₂²⋊C₄×C₃₁	—	2	2	no	—
147	C₂²⋊C₄×C₃₁	—	2	8	no	—
148	C₂²⋊C₄×C₃₁	—	4	8	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	4	4	no	—
152	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
154	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	#33359	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	8	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	4	no	—
162	C₂³×C₃₁	—	8	4	no	—
163	C₂³×C₃₁	—	8	2	no	—
164	C₂³×C₃₁	—	8	4	no	—
165	C₂³×C₃₁	—	8	2	no	—
166	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
168	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	4	1	no	—
172	Q₈×C₃₁	—	4	1	no	—
173	D₄×C₃₁	—	2	8	no	—
174	D₄×C₃₁	—	2	4	no	—
175	D₄×C₃₁	—	2	4	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	4	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	4	1	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	4	no	—
183	D₄×C₃₁	—	4	8	no	—
184	D₄×C₃₁	—	4	4	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	4	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	8	4	no	—
189	D₄×C₃₁	—	8	4	no	—
190	D₄×C₃₁	—	8	4	no	—
191	D₄×C₃₁	#99	1	4	yes	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	8	no	—
205	C₂×C₄×C₃₁	—	16	8	no	—
206	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
210	#3498	C₂×C₃₁	1	1	yes	characteristic
211	#2883	C₂×C₃₁	1	2	yes	complemented
212	#2128	C₂×C₃₁	1	2	yes	complemented
213	#2127	C₂×C₃₁	1	4	yes	complemented
214	#2126	C₂×C₃₁	1	2	yes	complemented
215	C₂×#353	C₂×C₃₁	1	4	yes	complemented
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	2	1	no	—
218	C₂²×C₃₁	—	2	4	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	—	4	4	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	4	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	—	8	8	no	—
228	C₂²×C₃₁	—	16	4	no	—
229	C₂²×C₃₁	—	16	2	no	—
230	C₂²×C₃₁	—	16	2	no	—
231	C₂²×C₃₁	C₂×#99	1	4	yes	—
232	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	2	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	1	no	—
239	C₄×C₃₁	—	4	1	no	—
240	C₄×C₃₁	—	8	8	no	—
241	C₄×C₃₁	C₂×#99	1	2	yes	—
242	C₂×H(5,𝔽₂)	C₄×C₃₁	1	1	yes	characteristic, complemented
243	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
244	#550	—	2	2	no	—
245	D₄²	—	2	4	no	—
246	D₄²	—	2	2	no	—
247	#535	C₄×C₃₁	1	1	yes	characteristic, complemented
248	#535	C₄×C₃₁	1	2	yes	complemented
249	#535	C₂²×C₃₁	1	1	yes	characteristic
250	#535	C₂²×C₃₁	1	2	yes	—
251	#532	C₄×C₃₁	1	2	yes	complemented
252	#532	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#127	C₄×C₃₁	1	1	yes	characteristic, complemented
254	C₂×#127	C₂²×C₃₁	1	2	yes	—
255	#518	—	2	2	no	—
256	C₄×#41	C₄×C₃₁	1	1	yes	characteristic, complemented
257	#410	—	2	2	no	—
258	#410	C₂²×C₃₁	1	4	yes	complemented
259	C₂×#99	—	2	4	no	—
260	C₂×#99	C₂²×C₃₁	1	8	yes	complemented
261	#354	C₂²×C₃₁	1	4	yes	complemented
262	#353	C₂²×C₃₁	1	4	yes	complemented
263	#353	C₂²×C₃₁	1	8	yes	complemented
264	C₂×C₃₁	—	2	1	no	—
265	C₂×C₃₁	—	2	4	no	—
266	C₂×C₃₁	—	4	2	no	—
267	C₂×C₃₁	—	4	4	no	—
268	C₂×C₃₁	—	4	1	no	—
269	C₂×C₃₁	—	8	1	no	—
270	C₂×C₃₁	—	8	2	no	—
271	C₂×C₃₁	C₂²×#99	1	1	yes	characteristic
272	H(5,𝔽₂)	—	4	4	no	—
273	H(5,𝔽₂)	—	4	4	no	—
274	C₂×#41	—	2	1	no	—
275	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
276	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
277	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
278	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
279	C₂×#41	D₄×C₃₁	1	2	yes	—
280	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×D₄	—	2	2	no	—
282	C₂²×D₄	—	2	4	no	—
283	C₂²×D₄	—	2	2	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
286	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
287	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
288	C₂²×D₄	D₄×C₃₁	1	4	yes	—
289	C₂²×D₄	D₄×C₃₁	1	2	yes	—
290	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
291	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
292	#127	—	2	2	no	—
293	#127	—	2	4	no	—
294	#127	C₂×C₄×C₃₁	1	2	yes	complemented
295	#127	C₂×C₄×C₃₁	1	2	yes	complemented
296	#127	C₂×C₄×C₃₁	1	2	yes	complemented
297	#127	C₂³×C₃₁	1	1	yes	characteristic
298	#127	C₂³×C₃₁	1	2	yes	—
299	#124	C₂×C₄×C₃₁	1	2	yes	complemented
300	#124	C₂×C₄×C₃₁	1	2	yes	complemented
301	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
302	#124	C₂³×C₃₁	1	1	yes	characteristic
303	#121	—	2	4	no	—
304	#121	—	2	2	no	—
305	#121	—	2	2	no	—
306	#121	—	2	4	no	—
307	#121	—	2	4	no	—
308	#121	—	2	2	no	—
309	#120	—	2	2	no	—
310	#120	—	2	4	no	—
311	#120	—	2	4	no	—
312	#120	—	2	2	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	4	no	—
317	C₄×D₄	—	2	2	no	—
318	C₄×D₄	—	4	8	no	—
319	#117	—	2	4	no	—
320	#117	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
321	#117	C₂×C₄×C₃₁	1	2	yes	complemented
322	C₂×C₂²⋊C₄	—	2	8	no	—
323	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
324	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	complemented
325	#99	—	2	4	no	—
326	#99	—	2	8	no	—
327	#99	C₂³×C₃₁	1	8	yes	—
328	C₃₁	#33359	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁴	—	2	1	no	—
330	C₂⁴	—	2	2	no	—
331	C₂⁴	C₂²⋊C₄×C₃₁	1	2	yes	—
332	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
333	#41	—	2	4	no	—
334	#41	—	4	2	no	—
335	#41	—	4	4	no	—
336	#41	—	4	4	no	—
337	#41	—	4	4	no	—
338	#41	—	4	2	no	—
339	#41	—	4	4	no	—
340	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
341	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
342	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	—	4	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	—	4	8	no	—
358	C₂×D₄	—	4	8	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	8	8	no	—
361	C₂×D₄	—	8	4	no	—
362	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
363	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
364	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
365	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
366	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
367	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
369	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
371	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	4	8	no	—
377	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
378	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	2	no	—
387	C₂²⋊C₄	—	2	4	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	4	8	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	4	4	no	—
396	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
397	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
398	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
399	C₂³	—	2	8	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	8	4	no	—
406	C₂³	—	8	2	no	—
407	C₂³	—	8	4	no	—
408	C₂³	—	8	2	no	—
409	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
410	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
411	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
412	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
413	Q₈	—	2	2	no	—
414	Q₈	—	4	1	no	—
415	Q₈	—	4	1	no	—
416	D₄	—	2	8	no	—
417	D₄	—	2	4	no	—
418	D₄	—	2	4	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	2	no	—
422	D₄	—	4	1	no	—
423	D₄	—	4	1	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	4	no	—
426	D₄	—	4	8	no	—
427	D₄	—	4	4	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	4	no	—
430	D₄	—	4	2	no	—
431	D₄	—	8	4	no	—
432	D₄	—	8	4	no	—
433	D₄	—	8	4	no	—
434	D₄	C₃₁×#99	1	4	yes	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	8	no	—
448	C₂×C₄	—	16	8	no	—
449	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
450	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
451	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
453	C₂²	—	2	2	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	1	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	8	8	no	—
465	C₂²	—	16	4	no	—
466	C₂²	—	16	2	no	—
467	C₂²	—	16	2	no	—
468	C₂²	C₂×C₃₁×#99	1	4	yes	—
469	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₄	—	2	2	no	—
471	C₄	—	2	1	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	8	no	—
478	C₄	C₂×C₃₁×#99	1	2	yes	—
479	C₂	—	2	1	no	—
480	C₂	—	2	4	no	—
481	C₂	—	4	2	no	—
482	C₂	—	4	4	no	—
483	C₂	—	4	1	no	—
484	C₂	—	8	1	no	—
485	C₂	—	8	2	no	—
486	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×#3498	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×#2128	C₂	semidirect
4. C₃₁×#2127	C₂	semidirect
5. C₃₁×#2126	C₂	semidirect
6. C₂×C₃₁×#353	C₂	semidirect
7. C₂×C₃₁×H(5,𝔽₂)	C₄	semidirect
12. C₃₁×#535	C₄	semidirect
13. C₃₁×#535	C₄	semidirect
16. C₃₁×#532	C₄	semidirect
18. C₂×C₃₁×#127	C₄	semidirect
21. C₄×#41×C₃₁	C₄	semidirect
23. C₃₁×#410	C₂²	semidirect
25. C₂×C₃₁×#99	C₂²	semidirect
26. C₃₁×#354	C₂²	semidirect
27. C₃₁×#353	C₂²	semidirect
28. C₃₁×#353	C₂²	semidirect
34. C₂×#41×C₃₁	C₂×C₄	semidirect
35. C₂×#41×C₃₁	C₂×C₄	semidirect
43. C₂²×D₄×C₃₁	C₂×C₄	semidirect
44. C₂²×D₄×C₃₁	C₂×C₄	semidirect
48. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
51. C₃₁×#127	C₂×C₄	semidirect
52. C₃₁×#127	C₂×C₄	semidirect
53. C₃₁×#127	C₂×C₄	semidirect
56. C₃₁×#124	C₂×C₄	semidirect
57. C₃₁×#124	C₂×C₄	semidirect
58. C₃₁×#124	C₂×C₄	semidirect
77. C₃₁×#117	C₂×C₄	semidirect
78. C₃₁×#117	C₂×C₄	semidirect
80. C₂×C₄²×C₃₁	C₂×C₄	semidirect
81. C₂×C₄²×C₃₁	C₂×C₄	semidirect
155. #33359	C₃₁	direct (N × H)
211. #2883	C₂×C₃₁	semidirect
212. #2128	C₂×C₃₁	semidirect
213. #2127	C₂×C₃₁	semidirect
214. #2126	C₂×C₃₁	semidirect
215. C₂×#353	C₂×C₃₁	semidirect
242. C₂×H(5,𝔽₂)	C₄×C₃₁	semidirect
247. #535	C₄×C₃₁	semidirect
248. #535	C₄×C₃₁	semidirect
251. #532	C₄×C₃₁	semidirect
253. C₂×#127	C₄×C₃₁	semidirect
256. C₄×#41	C₄×C₃₁	semidirect
258. #410	C₂²×C₃₁	semidirect
260. C₂×#99	C₂²×C₃₁	semidirect
261. #354	C₂²×C₃₁	semidirect
262. #353	C₂²×C₃₁	semidirect
263. #353	C₂²×C₃₁	semidirect
277. C₂×#41	C₂×C₄×C₃₁	semidirect
278. C₂×#41	C₂×C₄×C₃₁	semidirect
286. C₂²×D₄	C₂×C₄×C₃₁	semidirect
287. C₂²×D₄	C₂×C₄×C₃₁	semidirect
291. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
294. #127	C₂×C₄×C₃₁	semidirect
295. #127	C₂×C₄×C₃₁	semidirect
296. #127	C₂×C₄×C₃₁	semidirect
299. #124	C₂×C₄×C₃₁	semidirect
300. #124	C₂×C₄×C₃₁	semidirect
301. #124	C₂×C₄×C₃₁	semidirect
320. #117	C₂×C₄×C₃₁	semidirect
321. #117	C₂×C₄×C₃₁	semidirect
323. C₂×C₄²	C₂×C₄×C₃₁	semidirect
324. C₂×C₄²	C₂×C₄×C₃₁	semidirect
328. C₃₁	#33359	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×#3498	2
C₃₁×#2883	2
C₃₁×#2128	2
C₃₁×#2127	2
C₃₁×#2126	2
C₂×C₃₁×#353	2
#33359	31

Subgroups — C31×#33359 #21490118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33359 #21490118