Subgroups — C₃₁×#33756 #21490515

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2235	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2217	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2188	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2185	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#525	—	2	1	no	—
16	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
17	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	complemented
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
19	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#467	—	2	1	no	—
21	C₃₁×#467	—	2	1	no	—
22	C₃₁×#467	C₂²	1	1	yes	characteristic
23	C₃₁×#467	C₂²	1	2	yes	—
24	C₃₁×#467	C₂²	1	1	yes	characteristic
25	C₃₁×#467	C₂²	1	2	yes	—
26	C₃₁×#466	C₂²	1	1	yes	characteristic
27	C₃₁×#466	C₂²	1	1	yes	characteristic
28	C₃₁×#465	—	2	1	no	—
29	C₃₁×#465	—	2	1	no	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
32	C₃₁×#416	—	2	1	no	—
33	C₂×C₃₁×#103	—	2	1	no	—
34	C₃₁×#367	C₂²	1	2	yes	complemented
35	C₃₁×#366	C₂²	1	2	yes	complemented
36	C₃₁×#358	C₂²	1	2	yes	complemented
37	C₃₁×#358	C₂²	1	2	yes	complemented
38	C₃₁×#358	C₂²	1	2	yes	complemented
39	C₃₁×#357	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	—	2	1	no	—
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₃₁×#135	—	4	2	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	4	2	no	—
49	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×SD₁₆×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#123	—	4	1	no	—
61	C₃₁×C₂²⋊Q₈	—	2	1	no	—
62	C₃₁×C₂²⋊Q₈	—	2	1	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₃₁×C₂²⋊Q₈	—	2	2	no	—
65	C₃₁×C₂²⋊Q₈	—	4	1	no	—
66	C₃₁×C₂²⋊Q₈	—	4	1	no	—
67	C₃₁×#121	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₃₁×#120	—	4	1	no	—
70	C₄×D₄×C₃₁	—	4	1	no	—
71	C₄×D₄×C₃₁	—	4	1	no	—
72	C₃₁×#117	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
76	C₃₁×Q₃₂	—	4	2	no	—
77	C₃₁×SD₃₂	—	4	2	no	—
78	C₂×C₁₆×C₃₁	—	4	2	no	—
79	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
80	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
81	C₃₁×#107	—	2	1	no	—
82	C₃₁×#107	C₂³	1	2	yes	—
83	C₃₁×#107	C₂³	1	1	yes	characteristic
84	C₃₁×#107	C₂³	1	1	yes	characteristic
85	C₃₁×#106	—	2	1	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	2	1	no	—
89	C₃₁×#103	—	2	1	no	—
90	C₃₁×#102	—	2	1	no	—
91	C₃₁×#102	—	2	1	no	—
92	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
93	#41×C₃₁	—	4	2	no	—
94	#41×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	1	no	—
105	C₂²×C₄×C₃₁	—	4	1	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	—	4	1	no	—
108	C₂²×C₄×C₃₁	—	4	1	no	—
109	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	Q₁₆×C₃₁	—	2	2	no	—
116	Q₁₆×C₃₁	—	2	2	no	—
117	Q₁₆×C₃₁	—	2	2	no	—
118	Q₁₆×C₃₁	—	4	2	no	—
119	SD₁₆×C₃₁	—	4	2	no	—
120	D₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	4	1	no	—
125	C₂×C₈×C₃₁	—	4	1	no	—
126	C₂×C₈×C₃₁	—	4	1	no	—
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	1	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	1	no	—
138	C₄⋊C₄×C₃₁	—	4	1	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	1	no	—
142	C₂²⋊C₄×C₃₁	—	4	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	1	no	—
144	C₂²⋊C₄×C₃₁	—	4	1	no	—
145	C₂²⋊C₄×C₃₁	—	4	1	no	—
146	C₄²×C₃₁	—	4	1	no	—
147	C₁₆×C₃₁	—	2	2	no	—
148	C₁₆×C₃₁	—	2	2	no	—
149	#33756	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
155	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
156	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
157	Q₈×C₃₁	—	4	1	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	4	1	no	—
161	Q₈×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	4	1	no	—
163	D₄×C₃₁	—	4	1	no	—
164	D₄×C₃₁	—	8	2	no	—
165	D₄×C₃₁	—	8	2	no	—
166	D₄×C₃₁	—	8	2	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	8	2	no	—
182	C₂×C₄×C₃₁	—	8	2	no	—
183	C₂×C₄×C₃₁	—	8	2	no	—
184	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₈×C₃₁	—	2	1	no	—
189	C₈×C₃₁	—	2	1	no	—
190	C₈×C₃₁	—	2	1	no	—
191	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
192	#3050	C₂×C₃₁	1	1	yes	characteristic
193	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2237	C₂×C₃₁	1	2	yes	complemented
195	#2235	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#2232	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2217	C₂×C₃₁	1	2	yes	complemented
198	#2216	C₂×C₃₁	1	2	yes	complemented
199	#2189	C₂×C₃₁	1	2	yes	complemented
200	#2188	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#2185	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	4	4	no	—
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	#508	1	2	yes	—
215	C₂²×C₃₁	#510	1	2	yes	—
216	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	#534	1	1	yes	characteristic
226	C₄×C₃₁	#575	1	1	yes	characteristic
227	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
228	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	#525	—	2	1	no	—
231	C₂×C₂²⋊Q₈	—	2	1	no	—
232	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	complemented
233	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
234	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
235	#467	—	2	1	no	—
236	#467	—	2	1	no	—
237	#467	C₂²×C₃₁	1	1	yes	characteristic
238	#467	C₂²×C₃₁	1	2	yes	—
239	#467	C₂²×C₃₁	1	1	yes	characteristic
240	#467	C₂²×C₃₁	1	2	yes	—
241	#466	C₂²×C₃₁	1	1	yes	characteristic
242	#466	C₂²×C₃₁	1	1	yes	characteristic
243	#465	—	2	1	no	—
244	#465	—	2	1	no	—
245	#427	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
247	#416	—	2	1	no	—
248	C₂×#103	—	2	1	no	—
249	#367	C₂²×C₃₁	1	2	yes	complemented
250	#366	C₂²×C₃₁	1	2	yes	complemented
251	#358	C₂²×C₃₁	1	2	yes	complemented
252	#358	C₂²×C₃₁	1	2	yes	complemented
253	#358	C₂²×C₃₁	1	2	yes	complemented
254	#357	C₂²×C₃₁	1	2	yes	complemented
255	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
257	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
258	C₂×C₃₁	—	2	2	no	—
259	C₂×C₃₁	—	2	2	no	—
260	C₂×C₃₁	—	4	1	no	—
261	C₂×C₃₁	—	16	1	no	—
262	C₂×C₃₁	#3186	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
264	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
265	C₂×#41	—	2	1	no	—
266	C₂²×Q₈	—	2	1	no	—
267	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
268	#135	—	4	2	no	—
269	C₂×Q₁₆	—	2	1	no	—
270	C₂×Q₁₆	—	4	2	no	—
271	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
272	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×SD₁₆	—	2	1	no	—
274	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	—	4	2	no	—
276	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	#123	—	4	1	no	—
283	C₂²⋊Q₈	—	2	2	no	—
284	C₂²⋊Q₈	—	2	2	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	C₂²⋊Q₈	—	4	1	no	—
288	C₂²⋊Q₈	—	4	1	no	—
289	#121	—	2	1	no	—
290	#121	—	2	1	no	—
291	#120	—	4	1	no	—
292	C₄×D₄	—	4	1	no	—
293	C₄×D₄	—	4	1	no	—
294	#117	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	4	1	no	—
297	C₂×C₂²⋊C₄	—	4	1	no	—
298	Q₃₂	—	4	2	no	—
299	SD₃₂	—	4	2	no	—
300	C₂×C₁₆	—	4	2	no	—
301	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
302	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
303	#107	—	2	1	no	—
304	#107	C₂³×C₃₁	1	2	yes	—
305	#107	C₂³×C₃₁	1	1	yes	characteristic
306	#107	C₂³×C₃₁	1	1	yes	characteristic
307	#106	—	2	1	no	—
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	1	no	—
311	#103	—	2	1	no	—
312	#102	—	2	1	no	—
313	#102	—	2	1	no	—
314	C₃₁	#33756	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
316	#41	—	4	2	no	—
317	#41	—	4	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	4	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	—	4	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	Q₁₆	—	2	2	no	—
339	Q₁₆	—	2	2	no	—
340	Q₁₆	—	2	2	no	—
341	Q₁₆	—	4	2	no	—
342	SD₁₆	—	4	2	no	—
343	D₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	4	1	no	—
348	C₂×C₈	—	4	1	no	—
349	C₂×C₈	—	4	1	no	—
350	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₂²⋊C₄	—	4	1	no	—
366	C₂²⋊C₄	—	4	1	no	—
367	C₂²⋊C₄	—	4	1	no	—
368	C₂²⋊C₄	—	4	1	no	—
369	C₄²	—	4	1	no	—
370	C₁₆	—	2	2	no	—
371	C₁₆	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
377	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
378	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
379	Q₈	—	4	1	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	1	no	—
383	Q₈	—	8	2	no	—
384	D₄	—	4	1	no	—
385	D₄	—	4	1	no	—
386	D₄	—	8	2	no	—
387	D₄	—	8	2	no	—
388	D₄	—	8	2	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	8	2	no	—
404	C₂×C₄	—	8	2	no	—
405	C₂×C₄	—	8	2	no	—
406	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
410	C₈	—	2	1	no	—
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	C₃₁×#508	1	2	yes	—
426	C₂²	C₃₁×#510	1	2	yes	—
427	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	8	2	no	—
432	C₄	—	8	1	no	—
433	C₄	—	8	1	no	—
434	C₄	—	8	1	no	—
435	C₄	—	8	2	no	—
436	C₄	C₃₁×#534	1	1	yes	characteristic
437	C₄	C₃₁×#575	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	2	2	no	—
440	C₂	—	4	1	no	—
441	C₂	—	16	1	no	—
442	C₂	C₃₁×#3186	1	1	yes	characteristic
443	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
444	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#467	C₂	semidirect
3. C₃₁×#2237	C₂	semidirect
4. C₃₁×#2235	C₂	semidirect
5. C₃₁×#2232	C₂	semidirect
6. C₃₁×#2217	C₂	semidirect
7. C₃₁×#2216	C₂	semidirect
8. C₃₁×#2189	C₂	semidirect
9. C₃₁×#2188	C₂	semidirect
10. C₃₁×#2185	C₂	semidirect
11. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
17. C₂×C₃₁×Q₃₂	C₂²	semidirect
18. C₂×C₃₁×SD₃₂	C₂²	semidirect
34. C₃₁×#367	C₂²	semidirect
35. C₃₁×#366	C₂²	semidirect
36. C₃₁×#358	C₂²	semidirect
37. C₃₁×#358	C₂²	semidirect
38. C₃₁×#358	C₂²	semidirect
39. C₃₁×#357	C₂²	semidirect
149. #33756	C₃₁	direct (N × H)
193. C₂×#467	C₂×C₃₁	semidirect
194. #2237	C₂×C₃₁	semidirect
195. #2235	C₂×C₃₁	semidirect
196. #2232	C₂×C₃₁	semidirect
197. #2217	C₂×C₃₁	semidirect
198. #2216	C₂×C₃₁	semidirect
199. #2189	C₂×C₃₁	semidirect
200. #2188	C₂×C₃₁	semidirect
201. #2185	C₂×C₃₁	semidirect
202. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
232. C₂×Q₃₂	C₂²×C₃₁	semidirect
233. C₂×SD₃₂	C₂²×C₃₁	semidirect
249. #367	C₂²×C₃₁	semidirect
250. #366	C₂²×C₃₁	semidirect
251. #358	C₂²×C₃₁	semidirect
252. #358	C₂²×C₃₁	semidirect
253. #358	C₂²×C₃₁	semidirect
254. #357	C₂²×C₃₁	semidirect
314. C₃₁	#33756	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×#3050	2
C₂×C₃₁×#467	2
C₃₁×#2237	2
C₃₁×#2235	2
C₃₁×#2232	2
C₃₁×#2217	2
C₃₁×#2216	2
C₃₁×#2189	2
C₃₁×#2188	2
C₃₁×#2185	2
C₂×C₃₁×C₂²⋊C₁₆	2
#33756	31

Subgroups — C31×#33756 #21490515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33756 #21490515