Subgroups — C₂×C₃₁×#2223 #21490375

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2223	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#137	—	2	1	no	—
10	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×SD₃₂	—	2	1	no	—
15	C₂×C₃₁×SD₃₂	—	2	1	no	—
16	C₂×C₃₁×SD₃₂	—	2	2	no	—
17	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	complemented
18	C₃₁×#504	—	2	2	no	—
19	C₃₁×#504	C₂²	1	4	yes	complemented
20	C₂×C₃₁×M₃₂	—	2	1	no	—
21	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₃₁	—	2	1	no	—
23	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#473	—	2	2	no	—
25	C₃₁×#473	C₂²	1	4	yes	complemented
26	C₃₁×#472	—	2	2	no	—
27	C₃₁×#472	C₂²	1	4	yes	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#101	—	2	1	no	—
31	C₂×C₃₁×#100	—	2	1	no	—
32	C₃₁×#368	C₂²	1	4	yes	complemented
33	C₃₁×#362	C₂²	1	4	yes	complemented
34	C₃₁×#361	C₂²	1	4	yes	complemented
35	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₃₁×#137	—	2	4	no	—
39	C₃₁×#137	—	4	2	no	—
40	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
41	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
42	C₂×Q₁₆×C₃₁	—	2	1	no	—
43	C₂×Q₁₆×C₃₁	—	2	1	no	—
44	C₂×Q₁₆×C₃₁	—	2	1	no	—
45	C₂×Q₁₆×C₃₁	—	2	1	no	—
46	C₂×Q₁₆×C₃₁	—	2	1	no	—
47	C₂×Q₁₆×C₃₁	—	4	1	no	—
48	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₃₁	—	4	1	no	—
50	C₂×SD₁₆×C₃₁	—	4	1	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	4	1	no	—
57	C₂×D₈×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#131	—	2	1	no	—
59	C₃₁×#131	—	2	1	no	—
60	C₃₁×#131	D₄	1	2	yes	—
61	C₃₁×#131	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	—	2	1	no	—
63	C₂×M₁₆×C₃₁	—	2	1	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	—	2	1	no	—
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×SD₃₂	—	2	4	no	—
69	C₃₁×SD₃₂	—	2	4	no	—
70	C₃₁×M₃₂	—	2	1	no	—
71	C₃₁×M₃₂	—	2	4	no	—
72	C₃₁×M₃₂	C₂³	1	2	yes	—
73	C₂×C₁₆×C₃₁	—	2	1	no	—
74	C₂×C₁₆×C₃₁	—	2	4	no	—
75	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
78	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
79	C₃₁×#108	—	2	1	no	—
80	C₃₁×#108	—	2	1	no	—
81	C₃₁×#108	C₂³	1	2	yes	—
82	C₃₁×#108	C₂³	1	2	yes	—
83	C₃₁×#101	—	2	4	no	—
84	C₃₁×#100	—	2	4	no	—
85	C₂⁴×C₃₁	—	4	1	no	—
86	#41×C₃₁	—	2	1	no	—
87	#41×C₃₁	—	2	1	no	—
88	#41×C₃₁	D₈	1	2	yes	—
89	#41×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	4	1	no	—
92	C₂×Q₈×C₃₁	—	4	1	no	—
93	C₂×Q₈×C₃₁	—	4	1	no	—
94	C₂×Q₈×C₃₁	—	4	1	no	—
95	C₂×Q₈×C₃₁	—	4	1	no	—
96	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	1	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	4	1	no	—
104	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₃₁	—	2	2	no	—
109	Q₁₆×C₃₁	—	2	2	no	—
110	Q₁₆×C₃₁	—	4	4	no	—
111	Q₁₆×C₃₁	—	4	1	no	—
112	Q₁₆×C₃₁	—	4	1	no	—
113	SD₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	4	no	—
115	D₈×C₃₁	—	2	2	no	—
116	D₈×C₃₁	—	2	2	no	—
117	D₈×C₃₁	—	4	4	no	—
118	D₈×C₃₁	—	4	1	no	—
119	D₈×C₃₁	—	4	1	no	—
120	M₁₆×C₃₁	—	2	1	no	—
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	2	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	4	1	no	—
126	M₁₆×C₃₁	—	4	1	no	—
127	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	4	1	no	—
132	C₂×C₈×C₃₁	—	4	1	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
137	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
138	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
139	C₁₆×C₃₁	—	2	2	no	—
140	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
141	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×#2223	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	8	1	no	—
146	C₂³×C₃₁	—	8	1	no	—
147	C₂³×C₃₁	—	16	2	no	—
148	C₂³×C₃₁	#135	1	1	yes	characteristic
149	Q₈×C₃₁	—	2	1	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	8	1	no	—
153	Q₈×C₃₁	—	8	1	no	—
154	Q₈×C₃₁	C₂×D₈	1	2	yes	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	1	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	8	1	no	—
161	D₄×C₃₁	—	8	1	no	—
162	D₄×C₃₁	C₂×D₈	1	2	yes	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	8	1	no	—
168	C₂×C₄×C₃₁	—	8	1	no	—
169	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
170	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₈×C₃₁	—	2	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	#121	1	1	yes	characteristic
179	C₈×C₃₁	#121	1	1	yes	characteristic
180	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂×#504	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
186	#2223	C₂×C₃₁	1	8	yes	direct product factor
187	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂×#362	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	—	16	2	no	—
195	C₂²×C₃₁	—	16	2	no	—
196	C₂²×C₃₁	—	16	2	no	—
197	C₂²×C₃₁	#466	1	1	yes	characteristic
198	C₂²×C₃₁	#466	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	#466	1	1	yes	characteristic
203	C₄×C₃₁	#466	1	1	yes	characteristic
204	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
205	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
206	C₂×#137	—	2	1	no	—
207	C₂×AGL(1,ℤ/8)	—	2	1	no	—
208	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
209	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×SD₃₂	—	2	1	no	—
212	C₂×SD₃₂	—	2	1	no	—
213	C₂×SD₃₂	—	2	2	no	—
214	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	complemented
215	#504	—	2	2	no	—
216	#504	C₂²×C₃₁	1	4	yes	complemented
217	C₂×M₃₂	—	2	1	no	—
218	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
219	C₂²×C₁₆	—	2	1	no	—
220	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	#473	—	2	2	no	—
222	#473	C₂²×C₃₁	1	4	yes	complemented
223	#472	—	2	2	no	—
224	#472	C₂²×C₃₁	1	4	yes	—
225	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#101	—	2	1	no	—
228	C₂×#100	—	2	1	no	—
229	#368	C₂²×C₃₁	1	4	yes	complemented
230	#362	C₂²×C₃₁	1	4	yes	complemented
231	#361	C₂²×C₃₁	1	4	yes	complemented
232	C₂×C₃₁	—	2	1	no	—
233	C₂×C₃₁	—	2	1	no	—
234	C₂×C₃₁	—	4	2	no	—
235	C₂×C₃₁	—	16	2	no	—
236	C₂×C₃₁	#2223	1	2	yes	direct product factor
237	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
238	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×D₄	—	2	1	no	—
241	#137	—	2	4	no	—
242	#137	—	4	2	no	—
243	AGL(1,ℤ/8)	—	2	4	no	—
244	AGL(1,ℤ/8)	—	4	2	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	4	1	no	—
251	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
252	C₂×SD₁₆	—	4	1	no	—
253	C₂×SD₁₆	—	4	1	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	—	2	1	no	—
259	C₂×D₈	—	4	1	no	—
260	C₂×D₈	C₂³×C₃₁	1	2	yes	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	D₄×C₃₁	1	2	yes	—
264	#131	C₂³×C₃₁	1	2	yes	—
265	C₂×M₁₆	—	2	1	no	—
266	C₂×M₁₆	—	2	1	no	—
267	C₂×M₁₆	—	2	1	no	—
268	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	SD₃₂	—	2	4	no	—
272	SD₃₂	—	2	4	no	—
273	M₃₂	—	2	1	no	—
274	M₃₂	—	2	4	no	—
275	M₃₂	C₂³×C₃₁	1	2	yes	—
276	C₂×C₁₆	—	2	1	no	—
277	C₂×C₁₆	—	2	4	no	—
278	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
281	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
282	#108	—	2	1	no	—
283	#108	—	2	1	no	—
284	#108	C₂³×C₃₁	1	2	yes	—
285	#108	C₂³×C₃₁	1	2	yes	—
286	#101	—	2	4	no	—
287	#100	—	2	4	no	—
288	C₃₁	C₂×#2223	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	—	4	1	no	—
290	#41	—	2	1	no	—
291	#41	—	2	1	no	—
292	#41	D₈×C₃₁	1	2	yes	—
293	#41	C₂×D₄×C₃₁	1	2	yes	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	1	no	—
299	C₂×Q₈	—	4	1	no	—
300	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	4	1	no	—
305	C₂×D₄	—	4	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	Q₁₆	—	2	2	no	—
313	Q₁₆	—	2	2	no	—
314	Q₁₆	—	4	4	no	—
315	Q₁₆	—	4	1	no	—
316	Q₁₆	—	4	1	no	—
317	SD₁₆	—	4	4	no	—
318	SD₁₆	—	4	4	no	—
319	D₈	—	2	2	no	—
320	D₈	—	2	2	no	—
321	D₈	—	4	4	no	—
322	D₈	—	4	1	no	—
323	D₈	—	4	1	no	—
324	M₁₆	—	2	1	no	—
325	M₁₆	—	2	2	no	—
326	M₁₆	—	2	2	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	2	no	—
329	M₁₆	—	4	1	no	—
330	M₁₆	—	4	1	no	—
331	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
341	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
342	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₁₆	—	2	2	no	—
344	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
345	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
346	C₂³	—	2	1	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	8	1	no	—
349	C₂³	—	8	1	no	—
350	C₂³	—	16	2	no	—
351	C₂³	C₃₁×#135	1	1	yes	characteristic
352	Q₈	—	2	1	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	8	1	no	—
356	Q₈	—	8	1	no	—
357	Q₈	C₂×D₈×C₃₁	1	2	yes	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	2	1	no	—
361	D₄	—	4	2	no	—
362	D₄	—	4	2	no	—
363	D₄	—	8	1	no	—
364	D₄	—	8	1	no	—
365	D₄	C₂×D₈×C₃₁	1	2	yes	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	1	no	—
371	C₂×C₄	—	8	1	no	—
372	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
373	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
374	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₈	C₃₁×#121	1	1	yes	characteristic
382	C₈	C₃₁×#121	1	1	yes	characteristic
383	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
384	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	16	2	no	—
390	C₂²	—	16	2	no	—
391	C₂²	—	16	2	no	—
392	C₂²	C₃₁×#466	1	1	yes	characteristic
393	C₂²	C₃₁×#466	1	1	yes	characteristic
394	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	8	2	no	—
397	C₄	C₃₁×#466	1	1	yes	characteristic
398	C₄	C₃₁×#466	1	1	yes	characteristic
399	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
400	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	4	2	no	—
404	C₂	—	16	2	no	—
405	C₂	C₃₁×#2223	1	2	yes	direct product factor
406	C₂	C₂×C₃₁×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×SD₃₂	C₂	semidirect
2. C₂×C₃₁×#504	C₂	semidirect
3. C₂×C₃₁×#473	C₂	semidirect
5. C₃₁×#2223	C₂	direct (N × H)
6. C₂×C₃₁×#368	C₂	semidirect
7. C₂×C₃₁×#362	C₂	semidirect
8. C₂×C₃₁×#361	C₂	semidirect
17. C₂×C₃₁×SD₃₂	C₂²	semidirect
19. C₃₁×#504	C₂²	semidirect
25. C₃₁×#473	C₂²	semidirect
32. C₃₁×#368	C₂²	semidirect
33. C₃₁×#362	C₂²	semidirect
34. C₃₁×#361	C₂²	semidirect
142. C₂×#2223	C₃₁	direct (N × H)
182. C₂²×SD₃₂	C₂×C₃₁	semidirect
183. C₂×#504	C₂×C₃₁	semidirect
184. C₂×#473	C₂×C₃₁	semidirect
186. #2223	C₂×C₃₁	direct (N × H)
187. C₂×#368	C₂×C₃₁	semidirect
188. C₂×#362	C₂×C₃₁	semidirect
189. C₂×#361	C₂×C₃₁	semidirect
214. C₂×SD₃₂	C₂²×C₃₁	semidirect
216. #504	C₂²×C₃₁	semidirect
222. #473	C₂²×C₃₁	semidirect
229. #368	C₂²×C₃₁	semidirect
230. #362	C₂²×C₃₁	semidirect
231. #361	C₂²×C₃₁	semidirect
236. C₂×C₃₁	#2223	direct (N × H)
288. C₃₁	C₂×#2223	direct (N × H)
405. C₂	C₃₁×#2223	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₂×C₃₁×#504	2
C₂×C₃₁×#473	2
C₂×C₃₁×#472	2
C₃₁×#2223	2
C₂×C₃₁×#368	2
C₂×C₃₁×#362	2
C₂×C₃₁×#361	2
C₂×#2223	31

Subgroups — C2×C31×#2223 #21490375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2223 #21490375