Subgroups — C₂×C₃₁×#3291 #21517861

Order 7936.

2682 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#577	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3291	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#471	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#456	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#453	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#443	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
9	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	complemented
10	C₃₁×#577	—	2	16	no	—
11	C₃₁×#577	C₂²	1	32	yes	complemented
12	C₂×C₃₁×#137	—	2	4	no	—
13	C₂×C₃₁×#137	C₂²	1	4	yes	complemented
14	C₂×C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
15	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
16	C₂×C₃₁×#135	—	2	4	no	—
17	C₂×C₃₁×#135	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₃₁	—	2	4	no	—
19	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#131	C₂²	1	4	yes	complemented
21	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
23	C₃₁×#492	C₂²	1	8	yes	complemented
24	C₃₁×#471	C₂²	1	32	yes	complemented
25	C₃₁×#456	C₂²	1	16	yes	complemented
26	C₃₁×#453	C₂²	1	16	yes	complemented
27	C₃₁×#443	C₂²	1	16	yes	complemented
28	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
29	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
30	C₂×C₃₁×#101	C₂²	1	2	yes	complemented
31	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×2₋¹⁺⁴	—	2	8	no	—
34	C₃₁×2₋¹⁺⁴	—	4	2	no	—
35	C₃₁×2₋¹⁺⁴	C₂³	1	8	yes	—
36	C₃₁×H(5,𝔽₂)	—	2	8	no	—
37	C₃₁×H(5,𝔽₂)	—	4	2	no	—
38	C₃₁×H(5,𝔽₂)	C₂³	1	8	yes	complemented
39	C₂×#41×C₃₁	—	2	4	no	—
40	C₂×#41×C₃₁	—	2	4	no	—
41	C₂×#41×C₃₁	—	4	2	no	—
42	C₂×#41×C₃₁	C₂³	1	4	yes	—
43	C₂²×Q₈×C₃₁	—	2	4	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	4	no	—
46	C₂²×D₄×C₃₁	—	4	1	no	—
47	C₂²×D₄×C₃₁	—	4	2	no	—
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#137	—	2	8	no	—
50	C₃₁×#137	—	2	32	no	—
51	C₃₁×#137	C₂³	1	16	yes	complemented
52	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
53	C₃₁×AGL(1,ℤ/8)	—	2	32	no	—
54	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
55	C₃₁×#135	—	2	32	no	—
56	C₃₁×#135	D₄	1	8	yes	complemented
57	C₃₁×#135	C₂³	1	8	yes	—
58	C₂×Q₁₆×C₃₁	—	2	4	no	—
59	C₂×Q₁₆×C₃₁	D₄	1	2	yes	complemented
60	C₂×SD₁₆×C₃₁	—	2	4	no	—
61	C₂×SD₁₆×C₃₁	—	2	4	no	—
62	C₂×SD₁₆×C₃₁	—	2	32	no	—
63	C₂×SD₁₆×C₃₁	—	2	4	no	—
64	C₂×SD₁₆×C₃₁	—	2	4	no	—
65	C₂×SD₁₆×C₃₁	—	2	8	no	—
66	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
67	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	complemented
68	C₂×D₈×C₃₁	—	2	4	no	—
69	C₂×D₈×C₃₁	D₄	1	2	yes	complemented
70	C₃₁×#131	—	2	8	no	—
71	C₃₁×#131	C₂³	1	16	yes	complemented
72	C₂×M₁₆×C₃₁	—	2	4	no	—
73	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₃₁	—	2	4	no	—
75	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
76	C₃₁×C₄⋊Q₈	—	4	1	no	—
77	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	complemented
78	C₃₁×#127	—	4	1	no	—
79	C₃₁×#127	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#108	C₂³	1	8	yes	complemented
83	C₃₁×#104	C₂³	1	16	yes	complemented
84	C₃₁×#101	C₂³	1	8	yes	complemented
85	C₃₁×#100	C₂³	1	8	yes	complemented
86	C₄×C₈×C₃₁	C₂³	1	4	yes	complemented
87	C₂⁴×C₃₁	—	2	2	no	—
88	C₂⁴×C₃₁	—	4	2	no	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	2	4	no	—
91	#41×C₃₁	—	2	16	no	—
92	#41×C₃₁	—	2	16	no	—
93	#41×C₃₁	—	4	8	no	—
94	#41×C₃₁	—	4	8	no	—
95	#41×C₃₁	—	4	8	no	—
96	#41×C₃₁	—	4	8	no	—
97	#41×C₃₁	C₂×D₄	1	8	yes	—
98	#41×C₃₁	C₂⁴	1	8	yes	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	16	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	—	2	4	no	—
103	C₂×Q₈×C₃₁	—	4	2	no	—
104	C₂×Q₈×C₃₁	—	4	8	no	—
105	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
106	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	16	no	—
109	C₂×D₄×C₃₁	—	2	4	no	—
110	C₂×D₄×C₃₁	—	2	4	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	4	8	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	8	no	—
116	C₂×D₄×C₃₁	—	4	8	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	8	4	no	—
119	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
120	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	—	4	4	no	—
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	Q₁₆×C₃₁	—	2	16	no	—
128	Q₁₆×C₃₁	C₂×D₄	1	8	yes	complemented
129	SD₁₆×C₃₁	—	2	16	no	—
130	SD₁₆×C₃₁	—	2	16	no	—
131	SD₁₆×C₃₁	—	2	16	no	—
132	SD₁₆×C₃₁	—	2	16	no	—
133	SD₁₆×C₃₁	C₂×D₄	1	16	yes	complemented
134	D₈×C₃₁	—	2	16	no	—
135	D₈×C₃₁	C₂×D₄	1	8	yes	complemented
136	M₁₆×C₃₁	—	2	4	no	—
137	M₁₆×C₃₁	—	2	16	no	—
138	M₁₆×C₃₁	C₂⁴	1	8	yes	—
139	C₂×C₈×C₃₁	—	2	4	no	—
140	C₂×C₈×C₃₁	—	2	16	no	—
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
145	C₄⋊C₄×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	C₂⁴	1	2	yes	—
148	C₂×#3291	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	2	4	no	—
151	C₂³×C₃₁	—	4	8	no	—
152	C₂³×C₃₁	—	4	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	8	4	no	—
155	C₂³×C₃₁	—	16	4	no	—
156	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	4	no	—
158	Q₈×C₃₁	—	2	8	no	—
159	Q₈×C₃₁	—	4	8	no	—
160	Q₈×C₃₁	—	4	4	no	—
161	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
162	D₄×C₃₁	—	2	8	no	—
163	D₄×C₃₁	—	2	8	no	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	2	8	no	—
166	D₄×C₃₁	—	4	16	no	—
167	D₄×C₃₁	—	4	8	no	—
168	D₄×C₃₁	—	4	4	no	—
169	D₄×C₃₁	—	8	8	no	—
170	D₄×C₃₁	C₂²×D₄	1	8	yes	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	2	8	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	4	8	no	—
178	C₂×C₄×C₃₁	—	4	16	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
185	C₈×C₃₁	—	2	8	no	—
186	C₈×C₃₁	C₂²×D₄	1	4	yes	—
187	C₈×C₃₁	C₂²×D₄	1	4	yes	—
188	C₂×#577	C₂×C₃₁	1	4	yes	complemented
189	#3291	C₂×C₃₁	1	16	yes	direct product factor
190	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂×#471	C₂×C₃₁	1	4	yes	complemented
192	C₂×#456	C₂×C₃₁	1	2	yes	complemented
193	C₂×#453	C₂×C₃₁	1	2	yes	complemented
194	C₂×#443	C₂×C₃₁	1	2	yes	complemented
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	—	2	8	no	—
197	C₂²×C₃₁	—	4	8	no	—
198	C₂²×C₃₁	—	4	4	no	—
199	C₂²×C₃₁	—	8	4	no	—
200	C₂²×C₃₁	—	8	4	no	—
201	C₂²×C₃₁	—	16	2	no	—
202	C₂²×C₃₁	—	16	8	no	—
203	C₂²×C₃₁	D₄²	1	1	yes	characteristic
204	C₂²×C₃₁	D₄²	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
206	C₄×C₃₁	—	2	4	no	—
207	C₄×C₃₁	—	2	8	no	—
208	C₄×C₃₁	—	4	4	no	—
209	C₄×C₃₁	D₄²	1	2	yes	—
210	C₄×C₃₁	C₂³×D₄	1	2	yes	—
211	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
212	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	complemented
213	#577	—	2	16	no	—
214	#577	C₂²×C₃₁	1	32	yes	complemented
215	C₂×#137	—	2	4	no	—
216	C₂×#137	C₂²×C₃₁	1	4	yes	complemented
217	C₂×AGL(1,ℤ/8)	—	2	4	no	—
218	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
219	C₂×#135	—	2	4	no	—
220	C₂×#135	C₂²×C₃₁	1	2	yes	—
221	C₂²×SD₁₆	—	2	4	no	—
222	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
223	C₂×#131	C₂²×C₃₁	1	4	yes	complemented
224	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
226	#492	C₂²×C₃₁	1	8	yes	complemented
227	#471	C₂²×C₃₁	1	32	yes	complemented
228	#456	C₂²×C₃₁	1	16	yes	complemented
229	#453	C₂²×C₃₁	1	16	yes	complemented
230	#443	C₂²×C₃₁	1	16	yes	complemented
231	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
232	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
233	C₂×#101	C₂²×C₃₁	1	2	yes	complemented
234	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
235	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
236	C₂×C₃₁	—	2	1	no	—
237	C₂×C₃₁	—	2	1	no	—
238	C₂×C₃₁	—	4	8	no	—
239	C₂×C₃₁	—	8	4	no	—
240	C₂×C₃₁	#3291	1	2	yes	direct product factor
241	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
242	2₋¹⁺⁴	—	2	8	no	—
243	2₋¹⁺⁴	—	4	2	no	—
244	2₋¹⁺⁴	C₂³×C₃₁	1	8	yes	—
245	H(5,𝔽₂)	—	2	8	no	—
246	H(5,𝔽₂)	—	4	2	no	—
247	H(5,𝔽₂)	C₂³×C₃₁	1	8	yes	complemented
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	4	no	—
250	C₂×#41	—	4	2	no	—
251	C₂×#41	C₂³×C₃₁	1	4	yes	—
252	C₂²×Q₈	—	2	4	no	—
253	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
254	C₂²×D₄	—	2	4	no	—
255	C₂²×D₄	—	4	1	no	—
256	C₂²×D₄	—	4	2	no	—
257	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
258	#137	—	2	8	no	—
259	#137	—	2	32	no	—
260	#137	C₂³×C₃₁	1	16	yes	complemented
261	AGL(1,ℤ/8)	—	2	8	no	—
262	AGL(1,ℤ/8)	—	2	32	no	—
263	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
264	#135	—	2	32	no	—
265	#135	D₄×C₃₁	1	8	yes	complemented
266	#135	C₂³×C₃₁	1	8	yes	—
267	C₂×Q₁₆	—	2	4	no	—
268	C₂×Q₁₆	D₄×C₃₁	1	2	yes	complemented
269	C₂×SD₁₆	—	2	4	no	—
270	C₂×SD₁₆	—	2	4	no	—
271	C₂×SD₁₆	—	2	32	no	—
272	C₂×SD₁₆	—	2	4	no	—
273	C₂×SD₁₆	—	2	4	no	—
274	C₂×SD₁₆	—	2	8	no	—
275	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
276	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	complemented
277	C₂×D₈	—	2	4	no	—
278	C₂×D₈	D₄×C₃₁	1	2	yes	complemented
279	#131	—	2	8	no	—
280	#131	C₂³×C₃₁	1	16	yes	complemented
281	C₂×M₁₆	—	2	4	no	—
282	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
283	C₂²×C₈	—	2	4	no	—
284	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
285	C₄⋊Q₈	—	4	1	no	—
286	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	complemented
287	#127	—	4	1	no	—
288	#127	C₂³×C₃₁	1	4	yes	—
289	C₂×C₄⋊C₄	—	4	1	no	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	#108	C₂³×C₃₁	1	8	yes	complemented
292	#104	C₂³×C₃₁	1	16	yes	complemented
293	#101	C₂³×C₃₁	1	8	yes	complemented
294	#100	C₂³×C₃₁	1	8	yes	complemented
295	C₄×C₈	C₂³×C₃₁	1	4	yes	complemented
296	C₃₁	C₂×#3291	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	—	2	2	no	—
298	C₂⁴	—	4	2	no	—
299	#41	—	2	4	no	—
300	#41	—	2	4	no	—
301	#41	—	2	16	no	—
302	#41	—	2	16	no	—
303	#41	—	4	8	no	—
304	#41	—	4	8	no	—
305	#41	—	4	8	no	—
306	#41	—	4	8	no	—
307	#41	C₂×D₄×C₃₁	1	8	yes	—
308	#41	C₂⁴×C₃₁	1	8	yes	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	16	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	4	2	no	—
314	C₂×Q₈	—	4	8	no	—
315	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
316	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	16	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	4	8	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	4	no	—
325	C₂×D₄	—	4	8	no	—
326	C₂×D₄	—	4	8	no	—
327	C₂×D₄	—	4	2	no	—
328	C₂×D₄	—	8	4	no	—
329	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
330	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	4	no	—
336	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
337	Q₁₆	—	2	16	no	—
338	Q₁₆	C₂×D₄×C₃₁	1	8	yes	complemented
339	SD₁₆	—	2	16	no	—
340	SD₁₆	—	2	16	no	—
341	SD₁₆	—	2	16	no	—
342	SD₁₆	—	2	16	no	—
343	SD₁₆	C₂×D₄×C₃₁	1	16	yes	complemented
344	D₈	—	2	16	no	—
345	D₈	C₂×D₄×C₃₁	1	8	yes	complemented
346	M₁₆	—	2	4	no	—
347	M₁₆	—	2	16	no	—
348	M₁₆	C₂⁴×C₃₁	1	8	yes	—
349	C₂×C₈	—	2	4	no	—
350	C₂×C₈	—	2	16	no	—
351	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
354	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄²	—	2	1	no	—
357	C₄²	C₂⁴×C₃₁	1	2	yes	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	8	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	8	4	no	—
364	C₂³	—	16	4	no	—
365	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
366	Q₈	—	2	4	no	—
367	Q₈	—	2	8	no	—
368	Q₈	—	4	8	no	—
369	Q₈	—	4	4	no	—
370	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
371	D₄	—	2	8	no	—
372	D₄	—	2	8	no	—
373	D₄	—	2	4	no	—
374	D₄	—	2	8	no	—
375	D₄	—	4	16	no	—
376	D₄	—	4	8	no	—
377	D₄	—	4	4	no	—
378	D₄	—	8	8	no	—
379	D₄	C₂²×D₄×C₃₁	1	8	yes	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	16	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
393	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
394	C₈	—	2	8	no	—
395	C₈	C₂²×D₄×C₃₁	1	4	yes	—
396	C₈	C₂²×D₄×C₃₁	1	4	yes	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	8	no	—
399	C₂²	—	4	8	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	8	4	no	—
402	C₂²	—	8	4	no	—
403	C₂²	—	16	2	no	—
404	C₂²	—	16	8	no	—
405	C₂²	D₄²×C₃₁	1	1	yes	characteristic
406	C₂²	D₄²×C₃₁	1	1	yes	characteristic
407	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	C₄	—	2	4	no	—
409	C₄	—	2	8	no	—
410	C₄	—	4	4	no	—
411	C₄	D₄²×C₃₁	1	2	yes	—
412	C₄	C₂³×D₄×C₃₁	1	2	yes	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	4	8	no	—
416	C₂	—	8	4	no	—
417	C₂	C₃₁×#3291	1	2	yes	direct product factor
418	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#577	C₂	semidirect
2. C₃₁×#3291	C₂	direct (N × H)
3. C₂×C₃₁×#492	C₂	semidirect
4. C₂×C₃₁×#471	C₂	semidirect
5. C₂×C₃₁×#456	C₂	semidirect
6. C₂×C₃₁×#453	C₂	semidirect
7. C₂×C₃₁×#443	C₂	semidirect
9. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
11. C₃₁×#577	C₂²	semidirect
13. C₂×C₃₁×#137	C₂²	semidirect
19. C₂²×SD₁₆×C₃₁	C₂²	semidirect
20. C₂×C₃₁×#131	C₂²	semidirect
21. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
23. C₃₁×#492	C₂²	semidirect
24. C₃₁×#471	C₂²	semidirect
25. C₃₁×#456	C₂²	semidirect
26. C₃₁×#453	C₂²	semidirect
27. C₃₁×#443	C₂²	semidirect
28. C₂×C₃₁×#108	C₂²	semidirect
29. C₂×C₃₁×#104	C₂²	semidirect
30. C₂×C₃₁×#101	C₂²	semidirect
31. C₂×C₃₁×#100	C₂²	semidirect
32. C₂×C₄×C₈×C₃₁	C₂²	semidirect
38. C₃₁×H(5,𝔽₂)	C₂³	semidirect
51. C₃₁×#137	C₂³	semidirect
56. C₃₁×#135	D₄	semidirect
59. C₂×Q₁₆×C₃₁	D₄	semidirect
66. C₂×SD₁₆×C₃₁	D₄	semidirect
67. C₂×SD₁₆×C₃₁	C₂³	semidirect
69. C₂×D₈×C₃₁	D₄	semidirect
71. C₃₁×#131	C₂³	semidirect
77. C₃₁×C₄⋊Q₈	C₂³	semidirect
82. C₃₁×#108	C₂³	semidirect
83. C₃₁×#104	C₂³	semidirect
84. C₃₁×#101	C₂³	semidirect
85. C₃₁×#100	C₂³	semidirect
86. C₄×C₈×C₃₁	C₂³	semidirect
128. Q₁₆×C₃₁	C₂×D₄	semidirect
133. SD₁₆×C₃₁	C₂×D₄	semidirect
135. D₈×C₃₁	C₂×D₄	semidirect
148. C₂×#3291	C₃₁	direct (N × H)
188. C₂×#577	C₂×C₃₁	semidirect
189. #3291	C₂×C₃₁	direct (N × H)
190. C₂×#492	C₂×C₃₁	semidirect
191. C₂×#471	C₂×C₃₁	semidirect
192. C₂×#456	C₂×C₃₁	semidirect
193. C₂×#453	C₂×C₃₁	semidirect
194. C₂×#443	C₂×C₃₁	semidirect
212. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
214. #577	C₂²×C₃₁	semidirect
216. C₂×#137	C₂²×C₃₁	semidirect
222. C₂²×SD₁₆	C₂²×C₃₁	semidirect
223. C₂×#131	C₂²×C₃₁	semidirect
224. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
226. #492	C₂²×C₃₁	semidirect
227. #471	C₂²×C₃₁	semidirect
228. #456	C₂²×C₃₁	semidirect
229. #453	C₂²×C₃₁	semidirect
230. #443	C₂²×C₃₁	semidirect
231. C₂×#108	C₂²×C₃₁	semidirect
232. C₂×#104	C₂²×C₃₁	semidirect
233. C₂×#101	C₂²×C₃₁	semidirect
234. C₂×#100	C₂²×C₃₁	semidirect
235. C₂×C₄×C₈	C₂²×C₃₁	semidirect
240. C₂×C₃₁	#3291	direct (N × H)
247. H(5,𝔽₂)	C₂³×C₃₁	semidirect
260. #137	C₂³×C₃₁	semidirect
265. #135	D₄×C₃₁	semidirect
268. C₂×Q₁₆	D₄×C₃₁	semidirect
275. C₂×SD₁₆	D₄×C₃₁	semidirect
276. C₂×SD₁₆	C₂³×C₃₁	semidirect
278. C₂×D₈	D₄×C₃₁	semidirect
280. #131	C₂³×C₃₁	semidirect
286. C₄⋊Q₈	C₂³×C₃₁	semidirect
291. #108	C₂³×C₃₁	semidirect
292. #104	C₂³×C₃₁	semidirect
293. #101	C₂³×C₃₁	semidirect
294. #100	C₂³×C₃₁	semidirect
295. C₄×C₈	C₂³×C₃₁	semidirect
296. C₃₁	C₂×#3291	direct (N × H)
338. Q₁₆	C₂×D₄×C₃₁	semidirect
343. SD₁₆	C₂×D₄×C₃₁	semidirect
345. D₈	C₂×D₄×C₃₁	semidirect
417. C₂	C₃₁×#3291	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#577	2
C₃₁×#3291	2
C₂×C₃₁×#492	2
C₂×C₃₁×#471	2
C₂×C₃₁×#456	2
C₂×C₃₁×#453	2
C₂×C₃₁×#443	2
C₂×#3291	31

Subgroups — C2×C31×#3291 #21517861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3291 #21517861