Subgroups — C₂×C₃₁×#1378 #21469311

Order 7936.

586 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1378	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₂×C₃₁×M₃₂	—	2	1	no	—
10	C₂×C₃₁×M₃₂	—	2	1	no	—
11	C₂×C₃₁×M₃₂	—	2	1	no	—
12	C₂×C₃₁×M₃₂	—	2	1	no	—
13	C₂×C₃₁×M₃₂	—	2	1	no	—
14	C₂×C₃₁×M₃₂	—	2	1	no	—
15	C₂×C₃₁×M₃₂	C₄	1	2	yes	complemented
16	C₂×C₃₁×M₃₂	C₄	1	2	yes	complemented
17	C₂×C₃₁×M₃₂	C₄	1	2	yes	complemented
18	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
19	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₃₁	—	2	1	no	—
21	C₂²×C₁₆×C₃₁	C₄	1	2	yes	complemented
22	C₃₁×#405	—	2	1	no	—
23	C₃₁×#405	—	2	1	no	—
24	C₃₁×#405	C₄	1	2	yes	—
25	C₃₁×#405	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	—	2	1	no	—
28	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	C₈	1	2	yes	—
33	C₂×M₁₆×C₃₁	C₈	1	2	yes	—
34	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#117	—	2	1	no	—
51	C₃₁×#117	—	2	1	no	—
52	C₃₁×#117	C₈	1	2	yes	—
53	C₃₁×#117	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
56	C₂×C₄²×C₃₁	—	2	1	no	—
57	C₃₁×M₃₂	—	2	2	no	—
58	C₃₁×M₃₂	—	2	2	no	—
59	C₃₁×M₃₂	—	2	2	no	—
60	C₃₁×M₃₂	—	2	2	no	—
61	C₃₁×M₃₂	—	2	2	no	—
62	C₃₁×M₃₂	—	2	2	no	—
63	C₃₁×M₃₂	—	2	2	no	—
64	C₃₁×M₃₂	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆×C₃₁	—	2	2	no	—
66	C₂×C₁₆×C₃₁	—	2	2	no	—
67	C₂×C₁₆×C₃₁	—	2	2	no	—
68	C₂×C₁₆×C₃₁	—	2	2	no	—
69	C₂×C₁₆×C₃₁	—	2	2	no	—
70	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
71	C₃₁×#97	—	2	2	no	—
72	C₃₁×#97	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	—	2	2	no	—
75	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	M₁₆×C₃₁	—	2	2	no	—
92	M₁₆×C₃₁	—	2	2	no	—
93	M₁₆×C₃₁	—	2	2	no	—
94	M₁₆×C₃₁	C₂×C₈	1	4	yes	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₁₆×C₃₁	—	2	4	no	—
133	C₁₆×C₃₁	—	2	4	no	—
134	C₂×#1378	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	4	2	no	—
136	C₂³×C₃₁	C₄×C₈	1	1	yes	characteristic
137	C₂³×C₃₁	C₄×C₈	1	1	yes	characteristic
138	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
139	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
140	C₂³×C₃₁	C₄⋊C₈	1	1	yes	characteristic
141	C₂³×C₃₁	C₄⋊C₈	1	1	yes	characteristic
142	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	—	2	4	no	—
144	C₂×C₄×C₃₁	—	4	4	no	—
145	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
147	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
151	C₂×C₄×C₃₁	#97	1	1	yes	characteristic
152	C₂×C₄×C₃₁	#97	1	1	yes	characteristic
153	C₂×C₄×C₃₁	#97	1	1	yes	characteristic
154	C₂×C₄×C₃₁	#97	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
173	C₈×C₃₁	—	2	4	no	—
174	C₈×C₃₁	#95	1	1	yes	characteristic
175	C₈×C₃₁	#95	1	1	yes	characteristic
176	C₈×C₃₁	#95	1	1	yes	characteristic
177	C₈×C₃₁	#95	1	1	yes	characteristic
178	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
183	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
184	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
185	#1378	C₂×C₃₁	1	4	yes	direct product factor
186	C₂²×C₃₁	—	2	2	no	—
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	#336	1	1	yes	characteristic
194	C₂²×C₃₁	#336	1	1	yes	characteristic
195	C₂²×C₃₁	#336	1	1	yes	characteristic
196	C₂²×C₃₁	#336	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
198	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
200	C₄×C₃₁	—	4	4	no	—
201	C₄×C₃₁	#336	1	1	yes	characteristic
202	C₄×C₃₁	#336	1	1	yes	characteristic
203	C₄×C₃₁	#336	1	1	yes	characteristic
204	C₄×C₃₁	#336	1	1	yes	characteristic
205	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
206	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
207	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
208	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
209	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
210	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
212	C₂×M₃₂	—	2	1	no	—
213	C₂×M₃₂	—	2	1	no	—
214	C₂×M₃₂	—	2	1	no	—
215	C₂×M₃₂	—	2	1	no	—
216	C₂×M₃₂	—	2	1	no	—
217	C₂×M₃₂	—	2	1	no	—
218	C₂×M₃₂	—	2	1	no	—
219	C₂×M₃₂	C₄×C₃₁	1	2	yes	complemented
220	C₂×M₃₂	C₄×C₃₁	1	2	yes	complemented
221	C₂×M₃₂	C₄×C₃₁	1	2	yes	complemented
222	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
223	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
224	C₂²×C₁₆	—	2	1	no	—
225	C₂²×C₁₆	C₄×C₃₁	1	2	yes	complemented
226	#405	—	2	1	no	—
227	#405	—	2	1	no	—
228	#405	C₄×C₃₁	1	2	yes	—
229	#405	C₂²×C₃₁	1	2	yes	—
230	C₂×#97	—	2	1	no	—
231	C₂×C₄×C₈	—	2	1	no	—
232	C₂×C₃₁	—	2	1	no	—
233	C₂×C₃₁	—	2	1	no	—
234	C₂×C₃₁	—	2	1	no	—
235	C₂×C₃₁	—	2	1	no	—
236	C₂×C₃₁	—	2	1	no	—
237	C₂×C₃₁	—	2	1	no	—
238	C₂×C₃₁	#1378	1	2	yes	direct product factor
239	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
240	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂×M₁₆	—	2	1	no	—
242	C₂×M₁₆	—	2	1	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	C₈×C₃₁	1	2	yes	—
245	C₂×M₁₆	C₈×C₃₁	1	2	yes	—
246	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
260	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	#117	—	2	1	no	—
263	#117	—	2	1	no	—
264	#117	C₈×C₃₁	1	2	yes	—
265	#117	C₂×C₄×C₃₁	1	2	yes	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
268	C₂×C₄²	—	2	1	no	—
269	M₃₂	—	2	2	no	—
270	M₃₂	—	2	2	no	—
271	M₃₂	—	2	2	no	—
272	M₃₂	—	2	2	no	—
273	M₃₂	—	2	2	no	—
274	M₃₂	—	2	2	no	—
275	M₃₂	—	2	2	no	—
276	M₃₂	C₂×C₄×C₃₁	1	4	yes	complemented
277	C₂×C₁₆	—	2	2	no	—
278	C₂×C₁₆	—	2	2	no	—
279	C₂×C₁₆	—	2	2	no	—
280	C₂×C₁₆	—	2	2	no	—
281	C₂×C₁₆	—	2	2	no	—
282	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
283	#97	—	2	2	no	—
284	#97	—	2	2	no	—
285	C₄×C₈	—	2	2	no	—
286	C₄×C₈	—	2	2	no	—
287	C₃₁	C₂×#1378	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
304	M₁₆	—	2	2	no	—
305	M₁₆	—	2	2	no	—
306	M₁₆	—	2	2	no	—
307	M₁₆	C₂×C₈×C₃₁	1	4	yes	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
312	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
313	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
317	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
322	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
328	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
329	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	2	2	no	—
342	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₁₆	—	2	4	no	—
346	C₁₆	—	2	4	no	—
347	C₂³	—	4	2	no	—
348	C₂³	C₄×C₈×C₃₁	1	1	yes	characteristic
349	C₂³	C₄×C₈×C₃₁	1	1	yes	characteristic
350	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
351	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
352	C₂³	C₃₁×C₄⋊C₈	1	1	yes	characteristic
353	C₂³	C₃₁×C₄⋊C₈	1	1	yes	characteristic
354	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
358	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
359	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
360	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
361	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
362	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₃₁×#97	1	1	yes	characteristic
364	C₂×C₄	C₃₁×#97	1	1	yes	characteristic
365	C₂×C₄	C₃₁×#97	1	1	yes	characteristic
366	C₂×C₄	C₃₁×#97	1	1	yes	characteristic
367	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
368	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
369	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
370	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
371	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
372	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
373	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
374	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
375	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
376	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
377	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
378	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
379	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
384	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	C₃₁×#95	1	1	yes	characteristic
387	C₈	C₃₁×#95	1	1	yes	characteristic
388	C₈	C₃₁×#95	1	1	yes	characteristic
389	C₈	C₃₁×#95	1	1	yes	characteristic
390	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
391	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
392	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
393	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
394	C₂²	—	2	2	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	1	no	—
401	C₂²	C₃₁×#336	1	1	yes	characteristic
402	C₂²	C₃₁×#336	1	1	yes	characteristic
403	C₂²	C₃₁×#336	1	1	yes	characteristic
404	C₂²	C₃₁×#336	1	1	yes	characteristic
405	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
406	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
407	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
408	C₄	—	4	4	no	—
409	C₄	C₃₁×#336	1	1	yes	characteristic
410	C₄	C₃₁×#336	1	1	yes	characteristic
411	C₄	C₃₁×#336	1	1	yes	characteristic
412	C₄	C₃₁×#336	1	1	yes	characteristic
413	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
414	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
415	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
416	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	C₃₁×#1378	1	2	yes	direct product factor
424	C₂	C₂×C₃₁×#336	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	#336
prime radical	C₂⁴×C₃₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1378	C₂	direct (N × H)
15. C₂×C₃₁×M₃₂	C₄	semidirect
16. C₂×C₃₁×M₃₂	C₄	semidirect
17. C₂×C₃₁×M₃₂	C₄	semidirect
21. C₂²×C₁₆×C₃₁	C₄	semidirect
64. C₃₁×M₃₂	C₂×C₄	semidirect
70. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
134. C₂×#1378	C₃₁	direct (N × H)
185. #1378	C₂×C₃₁	direct (N × H)
219. C₂×M₃₂	C₄×C₃₁	semidirect
220. C₂×M₃₂	C₄×C₃₁	semidirect
221. C₂×M₃₂	C₄×C₃₁	semidirect
225. C₂²×C₁₆	C₄×C₃₁	semidirect
238. C₂×C₃₁	#1378	direct (N × H)
276. M₃₂	C₂×C₄×C₃₁	semidirect
282. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
287. C₃₁	C₂×#1378	direct (N × H)
423. C₂	C₃₁×#1378	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×M₃₂	2
C₂×C₃₁×#405	2
C₃₁×#1378	2
C₂×#1378	31

Subgroups — C2×C31×#1378 #21469311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1378 #21469311