Subgroups — C₂×C₃₁×#1806 #21477412

Order 7936.

1618 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#104	C₂	1	2	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1806	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#339	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₄×#41×C₃₁	—	2	2	no	—
10	C₄×#41×C₃₁	—	2	2	no	—
11	C₄×#41×C₃₁	C₄	1	4	yes	complemented
12	C₄×#41×C₃₁	C₂²	1	4	yes	—
13	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
14	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
18	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#117	—	2	1	no	—
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄²×C₃₁	—	2	1	no	—
23	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#423	—	2	2	no	—
26	C₃₁×#423	C₂²	1	4	yes	complemented
27	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
28	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
29	C₂×C₃₁×#104	—	2	8	no	—
30	C₂×C₃₁×#104	C₂²	1	8	yes	—
31	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
33	C₂×C₃₁×#95	—	2	1	no	—
34	C₃₁×#339	C₂²	1	8	yes	complemented
35	C₂×#41×C₃₁	—	2	1	no	—
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×#41×C₃₁	D₄	1	4	yes	—
41	C₂×#41×C₃₁	Q₈	1	4	yes	—
42	C₂×#41×C₃₁	C₂³	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	2	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	4	no	—
55	C₂×M₁₆×C₃₁	—	2	4	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
61	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	2	4	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₄×D₄×C₃₁	—	2	4	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
69	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	—	2	1	no	—
72	C₃₁×#117	—	2	4	no	—
73	C₃₁×#117	—	2	4	no	—
74	C₃₁×#117	C₂×C₄	1	2	yes	—
75	C₃₁×#117	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	—	2	1	no	—
88	C₂×C₄²×C₃₁	—	2	4	no	—
89	C₂×C₄²×C₃₁	—	2	4	no	—
90	C₂×C₄²×C₃₁	—	2	4	no	—
91	C₂×C₄²×C₃₁	—	2	4	no	—
92	C₂×C₄²×C₃₁	—	2	2	no	—
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
96	C₂×C₄²×C₃₁	D₄	1	2	yes	—
97	C₂×C₄²×C₃₁	D₄	1	2	yes	—
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
102	C₃₁×C₄⋊C₈	—	2	4	no	—
103	C₃₁×C₄⋊C₈	—	2	4	no	—
104	C₃₁×#104	—	2	16	no	—
105	C₃₁×#95	—	2	4	no	—
106	C₂⁴×C₃₁	—	2	1	no	—
107	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
108	#41×C₃₁	—	2	4	no	—
109	#41×C₃₁	—	2	4	no	—
110	#41×C₃₁	—	2	4	no	—
111	#41×C₃₁	—	2	4	no	—
112	#41×C₃₁	—	2	4	no	—
113	#41×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	C₄⋊C₄	1	8	yes	complemented
116	#41×C₃₁	C₂×D₄	1	4	yes	—
117	#41×C₃₁	C₂×Q₈	1	4	yes	—
118	C₂×Q₈×C₃₁	—	2	1	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
121	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	complemented
129	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	4	2	no	—
143	C₂²×C₄×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	4	4	no	—
145	C₂²×C₄×C₃₁	—	4	4	no	—
146	C₂²×C₄×C₃₁	—	4	4	no	—
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
162	C₂²×C₄×C₃₁	#41	1	2	yes	—
163	C₂²×C₄×C₃₁	#41	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	M₁₆×C₃₁	—	2	4	no	—
166	M₁₆×C₃₁	—	2	4	no	—
167	M₁₆×C₃₁	—	4	4	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	2	16	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	4	no	—
183	C₄²×C₃₁	—	2	4	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	2	4	no	—
187	C₄²×C₃₁	—	4	4	no	—
188	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
189	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
190	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
191	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
192	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
193	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
194	C₄²×C₃₁	C₂×D₄	1	2	yes	—
195	C₄²×C₃₁	C₂×D₄	1	2	yes	—
196	C₂×#1806	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	2	no	—
203	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂³×C₃₁	#121	1	2	yes	—
206	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
207	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₃₁	—	2	2	no	—
209	Q₈×C₃₁	—	2	4	no	—
210	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	4	no	—
213	D₄×C₃₁	—	2	8	no	—
214	D₄×C₃₁	—	2	2	no	—
215	D₄×C₃₁	—	2	4	no	—
216	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	complemented
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	8	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	8	no	—
233	C₂×C₄×C₃₁	—	4	16	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	4	4	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	2	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	#104	1	4	yes	complemented
240	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
248	C₂×C₄×C₃₁	#117	1	2	yes	—
249	C₂×C₄×C₃₁	#117	1	2	yes	—
250	C₂×C₄×C₃₁	#121	1	2	yes	—
251	C₂×C₄×C₃₁	#121	1	2	yes	—
252	C₂×C₄×C₃₁	#121	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
262	C₈×C₃₁	—	4	4	no	—
263	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
264	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
265	C₂²×#104	C₂×C₃₁	1	2	yes	—
266	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1806	C₂×C₃₁	1	8	yes	direct product factor
268	C₂×#339	C₂×C₃₁	1	2	yes	complemented
269	C₂²×C₃₁	—	2	4	no	—
270	C₂²×C₃₁	—	2	4	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	4	no	—
276	C₂²×C₃₁	—	4	8	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	4	2	no	—
279	C₂²×C₃₁	#380	1	2	yes	—
280	C₂²×C₃₁	#380	1	2	yes	—
281	C₂²×C₃₁	C₂×#104	1	2	yes	—
282	C₂²×C₃₁	#421	1	2	yes	—
283	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
284	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
285	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	4	no	—
290	C₄×C₃₁	—	4	8	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	#380	1	2	yes	—
294	C₄×C₃₁	#380	1	2	yes	—
295	C₄×C₃₁	C₂×#104	1	4	yes	complemented
296	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
297	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
298	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
299	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
300	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
301	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
302	C₄×#41	—	2	2	no	—
303	C₄×#41	—	2	2	no	—
304	C₄×#41	C₄×C₃₁	1	4	yes	complemented
305	C₄×#41	C₂²×C₃₁	1	4	yes	—
306	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
307	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
308	C₂×C₄×D₄	—	2	1	no	—
309	C₂×C₄×D₄	—	2	1	no	—
310	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
311	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
312	C₂×#117	—	2	1	no	—
313	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
314	C₂²×C₄⋊C₄	—	2	1	no	—
315	C₂²×C₄²	—	2	1	no	—
316	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
317	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
318	#423	—	2	2	no	—
319	#423	C₂²×C₃₁	1	4	yes	complemented
320	C₂×C₄⋊C₈	—	2	1	no	—
321	C₂×C₄⋊C₈	—	2	1	no	—
322	C₂×#104	—	2	8	no	—
323	C₂×#104	C₂²×C₃₁	1	8	yes	—
324	C₄×C₄⋊C₄	—	2	2	no	—
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
326	C₂×#95	—	2	1	no	—
327	#339	C₂²×C₃₁	1	8	yes	complemented
328	C₂×C₃₁	—	2	1	no	—
329	C₂×C₃₁	—	2	1	no	—
330	C₂×C₃₁	—	2	2	no	—
331	C₂×C₃₁	—	4	4	no	—
332	C₂×C₃₁	#1806	1	4	yes	direct product factor
333	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
334	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
335	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	2	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
341	C₂×#41	D₄×C₃₁	1	4	yes	—
342	C₂×#41	Q₈×C₃₁	1	4	yes	—
343	C₂×#41	C₂³×C₃₁	1	2	yes	—
344	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	2	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×M₁₆	—	2	2	no	—
355	C₂×M₁₆	—	2	4	no	—
356	C₂×M₁₆	—	2	4	no	—
357	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
358	C₂²×C₈	—	2	2	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
362	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	4	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
370	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	4	no	—
374	#117	—	2	4	no	—
375	#117	C₂×C₄×C₃₁	1	2	yes	—
376	#117	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄⋊C₄	—	2	8	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	4	no	—
381	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	—	2	4	no	—
390	C₂×C₄²	—	2	4	no	—
391	C₂×C₄²	—	2	4	no	—
392	C₂×C₄²	—	2	4	no	—
393	C₂×C₄²	—	2	2	no	—
394	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
397	C₂×C₄²	D₄×C₃₁	1	2	yes	—
398	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄²	D₄×C₃₁	1	2	yes	—
400	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
402	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
403	C₄⋊C₈	—	2	4	no	—
404	C₄⋊C₈	—	2	4	no	—
405	#104	—	2	16	no	—
406	#95	—	2	4	no	—
407	C₃₁	C₂×#1806	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	—	2	1	no	—
409	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
410	#41	—	2	4	no	—
411	#41	—	2	4	no	—
412	#41	—	2	4	no	—
413	#41	—	2	4	no	—
414	#41	—	2	4	no	—
415	#41	—	2	2	no	—
416	#41	—	2	2	no	—
417	#41	C₄⋊C₄×C₃₁	1	8	yes	complemented
418	#41	C₂×D₄×C₃₁	1	4	yes	—
419	#41	C₂×Q₈×C₃₁	1	4	yes	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	complemented
423	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	complemented
431	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
454	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
463	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
464	C₂²×C₄	#41×C₃₁	1	2	yes	—
465	C₂²×C₄	#41×C₃₁	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
467	M₁₆	—	2	4	no	—
468	M₁₆	—	2	4	no	—
469	M₁₆	—	4	4	no	—
470	C₂×C₈	—	2	4	no	—
471	C₂×C₈	—	4	4	no	—
472	C₄⋊C₄	—	2	16	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
478	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₄²	—	2	2	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	4	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	2	4	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	4	4	no	—
490	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
491	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
492	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
493	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
494	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
495	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
496	C₄²	C₂×D₄×C₃₁	1	2	yes	—
497	C₄²	C₂×D₄×C₃₁	1	2	yes	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
505	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
506	C₂³	C₃₁×#121	1	2	yes	—
507	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
508	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	Q₈	—	2	2	no	—
510	Q₈	—	2	4	no	—
511	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	complemented
512	D₄	—	2	4	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	8	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	4	no	—
517	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	complemented
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	—	4	16	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	C₃₁×#104	1	4	yes	complemented
541	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
543	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
544	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
545	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
549	C₂×C₄	C₃₁×#117	1	2	yes	—
550	C₂×C₄	C₃₁×#117	1	2	yes	—
551	C₂×C₄	C₃₁×#121	1	2	yes	—
552	C₂×C₄	C₃₁×#121	1	2	yes	—
553	C₂×C₄	C₃₁×#121	1	2	yes	—
554	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
555	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
556	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
557	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
563	C₈	—	4	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	4	8	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	C₃₁×#380	1	2	yes	—
575	C₂²	C₃₁×#380	1	2	yes	—
576	C₂²	C₂×C₃₁×#104	1	2	yes	—
577	C₂²	C₃₁×#421	1	2	yes	—
578	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
579	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
580	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	2	no	—
584	C₄	—	2	4	no	—
585	C₄	—	4	8	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	2	no	—
588	C₄	C₃₁×#380	1	2	yes	—
589	C₄	C₃₁×#380	1	2	yes	—
590	C₄	C₂×C₃₁×#104	1	4	yes	complemented
591	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
592	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
593	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
594	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
595	C₂	—	2	1	no	—
596	C₂	—	2	1	no	—
597	C₂	—	2	2	no	—
598	C₂	—	4	4	no	—
599	C₂	C₃₁×#1806	1	4	yes	direct product factor
600	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
601	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
602	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#423	C₂	semidirect
4. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#1806	C₂	direct (N × H)
6. C₂×C₃₁×#339	C₂	semidirect
11. C₄×#41×C₃₁	C₄	semidirect
13. C₂×C₄×Q₈×C₃₁	C₄	semidirect
14. C₂×C₄×Q₈×C₃₁	C₄	semidirect
17. C₂×C₄×D₄×C₃₁	C₄	semidirect
18. C₂×C₄×D₄×C₃₁	C₄	semidirect
26. C₃₁×#423	C₂²	semidirect
32. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
34. C₃₁×#339	C₂²	semidirect
60. C₄×Q₈×C₃₁	C₂×C₄	semidirect
61. C₄×Q₈×C₃₁	C₂×C₄	semidirect
68. C₄×D₄×C₃₁	C₂×C₄	semidirect
69. C₄×D₄×C₃₁	C₂×C₄	semidirect
115. #41×C₃₁	C₄⋊C₄	semidirect
120. C₂×Q₈×C₃₁	C₄⋊C₄	semidirect
128. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
196. C₂×#1806	C₃₁	direct (N × H)
210. Q₈×C₃₁	C₂×C₄⋊C₄	semidirect
216. D₄×C₃₁	C₂×C₄⋊C₄	semidirect
239. C₂×C₄×C₃₁	#104	semidirect
264. C₂×#423	C₂×C₃₁	semidirect
266. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
267. #1806	C₂×C₃₁	direct (N × H)
268. C₂×#339	C₂×C₃₁	semidirect
295. C₄×C₃₁	C₂×#104	semidirect
304. C₄×#41	C₄×C₃₁	semidirect
306. C₂×C₄×Q₈	C₄×C₃₁	semidirect
307. C₂×C₄×Q₈	C₄×C₃₁	semidirect
310. C₂×C₄×D₄	C₄×C₃₁	semidirect
311. C₂×C₄×D₄	C₄×C₃₁	semidirect
319. #423	C₂²×C₃₁	semidirect
325. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
327. #339	C₂²×C₃₁	semidirect
332. C₂×C₃₁	#1806	direct (N × H)
361. C₄×Q₈	C₂×C₄×C₃₁	semidirect
362. C₄×Q₈	C₂×C₄×C₃₁	semidirect
369. C₄×D₄	C₂×C₄×C₃₁	semidirect
370. C₄×D₄	C₂×C₄×C₃₁	semidirect
407. C₃₁	C₂×#1806	direct (N × H)
417. #41	C₄⋊C₄×C₃₁	semidirect
422. C₂×Q₈	C₄⋊C₄×C₃₁	semidirect
430. C₂×D₄	C₄⋊C₄×C₃₁	semidirect
511. Q₈	C₂×C₄⋊C₄×C₃₁	semidirect
517. D₄	C₂×C₄⋊C₄×C₃₁	semidirect
540. C₂×C₄	C₃₁×#104	semidirect
590. C₄	C₂×C₃₁×#104	semidirect
599. C₂	C₃₁×#1806	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×C₄×#41×C₃₁	2
C₂×C₃₁×#423	2
C₂²×C₃₁×#104	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1806	2
C₂×C₃₁×#339	2
C₂×#1806	31

Subgroups — C2×C31×#1806 #21477412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1806 #21477412