Subgroups — C₄×C₃₁×#421 #21477163

Order 7936.

1618 conjugacy classes of subgroups.

The subgroup lattice has 518 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×#421	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1759	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#104	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1733	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	—	2	2	no	—
11	C₄×#41×C₃₁	—	2	1	no	—
12	C₄×#41×C₃₁	—	2	1	no	—
13	C₄×#41×C₃₁	C₄	1	4	yes	complemented
14	C₄×#41×C₃₁	C₂²	1	2	yes	—
15	C₄×#41×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	complemented
17	C₂×C₄×D₄×C₃₁	—	2	2	no	—
18	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	complemented
19	C₂×C₃₁×#117	—	2	1	no	—
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	2	yes	—
22	C₂²×C₄²×C₃₁	—	2	1	no	—
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#421	C₄	1	16	yes	direct product factor
25	C₂×C₃₁×#104	C₂²	1	8	yes	—
26	C₄×M₁₆×C₃₁	—	2	1	no	—
27	C₄×M₁₆×C₃₁	—	2	1	no	—
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
30	C₂×C₃₁×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
34	C₃₁×#376	C₂²	1	2	yes	complemented
35	C₄³×C₃₁	C₂²	1	2	yes	complemented
36	C₃₁×#339	C₂²	1	8	yes	complemented
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂×#41×C₃₁	—	2	1	no	—
39	C₂×#41×C₃₁	—	2	4	no	—
40	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
41	C₂×#41×C₃₁	C₂×C₄	1	2	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	D₄	1	2	yes	—
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
53	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	4	no	—
61	C₄×D₄×C₃₁	—	2	4	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
66	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	—	2	1	no	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	2	4	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	—	2	2	no	—
73	C₃₁×#117	C₂×C₄	1	2	yes	—
74	C₃₁×#117	C₂×C₄	1	8	yes	complemented
75	C₃₁×#117	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	—	2	4	no	—
85	C₂×C₄²×C₃₁	—	2	2	no	—
86	C₂×C₄²×C₃₁	—	2	2	no	—
87	C₂×C₄²×C₃₁	—	2	4	no	—
88	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₃₁	D₄	1	2	yes	—
91	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×#104	C₂×C₄	1	32	yes	complemented
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₃₁×#95	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	1	no	—
105	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	8	no	—
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	—	2	4	no	—
110	#41×C₃₁	C₄²	1	8	yes	complemented
111	#41×C₃₁	C₂²×C₄	1	8	yes	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	C₄²	1	4	yes	complemented
115	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	8	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	C₄²	1	4	yes	complemented
122	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	8	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	4	8	no	—
133	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	#41	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₃₁	—	2	8	no	—
148	M₁₆×C₃₁	—	2	4	no	—
149	M₁₆×C₃₁	C₂²×C₄	1	8	yes	complemented
150	C₂×C₈×C₃₁	—	2	8	no	—
151	C₂×C₈×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	8	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	4	no	—
164	C₄²×C₃₁	—	2	4	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	4	no	—
167	C₄²×C₃₁	—	4	8	no	—
168	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
169	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
170	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
171	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
172	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
173	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
174	C₄²×C₃₁	C₂²×C₄	1	8	yes	complemented
175	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₄×#421	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	4	no	—
184	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂³×C₃₁	C₄×D₄	1	4	yes	—
187	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	2	4	no	—
190	Q₈×C₃₁	C₂×C₄²	1	4	yes	—
191	D₄×C₃₁	—	2	8	no	—
192	D₄×C₃₁	—	2	8	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	4	no	—
195	D₄×C₃₁	C₂×C₄²	1	4	yes	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	8	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	8	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	8	no	—
209	C₂×C₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	4	4	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
213	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
214	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₃₁	#117	1	2	yes	—
221	C₂×C₄×C₃₁	#117	1	2	yes	—
222	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
223	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
229	C₈×C₃₁	—	2	8	no	—
230	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
231	C₂×#421	C₂×C₃₁	1	2	yes	—
232	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1759	C₂×C₃₁	1	4	yes	complemented
235	C₄×#104	C₂×C₃₁	1	4	yes	complemented
236	#1733	C₂×C₃₁	1	2	yes	complemented
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	4	8	no	—
243	C₂²×C₃₁	—	4	4	no	—
244	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
245	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
246	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	4	no	—
252	C₄×C₃₁	—	4	8	no	—
253	C₄×C₃₁	—	4	4	no	—
254	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
255	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
256	C₄×C₃₁	#421	1	4	yes	direct product factor
257	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
258	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
259	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
260	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
261	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
262	C₄×#41	—	2	2	no	—
263	C₄×#41	—	2	1	no	—
264	C₄×#41	—	2	1	no	—
265	C₄×#41	C₄×C₃₁	1	4	yes	complemented
266	C₄×#41	C₂²×C₃₁	1	2	yes	—
267	C₄×#41	C₂²×C₃₁	1	2	yes	—
268	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	complemented
269	C₂×C₄×D₄	—	2	2	no	—
270	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	complemented
271	C₂×#117	—	2	1	no	—
272	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#117	C₂²×C₃₁	1	2	yes	—
274	C₂²×C₄²	—	2	1	no	—
275	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
276	#421	C₄×C₃₁	1	16	yes	direct product factor
277	C₂×#104	C₂²×C₃₁	1	8	yes	—
278	C₄×M₁₆	—	2	1	no	—
279	C₄×M₁₆	—	2	1	no	—
280	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
281	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
282	C₂×#97	—	2	1	no	—
283	C₂×C₄×C₈	—	2	1	no	—
284	C₄×C₄⋊C₄	—	2	2	no	—
285	C₄×C₂²⋊C₄	—	2	2	no	—
286	#376	C₂²×C₃₁	1	2	yes	complemented
287	C₄³	C₂²×C₃₁	1	2	yes	complemented
288	#339	C₂²×C₃₁	1	8	yes	complemented
289	C₂×C₃₁	—	2	2	no	—
290	C₂×C₃₁	—	2	2	no	—
291	C₂×C₃₁	—	2	2	no	—
292	C₂×C₃₁	—	4	4	no	—
293	C₂×C₃₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
294	C₂×C₃₁	C₂×#421	1	2	yes	—
295	C₂×#41	—	2	1	no	—
296	C₂×#41	—	2	1	no	—
297	C₂×#41	—	2	4	no	—
298	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
299	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
300	C₂×#41	C₂³×C₃₁	1	2	yes	—
301	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	D₄×C₃₁	1	2	yes	—
307	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	2	4	no	—
310	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
311	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
312	C₂²×C₈	—	2	2	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
316	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	4	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
324	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
325	#117	—	2	1	no	—
326	#117	—	2	1	no	—
327	#117	—	2	4	no	—
328	#117	—	2	4	no	—
329	#117	—	2	2	no	—
330	#117	—	2	2	no	—
331	#117	C₂×C₄×C₃₁	1	2	yes	—
332	#117	C₂×C₄×C₃₁	1	8	yes	complemented
333	#117	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	—	2	4	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	4	no	—
346	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
347	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₄²	D₄×C₃₁	1	2	yes	—
349	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
354	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#104	C₂×C₄×C₃₁	1	32	yes	complemented
357	#97	—	2	2	no	—
358	#97	—	2	2	no	—
359	C₄×C₈	—	2	2	no	—
360	C₄×C₈	—	2	2	no	—
361	#95	—	2	4	no	—
362	C₃₁	C₄×#421	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	2	1	no	—
364	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
365	#41	—	2	4	no	—
366	#41	—	2	8	no	—
367	#41	—	2	8	no	—
368	#41	—	2	4	no	—
369	#41	C₄²×C₃₁	1	8	yes	complemented
370	#41	C₂²×C₄×C₃₁	1	8	yes	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	C₄²×C₃₁	1	4	yes	complemented
374	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	C₄²×C₃₁	1	4	yes	complemented
381	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	—	4	8	no	—
392	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
394	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
395	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	#41×C₃₁	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	M₁₆	—	2	8	no	—
407	M₁₆	—	2	4	no	—
408	M₁₆	C₂²×C₄×C₃₁	1	8	yes	complemented
409	C₂×C₈	—	2	8	no	—
410	C₂×C₈	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	4	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	4	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	4	no	—
426	C₄²	—	4	8	no	—
427	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
428	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
429	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
430	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
431	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
432	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
433	C₄²	C₂²×C₄×C₃₁	1	8	yes	complemented
434	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂³	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	4	no	—
442	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂³	C₄×D₄×C₃₁	1	4	yes	—
445	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	4	no	—
448	Q₈	C₂×C₄²×C₃₁	1	4	yes	—
449	D₄	—	2	8	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	4	no	—
453	D₄	C₂×C₄²×C₃₁	1	4	yes	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	8	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
471	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
475	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
476	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₃₁×#117	1	2	yes	—
479	C₂×C₄	C₃₁×#117	1	2	yes	—
480	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
481	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
482	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
483	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
487	C₈	—	2	8	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	8	no	—
494	C₂²	—	4	4	no	—
495	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
496	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
497	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
498	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	8	no	—
504	C₄	—	4	4	no	—
505	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
506	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
507	C₄	C₃₁×#421	1	4	yes	direct product factor
508	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
509	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
510	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
511	C₂	—	2	2	no	—
512	C₂	—	2	2	no	—
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	C₂×C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
516	C₂	C₂×C₃₁×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
4. C₄×C₃₁×#117	C₂	semidirect
5. C₃₁×#1759	C₂	semidirect
6. C₄×C₃₁×#104	C₂	semidirect
7. C₃₁×#1733	C₂	semidirect
13. C₄×#41×C₃₁	C₄	semidirect
16. C₂×C₄×Q₈×C₃₁	C₄	semidirect
18. C₂×C₄×D₄×C₃₁	C₄	semidirect
24. C₃₁×#421	C₄	direct (N × H)
28. C₄×M₁₆×C₃₁	C₂²	semidirect
29. C₄×M₁₆×C₃₁	C₂²	semidirect
34. C₃₁×#376	C₂²	semidirect
35. C₄³×C₃₁	C₂²	semidirect
36. C₃₁×#339	C₂²	semidirect
52. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
74. C₃₁×#117	C₂×C₄	semidirect
98. C₃₁×#104	C₂×C₄	semidirect
110. #41×C₃₁	C₄²	semidirect
114. C₂×Q₈×C₃₁	C₄²	semidirect
121. C₂×D₄×C₃₁	C₄²	semidirect
149. M₁₆×C₃₁	C₂²×C₄	semidirect
174. C₄²×C₃₁	C₂²×C₄	semidirect
179. C₄×#421	C₃₁	direct (N × H)
232. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
233. C₄×#117	C₂×C₃₁	semidirect
234. #1759	C₂×C₃₁	semidirect
235. C₄×#104	C₂×C₃₁	semidirect
236. #1733	C₂×C₃₁	semidirect
256. C₄×C₃₁	#421	direct (N × H)
265. C₄×#41	C₄×C₃₁	semidirect
268. C₂×C₄×Q₈	C₄×C₃₁	semidirect
270. C₂×C₄×D₄	C₄×C₃₁	semidirect
276. #421	C₄×C₃₁	direct (N × H)
280. C₄×M₁₆	C₂²×C₃₁	semidirect
281. C₄×M₁₆	C₂²×C₃₁	semidirect
286. #376	C₂²×C₃₁	semidirect
287. C₄³	C₂²×C₃₁	semidirect
288. #339	C₂²×C₃₁	semidirect
310. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
332. #117	C₂×C₄×C₃₁	semidirect
356. #104	C₂×C₄×C₃₁	semidirect
362. C₃₁	C₄×#421	direct (N × H)
369. #41	C₄²×C₃₁	semidirect
373. C₂×Q₈	C₄²×C₃₁	semidirect
380. C₂×D₄	C₄²×C₃₁	semidirect
408. M₁₆	C₂²×C₄×C₃₁	semidirect
433. C₄²	C₂²×C₄×C₃₁	semidirect
507. C₄	C₃₁×#421	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×#421	2
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1759	2
C₄×C₃₁×#104	2
C₃₁×#1733	2
C₄×#421	31

Subgroups — C4×C31×#421 #21477163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#421 #21477163