Subgroups — C₂×C₃₁×#3398 #21518154

Order 7936.

1906 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3398	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#497	C₂	1	2	yes	maximal
5	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#464	C₂	1	2	yes	maximal
7	C₂×C₃₁×#462	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#441	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
13	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#557	—	2	4	no	—
19	C₃₁×#557	C₂²	1	8	yes	complemented
20	C₃₁×#556	—	2	4	no	—
21	C₃₁×#556	C₂²	1	8	yes	—
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#126	—	2	2	no	—
26	C₂×C₃₁×#125	—	2	1	no	—
27	C₂×C₃₁×#125	—	2	2	no	—
28	C₂×C₃₁×#124	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
30	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#117	—	2	1	no	—
37	C₃₁×#497	C₂²	1	16	yes	—
38	C₃₁×#494	C₂²	1	8	yes	complemented
39	C₃₁×#464	C₂²	1	16	yes	—
40	C₃₁×#462	C₂²	1	16	yes	complemented
41	C₃₁×#462	C₂²	1	8	yes	complemented
42	C₃₁×#461	C₂²	1	8	yes	—
43	C₃₁×#446	C₂²	1	8	yes	complemented
44	C₃₁×#441	C₂²	1	16	yes	complemented
45	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
46	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#103	C₂²	1	2	yes	—
51	C₂×C₃₁×#103	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂×Q₁₆×C₃₁	—	2	8	no	—
62	C₂×Q₁₆×C₃₁	—	4	4	no	—
63	C₂×Q₁₆×C₃₁	—	4	4	no	—
64	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
65	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
66	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
67	C₂×SD₁₆×C₃₁	—	2	4	no	—
68	C₂×SD₁₆×C₃₁	—	4	4	no	—
69	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×C₄⋊Q₈	—	2	2	no	—
75	C₃₁×C₄⋊Q₈	—	2	2	no	—
76	C₃₁×C₄⋊Q₈	—	2	4	no	—
77	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
80	C₃₁×#126	—	2	16	no	—
81	C₃₁×#125	—	2	8	no	—
82	C₃₁×#125	—	2	16	no	—
83	C₃₁×#124	—	2	4	no	—
84	C₃₁×#124	C₂³	1	8	yes	—
85	C₃₁×C₂²⋊Q₈	—	2	8	no	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×Q₈×C₃₁	—	2	4	no	—
88	C₄×Q₈×C₃₁	—	2	2	no	—
89	C₄×Q₈×C₃₁	—	2	4	no	—
90	C₄×Q₈×C₃₁	D₄	1	4	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	C₂³	1	4	yes	—
97	C₃₁×#117	—	2	8	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄²×C₃₁	—	2	2	no	—
115	C₂×C₄²×C₃₁	—	2	2	no	—
116	C₂×C₄²×C₃₁	—	2	1	no	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₃₁×#106	C₂³	1	4	yes	—
120	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
121	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
122	C₃₁×#103	C₂³	1	4	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	8	yes	—
125	C₃₁×#102	C₂³	1	4	yes	—
126	C₃₁×#97	C₂³	1	8	yes	—
127	C₄×C₈×C₃₁	C₂³	1	4	yes	—
128	C₂⁴×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	2	2	no	—
139	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
140	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
141	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
142	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	Q₁₆×C₃₁	—	2	16	no	—
160	Q₁₆×C₃₁	—	4	8	no	—
161	Q₁₆×C₃₁	—	4	8	no	—
162	SD₁₆×C₃₁	—	2	8	no	—
163	SD₁₆×C₃₁	—	4	8	no	—
164	C₂×C₈×C₃₁	—	2	4	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	#41	1	4	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
169	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂²⋊C₄×C₃₁	—	2	8	no	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₄²×C₃₁	—	2	8	no	—
189	C₄²×C₃₁	—	2	8	no	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	C₂×D₄	1	2	yes	—
192	C₄²×C₃₁	C₂×D₄	1	4	yes	—
193	C₄²×C₃₁	C₂⁴	1	2	yes	—
194	C₄²×C₃₁	C₂⁴	1	4	yes	—
195	C₂×#3398	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	4	2	no	—
198	C₂³×C₃₁	—	4	2	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₃₁	—	2	8	no	—
202	Q₈×C₃₁	—	2	2	no	—
203	Q₈×C₃₁	—	2	4	no	—
204	Q₈×C₃₁	—	2	4	no	—
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	2	4	no	—
207	Q₈×C₃₁	—	2	2	no	—
208	Q₈×C₃₁	—	4	4	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	Q₈×C₃₁	—	4	4	no	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	8	no	—
228	C₂×C₄×C₃₁	#137	1	4	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
235	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
236	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
237	C₈×C₃₁	—	2	4	no	—
238	C₈×C₃₁	—	2	2	no	—
239	C₈×C₃₁	C₂×#41	1	4	yes	—
240	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
242	#3398	C₂×C₃₁	1	16	yes	direct product factor
243	C₂×#497	C₂×C₃₁	1	2	yes	—
244	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂×#464	C₂×C₃₁	1	2	yes	—
246	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂×#462	C₂×C₃₁	1	2	yes	complemented
248	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
249	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂×#441	C₂×C₃₁	1	2	yes	complemented
251	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	8	4	no	—
256	C₂²×C₃₁	#550	1	2	yes	—
257	C₂²×C₃₁	C₂×#137	1	2	yes	—
258	C₂²×C₃₁	#578	1	2	yes	—
259	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	4	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	#550	1	2	yes	—
269	C₄×C₃₁	C₂×#137	1	4	yes	—
270	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
271	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
272	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
273	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
274	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
275	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
277	#557	—	2	4	no	—
278	#557	C₂²×C₃₁	1	8	yes	complemented
279	#556	—	2	4	no	—
280	#556	C₂²×C₃₁	1	8	yes	—
281	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
284	C₂×#126	—	2	2	no	—
285	C₂×#125	—	2	1	no	—
286	C₂×#125	—	2	2	no	—
287	C₂×#124	C₂²×C₃₁	1	2	yes	—
288	C₂×C₂²⋊Q₈	—	2	1	no	—
289	C₂×C₄×Q₈	—	2	1	no	—
290	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#117	—	2	1	no	—
296	#497	C₂²×C₃₁	1	16	yes	—
297	#494	C₂²×C₃₁	1	8	yes	complemented
298	#464	C₂²×C₃₁	1	16	yes	—
299	#462	C₂²×C₃₁	1	16	yes	complemented
300	#462	C₂²×C₃₁	1	8	yes	complemented
301	#461	C₂²×C₃₁	1	8	yes	—
302	#446	C₂²×C₃₁	1	8	yes	complemented
303	#441	C₂²×C₃₁	1	16	yes	complemented
304	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
305	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
308	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×#103	C₂²×C₃₁	1	2	yes	—
310	C₂×#103	C₂²×C₃₁	1	2	yes	—
311	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#97	C₂²×C₃₁	1	2	yes	—
313	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₃₁	—	8	2	no	—
315	C₂×C₃₁	#3398	1	4	yes	direct product factor
316	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
317	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
318	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
319	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
320	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
321	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
323	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	—	2	1	no	—
325	C₂×Q₁₆	—	2	8	no	—
326	C₂×Q₁₆	—	4	4	no	—
327	C₂×Q₁₆	—	4	4	no	—
328	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
329	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
330	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
331	C₂×SD₁₆	—	2	4	no	—
332	C₂×SD₁₆	—	4	4	no	—
333	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
334	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
335	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	4	no	—
341	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
342	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
343	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
344	#126	—	2	16	no	—
345	#125	—	2	8	no	—
346	#125	—	2	16	no	—
347	#124	—	2	4	no	—
348	#124	C₂³×C₃₁	1	8	yes	—
349	C₂²⋊Q₈	—	2	8	no	—
350	C₄×Q₈	—	2	8	no	—
351	C₄×Q₈	—	2	4	no	—
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	—	2	4	no	—
354	C₄×Q₈	D₄×C₃₁	1	4	yes	—
355	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
356	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
357	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
358	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	C₂³×C₃₁	1	4	yes	—
361	#117	—	2	8	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	—	2	2	no	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
383	#106	C₂³×C₃₁	1	4	yes	—
384	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
385	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
386	#103	C₂³×C₃₁	1	4	yes	—
387	#103	C₂³×C₃₁	1	8	yes	—
388	#103	C₂³×C₃₁	1	8	yes	—
389	#102	C₂³×C₃₁	1	4	yes	—
390	#97	C₂³×C₃₁	1	8	yes	—
391	C₄×C₈	C₂³×C₃₁	1	4	yes	—
392	C₃₁	C₂×#3398	1	1	yes	characteristic, direct product factor, Sylow
393	C₂⁴	—	2	1	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	4	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
405	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
406	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
407	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	4	no	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	Q₁₆	—	2	16	no	—
425	Q₁₆	—	4	8	no	—
426	Q₁₆	—	4	8	no	—
427	SD₁₆	—	2	8	no	—
428	SD₁₆	—	4	8	no	—
429	C₂×C₈	—	2	4	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	#41×C₃₁	1	4	yes	—
432	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
434	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
447	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
448	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
449	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₄²	—	2	8	no	—
454	C₄²	—	2	8	no	—
455	C₄²	—	2	4	no	—
456	C₄²	C₂×D₄×C₃₁	1	2	yes	—
457	C₄²	C₂×D₄×C₃₁	1	4	yes	—
458	C₄²	C₂⁴×C₃₁	1	2	yes	—
459	C₄²	C₂⁴×C₃₁	1	4	yes	—
460	C₂³	—	2	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	Q₈	—	2	8	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	4	no	—
468	Q₈	—	2	4	no	—
469	Q₈	—	2	4	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	2	2	no	—
472	Q₈	—	4	4	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	4	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	2	no	—
477	D₄	—	4	2	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	C₃₁×#137	1	4	yes	—
493	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
495	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
498	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
499	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
500	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
501	C₈	—	2	4	no	—
502	C₈	—	2	2	no	—
503	C₈	C₂×#41×C₃₁	1	4	yes	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	8	4	no	—
508	C₂²	C₃₁×#550	1	2	yes	—
509	C₂²	C₂×C₃₁×#137	1	2	yes	—
510	C₂²	C₃₁×#578	1	2	yes	—
511	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
512	C₄	—	2	4	no	—
513	C₄	—	2	4	no	—
514	C₄	—	2	2	no	—
515	C₄	—	4	4	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	4	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	C₃₁×#550	1	2	yes	—
521	C₄	C₂×C₃₁×#137	1	4	yes	—
522	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
523	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
524	C₂	—	8	2	no	—
525	C₂	C₃₁×#3398	1	4	yes	direct product factor
526	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
527	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
528	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3398	C₂	direct (N × H)
5. C₂×C₃₁×#494	C₂	semidirect
7. C₂×C₃₁×#462	C₂	semidirect
8. C₂×C₃₁×#462	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
11. C₂×C₃₁×#441	C₂	semidirect
19. C₃₁×#557	C₂²	semidirect
38. C₃₁×#494	C₂²	semidirect
40. C₃₁×#462	C₂²	semidirect
41. C₃₁×#462	C₂²	semidirect
43. C₃₁×#446	C₂²	semidirect
44. C₃₁×#441	C₂²	semidirect
195. C₂×#3398	C₃₁	direct (N × H)
240. C₂×#557	C₂×C₃₁	semidirect
242. #3398	C₂×C₃₁	direct (N × H)
244. C₂×#494	C₂×C₃₁	semidirect
246. C₂×#462	C₂×C₃₁	semidirect
247. C₂×#462	C₂×C₃₁	semidirect
249. C₂×#446	C₂×C₃₁	semidirect
250. C₂×#441	C₂×C₃₁	semidirect
278. #557	C₂²×C₃₁	semidirect
297. #494	C₂²×C₃₁	semidirect
299. #462	C₂²×C₃₁	semidirect
300. #462	C₂²×C₃₁	semidirect
302. #446	C₂²×C₃₁	semidirect
303. #441	C₂²×C₃₁	semidirect
315. C₂×C₃₁	#3398	direct (N × H)
392. C₃₁	C₂×#3398	direct (N × H)
525. C₂	C₃₁×#3398	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×#557	2
C₂×C₃₁×#556	2
C₃₁×#3398	2
C₂×C₃₁×#497	2
C₂×C₃₁×#494	2
C₂×C₃₁×#464	2
C₂×C₃₁×#462	2
C₂×C₃₁×#461	2
C₂×C₃₁×#446	2
C₂×C₃₁×#441	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3398	31

Subgroups — C2×C31×#3398 #21518154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3398 #21518154