Subgroups — C₂×C₃₁×#3522 #21519545

Order 7936.

7758 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3522	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₃₁×#559	C₂	1	3	yes	maximal, complemented
3	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#554	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#548	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#547	C₂	1	6	yes	maximal, complemented
8	C₂×C₃₁×#542	C₂	1	6	yes	maximal, complemented
9	C₂×C₃₁×#532	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#529	C₂	1	3	yes	maximal, complemented
11	C₂×C₃₁×#520	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	—
14	C₂²×#41×C₃₁	C₂²	1	6	yes	—
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#559	C₂²	1	48	yes	complemented
17	Q₈²×C₃₁	C₂²	1	16	yes	complemented
18	C₃₁×#554	C₂²	1	48	yes	complemented
19	C₃₁×#550	C₂²	1	16	yes	complemented
20	C₃₁×#548	C₂²	1	32	yes	complemented
21	C₃₁×#547	C₂²	1	96	yes	complemented
22	C₃₁×#542	C₂²	1	96	yes	complemented
23	C₃₁×#532	C₂²	1	48	yes	complemented
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	6	yes	—
26	C₂×C₃₁×#127	C₂²	1	3	yes	—
27	C₃₁×#529	C₂²	1	48	yes	complemented
28	C₂×C₃₁×#126	C₂²	1	6	yes	—
29	C₂×C₃₁×#125	C₂²	1	3	yes	—
30	C₂×C₃₁×#124	C₂²	1	3	yes	—
31	C₂×C₃₁×#124	C₂²	1	6	yes	—
32	C₂×C₃₁×#124	C₂²	1	6	yes	—
33	C₂×C₃₁×#123	C₂²	1	6	yes	—
34	C₂×C₃₁×#123	C₂²	1	12	yes	—
35	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	12	yes	—
37	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	—
38	C₂×C₃₁×#121	C₂²	1	3	yes	—
39	C₂×C₃₁×#121	C₂²	1	6	yes	—
40	C₂×C₃₁×#121	C₂²	1	6	yes	—
41	C₃₁×#520	C₂²	1	32	yes	complemented
42	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
44	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
46	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
47	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
48	C₂×C₄×Q₈×C₃₁	C₂²	1	16	yes	complemented
49	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
50	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
51	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
52	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
53	C₂×C₄×D₄×C₃₁	C₂²	1	6	yes	—
54	C₂×C₃₁×#117	C₂²	1	6	yes	—
55	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
56	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
57	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
58	C₂×#41×C₃₁	—	2	16	no	—
59	C₂×#41×C₃₁	—	4	24	no	—
60	C₂×#41×C₃₁	C₂³	1	16	yes	—
61	C₂×#41×C₃₁	C₂³	1	48	yes	—
62	C₂²×Q₈×C₃₁	—	2	8	no	—
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
77	C₃₁×C₄⋊Q₈	C₂³	1	24	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	48	yes	—
79	C₃₁×#127	C₂³	1	24	yes	—
80	C₃₁×#126	C₂³	1	48	yes	—
81	C₃₁×#125	C₂³	1	24	yes	—
82	C₃₁×#124	C₂³	1	24	yes	—
83	C₃₁×#124	C₂³	1	48	yes	—
84	C₃₁×#124	C₂³	1	48	yes	—
85	C₃₁×#123	C₂³	1	48	yes	—
86	C₃₁×#123	C₂³	1	96	yes	—
87	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	96	yes	—
89	C₃₁×C₂²⋊Q₈	C₂³	1	48	yes	—
90	C₃₁×#121	C₂³	1	24	yes	—
91	C₃₁×#121	C₂³	1	48	yes	—
92	C₃₁×#121	C₂³	1	48	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
96	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
97	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	24	yes	—
99	C₄×D₄×C₃₁	C₂³	1	48	yes	—
100	C₄×D₄×C₃₁	C₂³	1	48	yes	—
101	C₄×D₄×C₃₁	C₂³	1	48	yes	—
102	C₄×D₄×C₃₁	C₂³	1	24	yes	—
103	C₄×D₄×C₃₁	C₂³	1	48	yes	—
104	C₃₁×#117	C₂³	1	48	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	24	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
129	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
130	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
131	#41×C₃₁	—	2	64	no	—
132	#41×C₃₁	—	4	96	no	—
133	C₂×Q₈×C₃₁	—	2	32	no	—
134	C₂×Q₈×C₃₁	—	2	12	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	4	12	no	—
137	C₂×Q₈×C₃₁	#41	1	8	yes	complemented
138	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₃₁	C₂⁴	1	12	yes	—
140	C₂×Q₈×C₃₁	C₂⁴	1	24	yes	—
141	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
142	C₂×D₄×C₃₁	—	2	24	no	—
143	C₂×D₄×C₃₁	—	2	24	no	—
144	C₂×D₄×C₃₁	—	2	12	no	—
145	C₂×D₄×C₃₁	—	4	12	no	—
146	C₂×D₄×C₃₁	C₂⁴	1	12	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
148	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
149	C₂×D₄×C₃₁	C₂⁴	1	24	yes	—
150	C₂²×C₄×C₃₁	—	2	12	no	—
151	C₂²×C₄×C₃₁	—	2	12	no	—
152	C₂²×C₄×C₃₁	—	2	24	no	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	4	24	no	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	48	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
171	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
172	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
173	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	12	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	24	yes	—
179	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
180	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
181	C₂²⋊C₄×C₃₁	C₂⁴	1	8	yes	—
182	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
183	C₂²⋊C₄×C₃₁	C₂⁴	1	48	yes	—
184	C₂²⋊C₄×C₃₁	C₂⁴	1	24	yes	—
185	C₄²×C₃₁	C₂⁴	1	12	yes	—
186	C₄²×C₃₁	C₂⁴	1	12	yes	—
187	C₄²×C₃₁	C₂⁴	1	12	yes	—
188	C₄²×C₃₁	C₂⁴	1	12	yes	—
189	C₄²×C₃₁	C₂⁴	1	24	yes	—
190	C₂×#3522	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	12	no	—
194	C₂³×C₃₁	—	2	12	no	—
195	C₂³×C₃₁	—	2	12	no	—
196	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
197	C₂³×C₃₁	C₂⁵	1	12	yes	—
198	C₂³×C₃₁	C₂⁵	1	2	yes	—
199	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	24	no	—
201	Q₈×C₃₁	—	2	8	no	—
202	Q₈×C₃₁	—	4	24	no	—
203	Q₈×C₃₁	C₂×#41	1	16	yes	complemented
204	D₄×C₃₁	—	2	48	no	—
205	D₄×C₃₁	—	2	48	no	—
206	D₄×C₃₁	—	2	24	no	—
207	D₄×C₃₁	—	4	24	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	6	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	2	24	no	—
213	C₂×C₄×C₃₁	—	2	12	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	24	no	—
217	C₂×C₄×C₃₁	—	2	48	no	—
218	C₂×C₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	4	48	no	—
220	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
221	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
222	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
223	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
224	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
225	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
226	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
227	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
228	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
229	C₂×C₄×C₃₁	C₂⁵	1	6	yes	—
230	C₂×C₄×C₃₁	C₂⁵	1	2	yes	—
231	#3522	C₂×C₃₁	1	32	yes	direct product factor
232	C₂×#559	C₂×C₃₁	1	3	yes	complemented
233	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#554	C₂×C₃₁	1	3	yes	complemented
235	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#548	C₂×C₃₁	1	2	yes	complemented
237	C₂×#547	C₂×C₃₁	1	6	yes	complemented
238	C₂×#542	C₂×C₃₁	1	6	yes	complemented
239	C₂×#532	C₂×C₃₁	1	3	yes	complemented
240	C₂×#529	C₂×C₃₁	1	3	yes	complemented
241	C₂×#520	C₂×C₃₁	1	2	yes	complemented
242	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	8	no	—
246	C₂²×C₃₁	—	2	12	no	—
247	C₂²×C₃₁	—	2	12	no	—
248	C₂²×C₃₁	—	2	12	no	—
249	C₂²×C₃₁	—	4	24	no	—
250	C₂²×C₃₁	C₂²×#41	1	2	yes	—
251	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
252	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
254	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	6	no	—
257	C₄×C₃₁	—	2	6	no	—
258	C₄×C₃₁	—	2	6	no	—
259	C₄×C₃₁	—	2	12	no	—
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	4	no	—
262	C₄×C₃₁	C₂²×#41	1	12	yes	—
263	C₄×C₃₁	C₂×2₋¹⁺⁴	1	4	yes	—
264	C₂²×#41	C₂²×C₃₁	1	2	yes	—
265	C₂²×#41	C₂²×C₃₁	1	6	yes	—
266	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	#559	C₂²×C₃₁	1	48	yes	complemented
268	Q₈²	C₂²×C₃₁	1	16	yes	complemented
269	#554	C₂²×C₃₁	1	48	yes	complemented
270	#550	C₂²×C₃₁	1	16	yes	complemented
271	#548	C₂²×C₃₁	1	32	yes	complemented
272	#547	C₂²×C₃₁	1	96	yes	complemented
273	#542	C₂²×C₃₁	1	96	yes	complemented
274	#532	C₂²×C₃₁	1	48	yes	complemented
275	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
276	C₂×C₄⋊Q₈	C₂²×C₃₁	1	6	yes	—
277	C₂×#127	C₂²×C₃₁	1	3	yes	—
278	#529	C₂²×C₃₁	1	48	yes	complemented
279	C₂×#126	C₂²×C₃₁	1	6	yes	—
280	C₂×#125	C₂²×C₃₁	1	3	yes	—
281	C₂×#124	C₂²×C₃₁	1	3	yes	—
282	C₂×#124	C₂²×C₃₁	1	6	yes	—
283	C₂×#124	C₂²×C₃₁	1	6	yes	—
284	C₂×#123	C₂²×C₃₁	1	6	yes	—
285	C₂×#123	C₂²×C₃₁	1	12	yes	—
286	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	12	yes	—
288	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	—
289	C₂×#121	C₂²×C₃₁	1	3	yes	—
290	C₂×#121	C₂²×C₃₁	1	6	yes	—
291	C₂×#121	C₂²×C₃₁	1	6	yes	—
292	#520	C₂²×C₃₁	1	32	yes	complemented
293	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
295	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
297	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
299	C₂×C₄×Q₈	C₂²×C₃₁	1	16	yes	complemented
300	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
301	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
302	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
303	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
304	C₂×C₄×D₄	C₂²×C₃₁	1	6	yes	—
305	C₂×#117	C₂²×C₃₁	1	6	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
307	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
308	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
309	C₂×C₃₁	—	2	4	no	—
310	C₂×C₃₁	—	4	12	no	—
311	C₂×C₃₁	#3522	1	4	yes	direct product factor
312	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
313	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
315	C₂×#41	—	2	16	no	—
316	C₂×#41	—	4	24	no	—
317	C₂×#41	C₂³×C₃₁	1	16	yes	—
318	C₂×#41	C₂³×C₃₁	1	48	yes	—
319	C₂²×Q₈	—	2	8	no	—
320	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
322	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
323	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
324	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
326	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
327	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
328	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
329	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
330	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
331	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
334	C₄⋊Q₈	C₂³×C₃₁	1	24	yes	—
335	C₄⋊Q₈	C₂³×C₃₁	1	48	yes	—
336	#127	C₂³×C₃₁	1	24	yes	—
337	#126	C₂³×C₃₁	1	48	yes	—
338	#125	C₂³×C₃₁	1	24	yes	—
339	#124	C₂³×C₃₁	1	24	yes	—
340	#124	C₂³×C₃₁	1	48	yes	—
341	#124	C₂³×C₃₁	1	48	yes	—
342	#123	C₂³×C₃₁	1	48	yes	—
343	#123	C₂³×C₃₁	1	96	yes	—
344	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
345	C₂²⋊Q₈	C₂³×C₃₁	1	96	yes	—
346	C₂²⋊Q₈	C₂³×C₃₁	1	48	yes	—
347	#121	C₂³×C₃₁	1	24	yes	—
348	#121	C₂³×C₃₁	1	48	yes	—
349	#121	C₂³×C₃₁	1	48	yes	—
350	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
351	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
352	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
353	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
354	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
355	C₄×Q₈	C₂³×C₃₁	1	24	yes	—
356	C₄×D₄	C₂³×C₃₁	1	48	yes	—
357	C₄×D₄	C₂³×C₃₁	1	48	yes	—
358	C₄×D₄	C₂³×C₃₁	1	48	yes	—
359	C₄×D₄	C₂³×C₃₁	1	24	yes	—
360	C₄×D₄	C₂³×C₃₁	1	48	yes	—
361	#117	C₂³×C₃₁	1	48	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	6	yes	—
380	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
381	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
382	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
383	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
384	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
385	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
386	C₃₁	C₂×#3522	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
388	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
389	#41	—	2	64	no	—
390	#41	—	4	96	no	—
391	C₂×Q₈	—	2	32	no	—
392	C₂×Q₈	—	2	12	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	4	12	no	—
395	C₂×Q₈	#41×C₃₁	1	8	yes	complemented
396	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
397	C₂×Q₈	C₂⁴×C₃₁	1	12	yes	—
398	C₂×Q₈	C₂⁴×C₃₁	1	24	yes	—
399	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
400	C₂×D₄	—	2	24	no	—
401	C₂×D₄	—	2	24	no	—
402	C₂×D₄	—	2	12	no	—
403	C₂×D₄	—	4	12	no	—
404	C₂×D₄	C₂⁴×C₃₁	1	12	yes	—
405	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
406	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
407	C₂×D₄	C₂⁴×C₃₁	1	24	yes	—
408	C₂²×C₄	—	2	12	no	—
409	C₂²×C₄	—	2	12	no	—
410	C₂²×C₄	—	2	24	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	4	24	no	—
413	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
420	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	48	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
426	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
428	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
429	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
430	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
431	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
432	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
433	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
434	C₄⋊C₄	C₂⁴×C₃₁	1	12	yes	—
435	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
436	C₄⋊C₄	C₂⁴×C₃₁	1	24	yes	—
437	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
438	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
439	C₂²⋊C₄	C₂⁴×C₃₁	1	8	yes	—
440	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
441	C₂²⋊C₄	C₂⁴×C₃₁	1	48	yes	—
442	C₂²⋊C₄	C₂⁴×C₃₁	1	24	yes	—
443	C₄²	C₂⁴×C₃₁	1	12	yes	—
444	C₄²	C₂⁴×C₃₁	1	12	yes	—
445	C₄²	C₂⁴×C₃₁	1	12	yes	—
446	C₄²	C₂⁴×C₃₁	1	12	yes	—
447	C₄²	C₂⁴×C₃₁	1	24	yes	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	12	no	—
451	C₂³	—	2	12	no	—
452	C₂³	—	2	12	no	—
453	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
454	C₂³	C₂⁵×C₃₁	1	12	yes	—
455	C₂³	C₂⁵×C₃₁	1	2	yes	—
456	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
457	Q₈	—	2	24	no	—
458	Q₈	—	2	8	no	—
459	Q₈	—	4	24	no	—
460	Q₈	C₂×#41×C₃₁	1	16	yes	complemented
461	D₄	—	2	48	no	—
462	D₄	—	2	48	no	—
463	D₄	—	2	24	no	—
464	D₄	—	4	24	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	6	no	—
467	C₂×C₄	—	2	6	no	—
468	C₂×C₄	—	2	6	no	—
469	C₂×C₄	—	2	24	no	—
470	C₂×C₄	—	2	12	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	24	no	—
474	C₂×C₄	—	2	48	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	4	48	no	—
477	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
478	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
479	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
480	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
481	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
482	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
483	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
484	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
485	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
486	C₂×C₄	C₂⁵×C₃₁	1	6	yes	—
487	C₂×C₄	C₂⁵×C₃₁	1	2	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	2	12	no	—
492	C₂²	—	2	12	no	—
493	C₂²	—	2	12	no	—
494	C₂²	—	4	24	no	—
495	C₂²	C₂²×#41×C₃₁	1	2	yes	—
496	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
497	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
498	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
499	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	6	no	—
502	C₄	—	2	6	no	—
503	C₄	—	2	6	no	—
504	C₄	—	2	12	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	4	no	—
507	C₄	C₂²×#41×C₃₁	1	12	yes	—
508	C₄	C₂×C₃₁×2₋¹⁺⁴	1	4	yes	—
509	C₂	—	2	4	no	—
510	C₂	—	4	12	no	—
511	C₂	C₃₁×#3522	1	4	yes	direct product factor
512	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
513	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
514	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3522	C₂	direct (N × H)
2. C₂×C₃₁×#559	C₂	semidirect
3. C₂×Q₈²×C₃₁	C₂	semidirect
4. C₂×C₃₁×#554	C₂	semidirect
5. C₂×C₃₁×#550	C₂	semidirect
6. C₂×C₃₁×#548	C₂	semidirect
7. C₂×C₃₁×#547	C₂	semidirect
8. C₂×C₃₁×#542	C₂	semidirect
9. C₂×C₃₁×#532	C₂	semidirect
10. C₂×C₃₁×#529	C₂	semidirect
11. C₂×C₃₁×#520	C₂	semidirect
12. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
16. C₃₁×#559	C₂²	semidirect
17. Q₈²×C₃₁	C₂²	semidirect
18. C₃₁×#554	C₂²	semidirect
19. C₃₁×#550	C₂²	semidirect
20. C₃₁×#548	C₂²	semidirect
21. C₃₁×#547	C₂²	semidirect
22. C₃₁×#542	C₂²	semidirect
23. C₃₁×#532	C₂²	semidirect
27. C₃₁×#529	C₂²	semidirect
41. C₃₁×#520	C₂²	semidirect
48. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
137. C₂×Q₈×C₃₁	#41	semidirect
190. C₂×#3522	C₃₁	direct (N × H)
203. Q₈×C₃₁	C₂×#41	semidirect
231. #3522	C₂×C₃₁	direct (N × H)
232. C₂×#559	C₂×C₃₁	semidirect
233. C₂×Q₈²	C₂×C₃₁	semidirect
234. C₂×#554	C₂×C₃₁	semidirect
235. C₂×#550	C₂×C₃₁	semidirect
236. C₂×#548	C₂×C₃₁	semidirect
237. C₂×#547	C₂×C₃₁	semidirect
238. C₂×#542	C₂×C₃₁	semidirect
239. C₂×#532	C₂×C₃₁	semidirect
240. C₂×#529	C₂×C₃₁	semidirect
241. C₂×#520	C₂×C₃₁	semidirect
242. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
267. #559	C₂²×C₃₁	semidirect
268. Q₈²	C₂²×C₃₁	semidirect
269. #554	C₂²×C₃₁	semidirect
270. #550	C₂²×C₃₁	semidirect
271. #548	C₂²×C₃₁	semidirect
272. #547	C₂²×C₃₁	semidirect
273. #542	C₂²×C₃₁	semidirect
274. #532	C₂²×C₃₁	semidirect
278. #529	C₂²×C₃₁	semidirect
292. #520	C₂²×C₃₁	semidirect
299. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
311. C₂×C₃₁	#3522	direct (N × H)
386. C₃₁	C₂×#3522	direct (N × H)
395. C₂×Q₈	#41×C₃₁	semidirect
460. Q₈	C₂×#41×C₃₁	semidirect
511. C₂	C₃₁×#3522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3522	2
C₂×C₃₁×#559	2
C₂×Q₈²×C₃₁	2
C₂×C₃₁×#554	2
C₂×C₃₁×#550	2
C₂×C₃₁×#548	2
C₂×C₃₁×#547	2
C₂×C₃₁×#542	2
C₂×C₃₁×#532	2
C₂×C₃₁×#529	2
C₂×C₃₁×#520	2
C₂²×C₄×Q₈×C₃₁	2
C₂×#3522	31

Subgroups — C2×C31×#3522 #21519545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3522 #21519545