Subgroups — C₂²×C₃₁×#529 #21519422

Order 7936.

8670 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
socle	C₂⁴×C₃₁	C₂⁴
prime radical	C₂²×C₃₁×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#529	C₂	direct (N × H)
2. C₂²×C₃₁×#126	C₂	semidirect
3. C₂²×C₃₁×#125	C₂	semidirect
4. C₂²×C₃₁×#124	C₂	semidirect
5. C₂²×C₃₁×#123	C₂	semidirect
6. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
8. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
9. C₂²×C₄×D₄×C₃₁	C₂	semidirect
10. C₂²×C₄×D₄×C₃₁	C₂	semidirect
11. C₂²×C₃₁×#117	C₂	semidirect
12. C₂³×C₄²×C₃₁	C₂	semidirect
20. C₃₁×#529	C₂²	direct (N × H)
21. C₂×C₃₁×#126	C₂²	semidirect
22. C₂×C₃₁×#125	C₂²	semidirect
23. C₂×C₃₁×#124	C₂²	semidirect
24. C₂×C₃₁×#123	C₂²	semidirect
25. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
27. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
28. C₂×C₄×D₄×C₃₁	C₂²	semidirect
29. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₂×C₃₁×#117	C₂²	semidirect
47. C₂²×C₄²×C₃₁	C₂²	semidirect
73. C₃₁×#126	C₂³	semidirect
74. C₃₁×#125	C₂³	semidirect
75. C₃₁×#124	C₂³	semidirect
76. C₃₁×#123	C₂³	semidirect
77. C₃₁×C₂²⋊Q₈	C₂³	semidirect
79. C₄×Q₈×C₃₁	C₂³	semidirect
80. C₄×D₄×C₃₁	C₂³	semidirect
81. C₄×D₄×C₃₁	C₂³	semidirect
82. C₃₁×#117	C₂³	semidirect
99. C₂×C₄²×C₃₁	C₂³	semidirect
157. C₂²×#529	C₃₁	direct (N × H)
197. C₂×#529	C₂×C₃₁	direct (N × H)
198. C₂²×#126	C₂×C₃₁	semidirect
199. C₂²×#125	C₂×C₃₁	semidirect
200. C₂²×#124	C₂×C₃₁	semidirect
201. C₂²×#123	C₂×C₃₁	semidirect
202. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
204. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
205. C₂²×C₄×D₄	C₂×C₃₁	semidirect
206. C₂²×C₄×D₄	C₂×C₃₁	semidirect
207. C₂²×#117	C₂×C₃₁	semidirect
208. C₂³×C₄²	C₂×C₃₁	semidirect
216. C₂²×C₃₁	#529	direct (N × H)
238. #529	C₂²×C₃₁	direct (N × H)
239. C₂×#126	C₂²×C₃₁	semidirect
240. C₂×#125	C₂²×C₃₁	semidirect
241. C₂×#124	C₂²×C₃₁	semidirect
242. C₂×#123	C₂²×C₃₁	semidirect
243. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
245. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
246. C₂×C₄×D₄	C₂²×C₃₁	semidirect
247. C₂×C₄×D₄	C₂²×C₃₁	semidirect
248. C₂×#117	C₂²×C₃₁	semidirect
265. C₂²×C₄²	C₂²×C₃₁	semidirect
268. C₂×C₃₁	C₂×#529	direct (N × H)
297. #126	C₂³×C₃₁	semidirect
298. #125	C₂³×C₃₁	semidirect
299. #124	C₂³×C₃₁	semidirect
300. #123	C₂³×C₃₁	semidirect
301. C₂²⋊Q₈	C₂³×C₃₁	semidirect
303. C₄×Q₈	C₂³×C₃₁	semidirect
304. C₄×D₄	C₂³×C₃₁	semidirect
305. C₄×D₄	C₂³×C₃₁	semidirect
306. #117	C₂³×C₃₁	semidirect
323. C₂×C₄²	C₂³×C₃₁	semidirect
324. C₃₁	C₂²×#529	direct (N × H)
428. C₂²	C₃₁×#529	direct (N × H)
445. C₂	C₂×C₃₁×#529	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×#529	2
C₂²×C₃₁×#126	2
C₂²×C₃₁×#125	2
C₂²×C₃₁×#124	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₂²×C₃₁×#117	2
C₂³×C₄²×C₃₁	2
C₂²×#529	31

Subgroups — C22×C31×#529 #21519422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#529 #21519422