Subgroups — C₂²×C₃₁×#538 #21519430

Order 7936.

9566 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×#538	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×#126	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#124	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃₁×#121	C₂	1	2	yes	maximal, complemented
7	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₃₁×#120	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₂³×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁶×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
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	complemented
20	C₃₁×#538	C₂²	1	256	yes	direct product factor
21	C₂×C₃₁×#126	C₂²	1	48	yes	complemented
22	C₂×C₃₁×#124	C₂²	1	48	yes	complemented
23	C₂×C₃₁×#123	C₂²	1	48	yes	complemented
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
25	C₂×C₃₁×#121	C₂²	1	48	yes	complemented
26	C₂×C₃₁×#121	C₂²	1	24	yes	complemented
27	C₂×C₃₁×#120	C₂²	1	48	yes	complemented
28	C₂×C₄×D₄×C₃₁	C₂²	1	48	yes	complemented
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	24	yes	complemented
39	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
41	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
42	C₂⁵×C₃₁	—	2	8	no	—
43	C₂⁵×C₃₁	—	4	4	no	—
44	C₂⁵×C₃₁	—	4	4	no	—
45	C₂⁵×C₃₁	C₂³	1	2	yes	—
46	C₂⁵×C₃₁	C₂³	1	2	yes	—
47	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
48	C₂⁵×C₃₁	C₂³	1	12	yes	—
49	C₂²×Q₈×C₃₁	—	4	4	no	—
50	C₂²×Q₈×C₃₁	C₂³	1	12	yes	complemented
51	C₂²×D₄×C₃₁	—	2	16	no	—
52	C₂²×D₄×C₃₁	—	4	8	no	—
53	C₂²×D₄×C₃₁	—	4	8	no	—
54	C₂²×D₄×C₃₁	—	4	4	no	—
55	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
58	C₂²×D₄×C₃₁	C₂³	1	12	yes	complemented
59	C₂³×C₄×C₃₁	—	4	4	no	—
60	C₂³×C₄×C₃₁	—	4	4	no	—
61	C₂³×C₄×C₃₁	—	4	8	no	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	24	yes	complemented
71	C₃₁×#126	C₂³	1	128	yes	complemented
72	C₃₁×#124	C₂³	1	128	yes	complemented
73	C₃₁×#123	C₂³	1	128	yes	complemented
74	C₃₁×C₂²⋊Q₈	C₂³	1	64	yes	complemented
75	C₃₁×#121	C₂³	1	128	yes	complemented
76	C₃₁×#121	C₂³	1	64	yes	complemented
77	C₃₁×#120	C₂³	1	128	yes	complemented
78	C₄×D₄×C₃₁	C₂³	1	128	yes	complemented
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	complemented
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	complemented
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	complemented
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	complemented
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	64	yes	complemented
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	complemented
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	complemented
91	C₂×C₄²×C₃₁	C₂³	1	24	yes	complemented
92	C₂⁴×C₃₁	—	2	4	no	—
93	C₂⁴×C₃₁	—	2	4	no	—
94	C₂⁴×C₃₁	—	2	8	no	—
95	C₂⁴×C₃₁	—	2	48	no	—
96	C₂⁴×C₃₁	—	2	8	no	—
97	C₂⁴×C₃₁	—	2	8	no	—
98	C₂⁴×C₃₁	—	2	8	no	—
99	C₂⁴×C₃₁	—	2	8	no	—
100	C₂⁴×C₃₁	—	2	8	no	—
101	C₂⁴×C₃₁	—	4	24	no	—
102	C₂⁴×C₃₁	—	4	24	no	—
103	C₂⁴×C₃₁	—	4	64	no	—
104	C₂⁴×C₃₁	#41	1	8	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
107	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
108	C₂⁴×C₃₁	C₂⁴	1	16	yes	—
109	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈×C₃₁	—	4	24	no	—
111	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	complemented
112	C₂×D₄×C₃₁	—	2	96	no	—
113	C₂×D₄×C₃₁	—	4	48	no	—
114	C₂×D₄×C₃₁	—	4	48	no	—
115	C₂×D₄×C₃₁	—	4	24	no	—
116	C₂×D₄×C₃₁	C₂⁴	1	32	yes	—
117	C₂×D₄×C₃₁	C₂⁴	1	32	yes	—
118	C₂×D₄×C₃₁	C₂⁴	1	32	yes	—
119	C₂×D₄×C₃₁	C₂⁴	1	16	yes	complemented
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	8	no	—
126	C₂²×C₄×C₃₁	—	4	24	no	—
127	C₂²×C₄×C₃₁	—	4	24	no	—
128	C₂²×C₄×C₃₁	—	4	48	no	—
129	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	complemented
138	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	complemented
139	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	complemented
140	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	complemented
141	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	complemented
142	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
143	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
144	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
145	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
146	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
147	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	complemented
148	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	complemented
149	C₄²×C₃₁	C₂⁴	1	32	yes	complemented
150	C₂²×#538	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	32	no	—
152	C₂³×C₃₁	—	2	12	no	—
153	C₂³×C₃₁	—	2	12	no	—
154	C₂³×C₃₁	—	2	24	no	—
155	C₂³×C₃₁	—	2	32	no	—
156	C₂³×C₃₁	—	2	24	no	—
157	C₂³×C₃₁	—	2	24	no	—
158	C₂³×C₃₁	—	2	24	no	—
159	C₂³×C₃₁	—	2	24	no	—
160	C₂³×C₃₁	—	2	24	no	—
161	C₂³×C₃₁	—	4	16	no	—
162	C₂³×C₃₁	—	4	16	no	—
163	C₂³×C₃₁	—	4	192	no	—
164	C₂³×C₃₁	—	4	32	no	—
165	C₂³×C₃₁	—	4	32	no	—
166	C₂³×C₃₁	C₂×#41	1	24	yes	—
167	C₂³×C₃₁	C₂×#41	1	4	yes	—
168	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
169	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
170	C₂³×C₃₁	C₂⁵	1	8	yes	—
171	C₂³×C₃₁	C₂⁵	1	8	yes	—
172	C₂³×C₃₁	C₂⁵	1	4	yes	—
173	C₂³×C₃₁	C₂⁵	1	3	yes	—
174	Q₈×C₃₁	—	4	16	no	—
175	D₄×C₃₁	—	2	64	no	—
176	D₄×C₃₁	—	4	32	no	—
177	D₄×C₃₁	—	4	32	no	—
178	D₄×C₃₁	—	4	16	no	—
179	C₂×C₄×C₃₁	—	2	24	no	—
180	C₂×C₄×C₃₁	—	2	24	no	—
181	C₂×C₄×C₃₁	—	2	12	no	—
182	C₂×C₄×C₃₁	—	2	24	no	—
183	C₂×C₄×C₃₁	—	2	12	no	—
184	C₂×C₄×C₃₁	—	2	24	no	—
185	C₂×C₄×C₃₁	—	4	16	no	—
186	C₂×C₄×C₃₁	—	4	16	no	—
187	C₂×C₄×C₃₁	—	4	32	no	—
188	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
189	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
190	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
191	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
192	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
193	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
194	C₂×#538	C₂×C₃₁	1	48	yes	direct product factor
195	C₂²×#126	C₂×C₃₁	1	2	yes	complemented
196	C₂²×#124	C₂×C₃₁	1	2	yes	complemented
197	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
198	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×#121	C₂×C₃₁	1	2	yes	complemented
200	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×#120	C₂×C₃₁	1	2	yes	complemented
202	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
203	C₂³×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	2	4	no	—
206	C₂²×C₃₁	—	2	48	no	—
207	C₂²×C₃₁	—	2	8	no	—
208	C₂²×C₃₁	—	2	8	no	—
209	C₂²×C₃₁	—	2	8	no	—
210	C₂²×C₃₁	—	2	8	no	—
211	C₂²×C₃₁	—	2	8	no	—
212	C₂²×C₃₁	—	2	8	no	—
213	C₂²×C₃₁	—	4	64	no	—
214	C₂²×C₃₁	—	4	48	no	—
215	C₂²×C₃₁	—	4	48	no	—
216	C₂²×C₃₁	#538	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	C₂×H(5,𝔽₂)	1	6	yes	—
220	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	6	yes	—
221	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
222	C₄×C₃₁	—	2	8	no	—
223	C₄×C₃₁	—	2	8	no	—
224	C₄×C₃₁	—	2	4	no	—
225	C₄×C₃₁	—	2	8	no	—
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	2	8	no	—
228	C₂⁶	C₂²×C₃₁	1	1	yes	characteristic
229	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
230	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
231	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
232	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
233	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
234	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
235	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
236	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	complemented
237	#538	C₂²×C₃₁	1	256	yes	direct product factor
238	C₂×#126	C₂²×C₃₁	1	48	yes	complemented
239	C₂×#124	C₂²×C₃₁	1	48	yes	complemented
240	C₂×#123	C₂²×C₃₁	1	48	yes	complemented
241	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
242	C₂×#121	C₂²×C₃₁	1	48	yes	complemented
243	C₂×#121	C₂²×C₃₁	1	24	yes	complemented
244	C₂×#120	C₂²×C₃₁	1	48	yes	complemented
245	C₂×C₄×D₄	C₂²×C₃₁	1	48	yes	complemented
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
250	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
254	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
255	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	24	yes	complemented
256	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
257	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
258	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
259	C₂×C₃₁	—	2	8	no	—
260	C₂×C₃₁	—	4	8	no	—
261	C₂×C₃₁	—	4	8	no	—
262	C₂×C₃₁	C₂×#538	1	12	yes	direct product factor
263	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
264	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂⁵	—	2	8	no	—
267	C₂⁵	—	4	4	no	—
268	C₂⁵	—	4	4	no	—
269	C₂⁵	C₂³×C₃₁	1	2	yes	—
270	C₂⁵	C₂³×C₃₁	1	2	yes	—
271	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
272	C₂⁵	C₂³×C₃₁	1	12	yes	—
273	C₂²×Q₈	—	4	4	no	—
274	C₂²×Q₈	C₂³×C₃₁	1	12	yes	complemented
275	C₂²×D₄	—	2	16	no	—
276	C₂²×D₄	—	4	8	no	—
277	C₂²×D₄	—	4	8	no	—
278	C₂²×D₄	—	4	4	no	—
279	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
280	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
281	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
282	C₂²×D₄	C₂³×C₃₁	1	12	yes	complemented
283	C₂³×C₄	—	4	4	no	—
284	C₂³×C₄	—	4	4	no	—
285	C₂³×C₄	—	4	8	no	—
286	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
288	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
292	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
293	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
294	C₂³×C₄	C₂³×C₃₁	1	24	yes	complemented
295	#126	C₂³×C₃₁	1	128	yes	complemented
296	#124	C₂³×C₃₁	1	128	yes	complemented
297	#123	C₂³×C₃₁	1	128	yes	complemented
298	C₂²⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
299	#121	C₂³×C₃₁	1	128	yes	complemented
300	#121	C₂³×C₃₁	1	64	yes	complemented
301	#120	C₂³×C₃₁	1	128	yes	complemented
302	C₄×D₄	C₂³×C₃₁	1	128	yes	complemented
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	complemented
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	complemented
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	complemented
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	complemented
307	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₃₁	1	64	yes	complemented
313	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	complemented
314	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	complemented
315	C₂×C₄²	C₂³×C₃₁	1	24	yes	complemented
316	C₃₁	C₂²×#538	1	1	yes	characteristic, direct product factor, Sylow
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	8	no	—
320	C₂⁴	—	2	48	no	—
321	C₂⁴	—	2	8	no	—
322	C₂⁴	—	2	8	no	—
323	C₂⁴	—	2	8	no	—
324	C₂⁴	—	2	8	no	—
325	C₂⁴	—	2	8	no	—
326	C₂⁴	—	4	24	no	—
327	C₂⁴	—	4	24	no	—
328	C₂⁴	—	4	64	no	—
329	C₂⁴	#41×C₃₁	1	8	yes	—
330	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
331	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
332	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
333	C₂⁴	C₂⁴×C₃₁	1	16	yes	—
334	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂×Q₈	—	4	24	no	—
336	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	complemented
337	C₂×D₄	—	2	96	no	—
338	C₂×D₄	—	4	48	no	—
339	C₂×D₄	—	4	48	no	—
340	C₂×D₄	—	4	24	no	—
341	C₂×D₄	C₂⁴×C₃₁	1	32	yes	—
342	C₂×D₄	C₂⁴×C₃₁	1	32	yes	—
343	C₂×D₄	C₂⁴×C₃₁	1	32	yes	—
344	C₂×D₄	C₂⁴×C₃₁	1	16	yes	complemented
345	C₂²×C₄	—	2	8	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	8	no	—
351	C₂²×C₄	—	4	24	no	—
352	C₂²×C₄	—	4	24	no	—
353	C₂²×C₄	—	4	48	no	—
354	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
355	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
356	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
357	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
359	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
360	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
361	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
362	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	complemented
363	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	complemented
364	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	complemented
365	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	complemented
366	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	complemented
367	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
368	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
369	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
370	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
371	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
372	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	complemented
373	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	complemented
374	C₄²	C₂⁴×C₃₁	1	32	yes	complemented
375	C₂³	—	2	32	no	—
376	C₂³	—	2	12	no	—
377	C₂³	—	2	12	no	—
378	C₂³	—	2	24	no	—
379	C₂³	—	2	32	no	—
380	C₂³	—	2	24	no	—
381	C₂³	—	2	24	no	—
382	C₂³	—	2	24	no	—
383	C₂³	—	2	24	no	—
384	C₂³	—	2	24	no	—
385	C₂³	—	4	16	no	—
386	C₂³	—	4	16	no	—
387	C₂³	—	4	192	no	—
388	C₂³	—	4	32	no	—
389	C₂³	—	4	32	no	—
390	C₂³	C₂×#41×C₃₁	1	24	yes	—
391	C₂³	C₂×#41×C₃₁	1	4	yes	—
392	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
393	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
394	C₂³	C₂⁵×C₃₁	1	8	yes	—
395	C₂³	C₂⁵×C₃₁	1	8	yes	—
396	C₂³	C₂⁵×C₃₁	1	4	yes	—
397	C₂³	C₂⁵×C₃₁	1	3	yes	—
398	Q₈	—	4	16	no	—
399	D₄	—	2	64	no	—
400	D₄	—	4	32	no	—
401	D₄	—	4	32	no	—
402	D₄	—	4	16	no	—
403	C₂×C₄	—	2	24	no	—
404	C₂×C₄	—	2	24	no	—
405	C₂×C₄	—	2	12	no	—
406	C₂×C₄	—	2	24	no	—
407	C₂×C₄	—	2	12	no	—
408	C₂×C₄	—	2	24	no	—
409	C₂×C₄	—	4	16	no	—
410	C₂×C₄	—	4	16	no	—
411	C₂×C₄	—	4	32	no	—
412	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
413	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
414	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
415	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
416	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
417	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	48	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	2	8	no	—
423	C₂²	—	2	8	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	2	8	no	—
426	C₂²	—	2	8	no	—
427	C₂²	—	4	64	no	—
428	C₂²	—	4	48	no	—
429	C₂²	—	4	48	no	—
430	C₂²	C₃₁×#538	1	16	yes	direct product factor
431	C₂²	C₂²×#41×C₃₁	1	8	yes	—
432	C₂²	C₂²×#41×C₃₁	1	6	yes	—
433	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	6	yes	—
434	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	6	yes	—
435	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
436	C₄	—	2	8	no	—
437	C₄	—	2	8	no	—
438	C₄	—	2	4	no	—
439	C₄	—	2	8	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	8	no	—
442	C₂	—	2	8	no	—
443	C₂	—	4	8	no	—
444	C₂	—	4	8	no	—
445	C₂	C₂×C₃₁×#538	1	12	yes	direct product factor
446	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
447	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
448	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#538	C₂	direct (N × H)
2. C₂²×C₃₁×#126	C₂	semidirect
3. C₂²×C₃₁×#124	C₂	semidirect
4. C₂²×C₃₁×#123	C₂	semidirect
5. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₃₁×#121	C₂	semidirect
7. C₂²×C₃₁×#121	C₂	semidirect
8. C₂²×C₃₁×#120	C₂	semidirect
9. C₂²×C₄×D₄×C₃₁	C₂	semidirect
10. C₂³×C₂²⋊C₄×C₃₁	C₂	semidirect
12. C₂³×Q₈×C₃₁	C₂²	semidirect
16. C₂³×D₄×C₃₁	C₂²	semidirect
19. C₂⁴×C₄×C₃₁	C₂²	semidirect
20. C₃₁×#538	C₂²	direct (N × H)
21. C₂×C₃₁×#126	C₂²	semidirect
22. C₂×C₃₁×#124	C₂²	semidirect
23. C₂×C₃₁×#123	C₂²	semidirect
24. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
25. C₂×C₃₁×#121	C₂²	semidirect
26. C₂×C₃₁×#121	C₂²	semidirect
27. C₂×C₃₁×#120	C₂²	semidirect
28. C₂×C₄×D₄×C₃₁	C₂²	semidirect
29. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
31. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
32. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
38. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
39. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
40. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
41. C₂²×C₄²×C₃₁	C₂²	semidirect
50. C₂²×Q₈×C₃₁	C₂³	semidirect
58. C₂²×D₄×C₃₁	C₂³	semidirect
70. C₂³×C₄×C₃₁	C₂³	semidirect
71. C₃₁×#126	C₂³	semidirect
72. C₃₁×#124	C₂³	semidirect
73. C₃₁×#123	C₂³	semidirect
74. C₃₁×C₂²⋊Q₈	C₂³	semidirect
75. C₃₁×#121	C₂³	semidirect
76. C₃₁×#121	C₂³	semidirect
77. C₃₁×#120	C₂³	semidirect
78. C₄×D₄×C₃₁	C₂³	semidirect
79. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
80. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
81. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
82. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
88. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
89. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
90. C₂×C₂²⋊C₄×C₃₁	C₂³	semidirect
91. C₂×C₄²×C₃₁	C₂³	semidirect
111. C₂×Q₈×C₃₁	C₂⁴	semidirect
119. C₂×D₄×C₃₁	C₂⁴	semidirect
137. C₂²×C₄×C₃₁	C₂⁴	semidirect
138. C₄⋊C₄×C₃₁	C₂⁴	semidirect
139. C₄⋊C₄×C₃₁	C₂⁴	semidirect
140. C₄⋊C₄×C₃₁	C₂⁴	semidirect
141. C₄⋊C₄×C₃₁	C₂⁴	semidirect
147. C₂²⋊C₄×C₃₁	C₂⁴	semidirect
148. C₂²⋊C₄×C₃₁	C₂⁴	semidirect
149. C₄²×C₃₁	C₂⁴	semidirect
150. C₂²×#538	C₃₁	direct (N × H)
194. C₂×#538	C₂×C₃₁	direct (N × H)
195. C₂²×#126	C₂×C₃₁	semidirect
196. C₂²×#124	C₂×C₃₁	semidirect
197. C₂²×#123	C₂×C₃₁	semidirect
198. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
199. C₂²×#121	C₂×C₃₁	semidirect
200. C₂²×#121	C₂×C₃₁	semidirect
201. C₂²×#120	C₂×C₃₁	semidirect
202. C₂²×C₄×D₄	C₂×C₃₁	semidirect
203. C₂³×C₂²⋊C₄	C₂×C₃₁	semidirect
216. C₂²×C₃₁	#538	direct (N × H)
229. C₂³×Q₈	C₂²×C₃₁	semidirect
233. C₂³×D₄	C₂²×C₃₁	semidirect
236. C₂⁴×C₄	C₂²×C₃₁	semidirect
237. #538	C₂²×C₃₁	direct (N × H)
238. C₂×#126	C₂²×C₃₁	semidirect
239. C₂×#124	C₂²×C₃₁	semidirect
240. C₂×#123	C₂²×C₃₁	semidirect
241. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
242. C₂×#121	C₂²×C₃₁	semidirect
243. C₂×#121	C₂²×C₃₁	semidirect
244. C₂×#120	C₂²×C₃₁	semidirect
245. C₂×C₄×D₄	C₂²×C₃₁	semidirect
246. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
247. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
248. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
249. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
255. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
256. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
257. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
258. C₂²×C₄²	C₂²×C₃₁	semidirect
262. C₂×C₃₁	C₂×#538	direct (N × H)
274. C₂²×Q₈	C₂³×C₃₁	semidirect
282. C₂²×D₄	C₂³×C₃₁	semidirect
294. C₂³×C₄	C₂³×C₃₁	semidirect
295. #126	C₂³×C₃₁	semidirect
296. #124	C₂³×C₃₁	semidirect
297. #123	C₂³×C₃₁	semidirect
298. C₂²⋊Q₈	C₂³×C₃₁	semidirect
299. #121	C₂³×C₃₁	semidirect
300. #121	C₂³×C₃₁	semidirect
301. #120	C₂³×C₃₁	semidirect
302. C₄×D₄	C₂³×C₃₁	semidirect
303. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
304. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
305. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
306. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
312. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
313. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
314. C₂×C₂²⋊C₄	C₂³×C₃₁	semidirect
315. C₂×C₄²	C₂³×C₃₁	semidirect
316. C₃₁	C₂²×#538	direct (N × H)
336. C₂×Q₈	C₂⁴×C₃₁	semidirect
344. C₂×D₄	C₂⁴×C₃₁	semidirect
362. C₂²×C₄	C₂⁴×C₃₁	semidirect
363. C₄⋊C₄	C₂⁴×C₃₁	semidirect
364. C₄⋊C₄	C₂⁴×C₃₁	semidirect
365. C₄⋊C₄	C₂⁴×C₃₁	semidirect
366. C₄⋊C₄	C₂⁴×C₃₁	semidirect
372. C₂²⋊C₄	C₂⁴×C₃₁	semidirect
373. C₂²⋊C₄	C₂⁴×C₃₁	semidirect
374. C₄²	C₂⁴×C₃₁	semidirect
430. C₂²	C₃₁×#538	direct (N × H)
445. C₂	C₂×C₃₁×#538	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×#538	2
C₂²×C₃₁×#126	2
C₂²×C₃₁×#124	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₂²×C₃₁×#120	2
C₂²×C₄×D₄×C₃₁	2
C₂³×C₂²⋊C₄×C₃₁	2
C₂²×#538	31

Subgroups — C22×C31×#538 #21519430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#538 #21519430