Subgroups — C₂²×C₃₁×#542 #21519437

Order 7936.

7974 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#542	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#126	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃₁×#124	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
7	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#542	C₂²	1	256	yes	direct product factor
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	24	yes	complemented
21	C₂×C₃₁×#126	C₂²	1	48	yes	complemented
22	C₂×C₃₁×#124	C₂²	1	24	yes	complemented
23	C₂×C₃₁×#124	C₂²	1	48	yes	complemented
24	C₂×C₃₁×#123	C₂²	1	48	yes	complemented
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	48	yes	complemented
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	24	yes	complemented
27	C₂×C₃₁×#121	C₂²	1	24	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	24	yes	complemented
29	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	complemented
30	C₂×C₃₁×#117	C₂²	1	24	yes	complemented
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
42	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
43	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
44	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
45	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
46	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
47	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
48	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
49	C₂⁵×C₃₁	C₂³	1	2	yes	—
50	C₂²×Q₈×C₃₁	—	2	8	no	—
51	C₂²×Q₈×C₃₁	—	4	8	no	—
52	C₂²×Q₈×C₃₁	C₂³	1	12	yes	—
53	C₂²×Q₈×C₃₁	C₂³	1	24	yes	—
54	C₂²×D₄×C₃₁	—	2	8	no	—
55	C₂²×D₄×C₃₁	—	4	8	no	—
56	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
58	C₂³×C₄×C₃₁	—	2	8	no	—
59	C₂³×C₄×C₃₁	—	4	8	no	—
60	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊Q₈	C₂³	1	64	yes	complemented
72	C₃₁×#126	C₂³	1	128	yes	complemented
73	C₃₁×#124	C₂³	1	64	yes	complemented
74	C₃₁×#124	C₂³	1	128	yes	complemented
75	C₃₁×#123	C₂³	1	128	yes	complemented
76	C₃₁×C₂²⋊Q₈	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	C₃₁×#117	C₂³	1	64	yes	complemented
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	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	24	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	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×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	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	C₂⁴×C₃₁	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	8	no	—
101	C₂⁴×C₃₁	—	2	8	no	—
102	C₂⁴×C₃₁	—	2	4	no	—
103	C₂⁴×C₃₁	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	8	no	—
105	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
107	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₂×Q₈×C₃₁	—	2	48	no	—
109	C₂×Q₈×C₃₁	—	4	48	no	—
110	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	—
111	C₂×Q₈×C₃₁	C₂⁴	1	32	yes	—
112	C₂×D₄×C₃₁	—	2	48	no	—
113	C₂×D₄×C₃₁	—	4	48	no	—
114	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
115	C₂×D₄×C₃₁	C₂⁴	1	32	yes	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	48	no	—
125	C₂²×C₄×C₃₁	—	4	48	no	—
126	C₂²×C₄×C₃₁	#41	1	8	yes	—
127	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	12	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	6	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
138	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
139	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
140	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
141	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
142	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
143	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
144	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
145	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
146	C₂²⋊C₄×C₃₁	C₂⁴	1	16	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	32	yes	—
155	C₂²×#542	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	12	no	—
157	C₂³×C₃₁	—	2	24	no	—
158	C₂³×C₃₁	—	2	24	no	—
159	C₂³×C₃₁	—	2	12	no	—
160	C₂³×C₃₁	—	2	12	no	—
161	C₂³×C₃₁	—	2	24	no	—
162	C₂³×C₃₁	—	4	16	no	—
163	C₂³×C₃₁	—	4	32	no	—
164	C₂³×C₃₁	C₂×#41	1	4	yes	—
165	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
166	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
167	C₂³×C₃₁	C₂⁵	1	4	yes	—
168	C₂³×C₃₁	C₂⁵	1	8	yes	—
169	C₂³×C₃₁	C₂⁵	1	3	yes	—
170	Q₈×C₃₁	—	2	32	no	—
171	Q₈×C₃₁	—	4	32	no	—
172	D₄×C₃₁	—	2	32	no	—
173	D₄×C₃₁	—	4	32	no	—
174	C₂×C₄×C₃₁	—	2	12	no	—
175	C₂×C₄×C₃₁	—	2	24	no	—
176	C₂×C₄×C₃₁	—	2	12	no	—
177	C₂×C₄×C₃₁	—	2	24	no	—
178	C₂×C₄×C₃₁	—	2	24	no	—
179	C₂×C₄×C₃₁	—	2	12	no	—
180	C₂×C₄×C₃₁	—	2	12	no	—
181	C₂×C₄×C₃₁	—	2	12	no	—
182	C₂×C₄×C₃₁	—	2	32	no	—
183	C₂×C₄×C₃₁	—	4	32	no	—
184	C₂×C₄×C₃₁	C₂×#41	1	24	yes	—
185	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
186	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
187	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
188	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
189	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
190	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
191	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
192	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
193	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
194	C₂×#542	C₂×C₃₁	1	48	yes	direct product factor
195	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×#126	C₂×C₃₁	1	2	yes	complemented
197	C₂²×#124	C₂×C₃₁	1	2	yes	complemented
198	C₂²×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
200	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
201	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
203	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	2	4	no	—
207	C₂²×C₃₁	—	2	8	no	—
208	C₂²×C₃₁	—	2	8	no	—
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	8	no	—
212	C₂²×C₃₁	—	4	24	no	—
213	C₂²×C₃₁	—	4	48	no	—
214	C₂²×C₃₁	#542	1	16	yes	direct product factor
215	C₂²×C₃₁	C₂²×#41	1	6	yes	—
216	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	6	yes	—
217	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	6	yes	—
218	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	—	2	8	no	—
221	C₄×C₃₁	—	2	4	no	—
222	C₄×C₃₁	—	2	8	no	—
223	C₄×C₃₁	—	2	8	no	—
224	C₄×C₃₁	—	2	4	no	—
225	C₄×C₃₁	—	2	4	no	—
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	C₂²×#41	1	8	yes	—
228	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
230	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
231	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
232	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
234	#542	C₂²×C₃₁	1	256	yes	direct product factor
235	C₂×C₄⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
236	C₂×#126	C₂²×C₃₁	1	48	yes	complemented
237	C₂×#124	C₂²×C₃₁	1	24	yes	complemented
238	C₂×#124	C₂²×C₃₁	1	48	yes	complemented
239	C₂×#123	C₂²×C₃₁	1	48	yes	complemented
240	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	48	yes	complemented
241	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
242	C₂×#121	C₂²×C₃₁	1	24	yes	—
243	C₂×C₄×Q₈	C₂²×C₃₁	1	24	yes	complemented
244	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	complemented
245	C₂×#117	C₂²×C₃₁	1	24	yes	complemented
246	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
251	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
254	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
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	2	yes	—
263	C₂×C₃₁	—	4	4	no	—
264	C₂×C₃₁	—	4	8	no	—
265	C₂×C₃₁	C₂×#542	1	12	yes	direct product factor
266	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
267	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
269	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
270	C₂⁵	C₂³×C₃₁	1	2	yes	—
271	C₂²×Q₈	—	2	8	no	—
272	C₂²×Q₈	—	4	8	no	—
273	C₂²×Q₈	C₂³×C₃₁	1	12	yes	—
274	C₂²×Q₈	C₂³×C₃₁	1	24	yes	—
275	C₂²×D₄	—	2	8	no	—
276	C₂²×D₄	—	4	8	no	—
277	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
278	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
279	C₂³×C₄	—	2	8	no	—
280	C₂³×C₄	—	4	8	no	—
281	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
283	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
289	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
291	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
293	#126	C₂³×C₃₁	1	128	yes	complemented
294	#124	C₂³×C₃₁	1	64	yes	complemented
295	#124	C₂³×C₃₁	1	128	yes	complemented
296	#123	C₂³×C₃₁	1	128	yes	complemented
297	C₂²⋊Q₈	C₂³×C₃₁	1	128	yes	complemented
298	C₂²⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
299	#121	C₂³×C₃₁	1	64	yes	—
300	C₄×Q₈	C₂³×C₃₁	1	64	yes	complemented
301	C₄×D₄	C₂³×C₃₁	1	64	yes	complemented
302	#117	C₂³×C₃₁	1	64	yes	complemented
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
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	24	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
312	C₂×C₂²⋊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×C₄²	C₂³×C₃₁	1	12	yes	—
318	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	24	yes	—
320	C₃₁	C₂²×#542	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	4	no	—
322	C₂⁴	—	2	8	no	—
323	C₂⁴	—	2	8	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	8	no	—
327	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
328	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
329	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
330	C₂×Q₈	—	2	48	no	—
331	C₂×Q₈	—	4	48	no	—
332	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	—
333	C₂×Q₈	C₂⁴×C₃₁	1	32	yes	—
334	C₂×D₄	—	2	48	no	—
335	C₂×D₄	—	4	48	no	—
336	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
337	C₂×D₄	C₂⁴×C₃₁	1	32	yes	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	8	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	8	no	—
342	C₂²×C₄	—	2	8	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	48	no	—
347	C₂²×C₄	—	4	48	no	—
348	C₂²×C₄	#41×C₃₁	1	8	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
351	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
352	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
353	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
355	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
356	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
357	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
359	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
360	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
361	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
362	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
363	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
364	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
365	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
366	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
367	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
368	C₂²⋊C₄	C₂⁴×C₃₁	1	16	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	—
373	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
374	C₄²	C₂⁴×C₃₁	1	16	yes	—
375	C₄²	C₂⁴×C₃₁	1	16	yes	—
376	C₄²	C₂⁴×C₃₁	1	32	yes	—
377	C₂³	—	2	12	no	—
378	C₂³	—	2	24	no	—
379	C₂³	—	2	24	no	—
380	C₂³	—	2	12	no	—
381	C₂³	—	2	12	no	—
382	C₂³	—	2	24	no	—
383	C₂³	—	4	16	no	—
384	C₂³	—	4	32	no	—
385	C₂³	C₂×#41×C₃₁	1	4	yes	—
386	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
387	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
388	C₂³	C₂⁵×C₃₁	1	4	yes	—
389	C₂³	C₂⁵×C₃₁	1	8	yes	—
390	C₂³	C₂⁵×C₃₁	1	3	yes	—
391	Q₈	—	2	32	no	—
392	Q₈	—	4	32	no	—
393	D₄	—	2	32	no	—
394	D₄	—	4	32	no	—
395	C₂×C₄	—	2	12	no	—
396	C₂×C₄	—	2	24	no	—
397	C₂×C₄	—	2	12	no	—
398	C₂×C₄	—	2	24	no	—
399	C₂×C₄	—	2	24	no	—
400	C₂×C₄	—	2	12	no	—
401	C₂×C₄	—	2	12	no	—
402	C₂×C₄	—	2	12	no	—
403	C₂×C₄	—	2	32	no	—
404	C₂×C₄	—	4	32	no	—
405	C₂×C₄	C₂×#41×C₃₁	1	24	yes	—
406	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
407	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
408	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
409	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
410	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
411	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
412	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
413	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
414	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
415	C₂²	—	2	4	no	—
416	C₂²	—	2	8	no	—
417	C₂²	—	2	8	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	4	24	no	—
422	C₂²	—	4	48	no	—
423	C₂²	C₃₁×#542	1	16	yes	direct product factor
424	C₂²	C₂²×#41×C₃₁	1	6	yes	—
425	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	6	yes	—
426	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	6	yes	—
427	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
428	C₄	—	2	4	no	—
429	C₄	—	2	8	no	—
430	C₄	—	2	4	no	—
431	C₄	—	2	8	no	—
432	C₄	—	2	8	no	—
433	C₄	—	2	4	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	4	no	—
436	C₄	C₂²×#41×C₃₁	1	8	yes	—
437	C₂	—	4	4	no	—
438	C₂	—	4	8	no	—
439	C₂	C₂×C₃₁×#542	1	12	yes	direct product factor
440	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
441	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
442	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂²×C₃₁×#121	C₂
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×#542	C₂	direct (N × H)
2. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₃₁×#126	C₂	semidirect
4. C₂²×C₃₁×#124	C₂	semidirect
5. C₂²×C₃₁×#124	C₂	semidirect
6. C₂²×C₃₁×#123	C₂	semidirect
7. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
8. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
10. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
11. C₂²×C₄×D₄×C₃₁	C₂	semidirect
12. C₂²×C₃₁×#117	C₂	semidirect
19. C₃₁×#542	C₂²	direct (N × H)
20. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×#126	C₂²	semidirect
22. C₂×C₃₁×#124	C₂²	semidirect
23. C₂×C₃₁×#124	C₂²	semidirect
24. C₂×C₃₁×#123	C₂²	semidirect
25. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
26. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
28. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
29. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₂×C₃₁×#117	C₂²	semidirect
71. C₃₁×C₄⋊Q₈	C₂³	semidirect
72. C₃₁×#126	C₂³	semidirect
73. C₃₁×#124	C₂³	semidirect
74. C₃₁×#124	C₂³	semidirect
75. C₃₁×#123	C₂³	semidirect
76. C₃₁×C₂²⋊Q₈	C₂³	semidirect
77. C₃₁×C₂²⋊Q₈	C₂³	semidirect
79. C₄×Q₈×C₃₁	C₂³	semidirect
80. C₄×D₄×C₃₁	C₂³	semidirect
81. C₃₁×#117	C₂³	semidirect
155. C₂²×#542	C₃₁	direct (N × H)
194. C₂×#542	C₂×C₃₁	direct (N × H)
195. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
196. C₂²×#126	C₂×C₃₁	semidirect
197. C₂²×#124	C₂×C₃₁	semidirect
198. C₂²×#124	C₂×C₃₁	semidirect
199. C₂²×#123	C₂×C₃₁	semidirect
200. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
201. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
203. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
204. C₂²×C₄×D₄	C₂×C₃₁	semidirect
205. C₂²×#117	C₂×C₃₁	semidirect
214. C₂²×C₃₁	#542	direct (N × H)
234. #542	C₂²×C₃₁	direct (N × H)
235. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
236. C₂×#126	C₂²×C₃₁	semidirect
237. C₂×#124	C₂²×C₃₁	semidirect
238. C₂×#124	C₂²×C₃₁	semidirect
239. C₂×#123	C₂²×C₃₁	semidirect
240. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
241. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
243. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
244. C₂×C₄×D₄	C₂²×C₃₁	semidirect
245. C₂×#117	C₂²×C₃₁	semidirect
265. C₂×C₃₁	C₂×#542	direct (N × H)
292. C₄⋊Q₈	C₂³×C₃₁	semidirect
293. #126	C₂³×C₃₁	semidirect
294. #124	C₂³×C₃₁	semidirect
295. #124	C₂³×C₃₁	semidirect
296. #123	C₂³×C₃₁	semidirect
297. C₂²⋊Q₈	C₂³×C₃₁	semidirect
298. C₂²⋊Q₈	C₂³×C₃₁	semidirect
300. C₄×Q₈	C₂³×C₃₁	semidirect
301. C₄×D₄	C₂³×C₃₁	semidirect
302. #117	C₂³×C₃₁	semidirect
320. C₃₁	C₂²×#542	direct (N × H)
423. C₂²	C₃₁×#542	direct (N × H)
439. C₂	C₂×C₃₁×#542	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×#542	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×#126	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
C₂²×#542	31

Subgroups — C22×C31×#542 #21519437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#542 #21519437