Subgroups — C₂²×C₃₁×#547 #21519458

Order 7936.

12230 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#547	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#123	C₂	1	4	yes	maximal, complemented
5	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₁×#121	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₃₁	—	2	32	no	—
11	C₂²×#41×C₃₁	C₂²	1	48	yes	complemented
12	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₃₁	C₂²	1	4	yes	—
17	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#547	C₂²	1	256	yes	direct product factor
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	24	yes	complemented
21	C₂×C₃₁×#123	C₂²	1	96	yes	complemented
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	48	yes	complemented
23	C₂×C₃₁×#121	C₂²	1	48	yes	complemented
24	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	complemented
25	C₂×C₄×D₄×C₃₁	C₂²	1	24	yes	complemented
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	48	yes	complemented
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₂⁵×C₃₁	C₂³	1	2	yes	—
37	C₂⁵×C₃₁	C₂³	1	2	yes	—
38	C₂×#41×C₃₁	—	2	384	no	—
39	C₂×#41×C₃₁	C₂³	1	128	yes	complemented
40	C₂²×Q₈×C₃₁	—	2	16	no	—
41	C₂²×Q₈×C₃₁	C₂³	1	24	yes	—
42	C₂²×D₄×C₃₁	—	2	32	no	—
43	C₂²×D₄×C₃₁	—	2	8	no	—
44	C₂²×D₄×C₃₁	D₄	1	16	yes	complemented
45	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	48	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
48	C₂³×C₄×C₃₁	—	2	32	no	—
49	C₂³×C₄×C₃₁	—	2	16	no	—
50	C₂³×C₄×C₃₁	—	2	16	no	—
51	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	48	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	24	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	64	yes	complemented
61	C₃₁×#123	C₂³	1	256	yes	complemented
62	C₃₁×C₂²⋊Q₈	C₂³	1	128	yes	complemented
63	C₃₁×#121	C₂³	1	128	yes	complemented
64	C₄×D₄×C₃₁	C₂³	1	64	yes	complemented
65	C₄×D₄×C₃₁	C₂³	1	64	yes	complemented
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	128	yes	complemented
72	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
73	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
74	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
76	C₂⁴×C₃₁	—	2	8	no	—
77	C₂⁴×C₃₁	—	2	8	no	—
78	C₂⁴×C₃₁	—	2	8	no	—
79	C₂⁴×C₃₁	—	2	8	no	—
80	C₂⁴×C₃₁	—	2	8	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	16	yes	—
82	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
83	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
84	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
85	#41×C₃₁	—	2	512	no	—
86	C₂×Q₈×C₃₁	—	2	96	no	—
87	C₂×Q₈×C₃₁	C₂⁴	1	32	yes	—
88	C₂×D₄×C₃₁	—	2	192	no	—
89	C₂×D₄×C₃₁	—	2	48	no	—
90	C₂×D₄×C₃₁	C₂×D₄	1	96	yes	complemented
91	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
92	C₂×D₄×C₃₁	C₂⁴	1	64	yes	—
93	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
94	C₂²×C₄×C₃₁	—	2	8	no	—
95	C₂²×C₄×C₃₁	—	2	16	no	—
96	C₂²×C₄×C₃₁	—	2	192	no	—
97	C₂²×C₄×C₃₁	—	2	8	no	—
98	C₂²×C₄×C₃₁	—	2	4	no	—
99	C₂²×C₄×C₃₁	—	2	96	no	—
100	C₂²×C₄×C₃₁	—	2	96	no	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
102	C₂²×C₄×C₃₁	#41	1	8	yes	complemented
103	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
104	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
105	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	64	yes	—
109	C₂²×C₄×C₃₁	C₂⁴	1	32	yes	—
110	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
112	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
113	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
114	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
115	C₄⋊C₄×C₃₁	C₂⁴	1	16	yes	—
116	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
117	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
118	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
119	C₂²⋊C₄×C₃₁	C₂⁴	1	64	yes	—
120	C₄²×C₃₁	C₂⁴	1	16	yes	—
121	C₂²×#547	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	2	64	no	—
123	C₂³×C₃₁	—	2	24	no	—
124	C₂³×C₃₁	—	2	24	no	—
125	C₂³×C₃₁	—	2	24	no	—
126	C₂³×C₃₁	—	2	24	no	—
127	C₂³×C₃₁	—	2	24	no	—
128	C₂³×C₃₁	—	4	32	no	—
129	C₂³×C₃₁	C₂²×D₄	1	48	yes	—
130	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
131	C₂³×C₃₁	C₂×#41	1	4	yes	—
132	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
133	C₂³×C₃₁	C₂⁵	1	8	yes	—
134	C₂³×C₃₁	C₂⁵	1	8	yes	—
135	C₂³×C₃₁	C₂⁵	1	3	yes	—
136	Q₈×C₃₁	—	2	64	no	—
137	D₄×C₃₁	—	2	128	no	—
138	D₄×C₃₁	—	2	32	no	—
139	D₄×C₃₁	C₂²×D₄	1	64	yes	complemented
140	C₂×C₄×C₃₁	—	2	24	no	—
141	C₂×C₄×C₃₁	—	2	48	no	—
142	C₂×C₄×C₃₁	—	2	128	no	—
143	C₂×C₄×C₃₁	—	2	24	no	—
144	C₂×C₄×C₃₁	—	2	12	no	—
145	C₂×C₄×C₃₁	—	2	64	no	—
146	C₂×C₄×C₃₁	—	2	64	no	—
147	C₂×C₄×C₃₁	C₂²×D₄	1	24	yes	—
148	C₂×C₄×C₃₁	C₂×#41	1	24	yes	complemented
149	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
150	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
151	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
152	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
153	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
154	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
155	C₂³×#41	C₂×C₃₁	1	2	yes	complemented
156	C₂×#547	C₂×C₃₁	1	48	yes	direct product factor
157	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×#123	C₂×C₃₁	1	4	yes	complemented
159	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
160	C₂²×#121	C₂×C₃₁	1	2	yes	complemented
161	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
163	C₂³×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
164	C₂²×C₃₁	—	2	8	no	—
165	C₂²×C₃₁	—	2	8	no	—
166	C₂²×C₃₁	—	2	96	no	—
167	C₂²×C₃₁	—	2	8	no	—
168	C₂²×C₃₁	—	2	8	no	—
169	C₂²×C₃₁	—	2	8	no	—
170	C₂²×C₃₁	—	4	48	no	—
171	C₂²×C₃₁	#547	1	16	yes	direct product factor
172	C₂²×C₃₁	C₂³×D₄	1	16	yes	—
173	C₂²×C₃₁	C₂³×D₄	1	6	yes	—
174	C₂²×C₃₁	C₂²×#41	1	6	yes	—
175	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	6	yes	—
176	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
177	C₄×C₃₁	—	2	8	no	—
178	C₄×C₃₁	—	2	16	no	—
179	C₄×C₃₁	—	2	8	no	—
180	C₄×C₃₁	—	2	4	no	—
181	C₄×C₃₁	C₂³×D₄	1	8	yes	—
182	C₄×C₃₁	C₂²×#41	1	8	yes	complemented
183	C₂²×#41	—	2	32	no	—
184	C₂²×#41	C₂²×C₃₁	1	48	yes	complemented
185	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
186	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
187	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
188	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
189	C₂⁴×C₄	C₂²×C₃₁	1	4	yes	—
190	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
191	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
192	#547	C₂²×C₃₁	1	256	yes	direct product factor
193	C₂×C₄⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
194	C₂×#123	C₂²×C₃₁	1	96	yes	complemented
195	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	48	yes	complemented
196	C₂×#121	C₂²×C₃₁	1	48	yes	complemented
197	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	complemented
198	C₂×C₄×D₄	C₂²×C₃₁	1	24	yes	complemented
199	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
200	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
201	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
202	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
203	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
204	C₂²×C₄⋊C₄	C₂²×C₃₁	1	48	yes	complemented
205	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
206	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
207	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
208	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
209	C₂×C₃₁	—	2	16	no	—
210	C₂×C₃₁	—	4	8	no	—
211	C₂×C₃₁	C₂×#547	1	12	yes	direct product factor
212	C₂×C₃₁	C₂⁴×D₄	1	1	yes	characteristic
213	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
214	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
215	C₂⁵	C₂³×C₃₁	1	2	yes	—
216	C₂⁵	C₂³×C₃₁	1	2	yes	—
217	C₂×#41	—	2	384	no	—
218	C₂×#41	C₂³×C₃₁	1	128	yes	complemented
219	C₂²×Q₈	—	2	16	no	—
220	C₂²×Q₈	C₂³×C₃₁	1	24	yes	—
221	C₂²×D₄	—	2	32	no	—
222	C₂²×D₄	—	2	8	no	—
223	C₂²×D₄	D₄×C₃₁	1	16	yes	complemented
224	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
225	C₂²×D₄	C₂³×C₃₁	1	48	yes	—
226	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
227	C₂³×C₄	—	2	32	no	—
228	C₂³×C₄	—	2	16	no	—
229	C₂³×C₄	—	2	16	no	—
230	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
232	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
233	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
234	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
235	C₂³×C₄	C₂³×C₃₁	1	48	yes	—
236	C₂³×C₄	C₂³×C₃₁	1	24	yes	—
237	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
239	C₄⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
240	#123	C₂³×C₃₁	1	256	yes	complemented
241	C₂²⋊Q₈	C₂³×C₃₁	1	128	yes	complemented
242	#121	C₂³×C₃₁	1	128	yes	complemented
243	C₄×D₄	C₂³×C₃₁	1	64	yes	complemented
244	C₄×D₄	C₂³×C₃₁	1	64	yes	complemented
245	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
246	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
247	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
248	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
249	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
250	C₂×C₄⋊C₄	C₂³×C₃₁	1	128	yes	complemented
251	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
254	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
255	C₃₁	C₂²×#547	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	2	8	no	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	—	2	8	no	—
259	C₂⁴	—	2	8	no	—
260	C₂⁴	—	2	8	no	—
261	C₂⁴	C₂×D₄×C₃₁	1	16	yes	—
262	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
263	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
264	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
265	#41	—	2	512	no	—
266	C₂×Q₈	—	2	96	no	—
267	C₂×Q₈	C₂⁴×C₃₁	1	32	yes	—
268	C₂×D₄	—	2	192	no	—
269	C₂×D₄	—	2	48	no	—
270	C₂×D₄	C₂×D₄×C₃₁	1	96	yes	complemented
271	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
272	C₂×D₄	C₂⁴×C₃₁	1	64	yes	—
273	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
274	C₂²×C₄	—	2	8	no	—
275	C₂²×C₄	—	2	16	no	—
276	C₂²×C₄	—	2	192	no	—
277	C₂²×C₄	—	2	8	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	96	no	—
280	C₂²×C₄	—	2	96	no	—
281	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
282	C₂²×C₄	#41×C₃₁	1	8	yes	complemented
283	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
284	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
285	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
286	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
287	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
288	C₂²×C₄	C₂⁴×C₃₁	1	64	yes	—
289	C₂²×C₄	C₂⁴×C₃₁	1	32	yes	—
290	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
291	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
292	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
293	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
294	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
295	C₄⋊C₄	C₂⁴×C₃₁	1	16	yes	—
296	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
297	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
298	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
299	C₂²⋊C₄	C₂⁴×C₃₁	1	64	yes	—
300	C₄²	C₂⁴×C₃₁	1	16	yes	—
301	C₂³	—	2	64	no	—
302	C₂³	—	2	24	no	—
303	C₂³	—	2	24	no	—
304	C₂³	—	2	24	no	—
305	C₂³	—	2	24	no	—
306	C₂³	—	2	24	no	—
307	C₂³	—	4	32	no	—
308	C₂³	C₂²×D₄×C₃₁	1	48	yes	—
309	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
310	C₂³	C₂×#41×C₃₁	1	4	yes	—
311	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
312	C₂³	C₂⁵×C₃₁	1	8	yes	—
313	C₂³	C₂⁵×C₃₁	1	8	yes	—
314	C₂³	C₂⁵×C₃₁	1	3	yes	—
315	Q₈	—	2	64	no	—
316	D₄	—	2	128	no	—
317	D₄	—	2	32	no	—
318	D₄	C₂²×D₄×C₃₁	1	64	yes	complemented
319	C₂×C₄	—	2	24	no	—
320	C₂×C₄	—	2	48	no	—
321	C₂×C₄	—	2	128	no	—
322	C₂×C₄	—	2	24	no	—
323	C₂×C₄	—	2	12	no	—
324	C₂×C₄	—	2	64	no	—
325	C₂×C₄	—	2	64	no	—
326	C₂×C₄	C₂²×D₄×C₃₁	1	24	yes	—
327	C₂×C₄	C₂×#41×C₃₁	1	24	yes	complemented
328	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
329	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
330	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
331	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
332	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
333	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
334	C₂²	—	2	8	no	—
335	C₂²	—	2	8	no	—
336	C₂²	—	2	96	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	2	8	no	—
339	C₂²	—	2	8	no	—
340	C₂²	—	4	48	no	—
341	C₂²	C₃₁×#547	1	16	yes	direct product factor
342	C₂²	C₂³×D₄×C₃₁	1	16	yes	—
343	C₂²	C₂³×D₄×C₃₁	1	6	yes	—
344	C₂²	C₂²×#41×C₃₁	1	6	yes	—
345	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	6	yes	—
346	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
347	C₄	—	2	8	no	—
348	C₄	—	2	16	no	—
349	C₄	—	2	8	no	—
350	C₄	—	2	4	no	—
351	C₄	C₂³×D₄×C₃₁	1	8	yes	—
352	C₄	C₂²×#41×C₃₁	1	8	yes	complemented
353	C₂	—	2	16	no	—
354	C₂	—	4	8	no	—
355	C₂	C₂×C₃₁×#547	1	12	yes	direct product factor
356	C₂	C₂⁴×D₄×C₃₁	1	1	yes	characteristic
357	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
358	C₂	C₂²×C₃₁×2₋¹⁺⁴	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. C₂³×#41×C₃₁	C₂	semidirect
2. C₂×C₃₁×#547	C₂	direct (N × H)
3. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₂²×C₃₁×#123	C₂	semidirect
5. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₃₁×#121	C₂	semidirect
7. C₂²×C₄×D₄×C₃₁	C₂	semidirect
8. C₂²×C₄×D₄×C₃₁	C₂	semidirect
9. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₂²×#41×C₃₁	C₂²	semidirect
19. C₃₁×#547	C₂²	direct (N × H)
20. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×#123	C₂²	semidirect
22. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
23. C₂×C₃₁×#121	C₂²	semidirect
24. C₂×C₄×D₄×C₃₁	C₂²	semidirect
25. C₂×C₄×D₄×C₃₁	C₂²	semidirect
31. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
39. C₂×#41×C₃₁	C₂³	semidirect
44. C₂²×D₄×C₃₁	D₄	semidirect
60. C₃₁×C₄⋊Q₈	C₂³	semidirect
61. C₃₁×#123	C₂³	semidirect
62. C₃₁×C₂²⋊Q₈	C₂³	semidirect
63. C₃₁×#121	C₂³	semidirect
64. C₄×D₄×C₃₁	C₂³	semidirect
65. C₄×D₄×C₃₁	C₂³	semidirect
71. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
90. C₂×D₄×C₃₁	C₂×D₄	semidirect
102. C₂²×C₄×C₃₁	#41	semidirect
121. C₂²×#547	C₃₁	direct (N × H)
139. D₄×C₃₁	C₂²×D₄	semidirect
148. C₂×C₄×C₃₁	C₂×#41	semidirect
155. C₂³×#41	C₂×C₃₁	semidirect
156. C₂×#547	C₂×C₃₁	direct (N × H)
157. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
158. C₂²×#123	C₂×C₃₁	semidirect
159. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
160. C₂²×#121	C₂×C₃₁	semidirect
161. C₂²×C₄×D₄	C₂×C₃₁	semidirect
162. C₂²×C₄×D₄	C₂×C₃₁	semidirect
163. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
171. C₂²×C₃₁	#547	direct (N × H)
182. C₄×C₃₁	C₂²×#41	semidirect
184. C₂²×#41	C₂²×C₃₁	semidirect
192. #547	C₂²×C₃₁	direct (N × H)
193. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
194. C₂×#123	C₂²×C₃₁	semidirect
195. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
196. C₂×#121	C₂²×C₃₁	semidirect
197. C₂×C₄×D₄	C₂²×C₃₁	semidirect
198. C₂×C₄×D₄	C₂²×C₃₁	semidirect
204. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
211. C₂×C₃₁	C₂×#547	direct (N × H)
218. C₂×#41	C₂³×C₃₁	semidirect
223. C₂²×D₄	D₄×C₃₁	semidirect
239. C₄⋊Q₈	C₂³×C₃₁	semidirect
240. #123	C₂³×C₃₁	semidirect
241. C₂²⋊Q₈	C₂³×C₃₁	semidirect
242. #121	C₂³×C₃₁	semidirect
243. C₄×D₄	C₂³×C₃₁	semidirect
244. C₄×D₄	C₂³×C₃₁	semidirect
250. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
255. C₃₁	C₂²×#547	direct (N × H)
270. C₂×D₄	C₂×D₄×C₃₁	semidirect
282. C₂²×C₄	#41×C₃₁	semidirect
318. D₄	C₂²×D₄×C₃₁	semidirect
327. C₂×C₄	C₂×#41×C₃₁	semidirect
341. C₂²	C₃₁×#547	direct (N × H)
352. C₄	C₂²×#41×C₃₁	semidirect
355. C₂	C₂×C₃₁×#547	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₃₁	2
C₂×C₃₁×#547	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₂²×C₄×D₄×C₃₁	2
C₂³×C₄⋊C₄×C₃₁	2
C₂²×#547	31

Subgroups — C22×C31×#547 #21519458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#547 #21519458