Subgroups — C₃₁×#3552 #16843485

Order 3968.

1004 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#563	C₂	1	4	yes	maximal, complemented
2	C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#554	C₂	1	2	yes	maximal, complemented
5	C₃₁×#553	C₂	1	2	yes	maximal, complemented
6	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#548	C₂	1	2	yes	maximal, complemented
9	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#544	C₂	1	2	yes	maximal, complemented
11	C₃₁×#542	C₂	1	2	yes	maximal, complemented
12	C₃₁×#541	C₂	1	2	yes	maximal, complemented
13	C₃₁×#539	C₂	1	4	yes	maximal, complemented
14	C₃₁×#536	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
28	C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
29	C₃₁×#126	C₂²	1	2	yes	—
30	C₃₁×#126	C₂²	1	2	yes	—
31	C₃₁×#126	C₂²	1	4	yes	—
32	C₃₁×#126	C₂²	1	4	yes	—
33	C₃₁×#126	C₂²	1	4	yes	—
34	C₃₁×#125	C₂²	1	1	yes	characteristic
35	C₃₁×#125	C₂²	1	1	yes	characteristic
36	C₃₁×#125	C₂²	1	4	yes	—
37	C₃₁×#125	C₂²	1	4	yes	—
38	C₃₁×#125	C₂²	1	4	yes	—
39	C₃₁×#125	C₂²	1	2	yes	—
40	C₃₁×#124	C₂²	1	1	yes	characteristic
41	C₃₁×#124	C₂²	1	1	yes	characteristic
42	C₃₁×#124	C₂²	1	2	yes	—
43	C₃₁×#124	C₂²	1	4	yes	—
44	C₃₁×#123	C₂²	1	2	yes	—
45	C₃₁×#123	C₂²	1	2	yes	—
46	C₃₁×#123	C₂²	1	4	yes	—
47	C₃₁×#123	C₂²	1	4	yes	—
48	C₃₁×#123	C₂²	1	4	yes	—
49	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
50	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
51	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
52	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
53	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
54	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
55	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
56	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
57	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
58	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
59	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
60	C₃₁×#121	C₂²	1	1	yes	characteristic
61	C₃₁×#121	C₂²	1	1	yes	characteristic
62	C₃₁×#121	C₂²	1	4	yes	—
63	C₃₁×#121	C₂²	1	2	yes	—
64	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
67	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
68	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
69	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
71	C₄×D₄×C₃₁	C₂²	1	2	yes	—
72	C₄×D₄×C₃₁	C₂²	1	2	yes	—
73	C₄×D₄×C₃₁	C₂²	1	4	yes	—
74	C₄×D₄×C₃₁	C₂²	1	4	yes	—
75	C₄×D₄×C₃₁	C₂²	1	2	yes	—
76	C₃₁×#117	C₂²	1	2	yes	—
77	C₃₁×#117	C₂²	1	2	yes	—
78	C₃₁×#117	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
83	#41×C₃₁	—	4	4	no	—
84	C₂×Q₈×C₃₁	—	4	4	no	—
85	C₂×Q₈×C₃₁	—	4	4	no	—
86	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
89	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
90	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
91	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
92	C₂×D₄×C₃₁	C₂³	1	2	yes	—
93	C₂×D₄×C₃₁	C₂³	1	2	yes	—
94	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₃₁	C₂³	1	4	yes	—
97	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
103	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
104	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
105	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
137	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
139	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
140	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
141	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
143	C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₄²×C₃₁	C₂³	1	2	yes	—
148	C₄²×C₃₁	C₂³	1	2	yes	—
149	C₄²×C₃₁	C₂³	1	4	yes	—
150	C₄²×C₃₁	C₂³	1	4	yes	—
151	C₂³×C₃₁	C₂⁴	1	2	yes	—
152	C₂³×C₃₁	C₂⁴	1	2	yes	—
153	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	2	4	no	—
157	Q₈×C₃₁	—	2	2	no	—
158	Q₈×C₃₁	—	4	4	no	—
159	Q₈×C₃₁	—	4	4	no	—
160	D₄×C₃₁	—	2	4	no	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	2	2	no	—
164	D₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	4	4	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	—	4	4	no	—
173	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
174	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
175	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
187	#3552	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
197	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
211	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
212	#563	C₂×C₃₁	1	4	yes	complemented
213	#557	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#554	C₂×C₃₁	1	2	yes	complemented
216	#553	C₂×C₃₁	1	2	yes	complemented
217	D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
218	D₄×Q₈	C₂×C₃₁	1	2	yes	complemented
219	#548	C₂×C₃₁	1	2	yes	complemented
220	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#544	C₂×C₃₁	1	2	yes	complemented
222	#542	C₂×C₃₁	1	2	yes	complemented
223	#541	C₂×C₃₁	1	2	yes	complemented
224	#539	C₂×C₃₁	1	4	yes	complemented
225	#536	C₂×C₃₁	1	2	yes	complemented
226	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×C₃₁	—	2	2	no	—
230	C₂×C₃₁	—	4	2	no	—
231	C₂×C₃₁	—	4	2	no	—
232	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
234	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂×#41	C₂²×C₃₁	1	2	yes	—
236	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
237	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
238	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
239	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
240	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
243	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
244	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
245	C₄⋊Q₈	C₂²×C₃₁	1	4	yes	—
246	#126	C₂²×C₃₁	1	2	yes	—
247	#126	C₂²×C₃₁	1	2	yes	—
248	#126	C₂²×C₃₁	1	4	yes	—
249	#126	C₂²×C₃₁	1	4	yes	—
250	#126	C₂²×C₃₁	1	4	yes	—
251	#125	C₂²×C₃₁	1	1	yes	characteristic
252	#125	C₂²×C₃₁	1	1	yes	characteristic
253	#125	C₂²×C₃₁	1	4	yes	—
254	#125	C₂²×C₃₁	1	4	yes	—
255	#125	C₂²×C₃₁	1	4	yes	—
256	#125	C₂²×C₃₁	1	2	yes	—
257	#124	C₂²×C₃₁	1	1	yes	characteristic
258	#124	C₂²×C₃₁	1	1	yes	characteristic
259	#124	C₂²×C₃₁	1	2	yes	—
260	#124	C₂²×C₃₁	1	4	yes	—
261	#123	C₂²×C₃₁	1	2	yes	—
262	#123	C₂²×C₃₁	1	2	yes	—
263	#123	C₂²×C₃₁	1	4	yes	—
264	#123	C₂²×C₃₁	1	4	yes	—
265	#123	C₂²×C₃₁	1	4	yes	—
266	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
269	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
270	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
271	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
272	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
273	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
274	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
275	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
276	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
277	#121	C₂²×C₃₁	1	1	yes	characteristic
278	#121	C₂²×C₃₁	1	1	yes	characteristic
279	#121	C₂²×C₃₁	1	4	yes	—
280	#121	C₂²×C₃₁	1	2	yes	—
281	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
284	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
285	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
286	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
287	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
288	C₄×D₄	C₂²×C₃₁	1	2	yes	—
289	C₄×D₄	C₂²×C₃₁	1	2	yes	—
290	C₄×D₄	C₂²×C₃₁	1	4	yes	—
291	C₄×D₄	C₂²×C₃₁	1	4	yes	—
292	C₄×D₄	C₂²×C₃₁	1	2	yes	—
293	#117	C₂²×C₃₁	1	2	yes	—
294	#117	C₂²×C₃₁	1	2	yes	—
295	#117	C₂²×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
300	C₃₁	#3552	1	1	yes	characteristic, direct product factor, Sylow
301	#41	—	4	4	no	—
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
307	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
308	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
309	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
310	C₂×D₄	C₂³×C₃₁	1	2	yes	—
311	C₂×D₄	C₂³×C₃₁	1	2	yes	—
312	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×D₄	C₂³×C₃₁	1	4	yes	—
315	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
320	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
321	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
322	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
323	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
327	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
329	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
333	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
334	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
335	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
336	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
348	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
349	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
352	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
353	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
354	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
356	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
358	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
360	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
361	C₄²	C₂³×C₃₁	1	1	yes	characteristic
362	C₄²	C₂³×C₃₁	1	1	yes	characteristic
363	C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	C₄²	C₂³×C₃₁	1	2	yes	—
366	C₄²	C₂³×C₃₁	1	2	yes	—
367	C₄²	C₂³×C₃₁	1	4	yes	—
368	C₄²	C₂³×C₃₁	1	4	yes	—
369	C₂³	C₂⁴×C₃₁	1	2	yes	—
370	C₂³	C₂⁴×C₃₁	1	2	yes	—
371	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
372	Q₈	—	2	4	no	—
373	Q₈	—	2	2	no	—
374	Q₈	—	2	4	no	—
375	Q₈	—	2	2	no	—
376	Q₈	—	4	4	no	—
377	Q₈	—	4	4	no	—
378	D₄	—	2	4	no	—
379	D₄	—	2	4	no	—
380	D₄	—	2	2	no	—
381	D₄	—	2	2	no	—
382	D₄	—	4	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
392	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
393	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
394	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
398	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
401	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
403	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
414	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
415	C₄	—	2	2	no	—
416	C₄	—	2	1	no	—
417	C₄	—	2	4	no	—
418	C₄	—	2	4	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
428	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
429	C₂	—	2	2	no	—
430	C₂	—	4	2	no	—
431	C₂	—	4	2	no	—
432	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
433	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
434	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#563	C₂	semidirect
2. C₃₁×#557	C₂	semidirect
3. C₃₁×#556	C₂	semidirect
4. C₃₁×#554	C₂	semidirect
5. C₃₁×#553	C₂	semidirect
6. D₄×Q₈×C₃₁	C₂	semidirect
7. D₄×Q₈×C₃₁	C₂	semidirect
8. C₃₁×#548	C₂	semidirect
9. C₃₁×#547	C₂	semidirect
10. C₃₁×#544	C₂	semidirect
11. C₃₁×#542	C₂	semidirect
12. C₃₁×#541	C₂	semidirect
13. C₃₁×#539	C₂	semidirect
14. C₃₁×#536	C₂	semidirect
15. C₂×C₃₁×C₄⋊Q₈	C₂	semidirect
16. C₃₁×#529	C₂	semidirect
17. C₃₁×#529	C₂	semidirect
187. #3552	C₃₁	direct (N × H)
212. #563	C₂×C₃₁	semidirect
213. #557	C₂×C₃₁	semidirect
214. #556	C₂×C₃₁	semidirect
215. #554	C₂×C₃₁	semidirect
216. #553	C₂×C₃₁	semidirect
217. D₄×Q₈	C₂×C₃₁	semidirect
218. D₄×Q₈	C₂×C₃₁	semidirect
219. #548	C₂×C₃₁	semidirect
220. #547	C₂×C₃₁	semidirect
221. #544	C₂×C₃₁	semidirect
222. #542	C₂×C₃₁	semidirect
223. #541	C₂×C₃₁	semidirect
224. #539	C₂×C₃₁	semidirect
225. #536	C₂×C₃₁	semidirect
226. C₂×C₄⋊Q₈	C₂×C₃₁	semidirect
227. #529	C₂×C₃₁	semidirect
228. #529	C₂×C₃₁	semidirect
300. C₃₁	#3552	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×#563	2
C₃₁×#557	2
C₃₁×#556	2
C₃₁×#554	2
C₃₁×#553	2
D₄×Q₈×C₃₁	2
C₃₁×#548	2
C₃₁×#547	2
C₃₁×#544	2
C₃₁×#542	2
C₃₁×#541	2
C₃₁×#539	2
C₃₁×#536	2
C₂×C₃₁×C₄⋊Q₈	2
C₃₁×#529	2
#3552	31

Subgroups — C31×#3552 #16843485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3552 #16843485