Subgroups — C₃₁×#23362 #21480121

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3153	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1905	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1904	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1903	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1902	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1885	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1884	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1841	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#137	C₂²	1	2	yes	—
11	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
12	C₃₁×#568	C₂²	1	2	yes	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
15	C₃₁×#519	C₂²	1	1	yes	characteristic
16	C₃₁×#518	C₂²	1	1	yes	characteristic
17	C₃₁×#497	C₄	1	2	yes	complemented
18	C₃₁×#496	C₄	1	2	yes	complemented
19	C₃₁×#495	C₄	1	4	yes	complemented
20	C₃₁×#421	C₂²	1	2	yes	—
21	C₃₁×#421	C₂²	1	2	yes	—
22	C₃₁×#419	C₂²	1	2	yes	—
23	C₃₁×#418	C₂²	1	2	yes	—
24	C₂×C₃₁×#101	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
26	C₃₁×#410	C₂²	1	2	yes	—
27	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#344	C₂²	1	2	yes	—
29	C₃₁×#344	C₂²	1	2	yes	—
30	C₃₁×#339	C₂²	1	2	yes	—
31	C₃₁×#339	C₂²	1	2	yes	—
32	C₃₁×#338	C₂²	1	2	yes	—
33	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
34	C₂×#41×C₃₁	C₂³	1	2	yes	—
35	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#137	—	4	2	no	—
38	C₃₁×#137	—	4	2	no	—
39	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
40	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
41	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
42	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
44	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
45	C₃₁×#131	—	4	2	no	—
46	C₃₁×#131	—	4	2	no	—
47	C₂×M₁₆×C₃₁	—	4	2	no	—
48	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
52	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
53	C₃₁×#124	—	4	1	no	—
54	C₃₁×#124	C₂×C₄	1	2	yes	—
55	C₃₁×#124	C₂×C₄	1	2	yes	—
56	C₃₁×#124	C₂×C₄	1	2	yes	complemented
57	C₃₁×#121	—	4	1	no	—
58	C₄×Q₈×C₃₁	—	4	2	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₃₁×#117	—	4	1	no	—
62	C₃₁×#117	C₂³	1	2	yes	—
63	C₃₁×#117	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#104	—	4	2	no	—
68	C₃₁×#104	—	4	2	no	—
69	C₃₁×#103	—	4	2	no	—
70	C₃₁×#102	—	4	2	no	—
71	C₃₁×#101	—	4	1	no	—
72	C₃₁×#100	—	4	1	no	—
73	C₃₁×#99	—	4	2	no	—
74	C₃₁×#97	C₂×C₄	1	2	yes	complemented
75	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	complemented
76	C₂⁴×C₃₁	—	2	1	no	—
77	#41×C₃₁	—	2	2	no	—
78	#41×C₃₁	—	2	2	no	—
79	#41×C₃₁	—	2	2	no	—
80	#41×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	1	no	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	4	1	no	—
91	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₃₁	—	4	4	no	—
99	SD₁₆×C₃₁	—	4	4	no	—
100	SD₁₆×C₃₁	—	4	4	no	—
101	D₈×C₃₁	—	4	4	no	—
102	M₁₆×C₃₁	—	2	2	no	—
103	M₁₆×C₃₁	—	2	2	no	—
104	M₁₆×C₃₁	—	2	2	no	—
105	M₁₆×C₃₁	—	2	2	no	—
106	M₁₆×C₃₁	—	2	2	no	—
107	M₁₆×C₃₁	—	2	2	no	—
108	M₁₆×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	4	4	no	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	4	2	no	—
127	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	#23362	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	2	no	—
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	4	1	no	—
133	C₂³×C₃₁	—	4	1	no	—
134	C₂³×C₃₁	—	8	2	no	—
135	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
136	Q₈×C₃₁	—	2	2	no	—
137	Q₈×C₃₁	—	2	4	no	—
138	Q₈×C₃₁	—	2	2	no	—
139	Q₈×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	2	2	no	—
141	D₄×C₃₁	—	2	4	no	—
142	D₄×C₃₁	—	2	2	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	2	no	—
146	D₄×C₃₁	—	8	2	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	8	1	no	—
158	C₂×C₄×C₃₁	—	16	2	no	—
159	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
161	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
162	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₈×C₃₁	—	2	4	no	—
164	C₈×C₃₁	—	4	2	no	—
165	C₈×C₃₁	—	4	2	no	—
166	#3153	C₂×C₃₁	1	1	yes	characteristic
167	#1905	C₂×C₃₁	1	2	yes	complemented
168	#1904	C₂×C₃₁	1	2	yes	complemented
169	#1903	C₂×C₃₁	1	2	yes	complemented
170	#1902	C₂×C₃₁	1	2	yes	complemented
171	#1888	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#1885	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#1884	C₂×C₃₁	1	2	yes	complemented
174	#1841	C₂×C₃₁	1	2	yes	complemented
175	C₂²×C₃₁	—	4	1	no	—
176	C₂²×C₃₁	—	4	2	no	—
177	C₂²×C₃₁	—	4	2	no	—
178	C₂²×C₃₁	—	8	2	no	—
179	C₂²×C₃₁	—	8	2	no	—
180	C₂²×C₃₁	—	8	2	no	—
181	C₂²×C₃₁	—	16	1	no	—
182	C₂²×C₃₁	#518	1	1	yes	characteristic
183	C₂²×C₃₁	#560	1	1	yes	characteristic
184	C₂²×C₃₁	#560	1	1	yes	characteristic
185	C₄×C₃₁	—	4	1	no	—
186	C₄×C₃₁	—	4	1	no	—
187	C₄×C₃₁	—	4	2	no	—
188	C₄×C₃₁	—	4	2	no	—
189	C₄×C₃₁	—	8	2	no	—
190	C₄×C₃₁	—	8	2	no	—
191	C₄×C₃₁	#518	1	2	yes	—
192	C₄×C₃₁	#560	1	2	yes	—
193	C₂×#137	C₂²×C₃₁	1	2	yes	—
194	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
195	#568	C₂²×C₃₁	1	2	yes	—
196	#532	C₂²×C₃₁	1	1	yes	characteristic
197	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
198	#519	C₂²×C₃₁	1	1	yes	characteristic
199	#518	C₂²×C₃₁	1	1	yes	characteristic
200	#497	C₄×C₃₁	1	2	yes	complemented
201	#496	C₄×C₃₁	1	2	yes	complemented
202	#495	C₄×C₃₁	1	4	yes	complemented
203	#421	C₂²×C₃₁	1	2	yes	—
204	#421	C₂²×C₃₁	1	2	yes	—
205	#419	C₂²×C₃₁	1	2	yes	—
206	#418	C₂²×C₃₁	1	2	yes	—
207	C₂×#101	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
209	#410	C₂²×C₃₁	1	2	yes	—
210	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	#344	C₂²×C₃₁	1	2	yes	—
212	#344	C₂²×C₃₁	1	2	yes	—
213	#339	C₂²×C₃₁	1	2	yes	—
214	#339	C₂²×C₃₁	1	2	yes	—
215	#338	C₂²×C₃₁	1	2	yes	—
216	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
217	C₂×C₃₁	—	2	1	no	—
218	C₂×C₃₁	—	2	1	no	—
219	C₂×C₃₁	—	2	1	no	—
220	C₂×C₃₁	—	8	2	no	—
221	C₂×C₃₁	—	8	2	no	—
222	C₂×C₃₁	#2380	1	1	yes	characteristic
223	C₂×#41	C₂³×C₃₁	1	2	yes	—
224	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
226	#137	—	4	2	no	—
227	#137	—	4	2	no	—
228	AGL(1,ℤ/8)	—	4	2	no	—
229	AGL(1,ℤ/8)	—	4	2	no	—
230	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
231	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
232	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
233	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
234	#131	—	4	2	no	—
235	#131	—	4	2	no	—
236	C₂×M₁₆	—	4	2	no	—
237	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
238	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
239	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
240	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
241	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
242	#124	—	4	1	no	—
243	#124	C₂×C₄×C₃₁	1	2	yes	—
244	#124	C₂×C₄×C₃₁	1	2	yes	—
245	#124	C₂×C₄×C₃₁	1	2	yes	complemented
246	#121	—	4	1	no	—
247	C₄×Q₈	—	4	2	no	—
248	C₄×D₄	—	4	2	no	—
249	C₄×D₄	—	4	1	no	—
250	#117	—	4	1	no	—
251	#117	C₂³×C₃₁	1	2	yes	—
252	#117	C₂³×C₃₁	1	2	yes	—
253	C₂×C₂²⋊C₄	—	4	1	no	—
254	C₂×C₂²⋊C₄	—	4	1	no	—
255	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
256	#104	—	4	2	no	—
257	#104	—	4	2	no	—
258	#103	—	4	2	no	—
259	#102	—	4	2	no	—
260	#101	—	4	1	no	—
261	#100	—	4	1	no	—
262	#99	—	4	2	no	—
263	#97	C₂×C₄×C₃₁	1	2	yes	complemented
264	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	complemented
265	C₃₁	#23362	1	1	yes	characteristic, direct product factor, Sylow
266	C₂⁴	—	2	1	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
282	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
288	Q₁₆	—	4	4	no	—
289	SD₁₆	—	4	4	no	—
290	SD₁₆	—	4	4	no	—
291	D₈	—	4	4	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	2	2	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	4	4	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	4	no	—
302	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	4	2	no	—
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
318	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	2	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	8	2	no	—
324	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
325	Q₈	—	2	2	no	—
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	2	no	—
329	D₄	—	2	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	2	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	8	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	8	1	no	—
347	C₂×C₄	—	16	2	no	—
348	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
350	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
351	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	16	1	no	—
362	C₂²	C₃₁×#518	1	1	yes	characteristic
363	C₂²	C₃₁×#560	1	1	yes	characteristic
364	C₂²	C₃₁×#560	1	1	yes	characteristic
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	C₃₁×#518	1	2	yes	—
372	C₄	C₃₁×#560	1	2	yes	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	8	2	no	—
377	C₂	—	8	2	no	—
378	C₂	C₃₁×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2380
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3153	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1905	C₂	semidirect
3. C₃₁×#1904	C₂	semidirect
4. C₃₁×#1903	C₂	semidirect
5. C₃₁×#1902	C₂	semidirect
6. C₃₁×#1888	C₂	semidirect
7. C₃₁×#1885	C₂	semidirect
8. C₃₁×#1884	C₂	semidirect
9. C₃₁×#1841	C₂	semidirect
17. C₃₁×#497	C₄	semidirect
18. C₃₁×#496	C₄	semidirect
19. C₃₁×#495	C₄	semidirect
41. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
43. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
51. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
52. C₃₁×#127	C₂×C₄	semidirect
56. C₃₁×#124	C₂×C₄	semidirect
74. C₃₁×#97	C₂×C₄	semidirect
75. C₄×C₈×C₃₁	C₂×C₄	semidirect
129. #23362	C₃₁	direct (N × H)
167. #1905	C₂×C₃₁	semidirect
168. #1904	C₂×C₃₁	semidirect
169. #1903	C₂×C₃₁	semidirect
170. #1902	C₂×C₃₁	semidirect
171. #1888	C₂×C₃₁	semidirect
172. #1885	C₂×C₃₁	semidirect
173. #1884	C₂×C₃₁	semidirect
174. #1841	C₂×C₃₁	semidirect
200. #497	C₄×C₃₁	semidirect
201. #496	C₄×C₃₁	semidirect
202. #495	C₄×C₃₁	semidirect
230. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
232. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
240. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
241. #127	C₂×C₄×C₃₁	semidirect
245. #124	C₂×C₄×C₃₁	semidirect
263. #97	C₂×C₄×C₃₁	semidirect
264. C₄×C₈	C₂×C₄×C₃₁	semidirect
265. C₃₁	#23362	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×#3153	2
C₃₁×#1905	2
C₃₁×#1904	2
C₃₁×#1903	2
C₃₁×#1902	2
C₃₁×#1888	2
C₃₁×#1885	2
C₃₁×#1884	2
C₃₁×#1841	2
#23362	31

Subgroups — C31×#23362 #21480121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23362 #21480121