Subgroups — C₃₁×#22802 #21479561

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1974	C₂	1	2	yes	maximal
4	C₃₁×#1929	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1806	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1772	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
12	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
13	C₄×SD₁₆×C₃₁	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₃₁	C₄	1	2	yes	complemented
15	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
16	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#433	—	2	4	no	—
18	C₃₁×#432	—	2	2	no	—
19	C₃₁×#431	—	2	2	no	—
20	C₂×C₃₁×#108	C₂²	1	2	yes	—
21	C₂×C₃₁×#108	C₂²	1	2	yes	—
22	C₃₁×#427	C₂²	1	1	yes	characteristic
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#104	C₂²	1	4	yes	—
26	C₃₁×#416	C₂²	1	2	yes	—
27	C₃₁×#408	C₂²	1	4	yes	—
28	C₃₁×#405	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#339	C₂²	1	4	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₂×#41×C₃₁	C₂³	1	2	yes	—
35	C₃₁×#135	—	2	2	no	—
36	C₃₁×#135	—	2	2	no	—
37	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₃₁	—	2	2	no	—
47	C₄×D₄×C₃₁	—	2	2	no	—
48	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₃₁×#117	—	2	2	no	—
51	C₃₁×#117	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#108	—	2	4	no	—
57	C₃₁×#108	—	4	2	no	—
58	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
59	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#106	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
65	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
66	C₃₁×#104	—	4	4	no	—
67	C₃₁×#103	C₂×C₄	1	2	yes	—
68	C₃₁×#103	C₂×C₄	1	2	yes	—
69	C₃₁×#102	C₂×C₄	1	2	yes	—
70	C₃₁×#102	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊C₈	—	2	4	no	—
72	C₃₁×C₂²⋊C₈	—	2	4	no	—
73	C₃₁×#97	—	2	4	no	—
74	C₄×C₈×C₃₁	—	2	4	no	—
75	C₄×C₈×C₃₁	—	2	4	no	—
76	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₃₁×#95	—	2	1	no	—
81	#41×C₃₁	—	2	2	no	—
82	#41×C₃₁	—	2	2	no	—
83	#41×C₃₁	—	2	2	no	—
84	#41×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
95	Q₁₆×C₃₁	—	2	2	no	—
96	SD₁₆×C₃₁	—	2	4	no	—
97	D₈×C₃₁	—	2	2	no	—
98	M₁₆×C₃₁	—	2	4	no	—
99	M₁₆×C₃₁	—	2	4	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	4	no	—
105	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₃₁	#41	1	2	yes	—
112	C₂×C₈×C₃₁	#41	1	2	yes	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	#22802	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
133	Q₈×C₃₁	—	2	2	no	—
134	Q₈×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	2	2	no	—
136	D₄×C₃₁	—	2	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	4	no	—
150	C₂×C₄×C₃₁	—	8	2	no	—
151	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
161	C₈×C₃₁	—	2	1	no	—
162	C₈×C₃₁	—	2	1	no	—
163	C₈×C₃₁	—	2	2	no	—
164	C₈×C₃₁	—	2	2	no	—
165	C₈×C₃₁	—	4	4	no	—
166	C₈×C₃₁	#117	1	2	yes	—
167	C₈×C₃₁	#117	1	2	yes	—
168	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
169	#2925	C₂×C₃₁	1	2	yes	complemented
170	#1974	C₂×C₃₁	1	2	yes	—
171	#1929	C₂×C₃₁	1	4	yes	complemented
172	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#1806	C₂×C₃₁	1	2	yes	complemented
174	#1772	C₂×C₃₁	1	2	yes	complemented
175	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
176	C₂²×C₃₁	—	2	1	no	—
177	C₂²×C₃₁	—	2	1	no	—
178	C₂²×C₃₁	—	4	2	no	—
179	C₂²×C₃₁	—	4	2	no	—
180	C₂²×C₃₁	—	4	2	no	—
181	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂²×C₃₁	#555	1	1	yes	characteristic
183	C₂²×C₃₁	#556	1	1	yes	characteristic
184	C₄×C₃₁	—	2	1	no	—
185	C₄×C₃₁	—	2	1	no	—
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	4	4	no	—
188	C₄×C₃₁	—	4	2	no	—
189	C₄×C₃₁	—	4	2	no	—
190	C₄×C₃₁	—	4	2	no	—
191	C₄×C₃₁	#443	1	2	yes	—
192	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
193	C₄×C₃₁	#520	1	1	yes	characteristic
194	C₄×C₃₁	#547	1	1	yes	characteristic
195	C₄×C₃₁	#550	1	1	yes	characteristic
196	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
197	C₄×#41	C₂²×C₃₁	1	2	yes	—
198	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
199	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
200	C₄×SD₁₆	C₄×C₃₁	1	2	yes	complemented
201	C₄×SD₁₆	C₄×C₃₁	1	2	yes	complemented
202	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
203	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
204	#433	—	2	4	no	—
205	#432	—	2	2	no	—
206	#431	—	2	2	no	—
207	C₂×#108	C₂²×C₃₁	1	2	yes	—
208	C₂×#108	C₂²×C₃₁	1	2	yes	—
209	#427	C₂²×C₃₁	1	1	yes	characteristic
210	#423	C₂²×C₃₁	1	2	yes	—
211	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×#104	C₂²×C₃₁	1	4	yes	—
213	#416	C₂²×C₃₁	1	2	yes	—
214	#408	C₂²×C₃₁	1	4	yes	—
215	#405	C₂²×C₃₁	1	4	yes	—
216	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
219	#339	C₂²×C₃₁	1	4	yes	—
220	#336	C₂²×C₃₁	1	2	yes	—
221	C₂×C₃₁	—	2	1	no	—
222	C₂×C₃₁	—	2	1	no	—
223	C₂×C₃₁	—	8	2	no	—
224	C₂×C₃₁	#2354	1	1	yes	characteristic
225	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
226	C₂×C₃₁	#2957	1	1	yes	characteristic
227	C₂×#41	C₂³×C₃₁	1	2	yes	—
228	#135	—	2	2	no	—
229	#135	—	2	2	no	—
230	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
231	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
232	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
233	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
234	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
236	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
238	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
242	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
243	#117	—	2	2	no	—
244	#117	C₂³×C₃₁	1	2	yes	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄²	—	2	2	no	—
247	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	#108	—	2	4	no	—
250	#108	—	4	2	no	—
251	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
252	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
253	#106	C₂×C₄×C₃₁	1	2	yes	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
257	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
259	#104	—	4	4	no	—
260	#103	C₂×C₄×C₃₁	1	2	yes	—
261	#103	C₂×C₄×C₃₁	1	2	yes	—
262	#102	C₂×C₄×C₃₁	1	2	yes	—
263	#102	C₂×C₄×C₃₁	1	2	yes	—
264	C₂²⋊C₈	—	2	4	no	—
265	C₂²⋊C₈	—	2	4	no	—
266	#97	—	2	4	no	—
267	C₄×C₈	—	2	4	no	—
268	C₄×C₈	—	2	4	no	—
269	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
270	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
272	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
273	#95	—	2	1	no	—
274	C₃₁	#22802	1	1	yes	characteristic, direct product factor, Sylow
275	#41	—	2	2	no	—
276	#41	—	2	2	no	—
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
289	Q₁₆	—	2	2	no	—
290	SD₁₆	—	2	4	no	—
291	D₈	—	2	2	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	2	4	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
304	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
305	C₂×C₈	#41×C₃₁	1	2	yes	—
306	C₂×C₈	#41×C₃₁	1	2	yes	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
311	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
321	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
322	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂³	—	2	2	no	—
325	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
326	Q₈	—	2	2	no	—
327	Q₈	—	2	2	no	—
328	D₄	—	2	2	no	—
329	D₄	—	2	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
351	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
353	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	4	4	no	—
359	C₈	C₃₁×#117	1	2	yes	—
360	C₈	C₃₁×#117	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
367	C₂²	C₃₁×#555	1	1	yes	characteristic
368	C₂²	C₃₁×#556	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	4	4	no	—
373	C₄	—	4	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	2	no	—
376	C₄	C₃₁×#443	1	2	yes	—
377	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
378	C₄	C₃₁×#520	1	1	yes	characteristic
379	C₄	C₃₁×#547	1	1	yes	characteristic
380	C₄	C₃₁×#550	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	C₃₁×#2354	1	1	yes	characteristic
385	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
386	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2925	C₂	semidirect
4. C₃₁×#1929	C₂	semidirect
5. C₃₁×#1848	C₂	semidirect
6. C₃₁×#1806	C₂	semidirect
7. C₃₁×#1772	C₂	semidirect
8. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₄×Q₁₆×C₃₁	C₄	semidirect
12. C₄×Q₁₆×C₃₁	C₄	semidirect
13. C₄×SD₁₆×C₃₁	C₄	semidirect
14. C₄×SD₁₆×C₃₁	C₄	semidirect
15. C₄×D₈×C₃₁	C₄	semidirect
16. C₄×D₈×C₃₁	C₄	semidirect
130. #22802	C₃₁	direct (N × H)
169. #2925	C₂×C₃₁	semidirect
171. #1929	C₂×C₃₁	semidirect
172. #1848	C₂×C₃₁	semidirect
173. #1806	C₂×C₃₁	semidirect
174. #1772	C₂×C₃₁	semidirect
175. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
198. C₄×Q₁₆	C₄×C₃₁	semidirect
199. C₄×Q₁₆	C₄×C₃₁	semidirect
200. C₄×SD₁₆	C₄×C₃₁	semidirect
201. C₄×SD₁₆	C₄×C₃₁	semidirect
202. C₄×D₈	C₄×C₃₁	semidirect
203. C₄×D₈	C₄×C₃₁	semidirect
274. C₃₁	#22802	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#135	2
C₃₁×#2925	2
C₃₁×#1974	2
C₃₁×#1929	2
C₃₁×#1848	2
C₃₁×#1806	2
C₃₁×#1772	2
C₈×C₄⋊C₄×C₃₁	2
#22802	31

Subgroups — C31×#22802 #21479561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22802 #21479561