Subgroups — C₂×C₃₁×#1668 #21475871

Order 7936.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1668	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₃₁	—	2	2	no	—
8	C₂²×D₈×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	—	2	1	no	—
10	C₂×C₃₁×#127	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#493	—	2	2	no	—
13	C₃₁×#493	C₂²	1	4	yes	—
14	C₃₁×#460	—	2	4	no	—
15	C₃₁×#460	C₂²	1	8	yes	complemented
16	C₈×Q₈×C₃₁	—	2	2	no	—
17	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
18	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₃₁	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#335	C₂²	1	4	yes	complemented
25	C₃₁×#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	—	2	2	no	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	—	2	2	no	—
30	C₂×SD₁₆×C₃₁	—	2	8	no	—
31	C₂×SD₁₆×C₃₁	—	4	8	no	—
32	C₂×D₈×C₃₁	—	2	4	no	—
33	C₂×D₈×C₃₁	—	2	4	no	—
34	C₂×D₈×C₃₁	—	4	4	no	—
35	C₂×D₈×C₃₁	—	4	4	no	—
36	C₂²×C₈×C₃₁	—	2	1	no	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₃₁×#127	—	2	2	no	—
41	C₃₁×#127	—	2	2	no	—
42	C₃₁×#127	—	2	2	no	—
43	C₃₁×#127	C₂³	1	4	yes	—
44	C₄×Q₈×C₃₁	—	2	1	no	—
45	C₄×Q₈×C₃₁	—	2	1	no	—
46	C₄×Q₈×C₃₁	D₄	1	2	yes	—
47	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×C₄⋊C₈	—	2	4	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	—	2	2	no	—
55	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
56	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
57	C₃₁×#102	—	2	8	no	—
58	C₄×C₈×C₃₁	—	2	4	no	—
59	C₄×C₈×C₃₁	D₄	1	2	yes	—
60	C₄×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂⁴×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	2	1	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	D₈	1	4	yes	complemented
65	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₃₁	—	2	4	no	—
67	C₂×D₄×C₃₁	—	2	4	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	—	4	4	no	—
74	C₂×D₄×C₃₁	—	4	4	no	—
75	C₂×D₄×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	8	4	no	—
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
82	SD₁₆×C₃₁	—	4	16	no	—
83	D₈×C₃₁	—	4	8	no	—
84	D₈×C₃₁	—	4	8	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	4	4	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₃₁	#41	1	1	yes	characteristic
102	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂×#1668	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	4	4	no	—
105	C₂³×C₃₁	—	8	4	no	—
106	C₂³×C₃₁	—	8	4	no	—
107	C₂³×C₃₁	—	8	4	no	—
108	C₂³×C₃₁	#121	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	2	no	—
110	Q₈×C₃₁	—	2	4	no	—
111	Q₈×C₃₁	C₂×D₈	1	4	yes	complemented
112	D₄×C₃₁	—	4	4	no	—
113	D₄×C₃₁	—	4	4	no	—
114	D₄×C₃₁	—	4	4	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	8	4	no	—
117	D₄×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	8	4	no	—
119	D₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₃₁	#135	1	2	yes	—
131	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
135	C₈×C₃₁	—	2	2	no	—
136	C₈×C₃₁	—	4	4	no	—
137	C₈×C₃₁	—	4	2	no	—
138	C₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
139	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
140	C₂×#460	C₂×C₃₁	1	2	yes	complemented
141	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#1668	C₂×C₃₁	1	8	yes	direct product factor
143	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#331	C₂×C₃₁	1	2	yes	complemented
145	C₂²×C₃₁	—	8	4	no	—
146	C₂²×C₃₁	—	8	4	no	—
147	C₂²×C₃₁	—	8	4	no	—
148	C₂²×C₃₁	—	16	8	no	—
149	C₂²×C₃₁	#460	1	2	yes	—
150	C₂²×C₃₁	#466	1	2	yes	—
151	C₂²×C₃₁	#472	1	2	yes	—
152	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	4	2	no	—
157	C₄×C₃₁	#460	1	2	yes	—
158	C₄×C₃₁	#466	1	2	yes	—
159	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
161	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
162	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂²×SD₁₆	—	2	2	no	—
164	C₂²×D₈	—	2	1	no	—
165	C₂²×D₈	—	2	1	no	—
166	C₂×#127	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
168	#493	—	2	2	no	—
169	#493	C₂²×C₃₁	1	4	yes	—
170	#460	—	2	4	no	—
171	#460	C₂²×C₃₁	1	8	yes	complemented
172	C₈×Q₈	—	2	2	no	—
173	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
174	C₂×C₄⋊C₈	—	2	1	no	—
175	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
177	C₂×#102	—	2	2	no	—
178	C₂×C₄×C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	#335	C₂²×C₃₁	1	4	yes	complemented
181	#331	C₂²×C₃₁	1	8	yes	complemented
182	C₂×C₃₁	—	16	4	no	—
183	C₂×C₃₁	#1668	1	4	yes	direct product factor
184	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
187	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂×SD₁₆	—	2	8	no	—
192	C₂×SD₁₆	—	4	8	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	4	4	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
201	#127	—	2	2	no	—
202	#127	—	2	2	no	—
203	#127	—	2	2	no	—
204	#127	C₂³×C₃₁	1	4	yes	—
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	D₄×C₃₁	1	2	yes	—
208	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
217	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
218	#102	—	2	8	no	—
219	C₄×C₈	—	2	4	no	—
220	C₄×C₈	D₄×C₃₁	1	2	yes	—
221	C₄×C₈	C₂³×C₃₁	1	2	yes	—
222	C₃₁	C₂×#1668	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	—	4	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	D₈×C₃₁	1	4	yes	complemented
227	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	8	4	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
244	SD₁₆	—	4	16	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
253	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₄²	#41×C₃₁	1	1	yes	characteristic
264	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
265	C₂³	—	4	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	C₃₁×#121	1	1	yes	characteristic
270	Q₈	—	2	4	no	—
271	Q₈	—	2	2	no	—
272	Q₈	C₂×D₈×C₃₁	1	4	yes	complemented
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
289	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
291	C₂×C₄	C₃₁×#135	1	2	yes	—
292	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	4	4	no	—
298	C₈	—	4	2	no	—
299	C₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	16	8	no	—
304	C₂²	C₃₁×#460	1	2	yes	—
305	C₂²	C₃₁×#466	1	2	yes	—
306	C₂²	C₃₁×#472	1	2	yes	—
307	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	C₃₁×#460	1	2	yes	—
313	C₄	C₃₁×#466	1	2	yes	—
314	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
315	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
316	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
317	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂	—	16	4	no	—
319	C₂	C₃₁×#1668	1	4	yes	direct product factor
320	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
321	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
322	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#460	C₂	semidirect
3. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1668	C₂	direct (N × H)
5. C₂×C₃₁×#335	C₂	semidirect
6. C₂×C₃₁×#331	C₂	semidirect
15. C₃₁×#460	C₂²	semidirect
17. C₈×Q₈×C₃₁	C₂²	semidirect
24. C₃₁×#335	C₂²	semidirect
25. C₃₁×#331	C₂²	semidirect
64. C₂×Q₈×C₃₁	D₈	semidirect
90. C₂×C₈×C₃₁	SD₁₆	semidirect
103. C₂×#1668	C₃₁	direct (N × H)
111. Q₈×C₃₁	C₂×D₈	semidirect
138. C₈×C₃₁	C₂×SD₁₆	semidirect
140. C₂×#460	C₂×C₃₁	semidirect
141. C₂×C₈×Q₈	C₂×C₃₁	semidirect
142. #1668	C₂×C₃₁	direct (N × H)
143. C₂×#335	C₂×C₃₁	semidirect
144. C₂×#331	C₂×C₃₁	semidirect
171. #460	C₂²×C₃₁	semidirect
173. C₈×Q₈	C₂²×C₃₁	semidirect
180. #335	C₂²×C₃₁	semidirect
181. #331	C₂²×C₃₁	semidirect
183. C₂×C₃₁	#1668	direct (N × H)
222. C₃₁	C₂×#1668	direct (N × H)
226. C₂×Q₈	D₈×C₃₁	semidirect
252. C₂×C₈	SD₁₆×C₃₁	semidirect
272. Q₈	C₂×D₈×C₃₁	semidirect
299. C₈	C₂×SD₁₆×C₃₁	semidirect
319. C₂	C₃₁×#1668	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×#493	2
C₂×C₃₁×#460	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1668	2
C₂×C₃₁×#335	2
C₂×C₃₁×#331	2
C₂×#1668	31

Subgroups — C2×C31×#1668 #21475871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1668 #21475871