Subgroups — C₂×C₃₁×#2102 #21480509

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2102	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#344	C₂	1	2	yes	maximal
5	C₂×C₃₁×#339	C₂	1	2	yes	maximal
6	C₂×C₃₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#430	—	2	2	no	—
13	C₃₁×#430	C₂²	1	4	yes	—
14	C₂×C₃₁×#108	—	2	2	no	—
15	C₃₁×#423	—	2	2	no	—
16	C₃₁×#423	C₂²	1	4	yes	—
17	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
18	C₃₁×#344	C₂²	1	8	yes	—
19	C₃₁×#339	C₂²	1	8	yes	—
20	C₃₁×H(3,ℤ/4)	C₂²	1	4	yes	—
21	C₂³×C₄×C₃₁	—	2	1	no	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
35	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	—	2	1	no	—
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	D₄	1	2	yes	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
52	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#108	—	2	8	no	—
55	C₃₁×C₄⋊C₈	—	2	8	no	—
56	C₂⁴×C₃₁	#41	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	1	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	4	4	no	—
62	C₂²×C₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	#41	1	2	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	2	4	no	—
76	M₁₆×C₃₁	—	2	4	no	—
77	M₁₆×C₃₁	—	2	4	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	4	1	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂²⋊C₄×C₃₁	—	4	2	no	—
94	C₄²×C₃₁	—	2	1	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	1	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
101	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
102	C₂×#2102	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	4	2	no	—
104	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₃₁	#126	1	2	yes	—
107	C₂³×C₃₁	#126	1	2	yes	—
108	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	8	4	no	—
117	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#123	1	2	yes	—
122	C₂×C₄×C₃₁	#124	1	2	yes	—
123	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#125	1	2	yes	—
126	C₂×C₄×C₃₁	#125	1	2	yes	—
127	C₂×C₄×C₃₁	#125	1	2	yes	—
128	C₂×C₄×C₃₁	#125	1	2	yes	—
129	C₂×C₄×C₃₁	#126	1	2	yes	—
130	C₂×C₄×C₃₁	#126	1	2	yes	—
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
135	C₈×C₃₁	—	2	4	no	—
136	C₈×C₃₁	—	4	2	no	—
137	C₈×C₃₁	—	4	4	no	—
138	C₂×#430	C₂×C₃₁	1	1	yes	characteristic
139	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
140	#2102	C₂×C₃₁	1	8	yes	direct product factor
141	C₂×#344	C₂×C₃₁	1	2	yes	—
142	C₂×#339	C₂×C₃₁	1	2	yes	—
143	C₂×H(3,ℤ/4)	C₂×C₃₁	1	1	yes	characteristic
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	#400	1	1	yes	characteristic
150	C₂²×C₃₁	#400	1	2	yes	—
151	C₂²×C₃₁	#400	1	1	yes	characteristic
152	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₃₁	C₂×#126	1	2	yes	—
154	C₄×C₃₁	—	2	4	no	—
155	C₄×C₃₁	—	4	2	no	—
156	C₄×C₃₁	—	8	4	no	—
157	C₄×C₃₁	#400	1	1	yes	characteristic
158	C₄×C₃₁	#400	1	2	yes	—
159	C₄×C₃₁	#400	1	1	yes	characteristic
160	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
161	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
162	C₄×C₃₁	C₂×#125	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#117	C₂²×C₃₁	1	2	yes	—
167	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
168	#430	—	2	2	no	—
169	#430	C₂²×C₃₁	1	4	yes	—
170	C₂×#108	—	2	2	no	—
171	#423	—	2	2	no	—
172	#423	C₂²×C₃₁	1	4	yes	—
173	C₂×C₄⋊C₈	—	2	2	no	—
174	#344	C₂²×C₃₁	1	8	yes	—
175	#339	C₂²×C₃₁	1	8	yes	—
176	H(3,ℤ/4)	C₂²×C₃₁	1	4	yes	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	2	no	—
181	C₂×C₃₁	#2102	1	2	yes	direct product factor
182	C₂×C₃₁	C₂×#400	1	1	yes	characteristic
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
196	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
197	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
198	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	D₄×C₃₁	1	2	yes	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	C₂³×C₃₁	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
213	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
214	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
216	#108	—	2	8	no	—
217	C₄⋊C₈	—	2	8	no	—
218	C₃₁	C₂×#2102	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	#41×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₃₁	1	2	yes	—
232	C₂²×C₄	#41×C₃₁	1	2	yes	—
233	C₂²×C₄	#41×C₃₁	1	2	yes	—
234	C₂²×C₄	#41×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
264	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂³	C₃₁×#126	1	2	yes	—
269	C₂³	C₃₁×#126	1	2	yes	—
270	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
282	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
283	C₂×C₄	C₃₁×#123	1	2	yes	—
284	C₂×C₄	C₃₁×#124	1	2	yes	—
285	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#125	1	2	yes	—
288	C₂×C₄	C₃₁×#125	1	2	yes	—
289	C₂×C₄	C₃₁×#125	1	2	yes	—
290	C₂×C₄	C₃₁×#125	1	2	yes	—
291	C₂×C₄	C₃₁×#126	1	2	yes	—
292	C₂×C₄	C₃₁×#126	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
296	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₃₁×#400	1	1	yes	characteristic
306	C₂²	C₃₁×#400	1	2	yes	—
307	C₂²	C₃₁×#400	1	1	yes	characteristic
308	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
309	C₂²	C₂×C₃₁×#126	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₃₁×#400	1	1	yes	characteristic
314	C₄	C₃₁×#400	1	1	yes	characteristic
315	C₄	C₃₁×#400	1	2	yes	—
316	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
317	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
318	C₄	C₂×C₃₁×#125	1	2	yes	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	C₃₁×#2102	1	2	yes	direct product factor
324	C₂	C₂×C₃₁×#400	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	#400
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2102	C₂	direct (N × H)
102. C₂×#2102	C₃₁	direct (N × H)
140. #2102	C₂×C₃₁	direct (N × H)
181. C₂×C₃₁	#2102	direct (N × H)
218. C₃₁	C₂×#2102	direct (N × H)
323. C₂	C₃₁×#2102	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×#430	2
C₂×C₃₁×#423	2
C₃₁×#2102	2
C₂×C₃₁×#344	2
C₂×C₃₁×#339	2
C₂×C₃₁×H(3,ℤ/4)	2
C₂×#2102	31

Subgroups — C2×C31×#2102 #21480509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2102 #21480509