Subgroups — C₂×C₃₁×#2138 #21490135

Order 7936.

1190 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2138	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	—	2	1	no	—
8	C₂×C₃₁×#135	—	2	2	no	—
9	C₂×C₃₁×#135	—	2	4	no	—
10	C₂×C₃₁×#135	C₄	1	8	yes	complemented
11	C₂×C₃₁×#135	C₂²	1	4	yes	—
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	—	2	1	no	—
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	—	2	1	no	—
17	C₂²×C₁₆×C₃₁	—	2	4	no	—
18	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	complemented
20	C₃₁×#427	—	2	2	no	—
21	C₃₁×#427	C₂²	1	4	yes	complemented
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₂×C₃₁×#106	—	2	1	no	—
24	C₃₁×#358	C₂²	1	8	yes	complemented
25	C₃₁×#357	C₂²	1	8	yes	complemented
26	C₂×#41×C₃₁	—	2	8	no	—
27	C₂×#41×C₃₁	—	2	4	no	—
28	C₂×#41×C₃₁	—	4	2	no	—
29	C₂×#41×C₃₁	—	4	4	no	—
30	C₂²×Q₈×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	4	1	no	—
33	C₂³×C₄×C₃₁	—	4	1	no	—
34	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
35	C₃₁×#135	—	2	16	no	—
36	C₃₁×#135	—	2	8	no	—
37	C₃₁×#135	C₂×C₄	1	16	yes	complemented
38	C₂×Q₁₆×C₃₁	—	2	1	no	—
39	C₂×Q₁₆×C₃₁	—	2	2	no	—
40	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₃₁	—	2	8	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×D₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	D₄	1	2	yes	—
51	C₂²×C₈×C₃₁	D₄	1	4	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#117	—	2	4	no	—
55	C₃₁×#117	—	4	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
58	C₂×C₄²×C₃₁	—	4	1	no	—
59	C₂×C₁₆×C₃₁	—	2	4	no	—
60	C₂×C₁₆×C₃₁	—	2	8	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	C₂³	1	4	yes	—
64	C₃₁×#106	—	2	4	no	—
65	C₂⁴×C₃₁	—	4	1	no	—
66	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
67	#41×C₃₁	—	2	16	no	—
68	#41×C₃₁	—	4	16	no	—
69	#41×C₃₁	—	4	8	no	—
70	C₂×Q₈×C₃₁	—	2	4	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	4	1	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	2	4	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	4	8	no	—
77	C₂×D₄×C₃₁	—	4	4	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	—	4	8	no	—
82	C₂²×C₄×C₃₁	—	4	4	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₃₁	D₈	1	2	yes	—
86	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
90	Q₁₆×C₃₁	—	2	4	no	—
91	Q₁₆×C₃₁	—	2	2	no	—
92	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
93	SD₁₆×C₃₁	—	2	16	no	—
94	D₈×C₃₁	—	2	4	no	—
95	D₈×C₃₁	—	2	2	no	—
96	D₈×C₃₁	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₃₁	C₂²⋊C₄	1	8	yes	—
99	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
105	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	—	4	4	no	—
109	C₄²×C₃₁	—	4	4	no	—
110	C₁₆×C₃₁	—	2	4	no	—
111	C₂×#2138	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	2	4	no	—
113	C₂³×C₃₁	—	4	2	no	—
114	C₂³×C₃₁	—	4	4	no	—
115	C₂³×C₃₁	—	8	4	no	—
116	C₂³×C₃₁	#102	1	4	yes	—
117	C₂³×C₃₁	C₂×D₈	1	2	yes	—
118	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₃₁	—	2	4	no	—
120	Q₈×C₃₁	—	4	2	no	—
121	Q₈×C₃₁	—	4	4	no	—
122	D₄×C₃₁	—	2	4	no	—
123	D₄×C₃₁	—	4	16	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	—	4	16	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	#102	1	4	yes	—
132	C₂×C₄×C₃₁	#102	1	8	yes	—
133	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
141	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
142	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
145	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
146	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2138	C₂×C₃₁	1	8	yes	direct product factor
148	C₂×#358	C₂×C₃₁	1	2	yes	complemented
149	C₂×#357	C₂×C₃₁	1	2	yes	complemented
150	C₂²×C₃₁	—	2	4	no	—
151	C₂²×C₃₁	—	2	8	no	—
152	C₂²×C₃₁	—	4	4	no	—
153	C₂²×C₃₁	—	8	4	no	—
154	C₂²×C₃₁	—	8	8	no	—
155	C₂²×C₃₁	C₂×#102	1	4	yes	—
156	C₂²×C₃₁	C₂×#102	1	2	yes	—
157	C₂²×C₃₁	#508	1	4	yes	—
158	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₃₁	—	4	4	no	—
160	C₄×C₃₁	—	8	4	no	—
161	C₄×C₃₁	C₂×#102	1	4	yes	—
162	C₄×C₃₁	C₂×#102	1	2	yes	—
163	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂²×#41	—	2	1	no	—
166	C₂×#135	—	2	2	no	—
167	C₂×#135	—	2	4	no	—
168	C₂×#135	C₄×C₃₁	1	8	yes	complemented
169	C₂×#135	C₂²×C₃₁	1	4	yes	—
170	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
171	C₂²×SD₁₆	—	2	1	no	—
172	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#117	—	2	1	no	—
175	C₂²×C₁₆	—	2	4	no	—
176	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
177	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	complemented
178	#427	—	2	2	no	—
179	#427	C₂²×C₃₁	1	4	yes	complemented
180	C₂×#107	C₂²×C₃₁	1	2	yes	—
181	C₂×#106	—	2	1	no	—
182	#358	C₂²×C₃₁	1	8	yes	complemented
183	#357	C₂²×C₃₁	1	8	yes	complemented
184	C₂×C₃₁	—	2	4	no	—
185	C₂×C₃₁	—	8	4	no	—
186	C₂×C₃₁	#2138	1	4	yes	direct product factor
187	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
188	C₂×C₃₁	C₂×#508	1	2	yes	—
189	C₂×#41	—	2	8	no	—
190	C₂×#41	—	2	4	no	—
191	C₂×#41	—	4	2	no	—
192	C₂×#41	—	4	4	no	—
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	4	1	no	—
196	C₂³×C₄	—	4	1	no	—
197	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
198	#135	—	2	16	no	—
199	#135	—	2	8	no	—
200	#135	C₂×C₄×C₃₁	1	16	yes	complemented
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
204	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
205	C₂×SD₁₆	—	2	8	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
210	C₂×D₈	C₂³×C₃₁	1	2	yes	—
211	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
212	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₃₁	1	2	yes	—
214	C₂²×C₈	D₄×C₃₁	1	4	yes	—
215	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
217	#117	—	2	4	no	—
218	#117	—	4	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	4	1	no	—
221	C₂×C₄²	—	4	1	no	—
222	C₂×C₁₆	—	2	4	no	—
223	C₂×C₁₆	—	2	8	no	—
224	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
225	#107	—	2	2	no	—
226	#107	C₂³×C₃₁	1	4	yes	—
227	#106	—	2	4	no	—
228	C₃₁	C₂×#2138	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	—	4	1	no	—
230	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	16	no	—
233	#41	—	4	8	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	8	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
249	C₂²×C₄	D₈×C₃₁	1	2	yes	—
250	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
257	SD₁₆	—	2	16	no	—
258	D₈	—	2	4	no	—
259	D₈	—	2	2	no	—
260	D₈	C₂²×C₄×C₃₁	1	4	yes	—
261	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
262	C₂×C₈	C₂²⋊C₄×C₃₁	1	8	yes	—
263	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
269	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₄²	—	4	4	no	—
274	C₁₆	—	2	4	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	8	4	no	—
279	C₂³	C₃₁×#102	1	4	yes	—
280	C₂³	C₂×D₈×C₃₁	1	2	yes	—
281	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	16	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	16	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	C₃₁×#102	1	4	yes	—
295	C₂×C₄	C₃₁×#102	1	8	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
298	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
304	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
305	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
306	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂²	—	2	4	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	8	no	—
312	C₂²	C₂×C₃₁×#102	1	4	yes	—
313	C₂²	C₂×C₃₁×#102	1	2	yes	—
314	C₂²	C₃₁×#508	1	4	yes	—
315	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₂×C₃₁×#102	1	4	yes	—
319	C₄	C₂×C₃₁×#102	1	2	yes	—
320	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
321	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
322	C₂	—	2	4	no	—
323	C₂	—	8	4	no	—
324	C₂	C₃₁×#2138	1	4	yes	direct product factor
325	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic
326	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×D₈
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2138	C₂	direct (N × H)
5. C₂×C₃₁×#358	C₂	semidirect
6. C₂×C₃₁×#357	C₂	semidirect
10. C₂×C₃₁×#135	C₄	semidirect
19. C₂²×C₁₆×C₃₁	C₂²	semidirect
21. C₃₁×#427	C₂²	semidirect
24. C₃₁×#358	C₂²	semidirect
25. C₃₁×#357	C₂²	semidirect
37. C₃₁×#135	C₂×C₄	semidirect
111. C₂×#2138	C₃₁	direct (N × H)
145. C₂³×C₁₆	C₂×C₃₁	semidirect
146. C₂×#427	C₂×C₃₁	semidirect
147. #2138	C₂×C₃₁	direct (N × H)
148. C₂×#358	C₂×C₃₁	semidirect
149. C₂×#357	C₂×C₃₁	semidirect
168. C₂×#135	C₄×C₃₁	semidirect
177. C₂²×C₁₆	C₂²×C₃₁	semidirect
179. #427	C₂²×C₃₁	semidirect
182. #358	C₂²×C₃₁	semidirect
183. #357	C₂²×C₃₁	semidirect
186. C₂×C₃₁	#2138	direct (N × H)
200. #135	C₂×C₄×C₃₁	semidirect
228. C₃₁	C₂×#2138	direct (N × H)
324. C₂	C₃₁×#2138	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂³×C₁₆×C₃₁	2
C₂×C₃₁×#427	2
C₃₁×#2138	2
C₂×C₃₁×#358	2
C₂×C₃₁×#357	2
C₂×#2138	31

Subgroups — C2×C31×#2138 #21490135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2138 #21490135