Subgroups — C₂×C₃₁×#2234 #21490400

Order 7936.

638 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×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2234	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#121	—	2	1	no	—
12	C₂×C₃₁×#117	—	2	1	no	—
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#466	—	2	2	no	—
16	C₃₁×#466	C₂²	1	4	yes	—
17	C₃₁×#427	—	2	2	no	—
18	C₃₁×#427	C₂²	1	4	yes	complemented
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#106	—	2	1	no	—
23	C₂×C₃₁×#102	—	2	1	no	—
24	C₃₁×#367	C₂²	1	4	yes	complemented
25	C₃₁×#366	C₂²	1	4	yes	complemented
26	C₃₁×#357	C₂²	1	4	yes	complemented
27	C₃₁×#357	C₂²	1	4	yes	complemented
28	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	4	1	no	—
31	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
32	C₂×D₈×C₃₁	—	2	1	no	—
33	C₂×D₈×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	—	2	2	no	—
35	C₂×D₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#121	—	2	4	no	—
43	C₃₁×#121	—	4	2	no	—
44	C₃₁×#117	—	2	4	no	—
45	C₃₁×#117	—	4	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
51	C₂×C₄²×C₃₁	—	4	1	no	—
52	C₂×C₁₆×C₃₁	—	2	2	no	—
53	C₂×C₁₆×C₃₁	—	2	2	no	—
54	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#107	—	2	1	no	—
57	C₃₁×#107	—	2	1	no	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#107	C₂³	1	2	yes	—
60	C₃₁×#107	C₂³	1	2	yes	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#106	—	2	4	no	—
63	C₃₁×#102	—	2	4	no	—
64	C₂⁴×C₃₁	—	4	1	no	—
65	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	4	4	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	8	4	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	D₈	1	2	yes	—
78	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	2	no	—
83	D₈×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	#41	1	2	yes	—
91	C₂×C₈×C₃₁	#41	1	2	yes	—
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₄⋊C₄×C₃₁	—	4	1	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₃₁	—	4	4	no	—
100	C₂²⋊C₄×C₃₁	—	4	4	no	—
101	C₄²×C₃₁	—	4	4	no	—
102	C₁₆×C₃₁	—	2	2	no	—
103	C₁₆×C₃₁	—	2	2	no	—
104	C₂×#2234	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	8	2	no	—
110	C₂³×C₃₁	—	8	2	no	—
111	C₂³×C₃₁	—	8	2	no	—
112	C₂³×C₃₁	C₂×D₈	1	2	yes	—
113	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
114	D₄×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	8	8	no	—
117	D₄×C₃₁	—	8	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	—	8	2	no	—
127	C₂×C₄×C₃₁	—	8	2	no	—
128	C₂×C₄×C₃₁	#123	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#137	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	#123	1	2	yes	—
135	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
136	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#2234	C₂×C₃₁	1	8	yes	direct product factor
140	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#357	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂×#357	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂²×C₃₁	—	2	2	no	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	4	4	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	16	4	no	—
153	C₂²×C₃₁	#480	1	2	yes	—
154	C₂²×C₃₁	#508	1	2	yes	—
155	C₂²×C₃₁	#509	1	2	yes	—
156	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₃₁	—	2	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#480	1	2	yes	—
162	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
163	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
164	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#121	—	2	1	no	—
167	C₂×#117	—	2	1	no	—
168	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
169	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
170	#466	—	2	2	no	—
171	#466	C₂²×C₃₁	1	4	yes	—
172	#427	—	2	2	no	—
173	#427	C₂²×C₃₁	1	4	yes	complemented
174	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#106	—	2	1	no	—
178	C₂×#102	—	2	1	no	—
179	#367	C₂²×C₃₁	1	4	yes	complemented
180	#366	C₂²×C₃₁	1	4	yes	complemented
181	#357	C₂²×C₃₁	1	4	yes	complemented
182	#357	C₂²×C₃₁	1	4	yes	complemented
183	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
184	C₂×C₃₁	—	4	2	no	—
185	C₂×C₃₁	—	16	2	no	—
186	C₂×C₃₁	#2234	1	4	yes	direct product factor
187	C₂×C₃₁	C₂×#480	1	1	yes	characteristic
188	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
189	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
190	C₂²×D₄	—	2	1	no	—
191	C₂²×D₄	—	4	1	no	—
192	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	C₂³×C₃₁	1	2	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₃₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	#121	—	2	4	no	—
204	#121	—	4	2	no	—
205	#117	—	2	4	no	—
206	#117	—	4	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	4	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₄²	—	4	1	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
216	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	C₂³×C₃₁	1	2	yes	—
221	#107	C₂³×C₃₁	1	2	yes	—
222	#107	C₂³×C₃₁	1	2	yes	—
223	#106	—	2	4	no	—
224	#102	—	2	4	no	—
225	C₃₁	C₂×#2234	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	—	4	1	no	—
227	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	D₈×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	#41×C₃₁	1	2	yes	—
253	C₂×C₈	#41×C₃₁	1	2	yes	—
254	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₄²	—	4	4	no	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	C₂×D₈×C₃₁	1	2	yes	—
274	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₃₁×#123	1	2	yes	—
290	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₃₁×#137	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₃₁×#123	1	2	yes	—
296	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
297	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	C₃₁×#480	1	2	yes	—
307	C₂²	C₃₁×#508	1	2	yes	—
308	C₂²	C₃₁×#509	1	2	yes	—
309	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₃₁×#480	1	2	yes	—
315	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
316	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	C₃₁×#2234	1	4	yes	direct product factor
320	C₂	C₂×C₃₁×#480	1	1	yes	characteristic
321	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
322	C₂	C₂×C₃₁×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂×C₃₁×#427	C₂	semidirect
3. C₃₁×#2234	C₂	direct (N × H)
4. C₂×C₃₁×#367	C₂	semidirect
5. C₂×C₃₁×#366	C₂	semidirect
6. C₂×C₃₁×#357	C₂	semidirect
7. C₂×C₃₁×#357	C₂	semidirect
8. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
18. C₃₁×#427	C₂²	semidirect
24. C₃₁×#367	C₂²	semidirect
25. C₃₁×#366	C₂²	semidirect
26. C₃₁×#357	C₂²	semidirect
27. C₃₁×#357	C₂²	semidirect
28. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
104. C₂×#2234	C₃₁	direct (N × H)
138. C₂×#427	C₂×C₃₁	semidirect
139. #2234	C₂×C₃₁	direct (N × H)
140. C₂×#367	C₂×C₃₁	semidirect
141. C₂×#366	C₂×C₃₁	semidirect
142. C₂×#357	C₂×C₃₁	semidirect
143. C₂×#357	C₂×C₃₁	semidirect
144. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
173. #427	C₂²×C₃₁	semidirect
179. #367	C₂²×C₃₁	semidirect
180. #366	C₂²×C₃₁	semidirect
181. #357	C₂²×C₃₁	semidirect
182. #357	C₂²×C₃₁	semidirect
183. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
186. C₂×C₃₁	#2234	direct (N × H)
225. C₃₁	C₂×#2234	direct (N × H)
319. C₂	C₃₁×#2234	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×#466	2
C₂×C₃₁×#427	2
C₃₁×#2234	2
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×#357	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₂×#2234	31

Subgroups — C2×C31×#2234 #21490400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2234 #21490400