Subgroups — C₂×#117×D₃₁ #16858873

Order 3968.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758469	C₂	1	2	yes	maximal, complemented
4	C₂×#11758449	C₂	1	2	yes	maximal, complemented
5	C₂×#11758454	C₂	1	2	yes	maximal, complemented
6	#117×D₃₁	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₄⋊C₄×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₃₁	C₂	1	2	yes	maximal
9	C₂×C₄²×D₃₁	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₁	C₄	1	16	yes	complemented
15	C₂²×C₄×D₃₁	C₂²	1	4	yes	—
16	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₃₁	C₂²	1	8	yes	—
19	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#117	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
26	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
27	#11758503	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
29	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	complemented
30	C₂×C₄×Dic₃₁	C₂²	1	4	yes	complemented
31	C₂×C₄×Dic₃₁	C₂²	1	2	yes	complemented
32	#11758469	C₂²	1	16	yes	complemented
33	C₄⋊C₄×D₃₁	C₂²	1	16	yes	complemented
34	C₂²⋊C₄×D₃₁	C₂²	1	16	yes	—
35	#11758454	C₂²	1	16	yes	complemented
36	#11758449	C₂²	1	16	yes	complemented
37	C₄²×D₃₁	C₂²	1	16	yes	complemented
38	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
39	C₂³×D₃₁	—	2	8	no	—
40	C₂³×D₃₁	C₂×C₄	1	8	yes	—
41	C₂³×D₃₁	C₂³	1	4	yes	—
42	C₂³×D₃₁	C₂³	1	1	yes	characteristic
43	C₂³×D₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
45	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
49	C₂²×Dic₃₁	C₂×C₄	1	8	yes	complemented
50	C₂²×Dic₃₁	C₂³	1	4	yes	—
51	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₃₁	C₂³	1	4	yes	—
54	C₂×C₄×D₃₁	—	2	16	no	—
55	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	—
56	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	—
57	C₂×C₄×D₃₁	C₂×C₄	1	64	yes	complemented
58	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	complemented
59	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	complemented
60	C₂×C₄×D₃₁	C₂³	1	16	yes	—
61	C₂×C₄×D₃₁	C₂³	1	4	yes	—
62	C₂×C₄×D₃₁	C₂³	1	4	yes	—
63	C₂×C₄×D₃₁	C₂³	1	4	yes	—
64	C₂×C₄×D₃₁	C₂³	1	4	yes	—
65	C₄⋊C₄×C₃₁	C₂³	1	8	yes	complemented
66	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
67	C₄²×C₃₁	C₂³	1	8	yes	complemented
68	C₂²⋊Dic₃₁	C₂³	1	8	yes	—
69	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	—
70	C₄⋊Dic₃₁	C₂³	1	8	yes	complemented
71	Dic₃₁⋊C₄	C₂³	1	16	yes	complemented
72	C₄×Dic₃₁	C₂³	1	8	yes	complemented
73	C₄×Dic₃₁	C₂³	1	16	yes	complemented
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
76	C₂³×C₃₁	C₂⁴	1	2	yes	—
77	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂²×D₃₁	—	2	4	no	—
79	C₂²×D₃₁	—	2	32	no	—
80	C₂²×D₃₁	—	2	16	no	—
81	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
82	C₂²×D₃₁	C₂²×C₄	1	16	yes	—
83	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
84	C₂²×D₃₁	#41	1	8	yes	—
85	C₂²×D₃₁	C₂⁴	1	2	yes	—
86	C₂²×D₃₁	C₂⁴	1	2	yes	—
87	C₂×C₄×C₃₁	—	2	8	no	—
88	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
91	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
92	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
93	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
94	C₂×Dic₃₁	—	2	8	no	—
95	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
96	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
97	C₂×Dic₃₁	C₂²×C₄	1	16	yes	complemented
98	C₂×Dic₃₁	C₂⁴	1	8	yes	—
99	C₂×Dic₃₁	C₂⁴	1	2	yes	—
100	C₂×Dic₃₁	C₂⁴	1	2	yes	—
101	C₄×D₃₁	—	2	32	no	—
102	C₄×D₃₁	C₂²×C₄	1	16	yes	—
103	C₄×D₃₁	C₂²×C₄	1	16	yes	—
104	C₄×D₃₁	C₂²×C₄	1	16	yes	complemented
105	C₄×D₃₁	C₂²×C₄	1	16	yes	complemented
106	C₂²×#117	—	31	1	no	maximal, Sylow
107	C₂²×C₃₁	—	2	8	no	—
108	C₂²×C₃₁	—	2	4	no	—
109	C₂²×C₃₁	C₂³×C₄	1	4	yes	—
110	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
111	C₂²×C₃₁	C₂×#41	1	4	yes	—
112	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
113	C₂×D₃₁	—	2	8	no	—
114	C₂×D₃₁	—	2	4	no	—
115	C₂×D₃₁	—	2	32	no	—
116	C₂×D₃₁	#117	1	16	yes	direct product factor
117	C₂×D₃₁	C₂³×C₄	1	4	yes	—
118	C₂×D₃₁	C₂³×C₄	1	4	yes	—
119	C₂×D₃₁	C₂×#41	1	8	yes	—
120	C₄×C₃₁	—	2	8	no	—
121	C₄×C₃₁	C₂³×C₄	1	4	yes	—
122	C₄×C₃₁	C₂³×C₄	1	4	yes	—
123	Dic₃₁	—	2	8	no	—
124	Dic₃₁	C₂³×C₄	1	4	yes	—
125	Dic₃₁	C₂³×C₄	1	4	yes	—
126	C₂⁴×C₄	—	31	1	no	—
127	C₂×#117	—	31	1	no	—
128	C₂×#117	—	31	16	no	—
129	C₂×#117	—	31	2	no	—
130	C₂×#117	—	31	2	no	—
131	C₂×#117	—	31	2	no	—
132	C₂×#117	D₃₁	1	1	yes	characteristic, direct product factor
133	C₂²×C₄⋊C₄	—	31	2	no	—
134	C₂²×C₂²⋊C₄	—	31	2	no	—
135	C₂²×C₄²	—	31	2	no	—
136	C₂×C₃₁	—	2	4	no	—
137	C₂×C₃₁	C₂×#117	1	4	yes	complemented
138	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
139	C₂×C₃₁	C₂²×#41	1	2	yes	—
140	D₃₁	—	2	4	no	—
141	D₃₁	C₂×#117	1	8	yes	direct product factor
142	C₂⁵	—	31	1	no	—
143	C₂³×C₄	—	31	1	no	—
144	C₂³×C₄	—	31	16	no	—
145	C₂³×C₄	—	31	1	no	—
146	C₂³×C₄	—	31	1	no	—
147	C₂³×C₄	—	31	8	no	—
148	C₂³×C₄	—	31	1	no	—
149	C₂³×C₄	—	31	1	no	—
150	C₂³×C₄	—	31	4	no	—
151	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
152	#117	—	31	8	no	—
153	#117	—	31	16	no	—
154	#117	—	31	16	no	—
155	#117	—	31	16	no	—
156	#117	C₂×D₃₁	1	8	yes	direct product factor
157	C₂×C₄⋊C₄	—	31	4	no	—
158	C₂×C₄⋊C₄	—	31	2	no	—
159	C₂×C₄⋊C₄	—	31	16	no	—
160	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
161	C₂×C₂²⋊C₄	—	31	2	no	—
162	C₂×C₂²⋊C₄	—	31	4	no	—
163	C₂×C₂²⋊C₄	—	31	16	no	—
164	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
165	C₂×C₄²	—	31	4	no	—
166	C₂×C₄²	—	31	2	no	—
167	C₂×C₄²	—	31	16	no	—
168	C₂×C₄²	C₂×D₃₁	1	2	yes	complemented
169	C₃₁	C₂²×#117	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	—	31	1	no	—
171	C₂⁴	—	31	8	no	—
172	C₂⁴	—	31	4	no	—
173	C₂⁴	—	31	1	no	—
174	C₂⁴	—	62	8	no	—
175	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
176	C₂²×C₄	—	31	8	no	—
177	C₂²×C₄	—	31	1	no	—
178	C₂²×C₄	—	31	1	no	—
179	C₂²×C₄	—	31	4	no	—
180	C₂²×C₄	—	31	8	no	—
181	C₂²×C₄	—	31	8	no	—
182	C₂²×C₄	—	31	8	no	—
183	C₂²×C₄	—	31	8	no	—
184	C₂²×C₄	—	31	64	no	—
185	C₂²×C₄	—	31	4	no	—
186	C₂²×C₄	—	31	4	no	—
187	C₂²×C₄	—	31	4	no	—
188	C₂²×C₄	—	31	4	no	—
189	C₂²×C₄	—	31	4	no	—
190	C₂²×C₄	—	31	16	no	—
191	C₂²×C₄	—	62	16	no	—
192	C₂²×C₄	C₄×D₃₁	1	8	yes	complemented
193	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
194	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
197	C₄⋊C₄	—	31	16	no	—
198	C₄⋊C₄	—	31	8	no	—
199	C₄⋊C₄	C₂²×D₃₁	1	8	yes	complemented
200	C₂²⋊C₄	—	31	8	no	—
201	C₂²⋊C₄	—	31	16	no	—
202	C₂²⋊C₄	C₂²×D₃₁	1	8	yes	—
203	C₄²	—	31	16	no	—
204	C₄²	—	31	8	no	—
205	C₄²	C₂²×D₃₁	1	8	yes	complemented
206	C₂³	—	2	4	no	—
207	C₂³	—	31	4	no	—
208	C₂³	—	31	8	no	—
209	C₂³	—	31	2	no	—
210	C₂³	—	31	16	no	—
211	C₂³	—	31	4	no	—
212	C₂³	—	31	2	no	—
213	C₂³	—	62	16	no	—
214	C₂³	—	62	32	no	—
215	C₂³	—	62	4	no	—
216	C₂³	C₂×C₄×D₃₁	1	4	yes	—
217	C₂³	C₂³×D₃₁	1	2	yes	—
218	C₂³	C₂³×D₃₁	1	1	yes	characteristic
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	31	4	no	—
221	C₂×C₄	—	31	4	no	—
222	C₂×C₄	—	31	16	no	—
223	C₂×C₄	—	31	2	no	—
224	C₂×C₄	—	31	2	no	—
225	C₂×C₄	—	31	16	no	—
226	C₂×C₄	—	31	16	no	—
227	C₂×C₄	—	31	16	no	—
228	C₂×C₄	—	31	16	no	—
229	C₂×C₄	—	31	8	no	—
230	C₂×C₄	—	62	32	no	—
231	C₂×C₄	—	62	8	no	—
232	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
233	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
234	C₂×C₄	C₂×C₄×D₃₁	1	16	yes	complemented
235	C₂×C₄	C₂³×D₃₁	1	8	yes	—
236	C₂×C₄	C₂³×D₃₁	1	2	yes	—
237	C₂×C₄	C₂³×D₃₁	1	2	yes	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	31	4	no	—
241	C₂²	—	31	8	no	—
242	C₂²	—	31	16	no	—
243	C₂²	—	31	4	no	—
244	C₂²	—	62	32	no	—
245	C₂²	—	62	4	no	—
246	C₂²	—	62	8	no	—
247	C₂²	C₂²×C₄×D₃₁	1	4	yes	—
248	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
249	C₂²	#41×D₃₁	1	4	yes	—
250	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
251	C₄	—	2	8	no	—
252	C₄	—	31	4	no	—
253	C₄	—	31	4	no	—
254	C₄	—	62	8	no	—
255	C₄	C₂²×C₄×D₃₁	1	4	yes	—
256	C₄	C₂²×C₄×D₃₁	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	31	8	no	—
259	C₂	—	62	4	no	—
260	C₂	#117×D₃₁	1	4	yes	direct product factor
261	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
262	C₂	C₂×#41×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×D₃₁	C₂²
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	C₂²×#117
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#11758503	C₂	semidirect
3. C₂×#11758469	C₂	semidirect
4. C₂×#11758449	C₂	semidirect
5. C₂×#11758454	C₂	semidirect
6. #117×D₃₁	C₂	direct (N × H)
7. C₂×C₄⋊C₄×D₃₁	C₂	semidirect
9. C₂×C₄²×D₃₁	C₂	semidirect
10. C₂×C₃₁×#117	C₂	semidirect
14. C₂²×C₄×D₃₁	C₄	semidirect
21. C₃₁×#117	C₂²	semidirect
22. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
24. C₂×C₄²×C₃₁	C₂²	semidirect
27. #11758503	C₂²	semidirect
28. C₂×C₄⋊Dic₃₁	C₂²	semidirect
29. C₂×Dic₃₁⋊C₄	C₂²	semidirect
30. C₂×C₄×Dic₃₁	C₂²	semidirect
31. C₂×C₄×Dic₃₁	C₂²	semidirect
32. #11758469	C₂²	semidirect
33. C₄⋊C₄×D₃₁	C₂²	semidirect
35. #11758454	C₂²	semidirect
36. #11758449	C₂²	semidirect
37. C₄²×D₃₁	C₂²	semidirect
44. C₂²×C₄×C₃₁	C₂×C₄	semidirect
49. C₂²×Dic₃₁	C₂×C₄	semidirect
57. C₂×C₄×D₃₁	C₂×C₄	semidirect
58. C₂×C₄×D₃₁	C₂×C₄	semidirect
59. C₂×C₄×D₃₁	C₂×C₄	semidirect
65. C₄⋊C₄×C₃₁	C₂³	semidirect
67. C₄²×C₃₁	C₂³	semidirect
70. C₄⋊Dic₃₁	C₂³	semidirect
71. Dic₃₁⋊C₄	C₂³	semidirect
72. C₄×Dic₃₁	C₂³	semidirect
73. C₄×Dic₃₁	C₂³	semidirect
90. C₂×C₄×C₃₁	C₂²×C₄	semidirect
97. C₂×Dic₃₁	C₂²×C₄	semidirect
104. C₄×D₃₁	C₂²×C₄	semidirect
105. C₄×D₃₁	C₂²×C₄	semidirect
116. C₂×D₃₁	#117	direct (N × H)
132. C₂×#117	D₃₁	direct (N × H)
137. C₂×C₃₁	C₂×#117	semidirect
141. D₃₁	C₂×#117	direct (N × H)
156. #117	C₂×D₃₁	direct (N × H)
160. C₂×C₄⋊C₄	C₂×D₃₁	semidirect
168. C₂×C₄²	C₂×D₃₁	semidirect
169. C₃₁	C₂²×#117	semidirect
192. C₂²×C₄	C₄×D₃₁	semidirect
199. C₄⋊C₄	C₂²×D₃₁	semidirect
205. C₄²	C₂²×D₃₁	semidirect
234. C₂×C₄	C₂×C₄×D₃₁	semidirect
260. C₂	#117×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₃₁	2
C₂×#11758503	2
C₂×#11758469	2
C₂×#11758449	2
C₂×#11758454	2
#117×D₃₁	2
C₂×C₄⋊C₄×D₃₁	2
C₂×C₂²⋊C₄×D₃₁	2
C₂×C₄²×D₃₁	2
C₂×C₃₁×#117	2
C₂²×#117	31

Subgroups — C2×#117×D31 #16858873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#117×D₃₁ #16858873