Subgroups — C₄×#41×D₃₁ #16858888

Order 3968.

1728 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#92499	C₂	1	2	yes	maximal, complemented
4	C₄×#92495	C₂	1	3	yes	maximal, complemented
5	#14521517	C₂	1	3	yes	maximal, complemented
6	C₄×#92497	C₂	1	6	yes	maximal, complemented
7	C₄×Q₈×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×D₃₁	C₂	1	6	yes	maximal, complemented
9	#117×D₃₁	C₂	1	3	yes	maximal, complemented
10	C₂×C₄²×D₃₁	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	#41×D₃₁	C₄	1	16	yes	direct product factor
14	C₂×#92499	C₂²	1	1	yes	characteristic
15	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
16	C₂×#92497	C₂²	1	3	yes	—
17	C₂×D₄×D₃₁	C₂²	1	3	yes	—
18	C₂×#92495	C₂²	1	3	yes	—
19	C₂²×C₄×D₃₁	C₂²	1	6	yes	—
20	C₂²×C₄×D₃₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
22	C₄×D₄×C₃₁	C₂²	1	6	yes	complemented
23	C₃₁×#117	C₂²	1	3	yes	complemented
24	C₂×C₄²×C₃₁	C₂²	1	3	yes	complemented
25	Q₈×Dic₃₁	C₂²	1	2	yes	complemented
26	D₄×Dic₃₁	C₂²	1	6	yes	complemented
27	C₄×C₂²⋊D₃₁	C₂²	1	12	yes	complemented
28	#11758503	C₂²	1	3	yes	complemented
29	C₂×C₄×Dic₃₁	C₂²	1	6	yes	complemented
30	C₂×C₄×Dic₃₁	C₂²	1	3	yes	complemented
31	#11758470	C₂²	1	6	yes	complemented
32	#11758469	C₂²	1	6	yes	complemented
33	C₄⋊C₄×D₃₁	C₂²	1	6	yes	—
34	#11758464	C₂²	1	6	yes	complemented
35	(C₂²⋊D₃₁)⋊C₄	C₂²	1	12	yes	complemented
36	C₂²⋊C₄×D₃₁	C₂²	1	6	yes	—
37	#11758454	C₂²	1	6	yes	complemented
38	C₄×D₁₂₄	C₂²	1	6	yes	complemented
39	#11758449	C₂²	1	6	yes	complemented
40	C₄²×D₃₁	C₂²	1	1	yes	characteristic
41	C₄²×D₃₁	C₂²	1	3	yes	—
42	C₄²×D₃₁	C₂²	1	6	yes	—
43	C₄²×D₃₁	C₂²	1	6	yes	complemented
44	C₄²×D₃₁	C₂²	1	3	yes	complemented
45	C₄²×D₃₁	C₂²	1	3	yes	complemented
46	C₄×Dic₆₂	C₂²	1	6	yes	complemented
47	C₂³×D₃₁	C₂³	1	3	yes	—
48	#41×C₃₁	C₂×C₄	1	8	yes	complemented
49	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₃₁	C₂³	1	3	yes	—
51	C₂²×C₄×C₃₁	C₂³	1	6	yes	—
52	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
53	C₂×C₂²⋊D₃₁	C₂³	1	6	yes	—
54	C₂²×Dic₃₁	C₂³	1	6	yes	—
55	C₂²×Dic₃₁	C₂³	1	3	yes	—
56	#92499	C₂×C₄	1	8	yes	complemented
57	Q₈×D₃₁	C₂×C₄	1	8	yes	complemented
58	#92497	C₂×C₄	1	24	yes	complemented
59	D₄×D₃₁	C₂×C₄	1	24	yes	complemented
60	#92495	C₂×C₄	1	24	yes	complemented
61	C₂×D₁₂₄	C₂³	1	3	yes	—
62	C₂×C₄×D₃₁	—	2	24	no	—
63	C₂×C₄×D₃₁	C₂×C₄	1	24	yes	complemented
64	C₂×C₄×D₃₁	C₂³	1	6	yes	—
65	C₂×C₄×D₃₁	C₂³	1	6	yes	—
66	C₂×C₄×D₃₁	C₂³	1	6	yes	—
67	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₃₁	C₂³	1	3	yes	—
69	C₂×C₄×D₃₁	C₂³	1	3	yes	—
70	C₂×C₄×D₃₁	C₂³	1	3	yes	—
71	C₂×C₄×D₃₁	C₂³	1	2	yes	—
72	C₂×Dic₆₂	C₂³	1	3	yes	—
73	C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
74	C₂²⋊C₄×C₃₁	C₂³	1	6	yes	—
75	C₄²×C₃₁	C₂³	1	6	yes	—
76	C₄²×C₃₁	C₂³	1	3	yes	—
77	C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₃₁	C₂³	1	6	yes	—
79	(C₂×D₃₁)⋊C₄	C₂³	1	12	yes	—
80	C₄⋊Dic₃₁	C₂³	1	6	yes	—
81	Dic₃₁⋊C₄	C₂³	1	12	yes	—
82	C₄×Dic₃₁	C₂³	1	6	yes	—
83	C₄×Dic₃₁	C₂³	1	2	yes	—
84	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
85	C₄×Dic₃₁	C₂³	1	3	yes	—
86	C₄×Dic₃₁	C₂³	1	6	yes	—
87	C₄×Dic₃₁	C₂³	1	6	yes	—
88	C₄×Dic₃₁	C₂³	1	6	yes	—
89	C₂³×C₃₁	C₂⁴	1	3	yes	—
90	C₂²×D₃₁	—	2	12	no	—
91	C₂²×D₃₁	C₂²×C₄	1	12	yes	—
92	C₂²×D₃₁	C₂⁴	1	3	yes	—
93	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
94	Q₈×C₃₁	C₂²×C₄	1	4	yes	complemented
95	D₄×C₃₁	C₂²×C₄	1	12	yes	complemented
96	C₂×C₄×C₃₁	—	2	12	no	—
97	C₂×C₄×C₃₁	C₂²×C₄	1	12	yes	complemented
98	C₂×C₄×C₃₁	C₂⁴	1	6	yes	—
99	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂⁴	1	3	yes	—
101	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
102	C₂²⋊D₃₁	C₂²×C₄	1	24	yes	complemented
103	C₂×Dic₃₁	—	2	12	no	—
104	C₂×Dic₃₁	C₂²×C₄	1	12	yes	complemented
105	C₂×Dic₃₁	C₂⁴	1	6	yes	—
106	C₂×Dic₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₃₁	C₂⁴	1	3	yes	—
108	C₂×Dic₃₁	C₂⁴	1	2	yes	—
109	D₁₂₄	C₂²×C₄	1	12	yes	complemented
110	C₄×D₃₁	—	2	12	no	—
111	C₄×D₃₁	—	2	12	no	—
112	C₄×D₃₁	—	2	12	no	—
113	C₄×D₃₁	C₂²×C₄	1	4	yes	—
114	C₄×D₃₁	C₂²×C₄	1	12	yes	—
115	C₄×D₃₁	C₂²×C₄	1	12	yes	complemented
116	C₄×D₃₁	C₂²×C₄	1	12	yes	complemented
117	C₄×D₃₁	#41	1	8	yes	direct product factor
118	Dic₆₂	C₂²×C₄	1	12	yes	complemented
119	C₂×C₄×#41	—	31	1	no	maximal, Sylow
120	C₂²×C₃₁	—	2	6	no	—
121	C₂²×C₃₁	C₂³×C₄	1	6	yes	—
122	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
123	C₂×D₃₁	—	2	6	no	—
124	C₂×D₃₁	—	2	24	no	—
125	C₂×D₃₁	C₂³×C₄	1	6	yes	—
126	C₂×D₃₁	C₂³×C₄	1	2	yes	—
127	C₂×D₃₁	C₂×#41	1	4	yes	—
128	C₄×C₃₁	—	2	6	no	—
129	C₄×C₃₁	C₂³×C₄	1	6	yes	—
130	C₄×C₃₁	C₂³×C₄	1	2	yes	—
131	C₄×C₃₁	C₂×#41	1	4	yes	complemented
132	Dic₃₁	—	2	6	no	—
133	Dic₃₁	C₂³×C₄	1	6	yes	—
134	Dic₃₁	C₂³×C₄	1	2	yes	—
135	Dic₃₁	C₂×#41	1	4	yes	complemented
136	C₂²×#41	—	31	1	no	—
137	C₄×#41	—	31	1	no	—
138	C₄×#41	—	31	2	no	—
139	C₄×#41	—	31	6	no	—
140	C₄×#41	—	31	3	no	—
141	C₄×#41	—	31	3	no	—
142	C₄×#41	D₃₁	1	1	yes	characteristic, direct product factor
143	C₂×C₄×Q₈	—	31	2	no	—
144	C₂×C₄×D₄	—	31	6	no	—
145	C₂×#117	—	31	3	no	—
146	C₂²×C₄²	—	31	3	no	—
147	C₂×C₃₁	—	2	6	no	—
148	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
149	C₂×C₃₁	C₂²×#41	1	2	yes	—
150	D₃₁	—	2	6	no	—
151	D₃₁	C₄×#41	1	4	yes	direct product factor
152	C₂×#41	—	31	16	no	—
153	C₂×#41	—	31	1	no	—
154	C₂×#41	—	31	3	no	—
155	C₂×#41	—	31	3	no	—
156	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
157	C₂²×Q₈	—	31	1	no	—
158	C₂²×D₄	—	31	3	no	—
159	C₂³×C₄	—	31	3	no	—
160	C₂³×C₄	—	31	6	no	—
161	C₄×Q₈	—	31	2	no	—
162	C₄×Q₈	—	31	6	no	—
163	C₄×Q₈	—	31	6	no	—
164	C₄×Q₈	C₂×D₃₁	1	2	yes	complemented
165	C₄×D₄	—	31	6	no	—
166	C₄×D₄	—	31	12	no	—
167	C₄×D₄	—	31	6	no	—
168	C₄×D₄	—	31	12	no	—
169	C₄×D₄	—	31	6	no	—
170	C₄×D₄	C₂×D₃₁	1	6	yes	complemented
171	#117	—	31	3	no	—
172	#117	—	31	6	no	—
173	#117	—	31	6	no	—
174	#117	—	31	6	no	—
175	#117	C₂×D₃₁	1	3	yes	complemented
176	C₂×C₄⋊C₄	—	31	6	no	—
177	C₂×C₂²⋊C₄	—	31	6	no	—
178	C₂×C₄²	—	31	6	no	—
179	C₂×C₄²	—	31	3	no	—
180	C₂×C₄²	—	31	3	no	—
181	C₂×C₄²	—	31	6	no	—
182	C₂×C₄²	—	31	6	no	—
183	C₂×C₄²	—	31	3	no	—
184	C₂×C₄²	—	31	3	no	—
185	C₂×C₄²	—	31	1	no	—
186	C₂×C₄²	C₂×D₃₁	1	3	yes	complemented
187	C₃₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
188	C₂⁴	—	31	3	no	—
189	#41	—	31	8	no	—
190	#41	—	31	24	no	—
191	#41	—	31	24	no	—
192	#41	C₄×D₃₁	1	8	yes	direct product factor
193	C₂×Q₈	—	31	8	no	—
194	C₂×Q₈	—	31	3	no	—
195	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
196	C₂×D₄	—	31	6	no	—
197	C₂×D₄	—	31	24	no	—
198	C₂×D₄	—	31	3	no	—
199	C₂×D₄	C₂²×D₃₁	1	3	yes	—
200	C₂²×C₄	—	31	3	no	—
201	C₂²×C₄	—	31	24	no	—
202	C₂²×C₄	—	31	1	no	—
203	C₂²×C₄	—	31	2	no	—
204	C₂²×C₄	—	31	3	no	—
205	C₂²×C₄	—	31	3	no	—
206	C₂²×C₄	—	31	3	no	—
207	C₂²×C₄	—	31	6	no	—
208	C₂²×C₄	—	31	6	no	—
209	C₂²×C₄	—	31	6	no	—
210	C₂²×C₄	—	31	6	no	—
211	C₂²×C₄	—	62	24	no	—
212	C₂²×C₄	C₂²×D₃₁	1	6	yes	—
213	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
214	C₄⋊C₄	—	31	12	no	—
215	C₄⋊C₄	—	31	6	no	—
216	C₄⋊C₄	C₂²×D₃₁	1	6	yes	—
217	C₂²⋊C₄	—	31	6	no	—
218	C₂²⋊C₄	—	31	12	no	—
219	C₂²⋊C₄	C₂²×D₃₁	1	6	yes	—
220	C₄²	—	31	1	no	—
221	C₄²	—	31	2	no	—
222	C₄²	—	31	6	no	—
223	C₄²	—	31	3	no	—
224	C₄²	—	31	6	no	—
225	C₄²	—	31	6	no	—
226	C₄²	—	31	6	no	—
227	C₄²	C₂²×D₃₁	1	3	yes	—
228	C₄²	C₂²×D₃₁	1	6	yes	—
229	C₄²	C₂²×D₃₁	1	1	yes	characteristic
230	C₂³	—	31	1	no	—
231	C₂³	—	31	12	no	—
232	C₂³	—	31	3	no	—
233	C₂³	—	62	12	no	—
234	C₂³	C₂³×D₃₁	1	3	yes	—
235	Q₈	—	31	12	no	—
236	Q₈	C₂×C₄×D₃₁	1	4	yes	complemented
237	D₄	—	31	24	no	—
238	D₄	—	31	12	no	—
239	D₄	C₂×C₄×D₃₁	1	12	yes	complemented
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	31	12	no	—
242	C₂×C₄	—	31	1	no	—
243	C₂×C₄	—	31	2	no	—
244	C₂×C₄	—	31	3	no	—
245	C₂×C₄	—	31	4	no	—
246	C₂×C₄	—	31	12	no	—
247	C₂×C₄	—	31	12	no	—
248	C₂×C₄	—	31	12	no	—
249	C₂×C₄	—	31	8	no	—
250	C₂×C₄	—	31	6	no	—
251	C₂×C₄	—	62	12	no	—
252	C₂×C₄	—	62	12	no	—
253	C₂×C₄	—	62	12	no	—
254	C₂×C₄	—	62	12	no	—
255	C₂×C₄	C₂×C₄×D₃₁	1	12	yes	complemented
256	C₂×C₄	C₂³×D₃₁	1	6	yes	—
257	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂³×D₃₁	1	3	yes	—
259	C₂×C₄	C₂³×D₃₁	1	2	yes	—
260	C₂²	—	2	6	no	—
261	C₂²	—	31	2	no	—
262	C₂²	—	31	4	no	—
263	C₂²	—	31	6	no	—
264	C₂²	—	62	24	no	—
265	C₂²	—	62	6	no	—
266	C₂²	C₂²×C₄×D₃₁	1	6	yes	—
267	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
268	C₄	—	2	6	no	—
269	C₄	—	31	2	no	—
270	C₄	—	31	6	no	—
271	C₄	—	31	4	no	—
272	C₄	—	62	6	no	—
273	C₄	C₂²×C₄×D₃₁	1	2	yes	—
274	C₄	C₂²×C₄×D₃₁	1	6	yes	—
275	C₄	#41×D₃₁	1	4	yes	direct product factor
276	C₂	—	2	6	no	—
277	C₂	—	31	4	no	—
278	C₂	—	62	6	no	—
279	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
280	C₂	C₂×#41×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₄×#41×C₃₁	C₂
socle, cosocle	C₂²×C₃₁	C₂⁵
prime radical	C₂×#41×D₃₁	C₂
hypercenter	C₄×#41	D₃₁
hypocenter	C₃₁	C₂×C₄×#41
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #41×Dic₃₁	C₂	semidirect
3. C₄×#92499	C₂	semidirect
4. C₄×#92495	C₂	semidirect
5. #14521517	C₂	semidirect
6. C₄×#92497	C₂	semidirect
7. C₄×Q₈×D₃₁	C₂	semidirect
8. C₄×D₄×D₃₁	C₂	semidirect
9. #117×D₃₁	C₂	semidirect
10. C₂×C₄²×D₃₁	C₂	semidirect
11. C₄×#41×C₃₁	C₂	semidirect
13. #41×D₃₁	C₄	direct (N × H)
21. C₄×Q₈×C₃₁	C₂²	semidirect
22. C₄×D₄×C₃₁	C₂²	semidirect
23. C₃₁×#117	C₂²	semidirect
24. C₂×C₄²×C₃₁	C₂²	semidirect
25. Q₈×Dic₃₁	C₂²	semidirect
26. D₄×Dic₃₁	C₂²	semidirect
27. C₄×C₂²⋊D₃₁	C₂²	semidirect
28. #11758503	C₂²	semidirect
29. C₂×C₄×Dic₃₁	C₂²	semidirect
30. C₂×C₄×Dic₃₁	C₂²	semidirect
31. #11758470	C₂²	semidirect
32. #11758469	C₂²	semidirect
34. #11758464	C₂²	semidirect
35. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
37. #11758454	C₂²	semidirect
38. C₄×D₁₂₄	C₂²	semidirect
39. #11758449	C₂²	semidirect
43. C₄²×D₃₁	C₂²	semidirect
44. C₄²×D₃₁	C₂²	semidirect
45. C₄²×D₃₁	C₂²	semidirect
46. C₄×Dic₆₂	C₂²	semidirect
48. #41×C₃₁	C₂×C₄	semidirect
56. #92499	C₂×C₄	semidirect
57. Q₈×D₃₁	C₂×C₄	semidirect
58. #92497	C₂×C₄	semidirect
59. D₄×D₃₁	C₂×C₄	semidirect
60. #92495	C₂×C₄	semidirect
63. C₂×C₄×D₃₁	C₂×C₄	semidirect
94. Q₈×C₃₁	C₂²×C₄	semidirect
95. D₄×C₃₁	C₂²×C₄	semidirect
97. C₂×C₄×C₃₁	C₂²×C₄	semidirect
102. C₂²⋊D₃₁	C₂²×C₄	semidirect
104. C₂×Dic₃₁	C₂²×C₄	semidirect
109. D₁₂₄	C₂²×C₄	semidirect
115. C₄×D₃₁	C₂²×C₄	semidirect
116. C₄×D₃₁	C₂²×C₄	semidirect
117. C₄×D₃₁	#41	direct (N × H)
118. Dic₆₂	C₂²×C₄	semidirect
131. C₄×C₃₁	C₂×#41	semidirect
135. Dic₃₁	C₂×#41	semidirect
142. C₄×#41	D₃₁	direct (N × H)
151. D₃₁	C₄×#41	direct (N × H)
164. C₄×Q₈	C₂×D₃₁	semidirect
170. C₄×D₄	C₂×D₃₁	semidirect
175. #117	C₂×D₃₁	semidirect
186. C₂×C₄²	C₂×D₃₁	semidirect
187. C₃₁	C₂×C₄×#41	semidirect
192. #41	C₄×D₃₁	direct (N × H)
236. Q₈	C₂×C₄×D₃₁	semidirect
239. D₄	C₂×C₄×D₃₁	semidirect
255. C₂×C₄	C₂×C₄×D₃₁	semidirect
275. C₄	#41×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
C₂×#41×D₃₁	2
#41×Dic₃₁	2
C₄×#92499	2
C₄×#92495	2
#14521517	2
C₄×#92497	2
C₄×Q₈×D₃₁	2
C₄×D₄×D₃₁	2
#117×D₃₁	2
C₂×C₄²×D₃₁	2
C₄×#41×C₃₁	2
C₂×C₄×#41	31

Subgroups — C4×#41×D31 #16858888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#41×D₃₁ #16858888