Subgroups — #16854609

Order 3968.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521846	C₂	1	1	yes	maximal, characteristic
2	#131×D₃₁	C₂	1	2	yes	maximal, complemented
3	#14521171	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521170	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521168	C₂	1	2	yes	maximal, complemented
6	#14521167	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521164	C₂	1	2	yes	maximal, complemented
8	#14521096	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521064	C₂	1	2	yes	maximal, complemented
10	#14521003	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₃₁	C₂²	1	2	yes	—
13	C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₃₁×#131	C₂²	1	2	yes	complemented
15	#11758527	C₂²	1	2	yes	—
16	#11758525	C₂²	1	2	yes	complemented
17	#11758495	C₄	1	2	yes	complemented
18	#11758492	C₄	1	2	yes	complemented
19	#11758491	C₄	1	2	yes	complemented
20	#11758488	C₄	1	2	yes	complemented
21	#11758484	—	2	2	no	—
22	#11758484	C₂²	1	2	yes	complemented
23	M₁₆×D₃₁	—	2	2	no	—
24	M₁₆×D₃₁	C₂²	1	2	yes	—
25	#11758481	C₂²	1	1	yes	characteristic
26	#11758478	—	2	2	no	—
27	#11758478	C₂²	1	2	yes	complemented
28	C₂×#92469	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₃₁	—	2	2	no	—
30	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
31	#11758449	C₂²	1	1	yes	characteristic
32	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#104	C₂²	1	2	yes	complemented
34	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	#11758425	C₂²	1	2	yes	complemented
36	#11758413	C₂²	1	2	yes	complemented
37	#11758410	C₂²	1	2	yes	complemented
38	#11758407	C₂²	1	1	yes	characteristic, complemented
39	#11758403	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
41	#11758393	C₂²	1	2	yes	complemented
42	#11758391	C₂²	1	1	yes	characteristic, complemented
43	#41×C₃₁	C₂³	1	2	yes	—
44	#92499	C₂×C₄	1	2	yes	—
45	Q₈×D₃₁	C₂×C₄	1	2	yes	—
46	#92497	—	2	2	no	—
47	#92497	C₂×C₄	1	2	yes	—
48	D₄×D₃₁	—	2	2	no	—
49	D₄×D₃₁	C₂×C₄	1	2	yes	—
50	#92495	—	2	2	no	—
51	#92495	C₂³	1	2	yes	—
52	C₂×C₄×D₃₁	—	2	2	no	—
53	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
54	Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
55	SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
56	D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₃₁	—	2	2	no	—
58	M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	#92482	C₂×C₄	1	2	yes	—
64	#92481	C₂×C₄	1	2	yes	—
65	#92480	C₂×C₄	1	2	yes	—
66	#92479	C₂×C₄	1	2	yes	—
67	(C₂×D₃₁)⋊C₄	—	2	1	no	—
68	Dic₃₁⋊C₄	—	2	1	no	—
69	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
70	#92474	—	2	2	no	—
71	#92474	C₂³	1	2	yes	—
72	C₂×C₃₁⋊C₈	—	2	2	no	—
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	Dic₁₂₄	C₂×C₄	1	1	yes	characteristic
76	D₂₄₈	C₂×C₄	1	1	yes	characteristic
77	#92470	C₂×C₄	1	2	yes	—
78	#92469	—	2	2	no	—
79	#92469	—	2	2	no	—
80	#92469	—	2	2	no	—
81	#92469	C₂×C₄	1	2	yes	—
82	#92469	C₂×C₄	1	2	yes	—
83	C₈×D₃₁	—	2	2	no	—
84	C₈×D₃₁	—	2	2	no	—
85	C₈×D₃₁	—	2	2	no	—
86	C₈×D₃₁	D₄	1	2	yes	—
87	C₈×D₃₁	D₄	1	2	yes	—
88	C₂²×D₃₁	—	2	2	no	—
89	C₂²×D₃₁	#41	1	1	yes	characteristic
90	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
91	D₄×C₃₁	—	2	2	no	—
92	D₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₃₁	—	2	2	no	—
97	C₂²⋊D₃₁	—	2	2	no	—
98	C₂²⋊D₃₁	—	2	2	no	—
99	C₂×Dic₃₁	—	2	2	no	—
100	C₂×Dic₃₁	—	2	1	no	—
101	C₂×Dic₃₁	#41	1	1	yes	characteristic
102	D₁₂₄	—	2	2	no	—
103	D₁₂₄	C₂²×C₄	1	2	yes	—
104	C₄×D₃₁	—	2	2	no	—
105	C₄×D₃₁	—	2	2	no	—
106	C₄×D₃₁	—	2	2	no	—
107	C₄×D₃₁	C₂²×C₄	1	2	yes	—
108	C₄×D₃₁	C₂×D₄	1	2	yes	—
109	Dic₆₂	—	2	2	no	—
110	Dic₆₂	C₂²×C₄	1	2	yes	—
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₃₁⋊C₈	—	2	2	no	—
117	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
118	C₃₁⋊C₈	C₂×D₄	1	2	yes	—
119	#2955	—	31	1	no	maximal, Sylow
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
122	C₂×D₃₁	—	2	2	no	—
123	C₂×D₃₁	—	4	1	no	—
124	C₂×D₃₁	—	4	4	no	—
125	C₂×D₃₁	C₄×D₄	1	2	yes	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	4	1	no	—
128	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	Dic₃₁	—	2	2	no	—
131	Dic₃₁	—	4	1	no	—
132	Dic₃₁	C₄×D₄	1	2	yes	—
133	#575	—	31	1	no	—
134	C₂×#131	—	31	2	no	—
135	#444	—	31	1	no	—
136	#444	—	31	1	no	—
137	#444	—	31	2	no	—
138	#443	—	31	2	no	—
139	#443	—	31	1	no	—
140	#443	D₃₁	1	1	yes	characteristic, complemented
141	#430	—	31	1	no	—
142	#421	—	31	2	no	—
143	#405	—	31	1	no	—
144	C₂×C₃₁	—	2	1	no	—
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
147	D₃₁	—	2	2	no	—
148	D₃₁	—	4	2	no	—
149	C₂×#41	—	31	2	no	—
150	#137	—	31	2	no	—
151	#137	—	31	2	no	—
152	AGL(1,ℤ/8)	—	31	2	no	—
153	AGL(1,ℤ/8)	—	31	2	no	—
154	#135	—	31	2	no	—
155	#135	—	31	1	no	—
156	#135	C₂×D₃₁	1	1	yes	characteristic
157	#131	—	31	2	no	—
158	#131	—	31	2	no	—
159	#131	—	31	2	no	—
160	#131	—	62	2	no	—
161	#131	—	62	2	no	—
162	#131	C₂×D₃₁	1	2	yes	complemented
163	C₂×M₁₆	—	31	1	no	—
164	C₂×M₁₆	—	31	2	no	—
165	C₂×M₁₆	—	62	2	no	—
166	C₂²×C₈	—	31	1	no	—
167	C₂²×C₈	—	62	2	no	—
168	#117	—	31	1	no	—
169	#108	—	31	2	no	—
170	#108	—	31	1	no	—
171	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
172	#104	—	31	2	no	—
173	#104	—	31	2	no	—
174	#104	—	31	2	no	—
175	#104	C₂×D₃₁	1	2	yes	complemented
176	#97	—	31	1	no	—
177	#97	—	31	1	no	—
178	C₄×C₈	—	31	1	no	—
179	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
180	C₃₁	#2955	1	1	yes	characteristic, complemented, Sylow
181	#41	—	31	2	no	—
182	#41	—	31	2	no	—
183	#41	—	31	2	no	—
184	#41	—	62	2	no	—
185	#41	—	62	2	no	—
186	#41	C₂²×D₃₁	1	2	yes	—
187	C₂×Q₈	—	31	2	no	—
188	C₂×D₄	—	31	2	no	—
189	C₂×D₄	—	62	2	no	—
190	C₂²×C₄	—	31	1	no	—
191	C₂²×C₄	—	62	2	no	—
192	Q₁₆	—	31	2	no	—
193	Q₁₆	—	31	1	no	—
194	Q₁₆	C₄×D₃₁	1	1	yes	characteristic
195	SD₁₆	—	31	2	no	—
196	SD₁₆	—	31	2	no	—
197	SD₁₆	—	31	2	no	—
198	SD₁₆	C₄×D₃₁	1	2	yes	—
199	D₈	—	31	2	no	—
200	D₈	—	31	1	no	—
201	D₈	C₄×D₃₁	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	31	2	no	—
204	M₁₆	—	31	2	no	—
205	M₁₆	—	31	2	no	—
206	M₁₆	—	62	2	no	—
207	M₁₆	—	62	2	no	—
208	M₁₆	—	62	2	no	—
209	M₁₆	—	62	2	no	—
210	M₁₆	C₂²×D₃₁	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	31	1	no	—
213	C₂×C₈	—	31	1	no	—
214	C₂×C₈	—	31	2	no	—
215	C₂×C₈	—	31	2	no	—
216	C₂×C₈	—	62	2	no	—
217	C₂×C₈	—	62	2	no	—
218	C₂×C₈	—	62	2	no	—
219	C₂×C₈	—	62	2	no	—
220	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
222	C₄⋊C₄	—	62	1	no	—
223	C₂²⋊C₄	—	62	1	no	—
224	C₄²	—	31	1	no	—
225	C₄²	C₂²×D₃₁	1	1	yes	characteristic
226	C₂³	—	31	1	no	—
227	C₂³	—	62	2	no	—
228	Q₈	—	31	2	no	—
229	Q₈	—	62	2	no	—
230	Q₈	C₂×C₄×D₃₁	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	31	2	no	—
233	D₄	—	62	2	no	—
234	D₄	—	62	2	no	—
235	D₄	—	62	2	no	—
236	D₄	—	62	2	no	—
237	D₄	C₂×C₄×D₃₁	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	31	1	no	—
241	C₂×C₄	—	31	2	no	—
242	C₂×C₄	—	31	2	no	—
243	C₂×C₄	—	62	2	no	—
244	C₂×C₄	—	62	2	no	—
245	C₂×C₄	—	62	2	no	—
246	C₂×C₄	—	62	2	no	—
247	C₂×C₄	—	62	1	no	—
248	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	31	2	no	—
251	C₈	—	31	2	no	—
252	C₈	—	62	2	no	—
253	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
254	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
255	C₈	D₄×D₃₁	1	1	yes	characteristic
256	C₈	D₄×D₃₁	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	31	2	no	—
259	C₂²	—	62	2	no	—
260	C₂²	—	124	1	no	—
261	C₂²	—	124	4	no	—
262	C₂²	#41×D₃₁	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	31	2	no	—
266	C₄	—	62	2	no	—
267	C₄	—	124	1	no	—
268	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
269	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	62	2	no	—
273	C₂	—	124	2	no	—
274	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#443	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#14521846	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#443	D₃₁
hypocenter	C₃₁	#2955
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₃₁	C₂	semidirect
3. #14521171	C₂	semidirect
4. #14521170	C₂	semidirect
5. #14521168	C₂	semidirect
6. #14521167	C₂	semidirect
7. #14521164	C₂	semidirect
8. #14521096	C₂	semidirect
9. #14521064	C₂	semidirect
10. #14521003	C₂	semidirect
11. C₃₁×#443	C₂	semidirect
14. C₃₁×#131	C₂²	semidirect
16. #11758525	C₂²	semidirect
17. #11758495	C₄	semidirect
18. #11758492	C₄	semidirect
19. #11758491	C₄	semidirect
20. #11758488	C₄	semidirect
22. #11758484	C₂²	semidirect
27. #11758478	C₂²	semidirect
32. C₃₁×#108	C₂²	semidirect
33. C₃₁×#104	C₂²	semidirect
34. C₄×C₈×C₃₁	C₂²	semidirect
35. #11758425	C₂²	semidirect
36. #11758413	C₂²	semidirect
37. #11758410	C₂²	semidirect
38. #11758407	C₂²	semidirect
39. #11758403	C₂²	semidirect
40. C₈×Dic₃₁	C₂²	semidirect
41. #11758393	C₂²	semidirect
42. #11758391	C₂²	semidirect
140. #443	D₃₁	semidirect
162. #131	C₂×D₃₁	semidirect
171. #108	C₂×D₃₁	semidirect
175. #104	C₂×D₃₁	semidirect
179. C₄×C₈	C₂×D₃₁	semidirect
180. C₃₁	#2955	semidirect

Maximal subgroups

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

subgroup	index
#14521846	2
#131×D₃₁	2
#14521171	2
#14521170	2
#14521168	2
#14521167	2
#14521164	2
#14521096	2
#14521064	2
#14521003	2
C₃₁×#443	2
#2955	31