Subgroups — #16854539

Order 3968.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758526	C₂	1	1	yes	maximal, characteristic
2	#14521523	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521159	C₂	1	2	yes	maximal, complemented
4	#14521147	C₂	1	2	yes	maximal, complemented
5	C₄×#92481	C₂	1	2	yes	maximal, complemented
6	C₄×#92479	C₂	1	2	yes	maximal, complemented
7	#14521089	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11758397	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
15	C₄×D₄×C₃₁	—	2	2	no	—
16	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
17	C₃₁×#117	—	2	1	no	—
18	C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	#11758526	C₄	1	8	yes	complemented
22	C₂×#92481	C₂²	1	2	yes	—
23	C₂×#92479	C₂²	1	2	yes	—
24	#11758503	C₂²	1	1	yes	characteristic
25	C₂×#92474	C₂²	1	1	yes	characteristic
26	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	#11758470	—	2	1	no	—
28	#11758470	—	2	1	no	—
29	C₂²⋊C₄×D₃₁	—	2	2	no	—
30	C₄×D₁₂₄	C₂²	1	2	yes	—
31	C₄×D₁₂₄	C₂²	1	2	yes	—
32	#11758423	C₂²	1	1	yes	characteristic, complemented
33	#11758423	C₂²	1	1	yes	characteristic, complemented
34	#11758420	C₂²	1	1	yes	characteristic, complemented
35	#11758420	C₂²	1	1	yes	characteristic, complemented
36	#11758397	C₂²	1	2	yes	—
37	#11758397	C₂²	1	2	yes	—
38	#11758396	C₂²	1	2	yes	complemented
39	#11758395	C₂²	1	2	yes	complemented
40	#11758391	C₂²	1	2	yes	complemented
41	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
42	C₂³×D₃₁	—	2	1	no	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×D₁₂₄	—	2	1	no	—
53	C₂×D₁₂₄	—	2	2	no	—
54	C₂×D₁₂₄	C₂×C₄	1	4	yes	complemented
55	C₂×D₁₂₄	C₂³	1	2	yes	—
56	C₂×C₄×D₃₁	—	2	2	no	—
57	C₂×C₄×D₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂²⋊Dic₃₁	—	2	1	no	—
68	#92481	C₂×C₄	1	8	yes	complemented
69	#92479	C₂×C₄	1	8	yes	complemented
70	(C₂×D₃₁)⋊C₄	—	2	2	no	—
71	(C₂×D₃₁)⋊C₄	—	2	2	no	—
72	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
74	C₄×Dic₃₁	—	2	1	no	—
75	#92474	C₂×C₄	1	4	yes	complemented
76	C₂×C₃₁⋊C₈	—	2	2	no	—
77	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₃₁	—	2	1	no	—
81	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₃₁	—	2	2	no	—
83	C₂²×D₃₁	—	2	4	no	—
84	C₂²×D₃₁	—	8	1	no	—
85	C₂²×D₃₁	—	8	1	no	—
86	Q₈×C₃₁	—	2	1	no	—
87	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	2	1	no	—
90	D₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	8	2	no	—
95	C₂×C₄×C₃₁	—	8	1	no	—
96	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#41	1	2	yes	—
101	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₃₁	—	2	1	no	—
103	C₂×Dic₃₁	—	2	1	no	—
104	D₁₂₄	—	2	4	no	—
105	D₁₂₄	—	2	2	no	—
106	D₁₂₄	C₂²×C₄	1	4	yes	—
107	C₄×D₃₁	—	8	2	no	—
108	C₄×D₃₁	—	8	2	no	—
109	C₃₁⋊C₈	—	2	1	no	—
110	C₃₁⋊C₈	—	2	1	no	—
111	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
112	#2946	—	31	1	no	maximal, Sylow
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	4	1	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	C₄×D₄	1	2	yes	—
119	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₃₁	—	2	4	no	—
121	C₂×D₃₁	—	4	2	no	—
122	C₂×D₃₁	—	4	2	no	—
123	C₂×D₃₁	—	8	4	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	C₄×D₄	1	2	yes	—
129	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
131	Dic₃₁	—	4	1	no	—
132	Dic₃₁	—	4	1	no	—
133	C₂×AGL(1,ℤ/8)	—	31	1	no	—
134	#520	D₃₁	1	1	yes	characteristic, complemented
135	#518	—	31	1	no	—
136	#442	—	31	2	no	—
137	#440	—	31	2	no	—
138	C₄×SD₁₆	—	31	2	no	—
139	C₄×D₈	—	31	2	no	—
140	#428	—	31	1	no	—
141	#416	—	31	1	no	—
142	C₂×#102	—	31	1	no	—
143	#405	—	31	1	no	—
144	C₂×C₃₁	—	2	2	no	—
145	C₂×C₃₁	—	4	2	no	—
146	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₃₁	#576	1	1	yes	characteristic
148	C₂×C₃₁	#577	1	1	yes	characteristic
149	D₃₁	—	4	4	no	—
150	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
151	C₂²×D₄	—	31	1	no	—
152	AGL(1,ℤ/8)	—	31	8	no	—
153	C₂×SD₁₆	—	31	2	no	—
154	C₂×D₈	—	31	2	no	—
155	C₂×M₁₆	—	31	1	no	—
156	C₂²×C₈	—	31	1	no	—
157	C₄×Q₈	C₂×D₃₁	1	2	yes	complemented
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	31	2	no	—
160	C₄×D₄	—	31	2	no	—
161	C₄×D₄	—	62	1	no	—
162	C₄×D₄	—	62	1	no	—
163	C₄×D₄	C₂×D₃₁	1	2	yes	complemented
164	#117	—	2	1	no	—
165	#117	—	31	1	no	—
166	#117	C₂×D₃₁	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	62	2	no	—
170	#107	—	31	2	no	—
171	#106	—	31	2	no	—
172	#103	—	31	1	no	—
173	#103	—	31	1	no	—
174	#102	—	31	1	no	—
175	#102	—	31	1	no	—
176	#102	—	31	2	no	—
177	#102	—	31	2	no	—
178	#97	—	31	2	no	—
179	C₄×C₈	—	31	2	no	—
180	C₃₁	#2946	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	62	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₃₁	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	31	4	no	—
187	C₂×D₄	—	31	2	no	—
188	C₂×D₄	—	62	2	no	—
189	C₂×D₄	—	62	1	no	—
190	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	62	2	no	—
195	C₂²×C₄	—	62	2	no	—
196	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
197	SD₁₆	—	31	8	no	—
198	D₈	—	31	8	no	—
199	M₁₆	—	31	4	no	—
200	C₂×C₈	—	31	2	no	—
201	C₂×C₈	—	31	1	no	—
202	C₂×C₈	—	31	1	no	—
203	C₂×C₈	—	62	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	31	1	no	—
207	C₄⋊C₄	—	31	1	no	—
208	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂²⋊D₃₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	62	1	no	—
213	C₂²⋊C₄	—	62	2	no	—
214	C₂²⋊C₄	—	62	2	no	—
215	C₂²⋊C₄	C₂²⋊D₃₁	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	62	1	no	—
218	C₄²	C₂²×D₃₁	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	62	2	no	—
221	C₂³	—	62	4	no	—
222	C₂³	—	248	1	no	—
223	C₂³	—	248	1	no	—
224	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₃₁	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	31	4	no	—
230	D₄	—	62	4	no	—
231	D₄	—	62	2	no	—
232	D₄	C₂×C₄×D₃₁	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	62	1	no	—
239	C₂×C₄	—	62	1	no	—
240	C₂×C₄	—	248	2	no	—
241	C₂×C₄	—	248	2	no	—
242	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
243	C₂×C₄	#92495	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
248	C₈	—	31	4	no	—
249	C₈	—	62	1	no	—
250	C₈	—	62	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	62	4	no	—
257	C₂²	—	124	2	no	—
258	C₂²	—	124	2	no	—
259	C₂²	—	248	4	no	—
260	C₂²	C₄×C₂²⋊D₃₁	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	124	1	no	—
267	C₄	—	124	1	no	—
268	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
269	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
270	C₄	C₂×#92495	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	124	4	no	—
274	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic
275	C₂	#14521852	1	1	yes	characteristic
276	C₂	#14521857	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#520	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#11758526	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#520	D₃₁
hypocenter	C₃₁	#2946
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14521523	C₂	semidirect
3. #14521159	C₂	semidirect
4. #14521147	C₂	semidirect
5. C₄×#92481	C₂	semidirect
6. C₄×#92479	C₂	semidirect
7. #14521089	C₂	semidirect
8. #14521045	C₂	semidirect
9. C₂×#11758397	C₂	semidirect
10. #14521005	C₂	semidirect
11. C₃₁×#520	C₂	semidirect
14. C₄×Q₈×C₃₁	C₂²	semidirect
16. C₄×D₄×C₃₁	C₂²	semidirect
21. #11758526	C₄	semidirect
32. #11758423	C₂²	semidirect
33. #11758423	C₂²	semidirect
34. #11758420	C₂²	semidirect
35. #11758420	C₂²	semidirect
38. #11758396	C₂²	semidirect
39. #11758395	C₂²	semidirect
40. #11758391	C₂²	semidirect
41. C₄×C₃₁⋊C₈	C₂²	semidirect
44. #41×C₃₁	C₂×C₄	semidirect
54. C₂×D₁₂₄	C₂×C₄	semidirect
68. #92481	C₂×C₄	semidirect
69. #92479	C₂×C₄	semidirect
75. #92474	C₂×C₄	semidirect
134. #520	D₃₁	semidirect
157. C₄×Q₈	C₂×D₃₁	semidirect
163. C₄×D₄	C₂×D₃₁	semidirect
180. C₃₁	#2946	semidirect
183. #41	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758526	2
#14521523	2
#14521159	2
#14521147	2
C₄×#92481	2
C₄×#92479	2
#14521089	2
#14521045	2
C₂×#11758397	2
#14521005	2
C₃₁×#520	2
#2946	31