Subgroups — #16854619

Order 3968.

458 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	#14521846	C₂	1	1	yes	maximal, characteristic
2	#14521814	C₂	1	2	yes	maximal, complemented
3	#14521172	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521168	C₂	1	2	yes	maximal, complemented
5	#14521166	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521165	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	#104×D₃₁	C₂	1	2	yes	maximal, complemented
10	#14521001	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#444	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	—	2	2	no	—
29	C₂×#92469	C₂²	1	1	yes	characteristic
30	C₂×#92469	C₂²	1	1	yes	characteristic
31	C₄²×D₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#104	C₂²	1	2	yes	complemented
34	C₃₁×#97	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	C₈×Dic₃₁	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	—	2	1	no	—
54	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
55	Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
56	SD₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
57	D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
58	M₁₆×C₃₁	—	2	2	no	—
59	M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	#92482	C₂×C₄	1	2	yes	complemented
65	#92481	C₂×C₄	1	2	yes	complemented
66	#92480	C₂×C₄	1	2	yes	complemented
67	#92479	C₂×C₄	1	2	yes	complemented
68	C₄×Dic₃₁	—	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, complemented
76	D₂₄₈	C₂×C₄	1	1	yes	characteristic, complemented
77	#92470	C₂×C₄	1	2	yes	complemented
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	#92469	D₄	1	2	yes	—
84	#92469	D₄	1	2	yes	—
85	C₈×D₃₁	—	2	2	no	—
86	C₈×D₃₁	—	2	2	no	—
87	C₈×D₃₁	—	2	2	no	—
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	1	no	—
100	C₂×Dic₃₁	—	2	2	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	—	4	2	no	—
108	C₄×D₃₁	C₂²×C₄	1	2	yes	—
109	C₄×D₃₁	C₂×D₄	1	2	yes	—
110	Dic₆₂	—	2	2	no	—
111	Dic₆₂	C₂²×C₄	1	2	yes	—
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₃₁⋊C₈	—	2	2	no	—
117	C₃₁⋊C₈	C₂²×C₄	1	2	yes	—
118	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
120	#2956	—	31	1	no	maximal, Sylow
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
123	C₂×D₃₁	—	2	2	no	—
124	C₂×D₃₁	—	4	1	no	—
125	C₂×D₃₁	—	8	2	no	—
126	C₂×D₃₁	C₄×D₄	1	2	yes	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	4	1	no	—
129	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	Dic₃₁	—	2	2	no	—
132	Dic₃₁	—	2	2	no	—
133	Dic₃₁	C₄×D₄	1	2	yes	—
134	#575	—	31	1	no	—
135	#568	—	31	2	no	—
136	#444	—	31	2	no	—
137	#444	—	31	1	no	—
138	#444	D₃₁	1	1	yes	characteristic, complemented
139	#443	—	31	1	no	—
140	#443	—	31	2	no	—
141	#443	—	31	1	no	—
142	#430	—	31	1	no	—
143	C₂×#104	—	31	2	no	—
144	C₄×M₁₆	—	31	1	no	—
145	C₂×C₃₁	—	2	1	no	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
148	D₃₁	—	2	2	no	—
149	D₃₁	—	4	2	no	—
150	C₂×#41	—	31	2	no	—
151	#137	—	31	2	no	—
152	#137	—	31	2	no	—
153	AGL(1,ℤ/8)	—	31	2	no	—
154	AGL(1,ℤ/8)	—	31	2	no	—
155	#135	—	31	2	no	—
156	#135	—	31	1	no	—
157	#135	C₂×D₃₁	1	1	yes	characteristic
158	#131	—	31	2	no	—
159	#131	—	31	2	no	—
160	#131	—	31	2	no	—
161	#131	—	62	2	no	—
162	#131	—	62	2	no	—
163	#131	C₂×D₃₁	1	2	yes	complemented
164	C₂×M₁₆	—	31	1	no	—
165	C₂×M₁₆	—	31	1	no	—
166	C₂×M₁₆	—	31	2	no	—
167	C₂×M₁₆	—	62	2	no	—
168	C₂×M₁₆	—	62	2	no	—
169	C₂×C₄²	—	31	1	no	—
170	#108	—	31	2	no	—
171	#108	—	31	1	no	—
172	#108	C₂×D₃₁	1	1	yes	characteristic, complemented
173	#104	—	31	2	no	—
174	#104	—	31	2	no	—
175	#104	—	31	2	no	—
176	#104	C₂×D₃₁	1	2	yes	complemented
177	#97	—	31	1	no	—
178	#97	C₂×D₃₁	1	1	yes	characteristic, complemented
179	C₄×C₈	—	31	1	no	—
180	C₄×C₈	—	31	1	no	—
181	C₃₁	#2956	1	1	yes	characteristic, complemented, Sylow
182	#41	—	31	2	no	—
183	#41	—	31	2	no	—
184	#41	—	31	2	no	—
185	#41	—	62	2	no	—
186	#41	—	62	2	no	—
187	#41	C₂²×D₃₁	1	2	yes	—
188	C₂×Q₈	—	31	2	no	—
189	C₂×D₄	—	31	2	no	—
190	C₂×D₄	—	62	2	no	—
191	C₂²×C₄	—	31	1	no	—
192	C₂²×C₄	—	62	2	no	—
193	C₂²×C₄	—	62	1	no	—
194	Q₁₆	—	31	2	no	—
195	Q₁₆	—	31	1	no	—
196	Q₁₆	C₄×D₃₁	1	1	yes	characteristic, complemented
197	SD₁₆	—	31	2	no	—
198	SD₁₆	—	31	2	no	—
199	SD₁₆	—	31	2	no	—
200	SD₁₆	C₄×D₃₁	1	2	yes	complemented
201	D₈	—	31	2	no	—
202	D₈	—	31	1	no	—
203	D₈	C₄×D₃₁	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	31	2	no	—
206	M₁₆	—	31	2	no	—
207	M₁₆	—	31	2	no	—
208	M₁₆	—	31	2	no	—
209	M₁₆	—	31	2	no	—
210	M₁₆	—	62	2	no	—
211	M₁₆	—	62	2	no	—
212	M₁₆	—	62	2	no	—
213	M₁₆	—	62	2	no	—
214	M₁₆	C₂²×D₃₁	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	31	1	no	—
217	C₂×C₈	—	31	1	no	—
218	C₂×C₈	—	62	2	no	—
219	C₂×C₈	—	62	2	no	—
220	C₂×C₈	—	62	2	no	—
221	C₂×C₈	—	62	2	no	—
222	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
224	C₄²	—	31	1	no	—
225	C₄²	—	62	1	no	—
226	C₄²	C₂²×D₃₁	1	1	yes	characteristic
227	C₂³	—	31	1	no	—
228	C₂³	—	62	2	no	—
229	Q₈	—	31	2	no	—
230	Q₈	—	62	2	no	—
231	Q₈	C₂×C₄×D₃₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	31	2	no	—
234	D₄	—	62	2	no	—
235	D₄	—	62	2	no	—
236	D₄	—	62	2	no	—
237	D₄	—	62	2	no	—
238	D₄	C₂×C₄×D₃₁	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	31	1	no	—
242	C₂×C₄	—	31	2	no	—
243	C₂×C₄	—	31	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	—	62	2	no	—
249	C₂×C₄	—	124	2	no	—
250	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	31	2	no	—
253	C₈	—	31	1	no	—
254	C₈	—	31	1	no	—
255	C₈	—	62	2	no	—
256	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
257	C₈	C₂×C₄×D₃₁	1	1	yes	characteristic
258	C₈	D₄×D₃₁	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	31	2	no	—
261	C₂²	—	62	2	no	—
262	C₂²	—	124	1	no	—
263	C₂²	—	248	2	no	—
264	C₂²	#41×D₃₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	31	2	no	—
268	C₄	—	62	2	no	—
269	C₄	—	62	2	no	—
270	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
271	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	62	2	no	—
275	C₂	—	124	2	no	—
276	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#444	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#14521846	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#444	D₃₁
hypocenter	C₃₁	#2956
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #14521814	C₂	semidirect
3. #14521172	C₂	semidirect
4. #14521168	C₂	semidirect
5. #14521166	C₂	semidirect
6. #14521165	C₂	semidirect
7. #14521164	C₂	semidirect
8. #14521096	C₂	semidirect
9. #104×D₃₁	C₂	semidirect
10. #14521001	C₂	semidirect
11. C₃₁×#444	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. C₃₁×#97	C₂²	semidirect
35. #11758425	C₂²	semidirect
36. #11758413	C₂²	semidirect
37. #11758410	C₂²	semidirect
38. #11758407	C₂²	semidirect
39. C₈×Dic₃₁	C₂²	semidirect
40. C₈×Dic₃₁	C₂²	semidirect
41. #11758393	C₂²	semidirect
42. #11758391	C₂²	semidirect
55. Q₁₆×C₃₁	C₂×C₄	semidirect
56. SD₁₆×C₃₁	C₂×C₄	semidirect
57. D₈×C₃₁	C₂×C₄	semidirect
64. #92482	C₂×C₄	semidirect
65. #92481	C₂×C₄	semidirect
66. #92480	C₂×C₄	semidirect
67. #92479	C₂×C₄	semidirect
75. Dic₁₂₄	C₂×C₄	semidirect
76. D₂₄₈	C₂×C₄	semidirect
77. #92470	C₂×C₄	semidirect
138. #444	D₃₁	semidirect
163. #131	C₂×D₃₁	semidirect
172. #108	C₂×D₃₁	semidirect
176. #104	C₂×D₃₁	semidirect
178. #97	C₂×D₃₁	semidirect
181. C₃₁	#2956	semidirect
196. Q₁₆	C₄×D₃₁	semidirect
200. SD₁₆	C₄×D₃₁	semidirect
203. D₈	C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
#14521846	2
#14521814	2
#14521172	2
#14521168	2
#14521166	2
#14521165	2
#14521164	2
#14521096	2
#104×D₃₁	2
#14521001	2
C₃₁×#444	2
#2956	31