Subgroups — #16854618

Order 3968.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521847	C₂	1	1	yes	maximal, characteristic
2	#14521815	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521169	C₂	1	4	yes	maximal, complemented
4	#14521164	C₂	1	4	yes	maximal, complemented
5	#14521097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758425	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11758413	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92495	C₂²	1	1	yes	characteristic
12	C₃₁×#131	—	2	2	no	—
13	C₃₁×#131	C₂²	1	4	yes	complemented
14	C₂×M₁₆×C₃₁	—	2	2	no	—
15	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	#11758528	C₄	1	2	yes	complemented
18	#11758527	C₂²	1	4	yes	—
19	#11758526	C₄	1	2	yes	complemented
20	#11758519	C₄	1	2	yes	complemented
21	#11758509	C₄	1	2	yes	complemented
22	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92474	C₂²	1	1	yes	characteristic
24	C₂×#92474	C₂²	1	1	yes	characteristic
25	#11758484	—	2	2	no	—
26	M₁₆×D₃₁	—	2	2	no	—
27	#11758478	C₂²	1	4	yes	complemented
28	#11758425	C₂²	1	2	yes	complemented
29	#11758425	C₂²	1	2	yes	complemented
30	#11758413	C₂²	1	2	yes	complemented
31	#11758413	C₂²	1	2	yes	complemented
32	#11758410	C₂²	1	4	yes	complemented
33	#11758403	C₂²	1	2	yes	complemented
34	C₈×Dic₃₁	C₂²	1	2	yes	complemented
35	#41×C₃₁	—	2	1	no	—
36	#41×C₃₁	—	2	1	no	—
37	#41×C₃₁	C₂×C₄	1	2	yes	—
38	#41×C₃₁	C₂³	1	2	yes	—
39	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₃₁	—	2	1	no	—
41	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₁	—	2	1	no	—
45	C₂²×Dic₃₁	—	2	1	no	—
46	#92495	—	2	1	no	—
47	#92495	—	2	1	no	—
48	#92495	C₂×C₄	1	2	yes	—
49	#92495	C₂³	1	2	yes	—
50	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₁	—	2	1	no	—
52	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₃₁	—	2	1	no	—
54	M₁₆×C₃₁	—	2	4	no	—
55	M₁₆×C₃₁	D₄	1	2	yes	—
56	M₁₆×C₃₁	D₄	1	2	yes	—
57	M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	#92482	C₂×C₄	1	4	yes	complemented
62	#92481	C₂×C₄	1	4	yes	complemented
63	#92480	C₂×C₄	1	4	yes	complemented
64	#92479	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₃₁	—	2	1	no	—
66	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
67	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
68	#92474	—	2	1	no	—
69	#92474	C₂×C₄	1	2	yes	—
70	#92474	C₂×C₄	1	2	yes	—
71	#92474	C₂³	1	2	yes	—
72	C₂×C₃₁⋊C₈	—	2	1	no	—
73	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
74	#92469	—	2	4	no	—
75	C₈×D₃₁	—	2	4	no	—
76	C₂³×C₃₁	—	2	1	no	—
77	C₂³×C₃₁	#41	1	1	yes	characteristic
78	C₂²×D₃₁	—	2	1	no	—
79	Q₈×C₃₁	—	2	1	no	—
80	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
81	D₄×C₃₁	—	2	2	no	—
82	D₄×C₃₁	—	2	2	no	—
83	D₄×C₃₁	—	2	1	no	—
84	D₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₃₁	—	2	2	no	—
95	C₂²⋊D₃₁	—	2	2	no	—
96	C₂×Dic₃₁	—	2	1	no	—
97	C₂×Dic₃₁	—	2	1	no	—
98	C₂×Dic₃₁	—	2	1	no	—
99	C₂×Dic₃₁	—	4	2	no	—
100	D₁₂₄	—	2	1	no	—
101	D₁₂₄	C₂²×C₄	1	2	yes	—
102	C₄×D₃₁	—	2	2	no	—
103	C₄×D₃₁	—	2	2	no	—
104	Dic₆₂	—	2	1	no	—
105	Dic₆₂	C₂²×C₄	1	2	yes	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	C₂×D₄	1	4	yes	—
108	C₃₁⋊C₈	—	2	2	no	—
109	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
110	#2956	—	31	1	no	maximal, Sylow
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	4	1	no	—
113	C₂²×C₃₁	—	8	1	no	—
114	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₂×D₃₁	—	2	2	no	—
118	C₂×D₃₁	—	8	1	no	—
119	C₄×C₃₁	—	2	2	no	—
120	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
121	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
122	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	Dic₃₁	—	2	2	no	—
125	Dic₃₁	—	2	2	no	—
126	Dic₃₁	—	4	1	no	—
127	#575	—	31	1	no	—
128	#568	—	31	1	no	—
129	#568	D₃₁	1	1	yes	characteristic, complemented
130	#444	—	31	4	no	—
131	#443	—	31	4	no	—
132	#430	—	31	1	no	—
133	C₂×#104	—	31	1	no	—
134	C₂×#104	—	31	1	no	—
135	C₄×M₁₆	—	31	1	no	—
136	C₂×C₃₁	—	2	1	no	—
137	C₂×C₃₁	—	2	1	no	—
138	C₂×C₃₁	—	2	1	no	—
139	C₂×C₃₁	—	4	2	no	—
140	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
141	D₃₁	—	4	2	no	—
142	C₂×#41	—	31	1	no	—
143	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
144	#137	—	31	2	no	—
145	#137	—	31	2	no	—
146	AGL(1,ℤ/8)	—	31	2	no	—
147	AGL(1,ℤ/8)	—	31	2	no	—
148	#135	—	31	4	no	—
149	#131	—	2	2	no	—
150	#131	—	31	4	no	—
151	#131	—	62	2	no	—
152	#131	C₂×D₃₁	1	4	yes	complemented
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	31	1	no	—
155	C₂×M₁₆	—	31	1	no	—
156	C₂×M₁₆	—	62	2	no	—
157	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
158	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
159	C₂×C₄²	—	31	1	no	—
160	#108	—	31	4	no	—
161	#104	—	31	2	no	—
162	#104	—	31	2	no	—
163	#104	—	31	2	no	—
164	#104	—	31	2	no	—
165	#97	—	31	2	no	—
166	C₄×C₈	—	31	2	no	—
167	C₃₁	#2956	1	1	yes	characteristic, complemented, Sylow
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	31	2	no	—
171	#41	—	31	2	no	—
172	#41	—	62	1	no	—
173	#41	—	62	1	no	—
174	#41	C₄×D₃₁	1	2	yes	—
175	#41	C₂²×D₃₁	1	2	yes	—
176	C₂×Q₈	—	31	1	no	—
177	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	31	1	no	—
180	C₂×D₄	—	62	1	no	—
181	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	62	1	no	—
184	C₂²×C₄	—	62	1	no	—
185	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
186	Q₁₆	—	31	4	no	—
187	SD₁₆	—	31	4	no	—
188	SD₁₆	—	31	4	no	—
189	D₈	—	31	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	31	2	no	—
193	M₁₆	—	31	2	no	—
194	M₁₆	—	31	2	no	—
195	M₁₆	—	62	1	no	—
196	M₁₆	—	62	4	no	—
197	M₁₆	C₂²⋊D₃₁	1	2	yes	—
198	M₁₆	C₂²⋊D₃₁	1	2	yes	—
199	M₁₆	C₂²×D₃₁	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	31	2	no	—
203	C₂×C₈	—	62	1	no	—
204	C₂×C₈	—	62	4	no	—
205	C₂×C₈	C₂²×D₃₁	1	2	yes	—
206	C₄²	—	31	1	no	—
207	C₄²	—	31	1	no	—
208	C₄²	—	62	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	62	1	no	—
211	C₂³	#92495	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	31	2	no	—
214	Q₈	—	62	1	no	—
215	Q₈	C₂×C₄×D₃₁	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	31	2	no	—
220	D₄	—	62	2	no	—
221	D₄	—	62	2	no	—
222	D₄	—	62	1	no	—
223	D₄	C₂×C₄×D₃₁	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	62	1	no	—
228	C₂×C₄	—	62	1	no	—
229	C₂×C₄	—	62	2	no	—
230	C₂×C₄	—	62	2	no	—
231	C₂×C₄	—	62	1	no	—
232	C₂×C₄	—	124	2	no	—
233	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
235	C₂×C₄	#92495	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	31	4	no	—
241	C₈	—	62	2	no	—
242	C₈	C₂×C₂²⋊D₃₁	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	62	2	no	—
247	C₂²	—	248	1	no	—
248	C₂²	C₄×C₂²⋊D₃₁	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₃₁	1	1	yes	characteristic
250	C₂²	C₂×#92495	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	62	2	no	—
253	C₄	—	62	2	no	—
254	C₄	—	124	1	no	—
255	C₄	C₄×C₂²⋊D₃₁	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₃₁	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	124	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#568	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#14521847	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#568	D₃₁
hypocenter	C₃₁	#2956
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #14521815	C₂	semidirect
3. #14521169	C₂	semidirect
4. #14521164	C₂	semidirect
5. #14521097	C₂	semidirect
6. C₂×#11758425	C₂	semidirect
7. C₂×#11758413	C₂	semidirect
8. M₁₆×Dic₃₁	C₂	semidirect
9. C₃₁×#568	C₂	semidirect
13. C₃₁×#131	C₂²	semidirect
17. #11758528	C₄	semidirect
19. #11758526	C₄	semidirect
20. #11758519	C₄	semidirect
21. #11758509	C₄	semidirect
27. #11758478	C₂²	semidirect
28. #11758425	C₂²	semidirect
29. #11758425	C₂²	semidirect
30. #11758413	C₂²	semidirect
31. #11758413	C₂²	semidirect
32. #11758410	C₂²	semidirect
33. #11758403	C₂²	semidirect
34. C₈×Dic₃₁	C₂²	semidirect
61. #92482	C₂×C₄	semidirect
62. #92481	C₂×C₄	semidirect
63. #92480	C₂×C₄	semidirect
64. #92479	C₂×C₄	semidirect
129. #568	D₃₁	semidirect
152. #131	C₂×D₃₁	semidirect
167. C₃₁	#2956	semidirect

Maximal subgroups

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

subgroup	index
#14521847	2
#14521815	2
#14521169	2
#14521164	2
#14521097	2
C₂×#11758425	2
C₂×#11758413	2
M₁₆×Dic₃₁	2
C₃₁×#568	2
#2956	31