Subgroups — #16854610

Order 3968.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 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	C₂×#11758484	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758525	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521168	C₂	1	4	yes	maximal, complemented
5	#14521163	C₂	1	4	yes	maximal, complemented
6	#14521097	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521067	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521005	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#421	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	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#117	C₂²	1	1	yes	characteristic
14	#11758528	C₄	1	2	yes	complemented
15	#11758527	C₂²	1	4	yes	—
16	#11758526	C₄	1	2	yes	complemented
17	#11758525	—	2	1	no	—
18	#11758525	—	2	1	no	—
19	#11758525	C₂²	1	2	yes	complemented
20	#11758525	C₂²	1	2	yes	complemented
21	#11758519	C₄	1	2	yes	complemented
22	#11758509	C₄	1	2	yes	complemented
23	C₂×#92474	—	2	1	no	—
24	C₂×#92474	C₂²	1	1	yes	characteristic
25	C₂×#92474	C₂²	1	1	yes	characteristic
26	C₂²×C₃₁⋊C₈	—	2	1	no	—
27	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	#11758484	—	2	1	no	—
29	#11758484	—	2	1	no	—
30	#11758484	C₂²	1	2	yes	complemented
31	#11758484	C₂²	1	2	yes	complemented
32	C₂×#92469	—	2	1	no	—
33	C₂×C₈×D₃₁	—	2	1	no	—
34	C₃₁×#104	C₂²	1	4	yes	complemented
35	#11758410	C₂²	1	4	yes	complemented
36	#11758393	C₂²	1	4	yes	complemented
37	#11758391	C₂²	1	2	yes	complemented
38	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
39	#41×C₃₁	—	2	1	no	—
40	#41×C₃₁	—	2	1	no	—
41	#41×C₃₁	C₂×C₄	1	2	yes	—
42	#41×C₃₁	C₂³	1	2	yes	—
43	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₃₁	—	2	1	no	—
45	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₃₁	—	2	1	no	—
49	#92495	—	2	1	no	—
50	#92495	—	2	1	no	—
51	#92495	C₂×C₄	1	2	yes	—
52	#92495	C₂³	1	2	yes	—
53	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₁	—	2	1	no	—
55	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₃₁	—	2	1	no	—
57	M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂²⋊C₄×C₃₁	—	2	1	no	—
61	C₄²×C₃₁	C₂³	1	2	yes	—
62	#92482	C₂×C₄	1	4	yes	—
63	#92481	C₂×C₄	1	4	yes	—
64	#92480	C₂×C₄	1	4	yes	—
65	#92479	C₂×C₄	1	4	yes	—
66	#92474	—	2	1	no	—
67	#92474	—	2	2	no	—
68	#92474	—	2	2	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	—	2	2	no	—
74	C₂×C₃₁⋊C₈	—	2	2	no	—
75	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
76	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
77	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
78	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
79	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
80	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
82	#92469	—	2	2	no	—
83	#92469	—	2	2	no	—
84	C₈×D₃₁	—	2	2	no	—
85	C₈×D₃₁	—	2	2	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	#41	1	1	yes	characteristic
88	C₂²×D₃₁	—	2	1	no	—
89	Q₈×C₃₁	—	2	1	no	—
90	Q₈×C₃₁	C₂²×C₄	1	2	yes	—
91	D₄×C₃₁	—	2	2	no	—
92	D₄×C₃₁	—	2	2	no	—
93	D₄×C₃₁	—	2	1	no	—
94	D₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₃₁	—	2	2	no	—
106	C₂²⋊D₃₁	—	2	2	no	—
107	C₂×Dic₃₁	—	2	1	no	—
108	D₁₂₄	—	2	1	no	—
109	D₁₂₄	C₂²×C₄	1	2	yes	—
110	C₄×D₃₁	—	2	2	no	—
111	C₄×D₃₁	—	2	2	no	—
112	Dic₆₂	—	2	1	no	—
113	Dic₆₂	C₂²×C₄	1	2	yes	—
114	C₈×C₃₁	—	2	2	no	—
115	C₃₁⋊C₈	—	2	2	no	—
116	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
117	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
118	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
120	C₃₁⋊C₈	C₂×D₄	1	1	yes	characteristic
121	#2955	—	31	1	no	maximal, Sylow
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	—	4	1	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
128	C₂×D₃₁	—	2	2	no	—
129	C₂×D₃₁	—	4	2	no	—
130	C₄×C₃₁	—	2	2	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
134	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	Dic₃₁	—	2	2	no	—
137	#575	—	31	1	no	—
138	C₂×#131	—	31	1	no	—
139	C₂×#131	—	31	1	no	—
140	#444	—	31	4	no	—
141	#443	—	31	4	no	—
142	#430	—	31	1	no	—
143	#421	—	31	1	no	—
144	#421	D₃₁	1	1	yes	characteristic, complemented
145	#405	—	31	1	no	—
146	C₂×C₃₁	—	2	1	no	—
147	C₂×C₃₁	—	2	1	no	—
148	C₂×C₃₁	—	2	1	no	—
149	C₂×C₃₁	—	4	2	no	—
150	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
151	D₃₁	—	4	2	no	—
152	C₂×#41	—	31	1	no	—
153	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
154	#137	—	31	2	no	—
155	#137	—	31	2	no	—
156	AGL(1,ℤ/8)	—	31	2	no	—
157	AGL(1,ℤ/8)	—	31	2	no	—
158	#135	—	31	4	no	—
159	#131	—	31	2	no	—
160	#131	—	31	2	no	—
161	#131	—	31	2	no	—
162	#131	—	31	2	no	—
163	#131	—	62	1	no	—
164	#131	—	62	1	no	—
165	#131	—	62	1	no	—
166	#131	—	62	1	no	—
167	C₂×M₁₆	—	31	1	no	—
168	C₂×M₁₆	—	31	1	no	—
169	C₂×M₁₆	—	62	1	no	—
170	C₂×M₁₆	—	62	1	no	—
171	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
172	C₂²×C₈	—	31	1	no	—
173	C₂²×C₈	—	62	1	no	—
174	C₂²×C₈	—	62	1	no	—
175	#117	C₂×D₃₁	1	1	yes	characteristic
176	#108	—	31	4	no	—
177	#104	—	31	4	no	—
178	#104	C₂×D₃₁	1	4	yes	complemented
179	#97	—	31	2	no	—
180	C₄×C₈	—	31	2	no	—
181	C₃₁	#2955	1	1	yes	characteristic, complemented, Sylow
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	31	2	no	—
185	#41	—	31	2	no	—
186	#41	—	62	1	no	—
187	#41	—	62	1	no	—
188	#41	C₄×D₃₁	1	2	yes	—
189	#41	C₂²×D₃₁	1	2	yes	—
190	C₂×Q₈	—	31	1	no	—
191	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	31	1	no	—
194	C₂×D₄	—	62	1	no	—
195	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	62	1	no	—
198	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
199	Q₁₆	—	31	4	no	—
200	SD₁₆	—	31	4	no	—
201	SD₁₆	—	31	4	no	—
202	D₈	—	31	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	31	2	no	—
205	M₁₆	—	31	2	no	—
206	M₁₆	—	31	2	no	—
207	M₁₆	—	62	2	no	—
208	M₁₆	—	62	2	no	—
209	M₁₆	—	62	2	no	—
210	M₁₆	—	62	2	no	—
211	M₁₆	—	62	1	no	—
212	M₁₆	C₂²×D₃₁	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	31	1	no	—
215	C₂×C₈	—	31	1	no	—
216	C₂×C₈	—	31	1	no	—
217	C₂×C₈	—	31	1	no	—
218	C₂×C₈	—	31	1	no	—
219	C₂×C₈	—	31	2	no	—
220	C₂×C₈	—	31	1	no	—
221	C₂×C₈	—	62	2	no	—
222	C₂×C₈	—	62	2	no	—
223	C₂×C₈	—	62	2	no	—
224	C₂×C₈	—	62	2	no	—
225	C₂×C₈	—	62	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₄²	C₂²×D₃₁	1	2	yes	—
229	C₂³	—	2	1	no	—
230	C₂³	—	62	1	no	—
231	C₂³	#92497	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	31	2	no	—
234	Q₈	—	62	1	no	—
235	Q₈	C₂×C₄×D₃₁	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	31	2	no	—
240	D₄	—	62	2	no	—
241	D₄	—	62	2	no	—
242	D₄	—	62	1	no	—
243	D₄	C₂×C₄×D₃₁	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	62	1	no	—
249	C₂×C₄	—	62	2	no	—
250	C₂×C₄	—	62	2	no	—
251	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
253	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
254	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
255	C₂×C₄	#92497	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	31	4	no	—
259	C₈	—	31	1	no	—
260	C₈	—	31	1	no	—
261	C₈	—	31	1	no	—
262	C₈	—	31	1	no	—
263	C₈	—	62	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	62	2	no	—
268	C₂²	—	124	2	no	—
269	C₂²	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
270	C₂²	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
271	C₂²	C₂×#92497	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	62	2	no	—
275	C₄	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
276	C₄	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
278	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	124	2	no	—
284	C₂	C₂×(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#421	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
prime radical	#14521847	C₂
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#421	D₃₁
hypocenter	C₃₁	#2955
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. C₂×#11758484	C₂	semidirect
3. C₂×#11758525	C₂	semidirect
4. #14521168	C₂	semidirect
5. #14521163	C₂	semidirect
6. #14521097	C₂	semidirect
7. #14521067	C₂	semidirect
8. #14521005	C₂	semidirect
9. C₃₁×#421	C₂	semidirect
14. #11758528	C₄	semidirect
16. #11758526	C₄	semidirect
19. #11758525	C₂²	semidirect
20. #11758525	C₂²	semidirect
21. #11758519	C₄	semidirect
22. #11758509	C₄	semidirect
30. #11758484	C₂²	semidirect
31. #11758484	C₂²	semidirect
34. C₃₁×#104	C₂²	semidirect
35. #11758410	C₂²	semidirect
36. #11758393	C₂²	semidirect
37. #11758391	C₂²	semidirect
38. C₄×C₃₁⋊C₈	C₂²	semidirect
144. #421	D₃₁	semidirect
178. #104	C₂×D₃₁	semidirect
181. C₃₁	#2955	semidirect

Maximal subgroups

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

subgroup	index
#14521847	2
C₂×#11758484	2
C₂×#11758525	2
#14521168	2
#14521163	2
#14521097	2
#14521067	2
#14521005	2
C₃₁×#421	2
#2955	31