Subgroups — #30674800

Order 16768.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862276	C₂	1	1	yes	maximal, characteristic
2	#22553789	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553417	C₂	1	2	yes	maximal, complemented
4	#22553412	C₂	1	2	yes	maximal, complemented
5	C₄×#14523503	C₂	1	2	yes	maximal, complemented
6	C₄×#14523501	C₂	1	2	yes	maximal, complemented
7	#22553352	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553308	C₂	1	1	yes	maximal, characteristic
9	C₂×#16862169	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553268	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₆₂	C₂²	1	1	yes	characteristic
13	#16862276	C₄	1	8	yes	complemented
14	C₂×#14523503	C₂²	1	2	yes	—
15	C₂×#14523501	C₂²	1	2	yes	—
16	C₂×#14523497	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₆₂	C₂²	1	2	yes	—
19	C₄×Dic₂₆₂	C₂²	1	2	yes	—
20	#16862211	—	2	1	no	—
21	#16862211	—	2	1	no	—
22	#16862205	C₂²	1	1	yes	characteristic
23	#16862202	—	2	2	no	—
24	#16862179	C₂²	1	2	yes	—
25	#16862177	C₂²	1	2	yes	—
26	#16862170	C₂²	1	1	yes	characteristic
27	#16862170	C₂²	1	1	yes	characteristic
28	#16862169	C₂²	1	2	yes	—
29	#16862169	C₂²	1	2	yes	—
30	#16862167	C₂²	1	1	yes	characteristic
31	#16862167	C₂²	1	1	yes	characteristic
32	#16862157	C₂²	1	2	yes	—
33	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
36	C₄×D₄×C₁₃₁	—	2	2	no	—
37	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
38	#117×C₁₃₁	—	2	1	no	—
39	#117×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
42	C₂×Dic₂₆₂	—	2	1	no	—
43	C₂×Dic₂₆₂	—	2	2	no	—
44	C₂×Dic₂₆₂	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₂₆₂	C₂³	1	2	yes	—
46	C₂²×Dic₁₃₁	—	2	1	no	—
47	#14523503	C₂×C₄	1	8	yes	complemented
48	#14523501	C₂×C₄	1	8	yes	complemented
49	#14523497	C₂×C₄	1	4	yes	complemented
50	C₂×C₁₃₁⋊C₈	—	2	2	no	—
51	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
54	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
56	Dic₁₃₁⋊C₄	—	2	2	no	—
57	Dic₁₃₁⋊C₄	—	2	2	no	—
58	C₂²⋊Dic₁₃₁	—	2	1	no	—
59	C₄×Dic₁₃₁	—	2	1	no	—
60	C₄×Dic₁₃₁	—	2	2	no	—
61	C₄×Dic₁₃₁	—	2	2	no	—
62	#41×C₁₃₁	—	2	2	no	—
63	#41×C₁₃₁	C₂×C₄	1	4	yes	—
64	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₃₁	—	2	1	no	—
66	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₁	—	2	1	no	—
68	C₂²×C₄×C₁₃₁	—	2	1	no	—
69	C₂²×C₄×C₁₃₁	—	2	2	no	—
70	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₁	—	2	1	no	—
72	C₄⋊C₄×C₁₃₁	—	2	2	no	—
73	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
77	C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
78	C₄²×C₁₃₁	—	2	2	no	—
79	C₄²×C₁₃₁	C₂³	1	2	yes	—
80	Dic₂₆₂	—	2	4	no	—
81	Dic₂₆₂	—	2	2	no	—
82	Dic₂₆₂	C₂²×C₄	1	4	yes	—
83	C₂×Dic₁₃₁	—	2	4	no	—
84	C₂×Dic₁₃₁	—	2	2	no	—
85	C₂×Dic₁₃₁	—	2	1	no	—
86	C₂×Dic₁₃₁	—	2	1	no	—
87	C₁₃₁⋊C₈	—	2	1	no	—
88	C₁₃₁⋊C₈	—	2	1	no	—
89	C₁₃₁⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂³×C₁₃₁	—	2	1	no	—
91	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
92	Q₈×C₁₃₁	—	2	1	no	—
93	Q₈×C₁₃₁	C₂²×C₄	1	2	yes	—
94	D₄×C₁₃₁	—	2	4	no	—
95	D₄×C₁₃₁	—	2	1	no	—
96	D₄×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	—	2	1	no	—
98	C₂×C₄×C₁₃₁	—	2	4	no	—
99	C₂×C₄×C₁₃₁	—	2	2	no	—
100	C₂×C₄×C₁₃₁	—	8	2	no	—
101	C₂×C₄×C₁₃₁	—	8	1	no	—
102	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₁	#41	1	2	yes	—
107	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
108	C₂²×C₁₃₁	—	2	1	no	—
109	C₂²×C₁₃₁	—	2	1	no	—
110	C₂²×C₁₃₁	—	2	2	no	—
111	C₂²×C₁₃₁	—	4	1	no	—
112	C₂²×C₁₃₁	—	4	1	no	—
113	C₂²×C₁₃₁	C₄×D₄	1	2	yes	—
114	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₃₁	—	2	2	no	—
116	C₄×C₁₃₁	—	2	2	no	—
117	C₄×C₁₃₁	—	2	2	no	—
118	C₄×C₁₃₁	—	4	2	no	—
119	C₄×C₁₃₁	C₄×D₄	1	2	yes	—
120	C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
122	Dic₁₃₁	—	2	4	no	—
123	Dic₁₃₁	—	4	1	no	—
124	Dic₁₃₁	—	4	1	no	—
125	C₂×C₁₃₁	—	2	2	no	—
126	C₂×C₁₃₁	—	4	2	no	—
127	C₂×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₁₃₁	#577	1	1	yes	characteristic
129	C₂×C₁₃₁	#578	1	1	yes	characteristic
130	C₁₃₁	#2947	1	1	yes	characteristic, complemented, Sylow
131	#2947	—	131	1	no	maximal, Sylow
132	C₂×#137	—	131	1	no	—
133	#520	D₁₃₁	1	1	yes	characteristic
134	#519	—	131	1	no	—
135	#441	—	131	2	no	—
136	#440	—	131	2	no	—
137	C₄×Q₁₆	—	131	2	no	—
138	C₄×SD₁₆	—	131	2	no	—
139	#428	—	131	1	no	—
140	#416	—	131	1	no	—
141	C₂×#103	—	131	1	no	—
142	#405	—	131	1	no	—
143	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
144	C₂²×Q₈	—	131	1	no	—
145	#137	—	131	8	no	—
146	C₂×Q₁₆	—	131	2	no	—
147	C₂×SD₁₆	—	131	2	no	—
148	C₂×M₁₆	—	131	1	no	—
149	C₂²×C₈	—	131	1	no	—
150	C₄×Q₈	—	262	1	no	—
151	C₄×Q₈	—	262	1	no	—
152	C₄×Q₈	—	131	2	no	—
153	C₄×Q₈	—	131	2	no	—
154	C₄×Q₈	C₂×D₁₃₁	1	2	yes	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
157	#117	—	2	1	no	—
158	#117	—	262	2	no	—
159	#117	—	131	1	no	—
160	#117	C₂×D₁₃₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
163	#107	—	131	2	no	—
164	#106	—	131	2	no	—
165	#103	—	131	1	no	—
166	#103	—	131	1	no	—
167	#103	—	131	2	no	—
168	#103	—	131	2	no	—
169	#102	—	131	1	no	—
170	#102	—	131	1	no	—
171	#97	—	131	2	no	—
172	C₄×C₈	—	131	2	no	—
173	#41	—	2	2	no	—
174	#41	C₄×D₁₃₁	1	4	yes	—
175	C₂×Q₈	—	262	1	no	—
176	C₂×Q₈	—	262	2	no	—
177	C₂×Q₈	—	131	4	no	—
178	C₂×Q₈	—	131	2	no	—
179	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	262	1	no	—
186	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
187	Q₁₆	—	131	8	no	—
188	SD₁₆	—	131	8	no	—
189	M₁₆	—	131	4	no	—
190	C₂×C₈	—	262	2	no	—
191	C₂×C₈	—	131	1	no	—
192	C₂×C₈	—	131	1	no	—
193	C₂×C₈	—	131	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	262	2	no	—
197	C₄⋊C₄	—	262	2	no	—
198	C₄⋊C₄	—	131	1	no	—
199	C₄⋊C₄	—	131	1	no	—
200	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	262	1	no	—
205	C₂²⋊C₄	C₂²⋊D₁₃₁	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	262	2	no	—
208	C₄²	—	262	2	no	—
209	C₄²	—	262	1	no	—
210	C₄²	C₂²×D₁₃₁	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
213	Q₈	—	2	1	no	—
214	Q₈	—	262	2	no	—
215	Q₈	—	262	4	no	—
216	Q₈	—	131	4	no	—
217	Q₈	C₂×C₄×D₁₃₁	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂×C₄×D₁₃₁	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	262	2	no	—
225	C₂×C₄	—	262	4	no	—
226	C₂×C₄	—	262	1	no	—
227	C₂×C₄	—	262	1	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
234	C₂×C₄	#14523513	1	2	yes	—
235	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
236	C₈	—	262	1	no	—
237	C₈	—	262	1	no	—
238	C₈	—	131	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₄×C₂²⋊D₁₃₁	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	262	4	no	—
251	C₄	—	524	1	no	—
252	C₄	—	524	1	no	—
253	C₄	C₄×C₂²⋊D₁₃₁	1	2	yes	—
254	C₄	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
255	C₄	C₂×#14523513	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₁₃₁	1	1	yes	characteristic
259	C₂	#22554123	1	1	yes	characteristic
260	C₂	#22554127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₁
commutator	C₄×C₁₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#520×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#41×C₁₃₁	C₂²
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#520	D₁₃₁
hypocenter	C₁₃₁	#2947
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #22553789	C₂	semidirect
3. #22553417	C₂	semidirect
4. #22553412	C₂	semidirect
5. C₄×#14523503	C₂	semidirect
6. C₄×#14523501	C₂	semidirect
7. #22553352	C₂	semidirect
9. C₂×#16862169	C₂	semidirect
10. #22553268	C₂	semidirect
13. #16862276	C₄	semidirect
44. C₂×Dic₂₆₂	C₂×C₄	semidirect
47. #14523503	C₂×C₄	semidirect
48. #14523501	C₂×C₄	semidirect
49. #14523497	C₂×C₄	semidirect
130. C₁₃₁	#2947	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862276	2
#22553789	2
#22553417	2
#22553412	2
C₄×#14523503	2
C₄×#14523501	2
#22553352	2
#22553308	2
C₂×#16862169	2
#22553268	2
#520×C₁₃₁	2
#2947	131