Subgroups — #21454663

Order 7808.

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	#16840245	C₂	1	1	yes	maximal, characteristic
2	#16840240	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839869	C₂	1	2	yes	maximal, complemented
4	#16839863	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₆₁	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₆₁	C₂	1	2	yes	maximal, complemented
7	#16839804	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839758	C₂	1	1	yes	maximal, characteristic
9	C₂×#14519701	C₂	1	1	yes	maximal, characteristic, complemented
10	#16839720	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×#92418	C₂²	1	1	yes	characteristic
13	Q₈×Dic₆₁	—	2	1	no	—
14	Q₈×Dic₆₁	—	2	1	no	—
15	Q₈×Dic₆₁	C₂²	1	2	yes	—
16	Q₈×Dic₆₁	C₂²	1	2	yes	—
17	Q₈×Dic₆₁	C₂²	1	2	yes	—
18	D₄×Dic₆₁	—	2	2	no	—
19	D₄×Dic₆₁	C₂²	1	2	yes	—
20	#14519736	—	2	2	no	—
21	#14519736	—	2	1	no	—
22	#14519736	C₂²	1	1	yes	characteristic
23	#14519736	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₆₁	—	2	1	no	—
25	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
26	#14519711	C₂²	1	2	yes	—
27	#14519709	C₂²	1	2	yes	—
28	#14519701	C₂²	1	2	yes	—
29	#14519701	C₂²	1	2	yes	—
30	#14519701	C₂²	1	1	yes	characteristic
31	#14519701	C₂²	1	1	yes	characteristic
32	#14519698	C₂²	1	1	yes	characteristic
33	#14519698	C₂²	1	1	yes	characteristic
34	#14519687	C₂²	1	2	yes	—
35	C₈×Dic₆₁	C₂²	1	2	yes	—
36	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
38	#137×C₆₁	C₄	1	8	yes	complemented
39	C₂×Q₁₆×C₆₁	C₂²	1	2	yes	—
40	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	—
41	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	#41×C₆₁	—	2	2	no	—
43	#41×C₆₁	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₆₁	—	2	1	no	—
45	C₂×Q₈×C₆₁	—	2	2	no	—
46	C₂×Q₈×C₆₁	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₆₁	—	2	1	no	—
50	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	—	2	1	no	—
52	C₂²×C₄×C₆₁	—	2	1	no	—
53	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×Dic₆₁	—	2	1	no	—
55	C₂²×Dic₆₁	—	2	2	no	—
56	Q₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
57	SD₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
58	M₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₆₁	C₂³	1	2	yes	—
60	C₂²⋊Dic₆₁	—	2	1	no	—
61	C₂²⋊Dic₆₁	—	2	2	no	—
62	C₂²⋊Dic₆₁	D₄	1	2	yes	—
63	C₄⋊Dic₆₁	—	2	1	no	—
64	C₄⋊Dic₆₁	—	2	2	no	—
65	C₄⋊Dic₆₁	—	2	2	no	—
66	C₄⋊Dic₆₁	—	2	2	no	—
67	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
68	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
69	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₆₁	C₂³	1	2	yes	—
72	C₄×Dic₆₁	—	2	1	no	—
73	C₄×Dic₆₁	—	2	2	no	—
74	C₄×Dic₆₁	—	2	2	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	C₂³	1	2	yes	—
77	C₂×#92418	—	2	2	no	—
78	C₂×#92418	C₂³	1	1	yes	characteristic
79	C₂×#92418	C₂³	1	1	yes	characteristic
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₆₁	—	2	4	no	—
83	Q₈×C₆₁	—	2	2	no	—
84	Q₈×C₆₁	—	2	1	no	—
85	Q₈×C₆₁	C₂²×C₄	1	4	yes	—
86	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
87	D₄×C₆₁	—	2	4	no	—
88	D₄×C₆₁	—	2	1	no	—
89	D₄×C₆₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#41	1	2	yes	—
97	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₆₁	—	2	1	no	—
99	C₂×Dic₆₁	—	2	1	no	—
100	C₂×Dic₆₁	—	2	2	no	—
101	C₂×Dic₆₁	—	8	2	no	—
102	C₂×Dic₆₁	—	8	1	no	—
103	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
104	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
105	C₈×C₆₁	C₂²×C₄	1	4	yes	—
106	#92418	—	2	1	no	—
107	#92418	—	2	1	no	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	4	1	no	—
113	C₂²×C₆₁	C₄×D₄	1	2	yes	—
114	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	2	4	no	—
117	C₄×C₆₁	C₄×D₄	1	2	yes	—
118	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
120	Dic₆₁	—	2	2	no	—
121	Dic₆₁	—	2	2	no	—
122	Dic₆₁	—	4	1	no	—
123	Dic₆₁	—	4	2	no	—
124	Dic₆₁	—	4	1	no	—
125	#2947	—	61	1	no	maximal, Sylow
126	C₂×C₆₁	—	2	2	no	—
127	C₂×C₆₁	—	4	2	no	—
128	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₆₁	#577	1	1	yes	characteristic
130	C₂×C₆₁	#578	1	1	yes	characteristic
131	C₂×#137	D₆₁	1	1	yes	characteristic
132	#520	—	61	1	no	—
133	#519	—	61	1	no	—
134	#441	—	61	2	no	—
135	#440	—	61	2	no	—
136	C₄×Q₁₆	—	61	2	no	—
137	C₄×SD₁₆	—	61	2	no	—
138	#428	—	61	1	no	—
139	#416	—	61	1	no	—
140	C₂×#103	—	61	1	no	—
141	#405	—	61	1	no	—
142	C₆₁	#2947	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
144	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
145	#137	Dic₆₁	1	8	yes	complemented
146	C₂×Q₁₆	C₂×D₆₁	1	2	yes	—
147	C₂×SD₁₆	C₂×D₆₁	1	2	yes	—
148	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
149	C₂²×C₈	—	61	1	no	—
150	C₄×Q₈	—	61	2	no	—
151	C₄×Q₈	—	61	2	no	—
152	C₄×Q₈	—	61	2	no	—
153	C₄×Q₈	—	122	1	no	—
154	C₄×Q₈	—	122	1	no	—
155	C₄×D₄	—	61	2	no	—
156	C₄×D₄	—	122	2	no	—
157	#117	—	61	1	no	—
158	#117	—	61	1	no	—
159	#117	—	122	2	no	—
160	#117	—	122	1	no	—
161	C₂×C₄⋊C₄	—	61	1	no	—
162	C₂×C₄⋊C₄	—	122	1	no	—
163	#107	—	61	2	no	—
164	#106	—	61	2	no	—
165	#103	—	61	2	no	—
166	#103	—	61	2	no	—
167	#103	—	61	1	no	—
168	#103	—	61	1	no	—
169	#102	—	61	1	no	—
170	#102	—	61	1	no	—
171	#97	—	61	2	no	—
172	C₄×C₈	—	61	2	no	—
173	#41	—	2	2	no	—
174	#41	C₂×Dic₆₁	1	4	yes	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂×Dic₆₁	1	4	yes	complemented
178	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
179	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₆₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	122	2	no	—
185	C₂²×C₄	—	122	1	no	—
186	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
187	Q₁₆	C₂×Dic₆₁	1	8	yes	complemented
188	SD₁₆	C₂×Dic₆₁	1	8	yes	complemented
189	M₁₆	C₂×Dic₆₁	1	4	yes	complemented
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	1	no	—
192	C₂×C₈	—	122	2	no	—
193	C₂×C₈	C₂²×D₆₁	1	2	yes	—
194	C₄⋊C₄	—	61	1	no	—
195	C₄⋊C₄	—	61	1	no	—
196	C₄⋊C₄	—	61	1	no	—
197	C₄⋊C₄	—	61	1	no	—
198	C₄⋊C₄	—	61	2	no	—
199	C₄⋊C₄	—	122	2	no	—
200	C₄⋊C₄	—	122	2	no	—
201	C₄⋊C₄	—	122	2	no	—
202	C₄⋊C₄	—	122	1	no	—
203	C₂²⋊C₄	—	61	2	no	—
204	C₂²⋊C₄	—	122	2	no	—
205	C₂²⋊C₄	—	122	1	no	—
206	C₄²	—	61	2	no	—
207	C₄²	—	122	2	no	—
208	C₄²	—	122	2	no	—
209	C₄²	—	122	2	no	—
210	C₄²	—	122	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₆₁	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	1	no	—
215	Q₈	—	2	4	no	—
216	Q₈	C₂²×Dic₆₁	1	4	yes	—
217	Q₈	C₂²×Dic₆₁	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂²×Dic₆₁	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	61	2	no	—
226	C₂×C₄	—	61	2	no	—
227	C₂×C₄	—	122	1	no	—
228	C₂×C₄	—	122	1	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	488	2	no	—
231	C₂×C₄	—	488	1	no	—
232	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
233	C₂×C₄	#11758224	1	2	yes	—
234	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
235	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
236	C₈	—	122	1	no	—
237	C₈	—	122	1	no	—
238	C₈	C₂²×Dic₆₁	1	4	yes	—
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	D₄×Dic₆₁	1	2	yes	—
245	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	122	2	no	—
249	C₄	—	122	2	no	—
250	C₄	—	244	1	no	—
251	C₄	—	244	2	no	—
252	C₄	—	244	1	no	—
253	C₄	D₄×Dic₆₁	1	2	yes	—
254	C₄	C₂³×Dic₆₁	1	1	yes	characteristic
255	C₄	C₂×#11758224	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×D₄×Dic₆₁	1	1	yes	characteristic
259	C₂	#16840571	1	1	yes	characteristic
260	C₂	#16840578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×#137×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₂×#137	D₆₁
hypocenter	C₆₁	#2947
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. #16840240	C₂	semidirect
3. #16839869	C₂	semidirect
4. #16839863	C₂	semidirect
5. Q₁₆×Dic₆₁	C₂	semidirect
6. SD₁₆×Dic₆₁	C₂	semidirect
7. #16839804	C₂	semidirect
9. C₂×#14519701	C₂	semidirect
10. #16839720	C₂	semidirect
38. #137×C₆₁	C₄	semidirect
46. C₂×Q₈×C₆₁	C₂×C₄	semidirect
56. Q₁₆×C₆₁	C₂×C₄	semidirect
57. SD₁₆×C₆₁	C₂×C₄	semidirect
58. M₁₆×C₆₁	C₂×C₄	semidirect
142. C₆₁	#2947	semidirect
145. #137	Dic₆₁	semidirect
177. C₂×Q₈	C₂×Dic₆₁	semidirect
187. Q₁₆	C₂×Dic₆₁	semidirect
188. SD₁₆	C₂×Dic₆₁	semidirect
189. M₁₆	C₂×Dic₆₁	semidirect