Subgroups — #21451053

Order 7808.

380 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16839701	C₂	1	2	yes	maximal
2	#16839695	C₂	1	2	yes	maximal
3	#16839660	C₂	1	1	yes	maximal, characteristic
4	#16839660	C₂	1	1	yes	maximal, characteristic
5	#16839649	C₂	1	2	yes	maximal
6	#16839645	C₂	1	2	yes	maximal
7	#16839632	C₂	1	1	yes	maximal, characteristic
8	#16839628	C₂	1	1	yes	maximal, characteristic
9	#16839628	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
18	C₂×#11758196	C₂²	1	2	yes	—
19	C₂×#11758196	C₂²	1	2	yes	—
20	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
21	#14519684	C₂²	1	2	yes	—
22	#14519684	C₂²	1	2	yes	—
23	#14519684	C₂²	1	2	yes	—
24	#14519684	C₂²	1	2	yes	—
25	#14519684	C₂²	1	2	yes	—
26	#14519684	C₂²	1	2	yes	—
27	#14519684	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
33	#95×C₆₁	C₂²	1	1	yes	characteristic
34	#95×C₆₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂³	1	2	yes	—
43	C₂²×Dic₆₁	C₂³	1	2	yes	—
44	C₂²×Dic₆₁	C₂³	1	2	yes	—
45	C₂²×Dic₆₁	C₂³	1	2	yes	—
46	C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₄⋊C₄×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	Q₈	1	4	yes	—
51	C₄⋊Dic₆₁	—	2	4	no	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	C₄⋊Dic₆₁	—	2	2	no	—
55	C₄⋊Dic₆₁	—	2	2	no	—
56	C₄⋊Dic₆₁	—	4	2	no	—
57	#11758196	—	2	4	no	—
58	#11758196	—	4	2	no	—
59	C₄×Dic₆₁	—	4	2	no	—
60	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	2	1	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₆₁	#41	1	2	yes	—
79	C₂×C₄×C₆₁	#41	1	2	yes	—
80	C₂×C₄×C₆₁	#41	1	2	yes	—
81	C₂×C₄×C₆₁	#41	1	2	yes	—
82	C₂×Dic₆₁	—	2	2	no	—
83	C₂×Dic₆₁	—	2	2	no	—
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	—	2	2	no	—
87	C₂×Dic₆₁	—	2	2	no	—
88	C₂×Dic₆₁	—	2	2	no	—
89	C₂×Dic₆₁	—	2	2	no	—
90	C₂×Dic₆₁	—	2	2	no	—
91	C₂×Dic₆₁	—	2	2	no	—
92	C₂×Dic₆₁	—	2	2	no	—
93	C₂×Dic₆₁	—	2	2	no	—
94	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
96	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
97	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
98	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
99	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
101	C₄×C₆₁	—	2	2	no	—
102	C₄×C₆₁	—	2	2	no	—
103	C₄×C₆₁	—	2	2	no	—
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	—	4	1	no	—
106	C₄×C₆₁	—	4	1	no	—
107	C₄×C₆₁	#125	1	4	yes	—
108	Dic₆₁	—	4	2	no	—
109	Dic₆₁	—	4	2	no	—
110	Dic₆₁	—	4	2	no	—
111	Dic₆₁	—	4	2	no	—
112	#2570	—	61	1	no	maximal, Sylow
113	C₂×C₆₁	C₂×#125	1	1	yes	characteristic
114	C₂×C₆₁	#533	1	1	yes	characteristic
115	C₂×C₆₁	#544	1	1	yes	characteristic
116	C₂×C₆₁	#552	1	1	yes	characteristic
117	C₂×C₆₁	#553	1	1	yes	characteristic
118	C₂×C₆₁	#556	1	1	yes	characteristic
119	C₂×C₆₁	#559	1	1	yes	characteristic
120	#400	—	61	2	no	—
121	#398	—	61	2	no	—
122	#389	—	61	1	no	—
123	#389	—	61	1	no	—
124	#387	—	61	2	no	—
125	#387	—	61	2	no	—
126	#384	—	61	1	no	—
127	#382	—	61	1	no	—
128	#382	—	61	1	no	—
129	C₄×C₄⋊C₄	—	61	1	no	—
130	C₄×C₄⋊C₄	D₆₁	1	1	yes	characteristic
131	C₆₁	#2570	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊C₄	—	61	1	no	—
133	C₂×C₄⋊C₄	—	61	1	no	—
134	C₂×C₄⋊C₄	—	61	1	no	—
135	C₂×C₄⋊C₄	—	61	1	no	—
136	C₂×C₄⋊C₄	—	61	2	no	—
137	C₂×C₄⋊C₄	—	61	2	no	—
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₄⋊C₄	—	61	2	no	—
140	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
142	C₂×C₄²	—	61	2	no	—
143	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
146	#95	—	61	2	no	—
147	#95	—	61	2	no	—
148	#95	—	61	2	no	—
149	#95	—	61	2	no	—
150	#95	—	61	2	no	—
151	#95	—	61	2	no	—
152	#95	—	61	2	no	—
153	#95	C₂×D₆₁	1	1	yes	characteristic
154	#95	C₂×D₆₁	1	1	yes	characteristic
155	C₂²×C₄	—	61	2	no	—
156	C₂²×C₄	—	61	2	no	—
157	C₂²×C₄	—	61	2	no	—
158	C₂²×C₄	—	61	2	no	—
159	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	122	4	no	—
169	C₄⋊C₄	—	122	4	no	—
170	C₄⋊C₄	—	122	2	no	—
171	C₄⋊C₄	—	122	2	no	—
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	244	2	no	—
175	C₄⋊C₄	—	244	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	244	2	no	—
179	C₄²	Dic₁₂₂	1	4	yes	—
180	C₂³	C₂³×D₆₁	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	122	2	no	—
196	C₂×C₄	—	122	2	no	—
197	C₂×C₄	—	122	2	no	—
198	C₂×C₄	—	122	2	no	—
199	C₂×C₄	—	122	2	no	—
200	C₂×C₄	—	122	2	no	—
201	C₂×C₄	—	122	2	no	—
202	C₂×C₄	—	122	2	no	—
203	C₂×C₄	—	122	2	no	—
204	C₂×C₄	—	122	2	no	—
205	C₂×C₄	—	122	2	no	—
206	C₂×C₄	—	122	2	no	—
207	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
208	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
209	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
210	C₂×C₄	#11758222	1	2	yes	—
211	C₂×C₄	#11758222	1	2	yes	—
212	C₂×C₄	#11758224	1	2	yes	—
213	C₂×C₄	#11758226	1	2	yes	—
214	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
215	C₂²	C₂×#11758224	1	1	yes	characteristic
216	C₂²	C₂×#11758222	1	1	yes	characteristic
217	C₂²	C₂×#11758222	1	1	yes	characteristic
218	C₂²	C₂×#11758226	1	1	yes	characteristic
219	C₂²	#14519823	1	1	yes	characteristic
220	C₂²	#14519824	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	244	2	no	—
228	C₄	—	244	2	no	—
229	C₄	—	244	2	no	—
230	C₄	—	244	2	no	—
231	C₄	#14519768	1	4	yes	—
232	C₂	C₂×#14519768	1	1	yes	characteristic
233	C₂	#16840319	1	1	yes	characteristic
234	C₂	#16840390	1	1	yes	characteristic
235	C₂	#16840452	1	1	yes	characteristic
236	C₂	#16840458	1	1	yes	characteristic
237	C₂	#16840485	1	1	yes	characteristic
238	C₂	#16840497	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₄×C₄⋊C₄×C₆₁	C₂
hypercenter	C₄×C₄⋊C₄	D₆₁
hypocenter	C₆₁	#2570
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
131. C₆₁	#2570	semidirect