Subgroups — #21448679

Order 7808.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16839648	C₂	1	2	yes	maximal, complemented
2	#16839648	C₂	1	2	yes	maximal, complemented
3	#16839637	C₂	1	2	yes	maximal
4	#16839630	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839628	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839625	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×Dic₆₁	C₂	1	2	yes	maximal
9	#16839608	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14519684	C₂	1	1	yes	maximal, characteristic
11	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₆₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
18	#14519684	C₂²	1	1	yes	characteristic
19	#14519684	C₂²	1	1	yes	characteristic
20	#14519684	C₂²	1	2	yes	—
21	#14519684	C₂²	1	1	yes	characteristic
22	#14519684	C₂²	1	2	yes	—
23	#14519684	C₂²	1	2	yes	—
24	#14519684	C₂²	1	2	yes	—
25	#14519684	C₂²	1	1	yes	characteristic
26	#14519684	C₂²	1	2	yes	—
27	#14519684	C₂²	1	2	yes	—
28	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
29	#117×C₆₁	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
36	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	—	2	4	no	—
38	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
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×C₄×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₆₁	—	2	4	no	—
46	C₂²×Dic₆₁	—	2	4	no	—
47	C₂²×Dic₆₁	—	4	2	no	—
48	C₂²×Dic₆₁	C₂³	1	2	yes	—
49	C₂²×Dic₆₁	C₂³	1	2	yes	—
50	C₂²×Dic₆₁	C₂³	1	2	yes	—
51	C₂²×Dic₆₁	C₂³	1	2	yes	—
52	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	complemented
53	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	complemented
54	C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
55	C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
56	C₄²×C₆₁	C₂×C₄	1	4	yes	complemented
57	C₄²×C₆₁	C₂×C₄	1	4	yes	complemented
58	C₂²⋊Dic₆₁	—	2	4	no	—
59	C₄⋊Dic₆₁	—	2	4	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	—	2	4	no	—
62	C₄×Dic₆₁	—	4	2	no	—
63	C₂³×C₆₁	—	2	2	no	—
64	C₂³×C₆₁	—	2	1	no	—
65	C₂³×C₆₁	—	2	1	no	—
66	C₂³×C₆₁	C₂²×C₄	1	2	yes	—
67	C₂³×C₆₁	#41	1	4	yes	—
68	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	8	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
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	C₂×Dic₆₁	—	2	2	no	—
95	C₂×Dic₆₁	—	4	8	no	—
96	C₂×Dic₆₁	#41	1	4	yes	—
97	C₂²×C₆₁	—	2	2	no	—
98	C₂²×C₆₁	—	2	2	no	—
99	C₂²×C₆₁	—	2	4	no	—
100	C₂²×C₆₁	—	2	4	no	—
101	C₂²×C₆₁	#117	1	4	yes	—
102	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₆₁	C₂×#41	1	2	yes	—
104	C₂²×C₆₁	C₂×#41	1	2	yes	—
105	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₆₁	—	2	2	no	—
108	C₄×C₆₁	—	2	2	no	—
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	—	2	2	no	—
111	C₄×C₆₁	—	2	2	no	—
112	C₄×C₆₁	—	2	2	no	—
113	Dic₆₁	—	2	4	no	—
114	Dic₆₁	—	4	2	no	—
115	Dic₆₁	—	4	2	no	—
116	Dic₆₁	—	4	2	no	—
117	#2343	—	61	1	no	maximal, Sylow
118	C₂×C₆₁	—	2	4	no	—
119	C₂×C₆₁	C₂×#117	1	1	yes	characteristic
120	C₂×C₆₁	C₄×#41	1	1	yes	characteristic
121	C₂×C₆₁	#520	1	1	yes	characteristic
122	C₂×C₆₁	#551	1	2	yes	—
123	C₂×C₆₁	#552	1	2	yes	—
124	C₂×#117	D₆₁	1	1	yes	characteristic
125	#387	—	61	2	no	—
126	#387	—	61	2	no	—
127	#385	—	61	2	no	—
128	#383	—	61	1	no	—
129	#382	—	61	1	no	—
130	#381	—	61	1	no	—
131	C₄×C₄⋊C₄	—	61	1	no	—
132	C₄×C₂²⋊C₄	—	61	2	no	—
133	#376	—	61	1	no	—
134	C₂×#95	—	61	1	no	—
135	C₆₁	#2343	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	61	2	no	—
137	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
138	#117	Dic₆₁	1	8	yes	complemented
139	C₂×C₄⋊C₄	—	61	2	no	—
140	C₂×C₄⋊C₄	—	61	2	no	—
141	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	61	2	no	—
144	C₂×C₂²⋊C₄	C₂×D₆₁	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂×D₆₁	1	1	yes	characteristic
146	C₂×C₄²	—	61	2	no	—
147	C₂×C₄²	—	61	2	no	—
148	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
150	#95	—	61	1	no	—
151	#95	—	61	1	no	—
152	#95	—	61	2	no	—
153	#95	—	61	1	no	—
154	#95	—	61	2	no	—
155	#95	—	61	2	no	—
156	#95	—	61	2	no	—
157	#95	—	61	1	no	—
158	#95	—	61	2	no	—
159	#95	—	61	2	no	—
160	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	61	2	no	—
163	C₂²×C₄	—	61	2	no	—
164	C₂²×C₄	—	61	2	no	—
165	C₂²×C₄	—	61	2	no	—
166	C₂²×C₄	—	122	4	no	—
167	C₂²×C₄	—	122	4	no	—
168	C₂²×C₄	—	244	2	no	—
169	C₂²×C₄	C₂×Dic₆₁	1	4	yes	—
170	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
176	C₄⋊C₄	—	122	4	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	complemented
179	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	complemented
180	C₂²⋊C₄	—	122	4	no	—
181	C₂²⋊C₄	C₂×Dic₆₁	1	4	yes	—
182	C₂²⋊C₄	C₂×Dic₆₁	1	4	yes	—
183	C₄²	—	122	4	no	—
184	C₄²	—	244	2	no	—
185	C₄²	C₂×Dic₆₁	1	4	yes	complemented
186	C₄²	C₂×Dic₆₁	1	4	yes	complemented
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	#11758222	1	4	yes	—
191	C₂³	C₂²×Dic₆₁	1	2	yes	—
192	C₂³	C₂³×D₆₁	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	61	4	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	—	122	2	no	—
208	C₂×C₄	—	122	2	no	—
209	C₂×C₄	—	122	2	no	—
210	C₂×C₄	—	122	2	no	—
211	C₂×C₄	—	122	2	no	—
212	C₂×C₄	—	122	2	no	—
213	C₂×C₄	—	122	2	no	—
214	C₂×C₄	—	244	8	no	—
215	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
216	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
217	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
218	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
219	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
220	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	#14519736	1	4	yes	—
226	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
227	C₂²	#41×D₆₁	1	2	yes	—
228	C₂²	C₂×#11758222	1	2	yes	—
229	C₂²	#14519823	1	1	yes	characteristic
230	C₂²	#14519824	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	122	4	no	—
238	C₄	—	244	2	no	—
239	C₄	—	244	2	no	—
240	C₄	—	244	2	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#14519736	1	1	yes	characteristic
243	C₂	#41×Dic₆₁	1	1	yes	characteristic
244	C₂	#16840245	1	1	yes	characteristic
245	C₂	#16840436	1	2	yes	—
246	C₂	#16840445	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×#117×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₂×#117	D₆₁
hypocenter	C₆₁	#2343
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #16839648	C₂	semidirect
2. #16839648	C₂	semidirect
4. #16839630	C₂	semidirect
5. #16839628	C₂	semidirect
7. C₄×C₄⋊Dic₆₁	C₂	semidirect
9. #16839608	C₂	semidirect
29. #117×C₆₁	C₄	semidirect
52. C₄⋊C₄×C₆₁	C₂×C₄	semidirect
53. C₄⋊C₄×C₆₁	C₂×C₄	semidirect
56. C₄²×C₆₁	C₂×C₄	semidirect
57. C₄²×C₆₁	C₂×C₄	semidirect
135. C₆₁	#2343	semidirect
138. #117	Dic₆₁	semidirect
178. C₄⋊C₄	C₂×Dic₆₁	semidirect
179. C₄⋊C₄	C₂×Dic₆₁	semidirect
185. C₄²	C₂×Dic₆₁	semidirect
186. C₄²	C₂×Dic₆₁	semidirect