Subgroups — #29566623

Order 15616.

304 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445936	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#92418	—	2	1	no	—
5	C₂×C₄⋊#92418	C₄	1	2	yes	—
6	C₂×#14519688	—	2	1	no	—
7	#16839517	—	2	1	no	—
8	#16839517	—	2	1	no	—
9	#16839517	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
11	C₂×#97×C₆₁	—	2	1	no	—
12	C₄³×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#336	C₄	1	2	yes	—
14	C₂²×#92418	—	2	2	no	—
15	C₂²×#92418	—	2	2	no	—
16	C₄⋊#92418	—	2	4	no	—
17	C₄⋊#92418	—	4	2	no	—
18	#14519688	—	4	1	no	—
19	#14519688	—	4	1	no	—
20	C₂²×C₈×C₆₁	—	2	2	no	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	4	no	—
29	#97×C₆₁	—	4	1	no	—
30	#97×C₆₁	—	4	1	no	—
31	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₄²×C₆₁	—	2	1	no	—
42	C₄²×C₆₁	—	2	1	no	—
43	C₄²×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	2	2	no	—
45	C₄²×C₆₁	—	4	1	no	—
46	C₄²×C₆₁	—	4	1	no	—
47	C₄²×C₆₁	—	4	1	no	—
48	C₄²×C₆₁	D₈	1	1	yes	characteristic
49	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
50	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
51	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
52	C₂×#92418	—	2	4	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	2	no	—
56	C₂×#92418	—	4	2	no	—
57	C₂×#92418	—	4	2	no	—
58	C₂³×C₆₁	#95	1	1	yes	characteristic
59	C₂×C₄×C₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	—	2	1	no	—
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	#100	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#104	1	2	yes	—
81	C₂×C₄×C₆₁	#104	1	2	yes	—
82	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
84	C₈×C₆₁	—	4	4	no	—
85	#92418	—	4	4	no	—
86	#92418	—	8	2	no	—
87	#7049	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	#338	1	1	yes	characteristic
89	C₂²×C₆₁	#339	1	1	yes	characteristic
90	C₂²×C₆₁	#339	1	1	yes	characteristic
91	C₂²×C₆₁	#340	1	1	yes	characteristic
92	C₂²×C₆₁	#343	1	1	yes	characteristic
93	C₂²×C₆₁	#344	1	1	yes	characteristic
94	C₂²×C₆₁	#344	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	2	1	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	4	1	no	—
102	C₄×C₆₁	—	4	1	no	—
103	C₄×C₆₁	—	4	1	no	—
104	C₄×C₆₁	#329	1	2	yes	—
105	C₄×C₆₁	#330	1	2	yes	—
106	#1831	—	61	1	no	—
107	#1767	—	61	1	no	—
108	#1767	D₆₁	1	1	yes	characteristic
109	C₂×C₆₁	#1291	1	1	yes	characteristic
110	C₂×C₆₁	#1291	1	1	yes	characteristic
111	C₂×C₆₁	#1292	1	1	yes	characteristic
112	C₂×C₆₁	#1295	1	1	yes	characteristic
113	C₂×C₆₁	#1303	1	1	yes	characteristic
114	C₂×C₆₁	#1303	1	1	yes	characteristic
115	C₂×C₆₁	#1307	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₂×C₄⋊C₈	—	122	1	no	—
119	C₂×#97	—	2	1	no	—
120	C₂×#97	—	122	1	no	—
121	C₄³	C₂×D₆₁	1	1	yes	characteristic
122	#336	—	61	2	no	—
123	#336	—	122	1	no	—
124	#336	—	122	1	no	—
125	#336	Dic₆₁	1	2	yes	—
126	C₆₁	#7049	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	122	2	no	—
129	C₂²×C₈	—	122	2	no	—
130	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	122	4	no	—
139	C₄⋊C₈	—	244	2	no	—
140	#97	—	4	1	no	—
141	#97	—	4	1	no	—
142	#97	—	244	1	no	—
143	#97	—	244	1	no	—
144	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
145	C₂²×C₄	#11758196	1	1	yes	characteristic
146	C₂²×C₄	#11758196	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
148	C₂²×C₄	#11758198	1	1	yes	characteristic
149	C₂²×C₄	#11758198	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	122	4	no	—
155	C₂×C₈	—	244	2	no	—
156	C₂×C₈	—	244	2	no	—
157	C₂×C₈	—	244	2	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	—	244	2	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	#11758199	1	1	yes	characteristic
168	C₄²	#11758200	1	1	yes	characteristic
169	C₄²	#11758201	1	1	yes	characteristic
170	C₄²	#11758202	1	1	yes	characteristic
171	C₂³	#14519684	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
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	#14519693	1	1	yes	characteristic
188	C₂×C₄	#14519695	1	1	yes	characteristic
189	C₂×C₄	#14519697	1	1	yes	characteristic
190	C₂×C₄	#14519698	1	1	yes	characteristic
191	C₂×C₄	#14519700	1	1	yes	characteristic
192	C₂×C₄	#14519701	1	1	yes	characteristic
193	C₂×C₄	#14519703	1	2	yes	—
194	C₂×C₄	#14519705	1	2	yes	—
195	C₂×C₄	#14519708	1	1	yes	characteristic
196	C₂×C₄	#14519710	1	1	yes	characteristic
197	C₈	—	4	4	no	—
198	C₈	—	488	2	no	—
199	C₈	—	244	4	no	—
200	C₂²	#16839520	1	1	yes	characteristic
201	C₂²	#16839522	1	1	yes	characteristic
202	C₂²	#16839523	1	1	yes	characteristic
203	C₂²	#16839524	1	1	yes	characteristic
204	C₂²	#16839530	1	1	yes	characteristic
205	C₂²	#16839532	1	1	yes	characteristic
206	C₂²	#16839533	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	1	no	—
216	C₄	#16839501	1	2	yes	—
217	C₄	#16839503	1	2	yes	—
218	C₂	#21443691	1	1	yes	characteristic
219	C₂	#21443692	1	1	yes	characteristic
220	C₂	#21443693	1	1	yes	characteristic
221	C₂	#21443701	1	1	yes	characteristic
222	C₂	#21443720	1	1	yes	characteristic
223	C₂	#21443722	1	1	yes	characteristic
224	C₂	#21443729	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519684
commutator	C₂²×C₄×C₆₁	C₄²
Frattini	C₄³	C₂×D₆₁
Fitting	C₆₁×#1767	C₂
socle, prime radical	C₂³×C₆₁	#95
cosocle	C₄³×C₆₁	C₂²
hypercenter	#1767	D₆₁
hypocenter	C₆₁	#7049
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
126. C₆₁	#7049	semidirect

Maximal subgroups

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

subgroup	index
#21445936	2
#21445672	2
C₆₁×#1767	2
#7049	61