Finite Group Databasebeta

Subgroups — #29669342

Order 15616.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456564	C₂	1	3	yes	maximal, complemented
2	#21456194	C₂	1	3	yes	maximal
3	#21445479	C₂	1	6	yes	maximal, complemented
4	#21445431	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840319	C₂²	1	3	yes	—
7	C₂×#14519774	C₂²	1	1	yes	characteristic
8	#16840160	C₂²	1	3	yes	complemented
9	#16840142	C₂²	1	3	yes	complemented
10	#16840127	—	2	6	no	—
11	#16840113	C₂²	1	6	yes	complemented
12	#16840109	C₂²	1	6	yes	—
13	#16840109	C₂²	1	6	yes	—
14	#16840020	—	2	6	no	—
15	#16839870	—	2	6	no	—
16	#16839768	—	2	6	no	—
17	C₄×M₁₆×C₆₁	C₂²	1	3	yes	—
18	C₆₁×#322	C₂²	1	3	yes	complemented
19	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₂₄₄	—	2	3	no	—
21	C₂×Dic₂₄₄	—	2	6	no	—
22	#14519777	C₂³	1	6	yes	—
23	#14519774	—	2	3	no	—
24	#14519774	C₂³	1	2	yes	—
25	#14519769	—	2	3	no	—
26	#14519756	—	2	6	no	—
27	C₄×Dic₁₂₂	—	2	6	no	—
28	#14519736	—	2	3	no	—
29	#14519709	—	2	6	no	—
30	#14519702	—	2	12	no	—
31	#14519699	—	2	12	no	—
32	#14519699	—	2	12	no	—
33	C₂×M₁₆×C₆₁	D₄	1	6	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#97×C₆₁	D₄	1	6	yes	—
36	C₄×C₈×C₆₁	C₂³	1	3	yes	—
37	C₄×C₈×C₆₁	C₂³	1	3	yes	—
38	C₂³×D₆₁	—	4	1	no	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
40	C₂×D₂₄₄	—	2	6	no	—
41	C₂×D₂₄₄	—	4	12	no	—
42	C₂×D₂₄₄	—	4	3	no	—
43	C₂×Dic₁₂₂	—	2	6	no	—
44	M₁₆×C₆₁	—	4	6	no	—
45	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
46	C₂×C₈×C₆₁	C₂×D₄	1	6	yes	—
47	C₄²×C₆₁	C₂×D₄	1	3	yes	—
48	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₂²⋊Dic₆₁	—	4	3	no	—
50	C₄⋊Dic₆₁	—	2	6	no	—
51	C₄⋊Dic₆₁	—	2	6	no	—
52	#11758196	—	4	6	no	—
53	C₄×Dic₆₁	—	4	3	no	—
54	Dic₂₄₄	—	4	12	no	—
55	C₂³×C₆₁	#127	1	1	yes	characteristic
56	C₂²×D₆₁	—	4	2	no	—
57	C₂²×D₆₁	—	8	6	no	—
58	C₂×C₄×C₆₁	—	2	6	no	—
59	C₂×C₄×C₆₁	#127	1	3	yes	—
60	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
61	C₂×Dic₆₁	—	4	6	no	—
62	D₂₄₄	—	4	12	no	—
63	D₂₄₄	—	8	6	no	—
64	Dic₁₂₂	—	4	12	no	—
65	C₈×C₆₁	—	2	6	no	—
66	C₈×C₆₁	—	2	6	no	—
67	#19753	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	—	2	3	no	—
69	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
70	C₂×D₆₁	—	8	6	no	—
71	C₂×D₆₁	—	16	4	no	—
72	C₄×C₆₁	—	2	3	no	—
73	C₄×C₆₁	#575	1	6	yes	—
74	Dic₆₁	—	8	6	no	—
75	#3185	—	61	3	no	—
76	#3138	—	61	3	no	—
77	#1720	—	61	6	no	—
78	#1711	—	61	2	no	—
79	#1455	D₆₁	1	1	yes	characteristic, complemented
80	C₂×C₆₁	—	4	1	no	—
81	C₂×C₆₁	#3153	1	3	yes	—
82	D₆₁	—	16	2	no	—
83	#533	—	61	3	no	—
84	C₂×#127	—	61	1	no	—
85	#498	—	61	3	no	—
86	#494	—	61	3	no	—
87	#490	—	122	6	no	—
88	#487	—	61	6	no	—
89	#486	—	61	6	no	—
90	#486	—	61	6	no	—
91	#470	—	122	6	no	—
92	#441	—	122	6	no	—
93	#418	—	122	6	no	—
94	C₄×M₁₆	C₂×D₆₁	1	3	yes	—
95	#322	C₂×D₆₁	1	3	yes	complemented
96	C₈²	C₂×D₆₁	1	1	yes	characteristic, complemented
97	C₆₁	#19753	1	1	yes	characteristic, complemented, Sylow
98	C₂²×D₄	—	122	3	no	—
99	C₂×Q₁₆	—	122	6	no	—
100	C₂×M₁₆	D₂₄₄	1	6	yes	—
101	C₄⋊Q₈	—	61	6	no	—
102	#127	—	61	2	no	—
103	#127	—	122	3	no	—
104	#125	—	122	3	no	—
105	C₂²⋊Q₈	—	122	6	no	—
106	C₄×Q₈	—	122	6	no	—
107	#117	—	122	3	no	—
108	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
109	#106	—	122	6	no	—
110	#103	—	122	12	no	—
111	#102	—	122	12	no	—
112	#102	—	122	12	no	—
113	#97	D₂₄₄	1	6	yes	—
114	C₄×C₈	C₂²×D₆₁	1	3	yes	—
115	C₄×C₈	C₂²×D₆₁	1	3	yes	—
116	C₂⁴	—	244	1	no	—
117	C₂×Q₈	—	122	6	no	—
118	C₂×D₄	—	122	6	no	—
119	C₂×D₄	—	244	12	no	—
120	C₂×D₄	—	244	3	no	—
121	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
122	Q₁₆	—	244	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₂₄₄	1	6	yes	—
125	C₂×C₈	C₂×D₂₄₄	1	6	yes	—
126	C₄⋊C₄	—	122	6	no	—
127	C₄⋊C₄	—	122	6	no	—
128	C₄⋊C₄	—	244	6	no	—
129	C₂²⋊C₄	—	244	3	no	—
130	C₄²	—	244	3	no	—
131	C₄²	C₂×D₂₄₄	1	3	yes	—
132	C₄²	C₂³×D₆₁	1	1	yes	characteristic
133	C₂³	—	488	6	no	—
134	C₂³	—	244	2	no	—
135	C₂³	#14519774	1	1	yes	characteristic
136	Q₈	—	244	12	no	—
137	D₄	—	488	6	no	—
138	D₄	—	244	12	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	244	6	no	—
141	C₂×C₄	#14519774	1	3	yes	—
142	C₂×C₄	C₂²×D₂₄₄	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	976	4	no	—
147	C₂²	—	488	6	no	—
148	C₂²	C₂×#14519774	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	488	6	no	—
151	C₄	#16840562	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	976	2	no	—
154	C₂	#21456285	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂×#14519774
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1455	C₂
socle	C₂²×C₆₁	C₂×#127
prime radical	C₂×#14519774	C₂²
hypercenter	#1455	D₆₁
hypocenter	C₆₁	#19753
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
1. #21456564	C₂	semidirect
3. #21445479	C₂	semidirect
4. #21445431	C₂	semidirect
5. C₆₁×#1455	C₂	semidirect
8. #16840160	C₂²	semidirect
9. #16840142	C₂²	semidirect
11. #16840113	C₂²	semidirect
18. C₆₁×#322	C₂²	semidirect
19. C₈²×C₆₁	C₂²	semidirect
79. #1455	D₆₁	semidirect
95. #322	C₂×D₆₁	semidirect
96. C₈²	C₂×D₆₁	semidirect
97. C₆₁	#19753	semidirect

Maximal subgroups

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

subgroup	index
#21456564	2
#21456194	2
#21445479	2
#21445431	2
C₆₁×#1455	2
#19753	61