Finite Group Databasebeta

Subgroups — #30603720

Order 15616.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₄×#14519851	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840650	C₂	1	1	yes	maximal, characteristic
3	#21460737	C₂	1	4	yes	maximal, complemented
4	#21444103	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	C₄	1	2	yes	—
6	C₂²×#11758187	C₄	1	2	yes	—
7	#16840650	C₄	1	2	yes	—
8	#16840650	C₂²	1	2	yes	—
9	#16840650	C₂²	1	2	yes	—
10	C₄×#11758187	C₄	1	2	yes	—
11	#16839815	C₄	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839484	C₂²	1	2	yes	—
14	C₈×#92418	C₄	1	2	yes	—
15	C₆₁×#431	C₄	1	1	yes	characteristic
16	#14519851	C₈	1	8	yes	complemented
17	C₂×#11758187	—	2	4	no	—
18	C₂×#11758187	C₂×C₄	1	4	yes	—
19	C₂×#11758187	C₂×C₄	1	4	yes	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#92418	—	2	1	no	—
23	C₂²⋊#92418	—	2	1	no	—
24	C₄×#92418	C₂×C₄	1	2	yes	—
25	C₄×#92418	C₂×C₄	1	1	yes	characteristic
26	C₄×#92418	C₂×C₄	1	1	yes	characteristic
27	C₄×#92418	C₂×C₄	1	1	yes	characteristic
28	C₄×#92418	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	1	no	—
31	C₂²⋊C₈×C₆₁	—	2	1	no	—
32	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	C₄²	1	1	yes	characteristic
38	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
39	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×#92418	—	2	2	no	—
42	C₂×#92418	C₄²	1	1	yes	characteristic
43	C₂×#92418	C₄²	1	1	yes	characteristic
44	C₂×#92418	C₂×C₈	1	4	yes	—
45	C₂×#92418	M₁₆	1	4	yes	—
46	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
47	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
48	#11758187	—	2	4	no	—
49	#11758187	C₂×C₈	1	8	yes	—
50	C₂³×C₆₁	#97	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	4	1	no	—
52	C₂×C₄×C₆₁	—	4	1	no	—
53	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
55	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#97	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₆₁	—	4	2	no	—
62	#92418	—	4	2	no	—
63	#92418	C₄×C₈	1	2	yes	—
64	#92418	#97	1	2	yes	—
65	#92418	C₂²×C₈	1	2	yes	—
66	#92418	C₂×M₁₆	1	2	yes	—
67	#11406	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	—	2	1	no	—
69	C₂²×C₆₁	—	2	1	no	—
70	C₂²×C₆₁	#322	1	2	yes	—
71	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	#322	1	2	yes	—
77	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
79	C₄×M₃₂	—	61	1	no	—
80	C₂×#346	—	61	1	no	—
81	#1450	—	61	1	no	—
82	#1313	—	61	4	no	—
83	C₂×C₆₁	—	2	2	no	—
84	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
85	C₂×C₆₁	#2110	1	1	yes	characteristic
86	C₂×C₆₁	#2110	1	1	yes	characteristic
87	C₂×M₃₂	—	61	2	no	—
88	C₂²×C₁₆	—	61	2	no	—
89	#431	—	61	1	no	—
90	#431	#6923	1	1	yes	characteristic
91	C₂×C₄×C₈	—	61	1	no	—
92	#346	—	61	2	no	—
93	#346	—	61	2	no	—
94	#346	—	61	2	no	—
95	C₄×C₁₆	—	61	2	no	—
96	#322	—	61	2	no	—
97	C₈²	—	61	2	no	—
98	C₆₁	#11406	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	61	1	no	—
100	C₂²×C₈	—	61	1	no	—
101	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
102	M₃₂	—	61	8	no	—
103	C₂×C₁₆	—	61	4	no	—
104	C₂×C₁₆	—	61	4	no	—
105	C₂×C₁₆	—	122	4	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	122	1	no	—
108	C₂²⋊C₈	—	2	1	no	—
109	C₂²⋊C₈	—	122	1	no	—
110	C₄×C₈	—	61	1	no	—
111	C₄×C₈	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	—	61	2	no	—
115	C₄×C₈	C₂×#6923	1	2	yes	—
116	C₂²×C₄	#11758213	1	1	yes	characteristic
117	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
118	C₂²×C₄	#11758217	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	61	4	no	—
121	C₂×C₈	—	61	4	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	—	61	1	no	—
126	C₂×C₈	—	122	2	no	—
127	C₄²	#11758213	1	1	yes	characteristic
128	C₄²	C₄×#6923	1	1	yes	characteristic
129	C₄²	#11758217	1	1	yes	characteristic
130	C₄²	C₂²×#6923	1	1	yes	characteristic
131	C₁₆	—	61	8	no	—
132	C₁₆	—	122	4	no	—
133	C₂³	#14519832	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₈×#6923	1	2	yes	—
137	C₂×C₄	#14519831	1	2	yes	—
138	C₂×C₄	#14519832	1	1	yes	characteristic
139	C₂×C₄	#14519832	1	1	yes	characteristic
140	C₂×C₄	#14519832	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
142	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
143	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	61	2	no	—
146	C₈	—	61	2	no	—
147	C₈	—	61	2	no	—
148	C₈	—	61	2	no	—
149	C₈	—	244	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#92420⋊C₈	1	2	yes	—
153	C₂²	C₂×#14519832	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#92420⋊C₈	1	2	yes	—
159	C₄	C₂×C₈×#6923	1	1	yes	characteristic
160	C₄	C₂×#14519831	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#92420⋊C₈	1	1	yes	characteristic
163	C₂	#21462040	1	1	yes	characteristic
164	C₂	#21462042	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#431	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	#97
cosocle	C₄×#92418	C₂³
hypercenter	#431	#6923
hypocenter	C₆₁	#11406
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #21460737	C₂	semidirect
16. #14519851	C₈	semidirect
98. C₆₁	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519851	2
C₂×#16840650	2
#21460737	2
#21444103	2
#11406	61