Subgroups — C₂²×#16840654 #30623840

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519851	C₂	1	2	yes	maximal
2	C₂×#16840654	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14519688	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₄	1	12	yes	complemented
5	C₂×#14519851	C₂²	1	12	yes	—
6	C₂²×#11758187	C₄	1	4	yes	complemented
7	#16840654	C₂²	1	16	yes	direct product factor
8	C₂³×#92418	C₄	1	1	yes	characteristic
9	C₂³×#92418	C₂²	1	1	yes	characteristic
10	C₂×#14519688	C₄	1	6	yes	—
11	C₂×#14519688	C₂²	1	6	yes	—
12	C₂²×C₄²×C₆₁	C₄	1	1	yes	characteristic
13	#14519851	C₂×C₄	1	24	yes	complemented
14	#14519851	C₂³	1	8	yes	—
15	C₂×#11758187	C₂×C₄	1	24	yes	complemented
16	C₂²×#92418	—	2	3	no	—
17	C₂²×#92418	—	2	1	no	—
18	C₂²×#92418	C₂×C₄	1	6	yes	—
19	C₂²×#92418	C₂×C₄	1	3	yes	—
20	C₂²×#92418	C₂×C₄	1	2	yes	—
21	C₂²×#92418	C₂×C₄	1	6	yes	—
22	C₂²×#92418	C₂³	1	3	yes	—
23	#14519688	C₂×C₄	1	12	yes	—
24	#14519688	C₂³	1	4	yes	—
25	C₂³×C₄×C₆₁	—	2	1	no	—
26	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	—	2	3	no	—
28	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
29	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	—	2	6	no	—
31	C₂²×C₄×C₆₁	—	4	4	no	—
32	C₂²×C₄×C₆₁	C₄²	1	3	yes	—
33	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	M₁₆	1	3	yes	—
35	C₂²×C₄×C₆₁	M₁₆	1	3	yes	—
36	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
38	C₄²×C₆₁	—	2	6	no	—
39	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
40	C₂×#92418	—	2	6	no	—
41	C₂×#92418	—	2	6	no	—
42	C₂×#92418	C₄²	1	6	yes	—
43	C₂×#92418	C₄²	1	6	yes	—
44	C₂×#92418	C₂²×C₄	1	4	yes	—
45	C₂×#92418	C₂²×C₄	1	3	yes	—
46	C₂×#92418	C₂²×C₄	1	6	yes	—
47	C₂×#92418	C₂²×C₄	1	6	yes	—
48	C₂×#92418	C₂⁴	1	1	yes	characteristic
49	#11758187	C₂²×C₄	1	16	yes	complemented
50	C₂³×C₆₁	—	2	4	no	—
51	C₂³×C₆₁	#97	1	3	yes	—
52	C₂³×C₆₁	#97	1	1	yes	characteristic
53	C₂³×C₆₁	C₂×M₁₆	1	3	yes	—
54	C₂×C₄×C₆₁	—	2	4	no	—
55	C₂×C₄×C₆₁	—	4	12	no	—
56	C₂×C₄×C₆₁	#97	1	6	yes	—
57	C₂×C₄×C₆₁	#97	1	3	yes	—
58	C₂×C₄×C₆₁	#97	1	3	yes	—
59	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
62	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
65	#92418	—	2	4	no	—
66	#92418	C₂×C₄²	1	6	yes	—
67	#92418	C₂³×C₄	1	2	yes	—
68	C₂²×#347	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	—	2	6	no	—
70	C₂²×C₆₁	—	2	6	no	—
71	C₂²×C₆₁	#347	1	4	yes	complemented
72	C₂²×C₆₁	C₂×#97	1	3	yes	—
73	C₂²×C₆₁	C₂×#97	1	3	yes	—
74	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
75	C₄×C₆₁	—	4	4	no	—
76	C₄×C₆₁	C₂×#97	1	3	yes	—
77	C₄×C₆₁	C₂×#97	1	3	yes	—
78	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
79	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
80	C₂²×M₃₂	—	61	2	no	—
81	C₂²×#97	—	61	1	no	—
82	C₂×#347	—	61	12	no	—
83	C₂×C₆₁	—	2	1	no	—
84	C₂×C₆₁	—	2	3	no	—
85	C₂×C₆₁	C₂×#347	1	6	yes	complemented
86	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
87	C₂³×C₈	—	61	1	no	—
88	C₂³×C₈	—	61	1	no	—
89	C₂²×C₄²	#6923	1	1	yes	characteristic
90	C₂×M₃₂	—	61	12	no	—
91	C₂×M₃₂	—	61	12	no	—
92	C₂²×C₁₆	—	61	4	no	—
93	C₂×#97	—	61	6	no	—
94	C₂×#97	—	61	6	no	—
95	#347	—	61	16	no	—
96	C₆₁	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	2	1	no	—
98	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
99	C₂²×C₈	—	61	6	no	—
100	C₂²×C₈	—	61	2	no	—
101	C₂²×C₈	—	61	3	no	—
102	C₂²×C₈	—	61	3	no	—
103	C₂²×C₈	—	61	6	no	—
104	C₂²×C₈	—	122	1	no	—
105	C₂²×C₈	—	122	3	no	—
106	C₂×C₄²	—	2	3	no	—
107	C₂×C₄²	C₂×#6923	1	6	yes	—
108	M₃₂	—	61	24	no	—
109	M₃₂	—	61	8	no	—
110	C₂×C₁₆	—	61	24	no	—
111	#97	—	61	4	no	—
112	#97	—	61	12	no	—
113	C₂⁴	#11758217	1	1	yes	characteristic
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	#11758213	1	1	yes	characteristic
117	C₂²×C₄	#11758213	1	3	yes	—
118	C₂²×C₄	C₄×#6923	1	3	yes	—
119	C₂²×C₄	C₄×#6923	1	1	yes	characteristic
120	C₂²×C₄	#11758217	1	3	yes	—
121	C₂²×C₄	C₂²×#6923	1	3	yes	—
122	C₂×C₈	—	61	6	no	—
123	C₂×C₈	—	61	6	no	—
124	C₂×C₈	—	61	6	no	—
125	C₂×C₈	—	61	6	no	—
126	C₂×C₈	—	61	1	no	—
127	C₂×C₈	—	61	3	no	—
128	C₂×C₈	—	61	4	no	—
129	C₂×C₈	—	122	6	no	—
130	C₂×C₈	—	122	6	no	—
131	C₄²	—	2	6	no	—
132	C₄²	C₂²×#6923	1	4	yes	—
133	C₁₆	—	61	16	no	—
134	C₂³	—	2	4	no	—
135	C₂³	#14519832	1	3	yes	—
136	C₂³	#14519832	1	1	yes	characteristic
137	C₂³	C₂×#11758217	1	3	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	#14519832	1	6	yes	—
141	C₂×C₄	#14519832	1	3	yes	—
142	C₂×C₄	#14519832	1	3	yes	—
143	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
144	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
145	C₂×C₄	C₂×#11758217	1	3	yes	—
146	C₂×C₄	C₂×#11758213	1	3	yes	—
147	C₂×C₄	C₂×#11758213	1	3	yes	—
148	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
149	C₈	—	61	6	no	—
150	C₈	—	61	2	no	—
151	C₈	—	122	4	no	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#16840654	1	4	yes	direct product factor
155	C₂²	C₂×#14519832	1	3	yes	—
156	C₂²	C₂×#14519832	1	3	yes	—
157	C₂²	C₂²×#11758217	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×#14519832	1	3	yes	—
160	C₄	C₂×#14519832	1	3	yes	—
161	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
162	C₄	C₂²×#11758213	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#16840654	1	6	yes	direct product factor
166	C₂	C₂²×#14519832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#6923
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂²×C₄²×C₆₁	C₄
socle	C₂³×C₆₁	#97
prime radical	C₂⁴×C₆₁	M₁₆
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂²×C₄²	#6923
hypocenter	C₆₁	C₂²×#347
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16840654	C₂	direct (N × H)
4. C₂×#14519851	C₄	semidirect
6. C₂²×#11758187	C₄	semidirect
7. #16840654	C₂²	direct (N × H)
13. #14519851	C₂×C₄	semidirect
15. C₂×#11758187	C₂×C₄	semidirect
49. #11758187	C₂²×C₄	semidirect
71. C₂²×C₆₁	#347	semidirect
85. C₂×C₆₁	C₂×#347	semidirect
96. C₆₁	C₂²×#347	semidirect
154. C₂²	#16840654	direct (N × H)
165. C₂	C₂×#16840654	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519851	2
C₂×#16840654	2
C₂²×#14519688	2
C₂²×#347	61

Subgroups — C22×#16840654 #30623840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840654 #30623840